FLORIDA STATE UNIVERSITY
COLLEGE OF MUSIC
EARLY SEVENTEENTH-CENTURY HARMONIC PRACTICE:
A CORPUS STUDY OF TONALITY, MODALITY, AND HARMONIC FUNCTION IN
ITALIAN SECULAR SONG WITH BAROQUE GUITAR ACCOMPANIMENT IN
ALFABETO TABLATURE
By
DANIEL C. TOMPKINS
A Dissertation submitted to the
College of Music
in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
2017
Copyright © 2017 Daniel C. Tompkins. All Rights Reserved.
Daniel C. Tompkins defended this dissertation on March 6, 2017.
The members of the supervisory committee were:
Evan A. Jones
Professor Directing Dissertation
Charles E. Brewer
University Representative
Clifton Callender
Committee Member
Jane Piper Clendinning
Committee Member
The Graduate School has verified and approved the above-named committee members, and
certifies that the dissertation has been approved in accordance with university requirements.
ii
for my family
iii
ACKNOWLEDGMENTS
This dissertation would not be possible without the help and support of many people. I
would like to thank my committee: Evan Jones guided this project and saw potential in what
seemed tangential ideas at the time; Cliff Callender introduced me to machine learning, and
most of my computer programming abilities are thanks to his instruction; Jane Clendinning
helped me to always see the bigger picture; and Charles Brewer has been an endless source
of knowledge for early music.
I would like to thank Gary Boye for his years of work cataloging hundreds of sources
for the guitar, without which this dissertation would not have been able to be realized. I
had the pleasure of taking his guitar history and literature class, which introduced me to
this wonderful music. He was also kind enough to correspond with me many times during
the making of this dissertation. His knowledge and help was invaluable. My guitar teachers
through the years deserve a special thanks for cultivating my interest in such a wonderful
instrument: Marc Yaxley, Bob Teixeira, and Doug James.
I am also very thankful to Nori Jacoby for sharing some very helpful code for hidden
Markov models. Thanks also to Paolo Aluffi for mathematical encouragement and assistance with Italian translation. Furthermore, I would like to thank the enormous online
community that has taken time to write programs, guides, and answer questions. Without
their assistance, this dissertation would have taken much longer. I am thankful for all of my
colleagues at Florida State University, and I would like to give particular thanks to Daniel
Thompson, whose help was enormous and whose name is likely to get us confused with each
other, and Lewis Jeter for sharing his programming expertise.
I am extremely grateful to my parents, Charles and Linda, who gave me my first guitar
at age seven and for their years of love and support. Norm and Diane Peck also deserve
a special thanks. Lastly, I would like to thank my wife, Katherine and our two daughters,
Nora Grade and Adalia, for the years of love and support and for making life so much fun.
iv
TABLE OF CONTENTS
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
List of Musical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
1 Introduction and Methodology
1.1 Review of Literature . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Alfabeto Tablature . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Computational Approaches to Analyzing Harmony . . . .
1.2 Methodology and Philosophy . . . . . . . . . . . . . . . . . . . .
1.2.1 The Significance of the Alfabeto System for Computational
1
2
2
4
10
11
2 The Baroque Guitar, its Notation, and its Place in Music
2.1 Alfabeto Notation . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Limits of Alfabeto and its Extensions . . . . . . . . .
2.2 Impact on Triadic Thought . . . . . . . . . . . . . . . . . .
2.2.1 The Scalla di Musica and Alfabeto Performance . . .
2.2.2 Continuo: Guitar Accompaniment Beyond Alfabeto .
2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Making the Alfabeto Corpus
3.1 Composers and Publishers . .
3.1.1 Rome . . . . . . . . .
3.1.2 Naples . . . . . . . . .
3.1.3 Venice . . . . . . . . .
3.1.4 Parma . . . . . . . . .
3.2 Input and Conversion of Data
3.3 Conclusion . . . . . . . . . . .
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4 Questions of Mode and Key
4.1 Historical Notation . . . . . . . . . . . . . .
4.2 Previous Computational Approaches . . . .
4.3 Scale Degree Frequency Analysis . . . . . .
4.3.1 Chronological Comparison . . . . . .
4.3.2 Genre Comparison . . . . . . . . . .
4.3.3 Bass Harmonization Analysis . . . .
4.3.4 Measuring Idiomatic Chord Changes
4.4 Conclusion . . . . . . . . . . . . . . . . . . .
v
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Methods
Theory
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15
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48
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56
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83
90
5 Harmonic Motion
5.1 Harmonic Schema in Seventeenth-Century Guitar Music .
5.2 Machine-Learning Approaches to Finding Chord Functions
5.2.1 Some Philosophical Issues . . . . . . . . . . . . . .
5.2.2 Some Statistical Data of Alfabeto Chords . . . . . .
5.2.3 Chord Clusters and Harmonic Function . . . . . . .
5.2.4 Hierarchical Clustering . . . . . . . . . . . . . . . .
5.2.5 Information Bottleneck Method . . . . . . . . . . .
5.2.6 Hidden Markov Model . . . . . . . . . . . . . . . .
5.3 A Hidden Markov Model Analysis of Harmonic Function .
5.3.1 Determining the Number of Functions . . . . . . .
5.3.2 Visualizing HMM Analyses . . . . . . . . . . . . . .
5.3.3 Alfabeto Corpus . . . . . . . . . . . . . . . . . . .
5.3.4 Bach Corpus . . . . . . . . . . . . . . . . . . . . .
5.3.5 Palestrina Corpus . . . . . . . . . . . . . . . . . . .
5.3.6 Harmonic Analyses from the Alfabeto Corpus . . .
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .
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92
93
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108
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118
118
6 A Comparative Analysis within the Alfabeto Corpus
6.1 Kapsperger’s Harmonic Practices . . . . . . . . . . . . . . . .
6.1.1 Differences in Guitar and Theorbo Continuo Tablatures
6.2 Comparative Analysis of Kapsperger’s Villanellas (1610–1640)
6.2.1 Chord Frequency Analysis . . . . . . . . . . . . . . . .
6.2.2 N-Gram Analysis . . . . . . . . . . . . . . . . . . . . .
6.2.3 Harmonic Function . . . . . . . . . . . . . . . . . . . .
6.2.4 Kapsperger with the Alfabeto Corpus . . . . . . . . . .
6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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121
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126
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129
135
135
138
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7 Conclusion: The Baroque Guitar, its Notation, and its Place in Music
Theory, Revisited
140
7.1 Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Appendices
A Additional HMM Function Graphs
144
B List of Terms
157
Bibliography
160
Biographical Sketch
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
vi
LIST OF TABLES
2.1
Major and minor triads in Amat’s musical circle . . . . . . . . . . . . . . . . .
25
2.2
Parallel major and minor chords in Sanz’s alfabeto layout . . . . . . . . . . .
26
4.1
Mode possibilities in the durus/mollis system . . . . . . . . . . . . . . . . . .
49
4.2
Notated modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.3
Possible coalesced modes due to accidentals and/or musica ficta . . . . . . . .
50
4.4
Temperley’s key profiles, ordered from 1̂–7̂ (pitch classes 0–11) . . . . . . . . .
54
4.5
Frequency of bass note harmonization centroids of Figure 4.15 . . . . . . . . .
76
4.6
Frequency of bass note harmonization centroids from Figure 4.16, alfabeto corpus; all numbers sum to 100 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
4.7
Frequency of bass note harmonization centroids from Figure 4.18 . . . . . . .
80
4.8
Frequency of bass note harmonization centroids from Figure 4.20 . . . . . . .
83
4.9
Taxicab distances of common chord movements . . . . . . . . . . . . . . . . .
90
5.1
Pairwise histogram of the alfabeto corpus’s major-mode songs . . . . . . . . .
97
5.2
Three clusters from the dendrogram analysis . . . . . . . . . . . . . . . . . . . 100
5.3
Harmonic functions in the alfabeto corpus (major songs only) using the information bottleneck method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.4
Author-labeled functions, information bottleneck . . . . . . . . . . . . . . . . 103
5.5
Pair-wise histogram of sample progression functions . . . . . . . . . . . . . . . 103
5.6
HMM results in three clusters: major-mode alfabeto songs . . . . . . . . . . . 107
5.7
HMM results in three clusters: minor-mode alfabeto songs . . . . . . . . . . . 107
6.1
Distance matrix of all seven books of villanellas by Kapsperger
6.2
Book similarity measuring bigram frequency . . . . . . . . . . . . . . . . . . . 132
6.3
Book similarity measuring bigram frequency . . . . . . . . . . . . . . . . . . . 132
vii
. . . . . . . . 129
LIST OF FIGURES
2.1
Italian alfabeto system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.2
Roman alfabeto system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.3
Marini’s suspensions in the alfabeto system, 1622 . . . . . . . . . . . . . . . .
20
2.4
Foscarini’s alfabeto dissonente, 1640 . . . . . . . . . . . . . . . . . . . . . . . .
21
2.5
Rasgueado (strummed) style, from Abbatessa 1650 . . . . . . . . . . . . . . .
22
2.6
Amat’s circle of alfabeto chords . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.7
Sanz’s layout of alfabeto chords . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.8
Scale harmonizations in durus and mollis, printed in several alfabeto songbooks 28
3.1
Kapsperger’s “Felici gl’Animi” represented in Python syntax . . . . . . . . . .
46
3.2
A continuo representation of Kapsperger’s “Felici gl’Animi” represented in
Python syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.1
Heinichen’s musical circle (1711) in the common-practice system . . . . . . . .
50
4.2
Major and minor key profiles from (Temperley and Marvin 2008); the graph
shows the frequency (in percent) that each scale degree occurs in the Haydn/Mozart string quartet corpus . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
4.3
K-means clustering of pitch frequency in the Palestrina corpus . . . . . . . . .
58
4.4
NEATO reduction of notated mode averages for the Palestrina corpus . . . . .
59
4.5
K-means clustering of pitch frequency in the Bach Corpus . . . . . . . . . . .
60
4.6
NEATO reduction of notated mode averages for the Bach corpus . . . . . . .
61
4.7
K-means clustering of pitch frequency in the alfabeto corpus . . . . . . . . . .
63
4.8
NEATO reduction of notated mode averages for the alfabeto corpus . . . . . .
64
4.9
NEATO graph of key averages for Palestrina, Bach, and alfabeto corpora, labeled with “P,” “B,” “A,” respectively . . . . . . . . . . . . . . . . . . . . . .
65
4.10
K-means clustering of key profiles in the Franco-Flemish School corpus (Masses) 67
4.11
K-means clustering of key profiles in the Franco-Flemish School corpus (motets) 68
viii
4.12
K-means clustering of key profiles in the Franco-Flemish School corpus (secular
songs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
4.13
Scalla di musica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
4.14
Percent of bass harmonizations in the alfabeto corpus . . . . . . . . . . . . . .
73
4.15
K-means clustering of alfabeto letter frequency . . . . . . . . . . . . . . . . .
75
4.16
Clustering of bass harmonization in the alfabeto corpus . . . . . . . . . . . . .
77
4.17
NEATO graph of continuo harmonization in the alfabeto corpus . . . . . . . .
78
4.18
Clustering of bass harmonization: Bach corpus . . . . . . . . . . . . . . . . .
81
4.19
NEATO graph of continuo harmonization in the Bach corpus . . . . . . . . .
82
4.20
Clustering of bass harmonization: Palestrina corpus . . . . . . . . . . . . . . .
84
4.21
NEATO graph of continuo harmonization in the Palestrina corpus . . . . . . .
85
4.22
NEATO graph of continuo harmonization in all corpora; black lines connect
modal pairs; Bach, Palestrina, and alfabeto corpora are labeled with “B,” “P,”
and “A,” respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
4.23
Clustering of bass harmonization: Franco-Flemish Mass corpus . . . . . . . .
87
4.24
Clustering of bass harmonization: Franco-Flemish motet corpus . . . . . . . .
88
4.25
Clustering of bass harmonization: Franco-Flemish secular song corpus . . . . .
89
5.1
Bigram graph of chord motion in the alfabeto corpus . . . . . . . . . . . . . .
98
5.2
Dendrograms of major-mode alfabeto songs . . . . . . . . . . . . . . . . . . . 101
5.3
Information bottleneck graph: alfabeto corpus major . . . . . . . . . . . . . . 104
5.4
A comparison of silhouette scores for 2–15 clusters using different distance metric.110
5.5
Function Fitness of HMM Models . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.6
Three functions in the alfabeto corpus using hidden Markov model . . . . . . 116
5.7
Three functions in the alfabeto corpus using hidden Markov model . . . . . . 117
5.8
Three functions in the alfabeto corpus using hidden Markov model . . . . . . 119
6.1
Bass Harmonization from (Kapsperger [Kapsberger] 1626) in the durus system 124
6.2
Bass Harmonization from (Kapsperger [Kapsberger] 1626) in the mollis system 125
ix
6.3
“Fiorite Valli (Flowering Valleys)” from (Kapsperger [Kapsberger] 1610), first
four measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.4
MDS graph of Table 6.1. Each node represents a book, and distance corresponds to difference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.5
Bigram motion that makes up at least 2% of total motion (major keys only) . 131
6.6
NEATO graph of bigram frequency, Kapsperger corpus . . . . . . . . . . . . . 133
6.7
NEATO graph of trigram frequency, Kapsperger corpus . . . . . . . . . . . . . 134
6.8
Four functions in the major-key songs . . . . . . . . . . . . . . . . . . . . . . 136
6.9
NEATO graph of all books from the alfabeto corpus (major-key songs only),
chord frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.10
A dendrogram comparing all of the books from the alfabeto corpus, calculated
using the Ward method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
A.1
Two functions in the alfabeto corpus using hidden Markov model . . . . . . . 144
A.2
Four functions in the alfabeto corpus using hidden Markov model . . . . . . . 145
A.3
Five functions in the alfabeto corpus using hidden Markov model . . . . . . . 146
A.4
Six functions in the alfabeto corpus using hidden Markov model . . . . . . . . 147
A.5
Seven functions in the alfabeto corpus using hidden Markov model . . . . . . 148
A.6
Eight functions in the alfabeto corpus using hidden Markov model . . . . . . . 149
A.7
Nine functions in the alfabeto corpus using hidden Markov model . . . . . . . 150
A.8
Ten functions in the alfabeto corpus using hidden Markov model
A.9
Eleven functions in the alfabeto corpus using hidden Markov model . . . . . . 152
A.10
Twelve functions in the alfabeto corpus using hidden Markov model . . . . . . 153
A.11
Thirteen functions in the alfabeto corpus using hidden Markov model . . . . . 154
A.12
Fourteen functions in the alfabeto corpus using hidden Markov model . . . . . 155
A.13
Fifteen functions in the alfabeto corpus using hidden Markov model . . . . . . 156
x
. . . . . . . 151
LIST OF MUSICAL EXAMPLES
2.1
The mixed style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.1
“Felici gl’Animi” (Kapsperger 1623)
. . . . . . . . . . . . . . . . . . . . . . .
44
3.2
Transcription of “Felici gl’Animi” (Kapsperger 1623) . . . . . . . . . . . . . .
45
5.1
“Felici gl’Animi” with function emission probabilities (0–1) . . . . . . . . . . . 120
6.1
“Colascione” from Kapsperger’s Libro Quarto di Intavolatura d’Chitarrone . . 122
6.2
“Colascione” from Kapsperger’s Libro Quarto di Intavolatura d’Chitarrone (Transcription) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
xi
ABSTRACT
This dissertation investigates harmonic properties of Italian secular songs with Baroque
guitar accompaniment in alfabeto tablature, which was a system of letters and other symbols
that represented a chord shape to be strummed using all five courses. All of the resulting
chords form major or minor triads. These letters were printed in songbooks for guitarists
to accompany secular songs. This triadic accompaniment has significant implications for
understanding harmonic practices during the seventeenth century.
The purpose of this dissertation is to use computational methods to better understand the
harmonic features of the alfabeto song repertoire, which can perhaps be a window through
which we can understand the broader harmonic practices of secular genres in the early
seventeenth century. Machine learning algorithms are employed to understand the modal
framework and harmonic function of the alfabeto corpus. K-means clustering is used to
determine how many modes are present within the corpus, building upon the recent work
of Albrecht and Huron. To understand the harmonic function within the alfabeto corpus,
hidden Markov models analyze the number of functions and which chords belong to each of
those functions using a similar methodology to Quinn and White.
The analyses also compare the alfabeto corpus to corpora of other genres and eras such as
J.S. Bach, Palestrina, and Franco-Flemish composers. The alfabeto corpus is also compared
with itself to better understand its style and any changes over time. This dissertation
provides a fresh look at harmonic practice in the seventeenth century by using statistical
models that are informed by historical performance practice.
xii
CHAPTER 1
INTRODUCTION AND METHODOLOGY
The seventeenth century saw significant changes in harmonic practice with the rise of monody
and thoroughbass. During the first half of the seventeenth century, there was a wave of
secular songbooks published in Italy. Many of these songs were to be accompanied by the
five-course guitar,1 which had made its way to Italy from Spain. The guitar was relatively
inexpensive, easy to play, and portable, all of which lead to its popularity.
A notation system called “alfabeto” was developed for the guitar. Alfabeto tablature was
a system of letters and other symbols that represented a chord shape to be strummed using
all five courses. All the resulting chords form either major or minor triads. These letters were
printed in songbooks for guitarists to accompany secular songs. This triadic accompaniment
has significant implications for understanding harmonic practices during the seventeenth
century.
The purpose of this dissertation is to use computational methods to better understand the
harmonic features of the alfabeto song repertoire, which provides a valuable perspective on
the broader harmonic practices of secular genres in the early seventeenth century. Chapter 2
will give an overview of alfabeto tablature, how it developed, and the guitar’s role in the rise
of thoroughbass. Chapter 3 will provide a thorough overview of what will be referred to as
the “alfabeto corpus” including how alfabeto tablature was converted into a Python-readable
script, how harmonies are represented, and how other non-alfabeto corpora were processed.
Chapter 4 will use computational methods to determine the modal framework of the alfabeto
corpus—whether the songs are written in many modes or a major-minor paradigm. Given
the results of Chapter 4, Chapter 5 will use machine learning to find harmonic function
within the alfabeto corpus. The studies on modality and harmonic function will compare
1
A “course” is a pair of strings, tuned to the unison or octave, that are to be played together as one
string, much like the modern-day mandolin or twelve-string guitar. I will refer to the five-course guitar as
simply the “guitar” throughout this dissertation.
1
the alfabeto corpus with Palestrina and J.S. Bach, both corpora from Michael Cuthbert’s
music21 (Cuthbert and Ariza 2010). Chapter 6 will focus on analyzing the development
of harmonic practices within the alfabeto corpus given the results from the previous two
chapters.
1.1
Review of Literature
In addition to the analyses, this dissertation explores the harmonic practices in the alfabeto song repertoire and provides a descriptive overview of computational methods used.
For this reason the literature review will be split into sources that deal directly with the
five-course guitar and seventeenth-century issues and sources that deal with computational
approaches.
1.1.1
Alfabeto Tablature
The alfabeto literature and its practices have been well documented by Gary Boye (Boye
2013e; Boye 2013d) and Cory Gavito (Gavito 2006). In this dissertation, I analyzed 522 songs
with alfabeto accompaniment, all of which have been found thanks to their work. Alfabeto
has not received a lot of recent scholarly activity in music theory despite its prevalence in Italy
during the first half of the seventeenth century. However, Richard Hudson (Hudson 1970),
Thomas Christensen (Christensen 1992), Alexander Dean (Dean 2009), and Lex Eisenhardt
(Eisenhardt 2014; Eisenhardt 2015) have explored the alfabeto literature with particular
focus on harmony.
Richard Hudson looks at various harmonic schema that were commonly played on the
5-course guitar. He notes that the introduction of alfabeto notation signaled a merging of
popular and notated styles (Hudson 1970, p. 163). He focuses on the instrumental alfabeto
repertoire, which was typically notated by alfabeto symbols over rhythm slashes and bar
lines, but suggests that similar harmonic practices would be found when accompanying a
voice. While the instrumental notation lacks key signatures, Hudson divides the repertoire
into two “modes” that correspond to the third of the final chord: per B quadro (major triad)
and per B molle (minor triad). He then identifies a number of schemata that are typical
2
of the two given modes. Each schema is viewed as an expansion and alteration of a basic
ritornello chord movement: I→IV→V→I.
Hudson implies a concept of harmonic function within the repertoire, which can be seen
in his analysis of Milanuzzi’s Secondo Scherzo delle Ariose Vaghezze where ♭VII→♭III is
implied to be V→I (Hudson 1970, p. 168). Hudson’s concepts of I→IV→V→I being the
basic schema for each chord progression and of implied functions within the progressions will
be central to this dissertation’s chapter on harmonic function.
Inspired by Hudson, Thomas Christensen discussed the way in which the triadic nature
of the alfabeto system contributed to triadic theory in the early seventeenth century and
influenced later theorists such as François Campion (Christensen 1992, pp. 33–5). While
Christensen provides an example from the alfabeto song repertoire, he focuses more on
alfabeto treatises and how some of the concepts therein bear striking resemblances to tonal
practices over a century later. At the core of his argument is the idea that a triad was
its own entity apart from counterpoint, which allows for the alfabeto tablature system to
exist (Christensen 1992, p. 2). This can be seen in the treatises of Amat, Sanz, Velasco,
and Corbetta. Some of the significant theoretical concepts from these treatises include the
layout of chords in perfect fifths; the separation of chords based on their quality (major and
minor); circular-drawn chord cycles of seconds, thirds, and fifths; and the harmonization of
bass notes in a given key.
While Christensen pays particular attention to Spanish treatises, he acknowledges that
the alfabeto system, despite its Spanish origins, was most thoroughly developed in Italy
where the alfabeto songbooks were a popular phenomenon (Christensen 1992, p. 16). I will
be using Christensen’s narrative of the guitar as a vehicle for harmonic change in Europe to
position the alfabeto song repertoire into its appropriate historical context and to support
the ways in which chords are represented in computer language.
Unlike previous publications, Alexander Dean focuses on the alfabeto song repertoire
and will therefore be an important source for this dissertation. He describes how the oral
Spanish tradition of alfabeto fused with the Italian style of monody and documents the
transmission of alfabeto tablature. He also expands Christensen’s scope with a thorough
3
look at various genres and practices that includes alfabeto, although without the narrative of
alfabeto tablature leading to the rule of the octave (Dean 2009, pp. 16–7). He also documents
the many varieties of bass-note and scale harmonizations that were printed in several alfabeto
publications. I will use computational methods to compare such harmonizations with actual
practice of the alfabeto song repertoire. Rather than insisting that the alfabeto repertoire
points towards common-practice tonality or is somehow a less-than-perfect attempt at it,
Dean concludes that one should look at the repertoire within its historical context (Dean
2009, p. 319). His conclusion is my motivation for a corpus study on this repertoire; I will
be presenting the harmonic practice of the alfabeto repertoire rather than trying to show it
as a way station between the sixteenth and eighteenth century.
Lux Eisenhardt, a Baroque guitarist, has explored the issues that arise from alfabeto
symbols being a realized continuo line (Eisenhardt 2014). He traces the history of alfabeto
research and its increased legitimacy in academic writing (Eisenhardt 2014, p. 73). Likewise,
he traces the history of the strummed (rasgueado/battuta) style from the late sixteenth
century through the eighteenth century with particular focus on the guitar accompanying a
voice without a bass instrument.
In addition to his extensive lists of extant sources for alfabeto songs, Cory Gavito provides
a comprehensive summary of the public opinion of the guitar and the rise of the strummed
style in Italy. He also explores common themes in the titles, dedications, texts, and publishing
of alfabeto songbooks. Gavito makes a compelling case for taking alfabeto songs seriously
and provides a history of scholars dismissing the “light” genre (Gavito 2006, pp. 172–5).
1.1.2
Computational Approaches to Analyzing Harmony
In the following chapters I will use computational methods to determine the number of
modes in the alfabeto corpus and the harmonic function of the alfabeto chords. Computational methods are useful because they can find trends in large numbers of songs that would
be difficult to find otherwise. Of course, the computational results are not intended to be
the end result but rather a starting point from which further discussion about modality and
harmonic function in the alfabeto corpus can continue.
4
David Temperley created key profiles for major and minor modes that are based on a
corpus study of folk song melodies (Temperley 2007). Each key profile is a twelve-member
vector that represents the frequency of each pitch class with the tonic pitch for each song
represented as pitch-class interval 0. He compares these to the cognitively-derived studies
by Krumhansl and Kessler (Krumhansl and Kessler 1982). Other methods of key finding
have been explored,2 but I will be using Temperley’s method for reasons that will become
apparent.
Joshua Albrecht and David Huron use Temperley’s key profile idea to find the number
of “modes” that existed in a given historical period with hopes of finding a trend of many
modes moving to two (major and minor). They use k-means clustering to explore majorminor tonality from 1400–1750 (presented in reverse-chronological order), which was divided
into fifty-year epochs (Albrecht and Huron 2014). Approximately fifty scores were analyzed
per epoch. Only the first and last ten measures are considered, and the final bass note
of each piece is considered the tonic. Key profiles, like those Temperley presented, are
created for each epoch. The data for each song and each epoch are clustered using k-means
clustering, and the data was visualized using multidimensional scaling. Dendrograms were
also used as an alternative methodology to cluster the data, which used Ward’s method to
find distance between scores. The study shows that music prior to the seventeenth century
was built on a multi-modal system while the seventeenth century and later was built upon
a major-minor system. In the earlier conference version of this study, they argued that the
results showed the emergence of the major-minor tonal system (Albrecht and Huron 2012).
This was nuanced for the latter version to include a discussion about the differing styles
and practices. They also call for a larger corpus study that focuses on style. I intend for
my study to greatly increase sample size (more than 500 alfabeto songs) and to focus on a
specific genre.
Christopher White used k-means clustering as a way of grouping together similar composers from a chronological perspective (White 2014). Rather than focusing on pitch frequency, he used chordal n-grams, which are ordered collections of n chords. For example,
2
One significant example is Ian Quinn’s work (Quinn 2010).
5
consider the following progression: I→vi→IV→V→I. The trigrams in the progression would
be: {I→vi→IV}, {vi→IV→V}, {IV→V→I}. White created vectors of most-used trigrams
for chord progressions by several composers and analyzed the data using k-means clustering,
which he fit into a larger discussion of historical practice and tonal styles. He used Quinn’s
key-finding algorithm (Quinn 2010) rather than Temperley’s key profiles.
There are two methods I will be looking at to analyze harmonic function in the alfabeto
corpus. One is called the Information Bottleneck method (Jacoby, Tishby, and Tymoczko
2015); the other method is hidden Markov models (HMMs), which has been used to find
harmonic function by Christopher White and Ian Quinn (Quinn and White 2013; Quinn and
White 2015; White 2013b). While both methods are very similar in results, their underlying
methodologies are philosophically different.
Information bottleneck uses a pairwise histogram of antecedent-consequent chord data
to cluster similar-moving chords together. The information bottleneck is a method in information theory developed by Tishby (Tishby, Pereira, and Bialek 1999). Based on a pairwise
histogram, the algorithm assigns chords to functions and scores the result. Functions are reassigned and scored again. Better scores are retained, and ultimately the results go through a
computational “bottleneck” that squeezes each chord into its most likely function. Function
labels are assigned based on a score of complexity and accuracy as defined by the musical
surface. A good score is one that has a high accuracy score and low complexity score. Results are plotted on a graph along with a line of best fit that shows the maximum accuracy
score given complexity.
Nori Jacoby wrote a program in MATLAB that is available as a web applet.3 Users can
upload pairwise histograms of a corpus and find the harmonic functions from the information
bottleneck method. De Clercq and Temperley’s corpora (De Clercq and Temperley 2011)
and other classical corpora are available for comparison. Users can also submit their own
function label assignments and compare their scores with the computer’s.
The Information Bottleneck method uses concrete clustering, so there is no way for a
chord to belong to more than one function. It also looks at generalized antecedent-consequent
3
http://cluster.norijacoby.com
6
chord movement rather than viewing each chord in its context. The methodology defines
harmonic functions as clusters of similarly-moving chords. In other words, the behavior of
chords creates the functions.
Hidden Markov models are different in that they offer fuzzy clustering, and take each
chord in its local context. For example, a vi chord can belong to tonic or subdominant,
based on context, or even belong in a fuzzy manner to both and act as a transitional chord
from tonic to subdominant. Perhaps the biggest difference between HMMs and Information
Bottleneck is the definition of a function. For HMMs, functions are hidden states that
emit tokens (chords). This implies that chords are a byproduct of a hidden movement
of functions. Function assignments are made through a process of iteration that finds the
transition probability between chords as well as the transition probability between functions.
In addition to these transition probabilities, emission probabilities of functions to chords is
found.
HMMs can suffer from getting stuck in local minima which can either slow down computation or produce misleading results. This can be minimized by having different starting
points (estimated transition probabilities, often randomly assigned) and by rotating and averaging training and testing corpora.4 The problem is often greater with a larger number of
functions.
One of the prominent issues I will focus on is selecting the optimal number of functions.
The primary purpose of White’s and/or Quinn’s work is to find the optimal number of functions in a corpus. This is an interesting question because it can potentially test theoretical
constructions such as the three-function tonic-subdominant-dominant paradigm. There is no
widely-applied method for determining the number of functions (states) when they are not
known. For this section, I will be referring to methodology by Quinn and White (Quinn and
White 2013; Quinn and White 2015; White 2013b), and Deese (Deese 2016), who attempted
to duplicate Quinn and White’s Bach study (Quinn and White 2013). I will simply call these
examples the “Quinn and White method.”
4
For example, Quinn, White, and Deese divide the corpus into fifths, train 4/5 of the data and test the
results on the remaining 1/5. Afterwards, they rotate the fifths. My approach is similar except I randomize
the fifths each time.
7
Quinn and White attempt to find the number of states that produces the most consistent
results. To achieve this, they create several HMMs using a modified Monte Carlo method,5
which involves dividing the corpus into fifths. Eighty percent of the corpus (4/5) is used as
training data, which results in probability information for chord transition, function transition, and chord emission (from the possible functions). The remaining twenty percent (1/5)
is used as testing data, which uses the trained data to label (decode) each chord with a
function.6 This process is repeated 300 times.7 After that is completed, the fifths are shifted
and the process is repeated so that eventually every part of the corpus has been a part of the
training and testing portion. The result is that every chord in each training set is associated
with a 300-member vector that contains function labels. They then create a distance matrix
based on the number of differences the vectors for each chord. For example, consider the
following three possibilities of test results for the following chords:8
Chord:
I:
V:
IV:
ii:
vi:
T
D
S
S
T
Function
T T T
D D D
S S S
S D S
S S T
T
D
S
S
S
The chords I, V, and IV are all maximally different. IV and ii are similar 80% of the
time,9 and so on. Quinn and White use the squared number of differences to create their
difference matrices. For example, the difference between I and V would be 25, while the
difference between IV and ii would be 1.
The difference matrix is clustered using k-means clustering where k is equal to the number
of functions. The silhouette score finds the degree to which the clusters are separate and
5
Broadly speaking, Monte Carlo methods use repeated random sampling to determine a certain statistic
result or measurement. One simple application is to draw a square with a circle inside it, scatter dots over
the surface (pseudo-randomly), count the number of dots inside and outside the circle, and repeat. The
result is a useful way to approximate pi. For more information, see (Kalos and Whitlock 2008).
6
It also provides the likelihood of that choice, which can be interesting for chords that do not cleanly fit
into a single function such as vi.
7
This is true in (White 2013b), while (Quinn and White 2015) only repeats the process 100 times.
8
The actual labels are numbers, and those numbers change from test to test, but I have provided familiar
function labels to better illustrate the idea.
9
Note that these represent single chords in a particular song, not a generalization of such chords.
8
distinct. A high silhouette score would indicate that chords were consistently labeled the
same function. Optimal function numbers are determined by peaks in the silhouette score,
with a peak being defined as a score that is higher than the number of clusters that precedes
and follows it (White 2013b, pp. 194-8).
Quinn and White’s method aims to show the most predictively consistent number of
functions. While I think this is a good starting point, I do not think it necessarily diminishes
the usefulness of other functions. I will be exploring their method of selecting the optimal
number of functions as well as others that exist in the field of information theory.
Finding the “correct” number of states when the number of states is unknown is a current
issue with HMM research. Frequently used methods include the Bayesian information criterion (BIC) and Akaike information criterion (AIC). Both approaches score the likelihood of
the HMM with a penalty for higher numbers of states. These approaches may be problematic
for some solutions, and several others have been proposed in the field of information theory
(Gassiat, Rousseau, et al. 2014; Nam, Aston, and Johansen 2014; Celeux and Durand 2008;
Robert, Ryden, and Titterington 2000).
Panayotis Mavromatis has used HMMs extensively to study rhythm and meter (Mavromatis 2005a; Mavromatis 2009a; Mavromatis 2009b; Mavromatis 2012). He suggests using
the minimum description length (MDL) to find the optimal number of functions. While his
focus is on rhythm, MDL may be useful to determine the number of harmonic functions as
well. Like the information bottleneck method, the purpose of MDL is to reduce entropy so
that there is a balance of model simplicity (fewer functions) and accuracy.
In addition to the sources described above, I have used Kaufman and Rousseeuw’s introductory book for cluster analyses as a guide for creating and analyzing chord data (Kaufman
and Rousseeuw 2009). Readers who are interested in the mathematics of clustering algorithms, distance (difference) measurements, and data analysis are encouraged to refer to
their book.
9
1.2
Methodology and Philosophy
The specifics of my methodologies will be described in detail in the following chapters,
but I will give a brief overview of how the songs are computationally represented and the
philosophical issues therein. The alfabeto corpus and all of the computational methods,
with the exception of the Information Bottleneck, were encoded and executed in Python
3.5. Python modules that were used to create and analyze data include music21 (Cuthbert
and Ariza 2010), scikit-learn (Pedregosa et al. 2011), hmmlearn, NumPy, and SciPy (Jones,
Oliphant, Peterson, et al. 2015). Most of the graphs were made using NetworkX (S. C. North
2004), Graphviz (Ellson et al. 2001), and matplotlib. Some of the more intensive computing
for this project was performed on the High Performance Computing Cluster at the Research
Computing Center at Florida State University.
Robert Gjerdingen describes ways in which one can engage in a “historically informed”
corpus study. He suggests a formula for what constitutes a corpus study (Gjerdingen 2014,
p. 193):
1. define the set of elements believed to occur within the corpus,
2. calculate appropriate statistics on the time series of those elements, and
3. from those statistics, deduce pertinent norms or rules for the corpus.
The first stage often presents the researcher’s bias towards what Gjerdingen terms as “presentist” or “historicist” (Gjerdingen 2014, p. 193). The first of these terms suggests analyzing
with the assumption that the music is being analyzed through a modern lens and may suggest
a kind of evolution from historical music to modern music. The second term, as a contrast,
treats each historical style as its own style with its own rules. These terms are derived from
philosophies of performing “early” music.10
Rather than focusing on this dichotomy, Gjerdingen presents many particular problems in
representing data in corpus studies. Such problems include discerning compound melodies
and limiting the scope of statistics to be sensitive to specific practices (Gjerdingen 2014,
10
Christensen elaborates on the terms “historicist” and “presentist,” under the context of general music
analysis, and the philosophical problems of each approach (Christensen 1993).
10
pp. 194–6). Gjerdingen frequently draws parallels to linguistics to highlight the importance
of viewing musical elements within a context rather than by themselves.
My philosophical method also draws upon the discussion of historical performance practice. Gjerdingen invokes Richard Taruskin’s writing on the issues with “historicist” approaches to performance (Gjerdingen 2014, p. 192). Taruskin points out the intellectual
folly in the search for “authentic” performance practice, noting that our performances will be
modern regardless of the instruments we use. However, this does not diminish the importance
of understanding historical styles, practices, instruments, and theories—being “historically
informed.”
As someone who performs on the lute and theorbo, I am often reminded of Taruskin’s
claim when, despite having a period instrument in my hands, the music was printed from
a computer, lit by electric lights, and the instrument was likely tuned by a digital tuner.
Additionally, I have been surrounded by modern music, theories, and instruments and cannot
become a musician of the seventeenth century. While the methodologies in this dissertation
are undoubtedly “modern,” they are intended to also be historically informed.
I have attempted to adapt the same philosophy to the analytical methods of this dissertation; I will use modern techniques of computation and label chords with Roman numerals
while at the same time being sensitive to historical practices. For example, the label of a
chord that is a perfect fifth above the final cadence as V will not carry the assumption that
it will be followed by I. Likewise the decision to represent pitches in alfabeto chords as pitch
class integers is influenced by the historical reality that there was no single way of tuning
the strings of the guitar.11
1.2.1
The Significance of the Alfabeto System for Computational
Methods
For computational analyses of harmony, one must answer the following question: what
is a chord? Or more specifically, how can a computer determine what constitutes a chord?
These fundamental questions are difficult to answer, especially depending on repertoire. One
11
For a thorough discussion of tuning on the five-course guitar, see the sixth chapter, “Stringing Matters,”
from (Eisenhardt 2015, pp. 124–149).
11
approach is the salami-slice method where each vertical sonority is counted as a chord.12
However, few would argue that every harmonic instance is a discrete chord, not to mention
that every chord is of equal importance.
Previous approaches such as those by White (White 2013b; White 2015), Quinn (Quinn
and White 2013), and Tymoczko13 have some way of determining chords from other harmonies. Such methods include omitting harmonies with fewer than three notes, omitting
harmonies that do not make up a certain percent threshold of the entire corpus, and/or
selecting only tertian chords. If chords can be separated from non-chords, there is the additional problem of assigning function, if appropriate, to the chords. While this process is easy
for most analysts, it can become computationally difficult, as David Temperley explains:
For many musicians and certainly most theorists, performing harmonic analysis
is a trivial task, requiring little thought or effort. This might lead one to suppose
that the principles behind it are simple and straightforward. However, as work
in other areas of psychology (e.g., speech perception and vision) has shown, tasks
that are performed effortlessly by humans often prove to be highly subtle and
complex. (Temperley 1997)
Temperley provides an elegant solution, as do the other authors, but it is computationally
heavy and makes some epistemological assumptions of what ought to be a chord.
Another approach is for the analyst to analyze each piece, such as the approach by Trevor
de Clercq and Temperley (Temperley and Clercq 2013; De Clercq and Temperley 2011). This
results in a smaller corpus and more work on the part of the analysts. It also increases the
bias of the analyst and removes the consistent algorithm-based approach to finding chords.
Ian Quinn and Christopher White have used songs where the chords were labeled by other
people from the McGill-Billboard and HookTheory corpora (Quinn and White 2015). While
this method potentially allows for a larger and more easily built corpus, it typically relies on
untrained and anonymous musicians to label the chords.
12
In music21, the method for this is “chordify,” where an entire score is compressed into chords on a single
staff. Note durations, barlines, and register can be preserved. (Cuthbert and Ariza 2010)
13
Several unpublished chordal data can be found in (Jacoby, Tishby, and Tymoczko 2015).
12
All of the examples so far assume that the notes for each harmony are given. Consider
early basso continuo sources where Figures were scant, inconsistent, or altogether missing.
At times, omitting Figures was intentional so as not to offend the professional musician as
Milanuzzi stated:
...sharps have not been placed, nor numbers marked where expected in the basso
continuo, taking for granted the accomplishment and virtuosity of those who will
play them, [while] at the same time keeping an eye on the vocal line.14
This entails a lot of guesswork on behalf of the analyst. Accompaniment from a thoroughbass was very much embedded within a tradition of performance practice, and it would be
another century before any règle de l’octave came into fashion.15 An analyst would therefore
need to determine what harmonies, if any, should be paired with a given bass note and only
then answer the question regarding the presence of a chord.
Alfabeto notation bypasses most of these problems. Each symbol can be considered a
chord, and one that was written by the composer (or possibly publisher). While consideration is still needed to transform chords into Roman numerals or other functional labels (if
appropriate), the ontology of a chord is clear and not a result of human or computer analysis.
It also allows for a way to analyze the harmony of early continuo that has previously been
inaccessible because alfabeto chords provide an example of thorough-bass accompaniment.
However, this does not mean that the way a guitar would accompany a thorough-bass is the
same way a keyboard or even a different fretted instrument would.
Giovanni Kapsperger [Ger. Johannes Kapsberger],16 a German-born lutenist and theorbist who made his career in Italy and Rome, published two books of villanellas that include
alfabeto chords and theorbo tablature (Kapsperger [Kapsberger] 1610; Kapsperger [Kapsberger] 1619b). There is often disagreement among the two types of accompaniment: the
theorbo often adds chordal sevenths, has proper voice leading and inversions, and sometimes
14
Translation provided by Giulia Nuti (Nuti 2007, p. 36). Original: Come anco non si son posti di
Diesis, ne signati i Numeri ne lor proprii luoghi del Basso Continuo, presupponendosi l’accortezza, e virtuosa
maniera di couli, che le sonerá, havendo l’occhio alla parte che canta. (Milanuzzi 1622)
15
However, as will be shown in following chapters, a scale harmonization resembling what would become
the règle de l’octave existed and was printed in several alfabeto songbooks.
16
More information about Kapsperger can be found in Chapter 3.
13
plays a completely different chord or thins the texture to thirds while the guitar plays complete triads. It is likely that a keyboard player would harmonize the bass in a different way
altogether, although it is likely all three instruments would agree more than disagree. Sometimes harmonies would change from one instrument to another for the sake of the beauty of
the music, considering Milanuzzi again:
The affetto rendered by the chitarrone or the spinet being different from that of
the chitarra alla spagnola in playing these arias, in many places I have changed
the note given to the chitarra from the one that is assigned in the basso fondamentale, for the other instruments; all [of this is] to make it more beautiful.17
The differences in continuo realization, which will be further explored in Chapter 6, are
relatively small and isolated within the larger scope of the alfabeto corpus. It is therefore
unlikely that there would be a significant statistical change in overall harmonic practices
between any given instruments. I think other differences would also arise within the improvisatory tradition of seventeenth-century accompaniment; isolated harmonies, inversions,
and suspensions may change, but the overall harmonic framework would not.
It should also be noted the striking similarity between alfabeto tablature and the contemporary fakebook publications and internet guitar websites that place guitar chords on top of
popular songs. Similar problems also arise; the chords may have been placed on the music by
the publisher without direct input from the composer, the chords are likely simplifications
of a more complicated accompaniment pattern, but the chords should be similar enough to
the recorded version to at least be recognizable.
The results of the analyses in this dissertation should therefore be interpreted as follows:
the large-scale harmonic trends accurately reflect the corpus, but there is some wiggle room
in the small details.
The code created for this dissertation and the encoded alfabeto corpus is available for
download from GitHub: https://github.com/tompkinsguitar/ and will be updated if more
alfabeto sources become available.
17
English translation by Giulia Nuti (Nuti 2007). Original: Avvertendovi che per esser vario l’affetto, che
rende il Chitarrone ’o Spinetta, da quello della Chitarra alla Spagnola nel sonar queste Ariette, in molti
luoghi h’o variata la notta nella ditta Chitarra da quella, che ’e assignata nel Basso fondamentale, posta per
gl’altri stromenti, il tutto fatto per dar gli maggior vaghezza. (Milanuzzi 1622)
14
CHAPTER 2
THE BAROQUE GUITAR, ITS NOTATION,
AND ITS PLACE IN MUSIC THEORY
“The Guitarre was never so much in
use & credit as it is at this day ...
every body knows it to be an imperfect
Instrument & yet finding upon
experience how agreeable a part it
bears in a consort...”
Nicola Matteis, The False
Consonances of Music, Or Instructions
for Playing a True Basse Upon The
Guitarre... (Matteis 1682, a.2.)
One of the many fascinating aspects of the five-course guitar is how it transformed from
a folk instrument associated with dance music, taverns, and barbers in Spain to an integral
part of basso continuo and vocal accompaniment in courts throughout Europe (Dean 2009,
p. 4). In all such settings, the guitar was almost always strummed in what is known as
the rasgueado style.1 The guitar was thus almost exclusively a chordal instrument, unlike
lutes and other fretted instruments that were plucked and capable of executing multi-voice
counterpoint. While it was active in the Spanish oral tradition prior to the seventeenth
century, it was in seventeenth-century Italy where the rasgueado tradition became notated
and standardized.
The role of the guitar in musical circles is a complicated one. Alfabeto songs were mostly
considered low-brow vernacular music, but they were often commissioned and composed
by individuals who were also responsible for more “sophisticated” musical genres. Scipione
Cerreto captured this conflicted status of the guitar in his 1608 Dell’arbore musicale:
There are also players of the Spanish guitar, who belong to the same status [as
the players of the double harp and lute]; notwithstanding that [the guitar] was
1
For a detailed explanation of this style, see (Boye 2013d).
15
used by people of low status and those of little worth, not to mention Buffoons
who have made use of it at banquets.2
Cerreto was not alone in his assertion that the lute was a more sophisticated instrument
than the guitar. In 1628, Vincenzo Giustianiani lamented that the introduction of the guitar
and theorbo “conspired to disband the lute completely, and they have nearly succeeded, as
similarly the fashion of dressing in the Spanish manner prevails over all other styles in Italy”
(Gavito 2006, p. 3). Despite flourishing on its own in Italy and eventually other places, the
guitar was still associated with its Spanish oral linage, most often being called “la chitarra
alla spagnola.” The guitar did not enjoy a particularly high social status in Spain either, as
vihuelist Sebastián de Covarrubias lamented in 1611:
Until now, this instrument (the vihuela) was highly valued and attracted excellent
musicians, but since the guitar was invented, very few devote themselves to the
study of the vihuela. This has been a great loss, because all kinds of music
were played on it, whereas the guitar is no more than a cowbell, so easy to play,
especially in rasgueado, that there is not a stable boy who is not a musician on
the guitar.3
Such a reputation can also be found in popular fiction, as the character Sancho Panza
declares in the sequel to Don Quixote from 1614:
Ah, my donkey, how I remember...when you breathed in, you gave a gracious
syllable, answered by a low G from your hindquarters, which made damned
better music than did my village barber’s guitar as he sang passacalles by night!4
2
“Vi sono ancora gli Sonatore di Chitarra alla Spagnola, alliquali si ben li tocca l’istesso grado, nondimeno
per essere state usata de gente basse, e di poco valore non dico da Boffoni, liquali se ne hanno servito ne i
conviti.” from (Scipione 1608, p. 37) translated by Cory Gavito (Gavito 2006, p. 14).
3
“Este instrumento ha sido hasta nuestros tiempos muy estimado, y ha avido exelentissimos músicos;
pero dispués que se inventaron las guitarras, son muy pocos los que se dan al estudio de la vigùela. Ha sido
una gran pérdida, porque en ella se ponía todo género de música puntada, y aora la guitarra no es más que
un cencerro, tan fácil de nañer, especialmente en lo rasgado, que no ay moco de cavallos que no sea música
de guitarra.”Sebastián de Covarrubias, Tesoro del lengua Castellana a Española, (Madrid, 1611); translated
by Lux Eisenhardt (Eisenhardt 2015, p. 15).
4
“¡Ay, asino mio, y cómo tengo en la memoria que cuando te iba á echar de comer á la cabelleriza, en
viendo cerner la cebada, rebuznabas y reias con una gracia como si fueras persona; y cuando respirabas
hácia dentro, dabas un gracioso silbo, respondiendo por el órgano traserto con un gamut, que ¡mal año! para
la guitarra del barbero de mi lugar que mejor música haga cuando canta ei pasacalle de noche!” Alonso
Fernández de Avalleneda, Segundo tomo del ingenioso hidalgo Don Quixote de la Mancha (Tarragona, 1614);
translated by Alexander Dean (Dean 2009, p. 4)
16
Alexander Dean suggests that the five-course guitar’s origin in the Spanish oral tradition
remained influential in the way it was played and the harmonies that were used in Italy and
beyond (Dean 2009). Nevertheless, Italy is where the guitar’s harmonic idiom was standardized and codified so that even Spanish guitarist Gaspar Sanz traveled to Italy to learn about
the instrument (Christensen 1992, p. 16). The guitar also became popular in Paris including
many French composers such as Marin Marais, Jean Baptiste de Lully, Guillaume Nivers,
and Robert de Visée. Even Louis XIV played the guitar. It spread to England too after the
Restoration and nearly replaced the lute by the end of the seventeenth century (Christensen
1992, p. 17). In addition to the published songs and dances with Baroque guitar accompaniment, several treatises for playing the Baroque guitar were published in Italy, France,
England, and Spain.5
It should be noted that not every alfabeto publication was secular, although the vast
majority were. One example, which can be found in Gary Boye’s database, is Canzonette
spirituali, e morali, che si cantano nell’Oratorio di Chiavenna, eretto sotto la protettione
di S. Filippo Neri. Accomodate per cantar à 1. 2. 3. voci come più piace, con le lettere
della chitarra sopra arie communi e nuove date in luce per trattenimento spirituale d’ogni
persona, which was edited by Filippo Neri and published in Milan in 1657 (Neri 1657). This
publication is not included in the alfabeto corpus, but it would be interesting to see how a
largely secular genre is treated in a sacred setting.
2.1
Alfabeto Notation
Music for the lute, four-course guitar, and vihuela used tablature systems for their primary notation in which horizontal lines represented a string or course on the instrument while
letters or numbers represented fret placement.6 Tablature was useful because it showed the
5
A table of these treatises by country can be found in Table 1 from (Christensen 1992, pp. 22-3).
Italian tablature used numbers while French tablature used letters. An early tablature referred to as
Neapolitan tablature used numbers except without zero being used for open strings, which means the first
fret would be numbered “2.” This was because the number zero had yet to catch on in Europe (Seife
2000, p. 81). Yet another tablature system referred to as German tablature used a different letter for each
string/fret combination, so there was no need for horizontal lines representing strings. However, the system
was complicated and not used very extensively, especially outside Germany.
6
17
player exactly where to put his or her fingers. Standard notation was more difficult because
a single pitch often exists in multiple places on the instrument; tablature eliminates the
unnecessary number of choices.
The five-course guitar was not typically a melodic or contrapuntal instrument but a
strummed, accompaniment instrument. All five courses would be strummed while the other
hand held down a chord in a technique called the strummed or rasgueado style. Since all
five courses were always used, a different system of notation was developed that eliminated
the need for the horizontal lines. That notation was called alfabeto tablature—a system of
letters and sometimes other symbols, each of which represented a specific hand shape that
made a major or minor triad. These letters would be placed above a vocal part to accompany
one to four singers or small ensemble.
Figure 2.1 shows a decoder of the alfabeto systems that was commonly placed in Italian
songbooks. Each letter is accompanied with its fingering in Italian tablature below.7 Figure 2.2 shows an alfabeto decoder from Kapsperger’s second book of villanellas (Kapsperger
[Kapsberger] 1619a), which was typical of songbooks published in Rome. There are some
slight differences between the two systems, the most notable of which is the “+” symbol
from the Italian decoder is the “X” symbol in the Roman decoder. The other differences
exist in the latter part of the system and are rarely used.8
2.1.1
Limits of Alfabeto and its Extensions
Although alfabeto tablature was groundbreaking in its exclusive use of triads, it was a
very limited system. There were no chords apart from major and minor triads, which could
pose some issues when trying to harmonize 7̂. This may have eventually led to the demise
of the entire system, but before it became obsolete, several composers extended the system
to keep up with more complex harmonies that were developing in the seventeenth century.
Guitarists were not completely restricted to the alfabeto symbols. Numbers could be
placed above or beside a shape, which would indicate to a guitarist to move the notated
7
Note that for Italian tablature, the highest-sounding course is represented by the bottom line, which is
upside-down from modern tablature and French tablature.
8
The complete differences between the Italian and Roman systems are documented by Cory Gavito
(Gavito 2006, pp. 78–9).
18
(a) Typical Italian alfabeto instructions, this from (Milanuzzi 1622)
Symbol
Chord
Sym.
Crd.
+ A
e G
O P
g f
B C D E
C D a d
Q R S T
F♯ B E A
F G
E F
V X
f♯ b
H I K L M N
B♭ A b♭ c E♭ A♭
Y Z & 9
G C C♯ E F
(b) Alfabeto symbols in modern chord format
Figure 2.1: Alfabeto letters decoded in Italian tablature (horizontal lines represent strings,
numbers represent frets); string layout from top line to bottom is A–D–G–B–E
Figure 2.2: Alfabeto instructions from (Kapsperger [Kapsberger] 1619a), typical of all Roman
sources
19
Figure 2.3: Marini’s suspensions in the alfabeto system, 1622
shape up to the indicated fret number.9 While this practice was common for the solo guitar
literature, it is only present in a few sources in the alfabeto corpus, all of which were composed
by Giovanni Stefani (Stefani 1621; Stefani 1622; Stefani 1623). Another practice known as
alfabeto dissonante, alfabeto falso, or lettere tagliate allowed for dissonant harmonies. An
early example of these are used by Biagio Marini, as seen in Figure 2.3 (Marini 1622), but
only as suspensions that resolve to the next chord. For example, a D chord (alfabeto symbol
C) may be preceded by alfabeto shape C*. These two together make a 4-3 suspension. These
are almost exclusively reserved for cadences and are only used when idiomatic to the guitar.
Marini also included a seventh for the G chord (alfabeto symbol A).
Giovanni Foscarini’s Li cinque libri della chitarra alla spagnola (Rome, 1640) illustrates
several alfabeto dissonante chords that go beyond suspensions (see Figure 2.4). The music,
however, is only notated for guitar.10 None of the songs in the alfabeto corpus use Foscarini’s
alfabeto dissonante, and just like the transposed alfabeto symbols, the alfabeto chords for
9
These were only notated for closed (barre) chords where all strings were fretted. The number typically
means to move the lowest-fretted finger to the fret that corresponds to that number. In terms of semitone
transposition, it can be thought of the shape being transposed n − 1 semitones.
10
For a very thorough description of Foscarini’s book, see (Boye 2013b).
20
Figure 2.4: Foscarini’s alfabeto dissonente, 1640
vocal accompaniment are far less adventurous than those for solo guitar. Foscarini’s book
was published as the popularity of alfabeto accompaniment for voices was slowing down. It
is of course quite possible that guitarists used some of these embellishing chords and that
the notated alfabeto symbols were just a skeletal structure of possible accompaniment—easy
enough for amateurs and a useful outline from which experts can improvise more complex
accompaniment.11
Although the alfabeto system was an economical replacement for tablature, it eventually
found itself inserted back into the tablature system in what is known as the mixed style,12 ,
which can be seen in Figure 2.1. The result is a fusion of Italian tablature and the alfabeto
system, which appeared in the 1630s. This hybrid style of tablature allowed guitarists to
play both melody and accompaniment as a soloist. The horizontal lines and numbers are
identical to Italian tablature described earlier, and the rhythm markings show how long to
hold the note underneath it.13 The alfabeto symbols indicate places where a chord is to
be strummed and simplifies the tablature, because the publisher no longer needs to specify
the fret placement for each string. This undoubtedly helped guitarists too. The strumming
direction was indicated by vertical dashes above and below the bottom horizontal line; dashes
below the bottom line indicate down strums, and those above the bottom line indicate up
11
From an analytical point of view, the alfabeto dissonante serve to embellish the triads, even thought the
voice leading is often very erratic. One could argue that the regular alfabeto chords are more structurally
important than alfabeto dissonante, although this issue is inconsequential for my current purposes.
12
It. battute e pizzicate which literally means strummed and plucked (Boye 2013c).
13
A rhythm marking lasts until another one is present. The eighth note after the B chord applies to all of
the notes until the D chord.
21
Figure 2.5: Rasgueado (strummed) style, from Abbatessa 1650
Example 2.1: An example of the mixed style, from Bartolotti (from (Boye 2013c))
strums.14 This mixed style allowed the solo literature for guitar to blossom; guitarists could
play melodies and provide their own chordal accompaniment. The style was not used for vocal
accompaniment, but it can provide a glimpse into how guitarists may have embellished the
chordal framework, and it likely influenced how guitarists accompanied figured bass, which
is discussed at the end of this chapter.
2.2
Impact on Triadic Thought
As mentioned above, each alfabeto symbol makes a major or minor triad. There are
no diminished chords, suspensions, or seventh chords, and the inversion of each chord was
inconsequential and could change from guitar to guitar because tuning was not standardized.
This meant that guitarists had some understanding of chordal roots. The limitations of five
courses and that they must all be strummed for each chord invited a system where chords
could be understood as building blocks of composition rather than a result of counterpoint.
14
Boye again provides a comprehensive list of sources that use the mixed style as well as instructions for
how to read it (Boye 2013c).
22
If one follows the voice leading of a given chord progression (and chooses one of many possible
tunings), the voice leading is fraught with parallel fifths and octaves and unresolved leading
tones, not to mention the issue of inversions.
Thomas Christensen wrote about the significance of the triadic theory in several alfabeto
treatises and traces a lineage from early alfabeto sources to Campion’s Rule of the Octave,
which paves the way for Rameau’s triadic theories (Christensen 1992). His focus is primarily
on Spanish treatises, because while Italy was the main hub of alfabeto song publications,
almost all of the theoretical writing about the alfabeto system came from Spain.
The first known publication of alfabeto tablature was in Joan Carlos Amat’s Guitarra
Española (Amat c. 1761), which was originally published in 1596 and was reprinted through
the beginning of the nineteenth century (Eisenhardt 2015, p. 1). Guitarra Española is a
remarkable treatise for several reasons, but perhaps the most striking is the circle of chords
laid out in fifths, which can be seen in Figure 2.6 and is transcribed into modern chord
notation in Table 2.1.
While Amat’s circle may look very similar to Heinichen’s circle of fifths, it is important to
note that Amat’s circle is only showing chord relationships, not key relationships. Nevertheless, the triadic focus of the circle seems very forward thinking. Amat also claims to be able
to accompany a five-part piece by Palestrina using his numerical chord system (Christensen
1992, p. 26).
It is also important to note the differences between Amat’s alfabeto notation and the
Italian style. Amat used numbers—1–12 major or minor triads—rather than letters. His
numbers were also laid out in fifths while the Italian system was likely arranged by frequency
of use. Amat’s notation did not catch on even in Spain, where the Italian alfabeto notation
became standard despite the tradition originating in Spain.
Spanish guitarist Gaspar Sanz, who studied the guitar in Italy and showed a strong command of its practice, provided a different alfabeto chord layout than the Italian sources.
While each symbol represents the same fingering as the Italian alfabeto system (see Figure 2.1), he changes the layout significantly. The Italian layout is probably ordered by most
used—the chords on the far right are rarely, if ever, used. Sanz’s layout, which can be seen
23
Figure 2.6: Amat’s circle of alfabeto chords
24
Table 2.1: Major and minor triads in Amat’s musical circle
Number
N
B
1
E
e
2
A
a
3
D
d
4
G
g
5
C
c
6
F
f
7
B♭
b♭
8
E♭
e♭
9
A♭
a♭
10
D♭/C♯ d♭/c♯
11
F♯
f♯
12
B
b
in Figure 2.7 and is translated into chord symbols in Table 2.2, is in a line of perfect fifths
with parallel major and minor chords grouped together. It is an elegant blend of Amat’s
layout and the Italian tradition of alfabeto letters (rather than Amat’s numbers). Sanz also
shows the transposed chord shapes as relating to their non-transposed types; for example,
F and M2 (seen on the right side of the 1 column in Figure 2.7) are different fingerings of
the same chord with the latter being on the upper frets of the guitar and the former on the
lower. He also displays an understanding of how to change the mode of a chord. Column
9 of Figure 2.7 displays chords N and N with a “+” sign (cruce) on top. The cruce, likely
adapted from the E minor chord in column 1, indicates to turn a major chord minor. This
mode-changing cruce is mixed with transposition numbers in many other chords in his chart.
2.2.1
The Scalla di Musica and Alfabeto Performance
Perhaps in order to assist amateur guitarists accompany a bass line with alfabeto chords,
a simple guide was often printed at the beginning of alfabeto songbooks called the scalla di
musica (see Figure 2.8). It pairs a bass note with an alfabeto chord. The scalla includes
harmonizations in two signatures, durus and mollis, but does not acknowledge the many
different finals—or even any in a minor mode. The durus scale is in G while the mollis is
in F. Some of the harmonizations also seem unusual, such as the major triad (E) above the
25
Figure 2.7: Sanz’s layout of alfabeto chords
Table 2.2: Parallel major and minor chords in Sanz’s alfabeto layout
Number Chords
1
e, E
2
a, A
3
d, D
4
g, G
5
c, C
6
f, F
7
b♭, B♭
8
e♭, E♭
9
g♯, A♭
10
c♯, D♭
11
f♯, F♯
12
b, B
26
sixth scale degree in the durus scale. It is my hypothesis that the harmonizations are general
suggestions for what one should do with a bass note in a given system (durus/mollis). This
hypothesis will be tested in Chapter 4.
The inclusion of the scalla di musica in several alfabeto sources is what Eisenhardt,
Dean, and Christensen view as pointing towards Campion’s rule of the octave a century
later, although the authors agree that the intent of the scalla is different from Campion’s
rule. Nevertheless, the scalla clearly implies that the guitar was indeed considered a continuo
instrument associated with the bass rather than only with the melody. This is not to say
that alfabeto chords disregarded the melody as the scalla would imply; Milanuzzi indicated
that he altered some of the alfabeto symbols to better fit the charm of the music (Eisenhardt
2015, p. 56).
All of the alfabeto songbook publications that have a printed scalla refer to durus and
mollis as the signature of no flats or one flat. However, some other guitar sources refer to
durus and mollis as alfabeto chords with a major or minor third, respectively. This method
teaches the guitarist how to match a major or minor harmony to a given bass note with no
regard to “key.”
One question that has to be considered is how much performers would have relied on
the alfabeto symbols. Dean suggests three levels of alfabeto accompaniment (Dean 2009,
pp. 224–5):
1. “Simple alfabeto”: the performer would rely on the alfabeto tablature and perhaps
melody with no regard to the continuo line;
2. “Informed alfabeto”: the performer would read from the alfabeto tablature but would
be able to add suspensions and other things from the continuo; and
3. “Complete realization”: the performer would read completely from the continuo line
without regard to the alfabeto tablature.
The scalla may have been a way of helping a performer move to different levels of alfabeto
accompaniment by associating bass notes with alfabeto chords, even if the method is rather
simplistic. While the scalla is clearly an imperfect method for realizing continuo, it serves
27
(a) Original from (Obizzi 1627)
G
Am
5
3
Scalla di Musica per B. Quadro.
5
3
F
C
D
E
F
G
6
3
5
3
5
3
5
3
5
3
5
3
5
3
Scalla di Musica per B. Molle.
Gm
5
3
G
5
3
Am
F
Am
B♭
C
Dm
E
Gm
5
3
5
3
5
3
5
3
5
3
5
3
5
3
(b) In modern notation
Figure 2.8: Scale harmonizations in durus and mollis, printed in several alfabeto songbooks
28
as an important relic of the development of triadic theory. It also, like alfabeto tablature,
shows how amateurs may have learned to harmonize a bass line.
2.2.2
Continuo: Guitar Accompaniment Beyond Alfabeto
The alfabeto accompaniment of vocal music is an early example of realized basso continuo. Each chord matches a bass note, and the bass and guitar provide the accompaniment
for the vocalist(s). Given the scant and non-standardized figures from the early seventeenthcentury bass lines, the alfabeto corpus can answer important questions on how various bass
progressions would be harmonized—or at least how a guitarist would do it. Alfabeto tablature was not included in many continuo sources after mid-seventeenth century, but the guitar
continued to be used in continuo settings well into the eighteenth century.15 It is likely that
more skilled guitarists were already reading from the bass line rather than alfabeto symbols,
which were used by amateurs.
The most comprehensive treatise for using the guitar for continuo realization is Nicola
Matteis’s The False Consonances of Musick or Instructions for Playing the True Base Upon
the Guitarre... (Matteis 1682). While acknowledging the guitar as an imperfect instrument
for continuo realization, he uses alfabeto tablature and continuo notation to instruct guitarists in the art of realizing a figured bass.16 Perhaps most striking in his treatise is how he
insists that the guitarist must play both the bass and the harmony even though the guitar
only contained five courses and was not well equipped to play bass lines. Also unlike the
alfabeto tradition, Matteis recommends proper voice leading and implies that most guitarists
execute it incorrectly.
The guitar was a common continuo instrument throughout Europe, and it is plausible
that the alfabeto system and the scalla di musica would have been a guitarist’s way of
learning more advanced continuo. In such a case, it is not just that the alfabeto system was
15
Eisenhardt gives an overview of the transition from rasgueado to continuo.
His examples contain much more detailed and complicated figures than any that exist in the alfabeto
corpus. This is partly because Matteis’s treatise was published decades later and also because the songs in
the alfabeto corpus are simple and do not rely on counterpoint.
16
29
an example of early realized continuo, but that it directly influenced the development and
performance of continuo.17
2.3
Conclusion
The five-course guitar and its notation have significant implications for understanding
harmonic practices in the seventeenth century, and the sheer popularity of alfabeto songbooks
indicates that guitar songs were very much in the air—its harmonies unavoidable in Italy.
Any theoretical writing in the guitar publications—such as the scalla di musica and Amat’s
chord chart—were included for practical purposes, yet it is remarkable in hindsight how much
those practical writings can speak to performance practice, pedagogy, and triadic theory.
17
Dean makes the case that the rasgueado style of guitar accompaniment influenced not only plucked-string
accompaniment but keyboard accompaniment as well (Dean 2009, pp. 265–7).
30
CHAPTER 3
MAKING THE ALFABETO CORPUS
“These are my songs, now to appear in
the theater of the world...”
Crescentio Salzilli, 16161
Italian secular songs with alfabeto accompaniment may seem like a rather obscure corner
of music history, but such publications accounted for over half of all published secular music
in many of the years between 1620 and 1640 (Eisenhardt 2015, p. 20). Given its popularity
and prolific spread through Europe, the guitar undoubtedly had an impact on how music
was written, understood, and performed. Perhaps most importantly, the alfabeto tablature
provides a realized continuo harmonization during a time when rules were not codified. The
guitar fully separated counterpoint from harmony, and its topology helped separate bass
notes from chord roots. This chapter presents the composers, publishers, cities, and genres
that are used for this project.
The alfabeto corpus consists of over five hundred songs published in Italy between 1610
and 1657—the majority in the 1620s. To be included in the alfabeto corpus, each song must:
1. have been published in Italy in the seventeenth century;
2. have at least one vocal part;
3. have a bass part (with or without figures);
4. have alfabeto symbols on the score;
5. be complete and readable; and
6. have a distinguishable final cadence and key signature (mollis or durus).
1
(Salzilli 1616b), “Queste mie canzonette, che hora compariscono nel teatro del mondo...”
31
This list leaves out the entire body of instrumental alfabeto guitar works as well as
the few instrumental ensemble publications with alfabeto tablature. This is because the
instrumental publications do not include key signatures or bass parts and will therefore be
unable to contribute to the study of basso continuo and modes in the same way the vocal
repertoire can.
A complete listing of alfabeto songbooks published in Italy (including Rome) can be found
in Cory Gavito’s dissertation (Gavito 2006, Appendix 1). Boye also maintains an online
database of tablature for guitar, lute, and theorbo, and was the primary source from which
I was able to locate the sources for the alfabeto corpus (Boye 2013e). Several sources listed
in Boye’s database are lost, missing, or unreadable, but the majority met the requirements
above and are included in the alfabeto corpus. A complete list of the corpus can be found
in the bibliography.
3.1
Composers and Publishers
There is some debate as to who placed the alfabeto tablature in the Italian songbooks.
It has been asserted that alfabeto chords were placed by publishers rather than composers,
or that perhaps only composers who were also guitarists had a hand in including alfabeto
(Nuti 2007). Eisenhardt questions this assumption based on what he calls the “amateurish
approach to basso continuo” that is found in the alfabeto chords (Eisenhardt 2015, p. 58).
As an example, he presents the differences between the continuo realizations of the theorbo
and guitar in Kapsperger’s third book of villanellas, but it is possible this is due to differing
idiomatic movements of the respective instruments. It is also possible that the theorbo and
alfabeto tablatures were intended to be options for harmonization rather than to be played
together.2
Alfabeto tablature was met with some sarcasm from seventeenth-century lutenist Bellerofonte Castaldi:
Please do not turn away because the Author, most well knowing how to do it, did
not place the A.B.Cs of the Spanish Guitar above each one of these Airs, as one
2
This discrepancy will be discussed in more detail in Chapter 6.
32
does according to current usage. This would have been done if one had not seen
that such pedantry is of little use to those who don’t know (if the letters are not
discarded) of the innumerable errors that occur at the cadences because of the
aforementioned hieroglyphs. He who knows how does not need to be taught.3
Regardless of who put the alfabeto tablature on the page, songbooks with alfabeto tablature were extremely popular in Italy. I do not think the significance of alfabeto tablature
is diminished if it was placed by the publisher rather than composer, but I will instead be
sure to acknowledge the publishers when providing an overview of the alfabeto corpus.
The method of how composers and/or publishers chose alfabeto chords for accompaniment has been debated. Eisenhardt suggests three possible strategies (Eisenhardt 2014,
p. 77):
1. use a scale triad system such as the scalla di musica;
2. try to fit standard vernacular chord progressions, such as the passacaglia; onto the
melody
3. use rules of counterpoint.
The analyses in later chapters will compare the scalla di musica with actual alfabeto
compositional practice and will also discuss the other two options. Overall, the alfabeto fits
the continuo line well enough considering the limits of the system.
The following sections provide an overview of all publications that were included in the
alfabeto corpus, presented by place of publication. Any significant irregularities found in the
process of inputting a publication will be addressed. Also, any books that seem to realize
continuo much better or worse than the alfabeto corpus as a whole will be discussed. A more
compact list of the alfabeto corpus, ordered by composer, is provided in a separate section
of the bibliography.
3
(Castaldi 1623); translated in (Gavito 2006, p. 148); also cited in (Eisenhardt 2015, p. 58). “Il quale
digratia non si torca, perche l’Autore, come benissimo sà fare, non habbia messo l’A.B.C. della Chitarra
Spagnolissima sopra ciascheduna di quest’ Arie che si faria pur anch’egli lasciato poortare a seconda dal uso
moderno, s’ei non si fosse accorto che poco serve simil Pedanteria a chi non sà se non scartazzare, per mille
spropositi che ne le cadenze occorrono mediante il geroglifico sudetto, e colui che sà non ha bisogno che se
gl’insegni.”
33
3.1.1
Rome
Publications from Rome span from the first publication included in the alfabeto corpus
in 1610 to the last in 1652. As discussed in Chapter 2, the publications from Rome had a
slightly different alfabeto decoder than those from other cities. This does not likely affect the
chord progression but it is important to know about when converting symbols from alfabeto
tablature to chords, else all intended E minor chords would become B minor chords.4 Below
is an overview of all of the songbooks in the alfabeto corpus that were published in Rome.
Kapsperger. Giovanni Girolamo Kapsperger [Ger. Kapsberger] was a German born
composer who made his career in Rome as a composer, lutenist, theorbist, and guitarist
who was hailed by Kircher as the successor to Monteverdi and worked with other esteemed
composers such as Frescobaldi (Coelho 2016). He was a prolific publisher in sacred, dramatic,
and secular genres and developed virtuosic repertoire for solo theorbo in four publications
of toccatas and dances.
He published seven books of villanellas for voice, basso continuo, and guitar with alfabeto
accompaniment. His first book of villanellas from 1610 is the first extant publication of
alfabeto songs, and his last (1640) was published when the flourish of alfabeto songbooks
was fading. Since Kapsperger was a guitarist himself, it is likely he at least had a strong
input on the alfabeto letters that were published, if not writing them himself. In the foreword
to his second book of villanellas, lyricist Remigio Romano reflects on both the popularity of
Kapsperger’s villanellas as well as its association with lower classes:
The villanellas, which sprang with such grace and beauty from your [Kapsperger’s]
pen, are now being passed around in a thousand ragged and impoverished disguises, stripped of their glory. In an act of sympathy and love I have now gathered
them all together and, as far as possible, I have reclothed them in their original
splendor.5
Kapsperger’s first and third books of villanellas—the only books in manuscript format
in the entire alfabeto corpus—include theorbo tablature in addition to alfabeto tablature
4
5
While there are other differences betweent the systems, these two chords are the most commonly used.
Translated in (Leopold 1983, p. 20).
34
and a bass line. These books provide a unique insight into how a continuo line may have
been realized differently based on instrument choice, because there are many moments of
disagreement between the alfabeto chord and theorbo tablature. Theorbos have the ability
to play the bass line and its harmonization, while the guitar can only successfully execute
the harmonization.
The Italian tablature of the theorbo allowed for a wider range of harmonies including
diminished triads, seventh chords, and suspensions. It also allows for harmonization in
two voices rather than a triad. While the guitar is free from rules of inversion and voice
leading, Kapsperger’s theorbo tablature features proper voice leading and inversions where
necessary. Kapsperger includes an extensive appendix for how to realize basso continuo
on the theorbo in his third book of solo theorbo tablature, which is discussed further in
Chapter 6 (Kapsperger [Kapsberger] 1626, p. 46). This makes for an interesting comparison
with the harmonized scales in several alfabeto sources, although Kapsperger did not include
such scales in his books.
Kapsperger’s seven books of villanellas will be analyzed further in the final chapter of
this dissertation. His use of harmony over the thirty years of publication will be compared
as well as specific differences between the continuo realizations of the guitar and theorbo.
Rontaini. Raffaello Rontani (d.1622) worked for the Medici in Florence from 1610 and
from 1616 until his death was maestro di cappella of S Giovanni dei Fiorentini in Rome. He
gained popularity for his secular vocal music, most of which was published in Rome (Fortune
and Carter 2016).
The alfabeto corpus includes four publications by Rontani, which are four volumes of his
Le Varie Musiche that included songs for one to three voices.6 His first book was published
in 1619, the second and third in 1620, and the fourth in 1623. His first book was published
by Luca Antonio Soldi while the rest were published by Antonio Poggioli. The second book
contains several torn pages, so not all of the songs are included in the alfabeto corpus.
6
Almost all of the songs are for one or two voices rather than three, and only his first and fourth book
have any songs for three voices.
35
Robletti. Giovanni Battista Robletti (fl 1609–50) was an Italian music printer (Ajani
2016). His 1622 publication, Vezzosetti fiori di varii eccellenti autori, cioe, madrigali, ottave,
dialoghi, arie, et villanelle, a una, e due voci. Da cantarsi con il cembalo, tiorba, chitarra
spagnola, &c., is neatly printed and includes lower-case alfabeto symbols unlike most of the
alfabeto corpus. There are only four songs that have alfabeto tablature that are included in
the alfabeto corpus.
Sabbatini. Pietro Paolo Sabbatini made his entire career in Rome and primarily published short, strophic songs (Whenham 2016). Only one of his books is included in the
alfabeto corpus: Prima scelta di Villanelle a due voci ... da onarsi in qualfivoglia Intromento con le lettere accomodate alla Chitarra Spagnola in quelle più à proposito, 1652. This
book was the last one published of the entire alfabeto corpus. While the style is the same
as songs of previous decades, the continuo has several more figures, including suspensions,
seventh chords, and ascending
6
3
sequences.
The alfabeto chords, limited to triads, do not accommodate suspensions or seventh chords
and also do not seem to agree with the figures in the bass. Several of the final cadences end
with ii6 →V→I in the figured bass but are realized as IV→V→I. Other books in the alfabeto
corpus do not have so much disagreement with the continuo line, although most books do
not have as many figures.
3.1.2
Naples
The publications from Naples were among the earliest and ended as the alfabeto songs
were reaching their peak in Venice. Alfabeto tablature was part of a wave of Spanish influence
and style that first came to Naples and then spread throughout the rest of Europe. The
publications are not as neatly printed as the Venetian or Roman books, but they nevertheless
serve as an important collection of early alfabeto songbooks.
Montesardo. Girolamo Montesardo (fl 1606–c1620) was an important figure for the
guitar and for alfabeto tablature. His 1606 publication, Nuova inventione d’intavolatura
was the first printed book in alfabeto tablature (Boye 2013a). His layout of the letters was
the framework for nearly all other Italian sources for decades. This book does not contain
36
any vocal music but is an instructional book for aspiring guitarists and includes several
instrumental dances.
Montesardo addresses the tuning of the guitar, which highlights the non-standard tuning
of the day—particularly when playing without any other instruments. Montesardo instructs
the guitarist to tune the lowest course to a “convenient tone” and the others accordingly.
The result, if the “convenient tone” is A, would be A–D–G–B–E with bourdons (octaves) on
the A and D strings and unison pairs on the others (Boye 2013a). This method of tuning a
string until suitable is similar to John Dowland’s instructions for tuning the lute in his son
Robert’s Varitie of lute lessons (1610):
Wherefore first have consideration to the greatnesse or smalnesse of the Instrument, and thereby proportionably size your strings, appointing for the bigger
Lute the greater strings, and for the lesser Lute the smaller strings, which being
so thought on, first set on your Trebles, which must be strayned neither too stiffe
nor too slacke, but of such a reasonable height that they may deliver a pleasant
sound, and also (as Musitions call it) play too and fro after the strokes thereon.7
As discussed in the previous chapter, Alfabeto tablature was used prior to Montesardo’s
1606 book in Amat’s Guitarra Espaǹola, which was notated with numbers, and in several
Italian and Spanish poems in letter format.8 What was new, however, Montesardo’s rhythmic
notation was indeed new. He devised a system of placing alfabeto letters in uppercase and
lowercase as well as above and below a horizontal line that would serve to specify the rhythm,
meter, and strumming direction. The result is a system that is difficult to understand, and
it unsurprisingly did not catch on (Boye 2013a).
Six years later, Montesardo’s first book of songs with alfabeto accompaniment, I lieti
giorni di Napoli, concertini italiani in aria spaguola a due, e tre voci con la lettere dell’alfabeto
per la chitarra; madrigaletti, et arie gravi passagiate a una, e due voci per cantare alla tiorba,
gravecimbalo, arpa doppia, et altri istrumenti . . . [op. 11], was published by Giovanni
Battista Gargano and Lucrezia Nucci. The songs are all simple and short, like most of those
7
8
From John Dowland’s introduction to his son Robert’s book, (Dowland 1610); spelling follows the source.
These sources, often fragmentary, are covered in detail by Daniel Zuluaga (Zuluaga 2014).
37
in the alfabeto corpus, and consist of scherzi, madrigali, and a few songs in dialog (dialogo)
format. Several of the songs are printed in part format, and the alfabeto tablature is only
printed on the bass part, which may or may not have also been a vocal part, indicating that
the alfabeto tablature was intended as a continuo realization.
Montesardo seemed to understand chordal inversions, particularly over bass notes with
accidentals. One should be careful, however, because a sharp added before a bass note
may indicate that the note should be raised or that it should be harmonized with a major
triad; musical context and alfabeto letters can help one quickly distinguish one from another.
Unlike most of the alfabeto corpus, Montesardo specifically calls for a theorbo to realize the
bass line. Perhaps another interesting feature of Montesardo’s book is the use of barlines;
some songs contain regular barlines while others may include only barlines at the end of
phrases—perhaps more reminiscent of an earlier style of notation.
Salzilli. Crescentio Salzilli (c.1580–after 1621) was an Italian composer and lutenist
who worked at the Santissima Annunziata in Naples (Larson 2016[b]). Two of his publications
are included in the alfabeto corpus, both of which were published in 1616: La sirena. Libro
secondo delle canzonette a tre voci, published by Giovanni Battista Gargano and Lucrezia
Nucci (Salzilli 1616b), and Amarille. Libro terzo delle canzonette a tre voci, published by
Lucretio Nucci (Salzilli 1616a).
In the first book, La Sirena, the alfabeto chords are placed above the text, which is
written in stanza format rather than being placed under the notation. This is similar to the
early alfabeto sources that are written over poetry without notation.9 However, it is fairly
easy to match words to the notated pitches. The second book, Amarille, is badly torn, so
only a few of the songs were complete enough to be included in the alfabeto corpus.
Giaccio. Orazio Giaccio (c.1590–c.1660) was an Italian composer and singer who, like
Salzilli, was employed as a bass singer at the Santissima Annunziata in Naples as well as
some cathedrals (Larson 2016[a]). Two of his books are included in the alfabeto corpus,
published in 1618 and 1620, respectively:10 Laberinto amoroso, canzonette a tre voci . . .
9
10
For a thorough description of these early sources, see Daniel Zuluaga’s dissertation (Zuluaga 2014).
The second book has a date discrepancy, described below.
38
libro terzo, published by Pietro Paolo Ricci (Giaccio 1618b), and Armoniose voci, canzonette
in aria spagnola, et italiana, a tre voci . . . libro primo, published by Battista Gargano and
Matteo Nucci (Giaccio 1618a).
The first book was written in part book format, and only the bass part is included, which
also has alfabeto chords. Alfabeto chords in the second book are placed above the text, much
like the first Salzilli book, and again can be easily aligned with the bass. There is a date
discrepancy in the second book; the title page is dated as 1620 while the dedication is dated
1618. It is listed under the 1620 date in Boye’s list (Boye 2013e).
3.1.3
Venice
The alfabeto songbook became enormously popular in Venice. Alessandro Vincenti published many books with alfabeto tablature for the publications described below. His method
of printing provided clear alfabeto symbols above the melody and bass, which made input
of the data simple.
Landi. Stefano Landi (c.1587–1639) was a composer, singer, and teacher in Italy. He
worked for the church and published many compositions including motets, madrigals, and
an opera (Murata 2016). Two songbooks with alfabeto tablature by Landi are included in
the alfabeto corpus, published in 1620 and 1637: Arie a una voce... (Landi 1620), and Il
quinto libro d’arie da cantarsi ad una voce, con la spinetta & con le littere per la chitara
(Landi 1637), both of which were published by Bartolomeo Mangi.
D’India. Sigismondo D’India (c.1582–1629) was a very distinguished Italian singer and
composer of monody. John Joyce and Glenn Watkins write that: “He was perhaps second
only to Monteverdi as the most distinguished composer of secular vocal music, especially
monody, in the early seventeenth-century Italy” (Joyce and Watkins 2016). Two of his
publications are included in the alfabeto corpus, published in 1621 and 1623: Le musiche .
. . a una et due voci da cantarsi nel chitarrone, clavicembalo, arpa doppia et altri stromenti
da corpo, con alcune arie, con l’alfabetto per la chitarra alla spagnola . . . libro quarto and
Le musiche . . . da cantarsi nel chitarrone, clavicembalo, arpa doppia & altri stromenti da
39
corpo, con alcune arie, con l’alfabetto per la chitarra alla spagnola . . . libro quinto, both of
which were published by Alessandro Vincenti.
Not every song in the books includes alfabeto tablature, but all that do are included in
the alfabeto corpus. It is noteworthy that there are some figures in the continuo part and
that, like some other seventeenth-century sources, D’India distinguishes between 11–10 and
4–3 suspensions.
Stefani. Giovanni Stefani (fl.1618–26) was an Italian music editor, and his publications
incorporate pieces from Johannes Kapsperger, Nicolò Borboni, Jacopo Peri, and Francesco
Monteverdi. His collections, which included alfabeto tablature, were enormously popular,
with the first reprinted four times. Roark Miller writes that the alfabeto chords were not
accurately placed and that some chords were in a different “mode” or “key” than the vocal
part (Miller 2016). While inputting all of the songs, I found the alfabeto chords matched the
continuo very well in the style of other books. The songs in the wrong key were presumably
written for a guitar tuned down a whole step, and I created a Python script that raised such
chords back up a whole step. A similar issue arises with the publications of Obizzi.
Three of Stefani’s publications are included in the alfabeto corpus, published in 1621,
1622, and 1623: Affetti amorosi canzonette ad una voce sola poste in musica da diversi
con la parte del basso, & le lettere dell’alfabetto per la chitara alla spagnola raccolte da
Giovanni Stefani con tre arie siciliane, & due vilanelle spagnole. Novamente in questa terza
impressione ristampate (Stefani 1621), Scherzi amorosi canzonette ad una voce sola poste in
musica da diversi, e raccolte da Giovanni Stefani con le lettere dell’alfabeto per la chitarra alla
spagnuola. . . . Libro secondo. Novamente in quest terza impressione corretti et ristampati
(Stefani 1622), and Affetti amorosi canzonette ad una voce sola poste in musica da diversi
autori con la parte del basso, & le lettere dell’alfabetto per la chitarra alla spagnola raccolte
da Giovanni Stefani con tre arie siciliane, & due vilanelle spagnole. Novamente in questa
quarta impressione ristampate (Stefani 1623), each of which was published by Alessandro
Vincenti.
40
It is notable that Stefani’s publications are the only ones to include transposition numbers—
a common occurrence in the instrumental alfabeto pieces, which are more virtuosic. For a
description of how transposition numbers work, see Chapter 2.
Milanuzzi. Carlo Milanuzzi (d. c.1647) was an Italian composer and organist and
worked for a number of cathedrals. He published at least nine books of “ariose vaghezze,”
which became popular in northern Italy (Roche and Miller 2016). Second only to Kapsperger’s
publications, five of his publications are included in the alfabeto corpus, published in 1622,
1625, 1628, 1630, and 1635: Primo scherzo delle ariose vaghezze, commode da cantarsi a
voce sola nel clavicembalo, chitarrone, arpa doppia, & altro simile stromenti, con le littere
dell’alfabetto, con l’intavolatura, e con la scala di musica per la chitarra alla spagnola . .
. [op. 7] (Milanuzzi 1622). Secondo Scherzo delle ariose vaghezze, commode da cantarsi a
voce sola nel clavicembalo, chitarrone, arpa doppia, & altro simile stromenti, con le littere
dell’alfabetto, con l’intavolatura, e con la scala di musica per la chitarra alla spagnola . .
. aggiontovi . . .aclune sonate facili . . . [op. 8] (Milanuzzi 1625). Sesto libro delle
ariose vaghezze, comode da cantarsi a voce sola nel clavicembalo, chitarrone, o altro simile
stromento, con le lettere dell’alfabetto per la chitarra alla spagnola . . . [op. 15] (Milanuzzi
1628), Settimo libro delle ariose vaghezze, comode da cantarsi a voce sola, con le lettere
dell’alfabetto per la chitarra alla spagnola, aggiuntavi un’ arietta a due voci con sinfonie di
due violini, se piace . . . [op. 17] (Milanuzzi 1630), and Ottavo libro delle ariose vaghezze,
comode da cantarsi a voce sola nel clavicembalo, chitarrone, o altro simile stromento . . .
[op. 18], each of which were published by Alessandro Vincenti.
All of his publications are neatly printed and include an alfabeto decoder as well as the
scalla di musica. For a description of these, see Chapter 2.
Ghizzolo. Giovanni Ghizzolo (d. c.1625) was an Italian composer who worked for
Prince Siro of Correggio and several cathedrals (Roche 2016). Only one of his publications is
included in the alfabeto corpus, published in 1623: Frutti d’amore in vaghe & variate arie,
da cantarsi co’l chittarone, clavicembalo, o altro simile stromento, accomodatovi l’alfabetto
con le letter per la chitarra spagnola . . . libro quinto, et [op. 21] (Ghizzolo 1623), published
by Alessandro Vincenti.
41
Obizzi. Domenico Obizzi (c.1611–after 1630) was an Italian composer and singer who
was employed at Saint Mark’s in Venice (Timms and Miller 2016). Only one of his publications is included in the alfabeto corpus: Madrigali et arie a voce sola . . . da cantarsi in
chittarone, clavicimbalo, o altre sorte d’instromenti, con l’alfabetto all’ariette per la chitarra
alla spagnola . . . libro primo, [op. 2] (Obizzi 1627), which was published by Alessandro
Vincenti in 1627. Only the second half of the book contains alfabeto chords, and some of the
songs include alfabeto letters that are for a guitar tuned a whole step below the “standard”
tuning, similar to some of Stefani’s songs. Again, a Python script was created to transpose
the chords to match the bass notes.
Marini. Biagio Marini (1594–1663) was an Italian composer, violinist, and singer whose
father was a virtuosic theorbist in Warsaw (Dunn 2016). Two of his publications are included in the alfabeto corpus, published in 1622 and 1635. However only the latter of these,
Madrigaletti a una, due, tre, e quatro voci, con alcune vilanelle per cantare nella chitariglia
spagnola chitarone . . . con il suo basso continuo, libro quinto, [op. 9] (Marini 1635), was
published in Venice by Bartolomeo Mangi.
3.1.4
Parma
Biagio Marini’s earlier alfabeto publication (1622), Scherzi, e canzonette a una, e due
voci . . . accommodate da cantarsi nel chitarone, chitariglia, et altri stromenti simili . . .
[op. 5], was published in Parma by Anteo Viotti and is the only publication from Parma in
the alfabeto corpus.
3.2
Input and Conversion of Data
My priority in inputting the corpus was to make it large, accurate, efficient for computation, and easy to input. For these reasons I did not convert each song into a digital score
but rather came up with a shorthand notation that aligns the bass notes with the alfabeto
chords with barlines. Each song is also associated with a key signature and the final bass
note. Of course, rhythm and melody are both lost with this approach, but neither will be
necessary for the given analyses that focus on harmonic accompaniment and modality.
42
For an example of how the music is converted, I will be using Kapsperger’s “Felice
gl’Animi” from his fourth book of villanellas (Kapsperger [Kapsberger] 1623). The original
is provided in Example 3.1, and a transcription is provided in Example 3.2. The transcription uses modern chord symbols and Roman numerals rather than alfabeto symbols. The
shorthand, described above, of this song is shown in Figure 3.1.
As can be seen in the encoding11 of “Felici gl’Animi,” bass notes are aligned with an
alfabeto chord. Adjacent duplicate alfabeto chords are eventually removed, and any bass
note without a pair is paired with the last written alfabeto chord. Bass notes are represented
as pitch-class integers, which will better match what will happen to the alfabeto chords,
discussed below. The “data” of each song contains the key signature (“n” for durus and
“f” for mollis). This allows each song to be associated with its key even if the notes are
transposed.
The tuning of the guitar was A–D–G–B–E from the course on top of the instrument to
the course at the bottom. However, the octave of these courses were subject to change. For
example, the courses may have been tuned to unisons or octaves, and depending on this,
the A, D, or G course may sound the lowest and may or may not have an octave pair that
sounds above the others. A thorough documentation of tuning possibilities can be found in
Alexander Dean’s dissertation (Dean 2009). Because of this unstandardized tuning, I have
used pitch-class integers to represent all of the pitches. The tuning of the guitar is therefore
[9, 2, 7, 11, 4].
To convert an alfabeto symbol to a chord, the fingering vector for the symbol is added
to the tuning vector, modulo 12. A vector in this instance is a shorthand for tablature. For
example, let’s look at the first chord of “Felici gl’Animi”. A G chord (alfabeto symbol A) has
a fingering vector of [2, 0, 0, 3, 3]; each number spot represents a string (the top string is on
the left, the bottom on the right), and the number corresponds to the fret placement. When
the tuning vector is added to the fingering vector of the first chord in “Felici gl’Animi,” the
resulting chord is found:
11
The syntax has been altered from the original to fit within the margins of this page.
43
Example 3.1: “Felici gl’Animi” (Kapsperger 1623)
44
Felici gl'Animi
G
G
Fe li ci
gl'an
7
i mi Che
G
greg gie gui
ses pia ce
uo
li, I gior ni
14
O
pia cer
sta
bi le
bVII
pia
bVII
cer
sta
bi
le
D
mean an o E
C
G
i mi Nel
pet
ro anni
V
Am
da
IV
I
A
dil et teu
o
C
E
li
l' core se
ren an o
IV
ii
pur
Em
er
ta
d'
Dm
am a
IV
v
D
C
V
D
o
pou
er
ta
d'a
ma
vi
v
IV
G
F
bi
le
o
I
II
VI
IV
Em Dm C
vi
fan
V
O
D
no Ne cure e
19
V
bVII vi
F
da
F Em
Di
C
G
no,
D
I
G. G. Kapsperger [Ger. Kapsberger]
transcribed by Daniel Tompkins
V
bVII
G
bi
le
I
Example 3.2: Transcription of “Felici gl’Animi” (Kapsperger 1623)
45
( c o n tin u o , a l f a b e t o symbol , data )
7 | 7 | 7 2 | 2 | 0 | 2 7 | 7 | 7 | 7 5 4 | 2 | 0 e9e | 0 2 4 | 9 | | 5 | 5 5 | 4 2 0 | 0 2 | 7 5 | 5 | 5 4 2 | 0 0 | 2 7
A| |AC|C| B |CA|A| | G+|C| B D | BF | I | | G| |+EB|BC|AG|G|G+E | B |CA
key : n , f i n a l : 7
Figure 3.1: Kapsperger’s “Felici gl’Animi” represented in Python syntax
[9, 2, 7, 11, 4]
+ [2, 0, 0, 3, 3](mod12)
=
[11, 2, 7, 2, 7]
(G Major)
Duplicate pitches are then removed, and the pitch classes are ordered from smallest to
largest number. After all of the chords are converted, all of the chords are transposed by so
that the final bass note is pitch-class 0 (C).12 These are then converted to Roman numerals
with no inversion figures. Songs can now be compared based on the frequency of chord
use and chord movement. However, bass notes are still unaccounted for. To have the bass
notes be a part of the chord, which is necessary for the continuo-based studies later on,
the alfabeto chords become intervals above the bass note they are associated with. To do
this, the pitch classes of the alfabeto chord are subtracted from the bass note, again modulo
12. For example, a G chord with a G bass will be written as [7, [0, 4, 7]], meaning a G
bass and notes 0, 4, and 7 semitones above that bass (i.e. major and root position). The
same chord with a B in the bass is written [11, [0, 3, 8]], meaning a G major chord in first
inversion.13 After all of the conversions, each song is represented as a string of bass-interval
chords. Adjacent duplicates are removed. Figure 3.2 shows what a continuo reduction looks
like for “Felici gl’Animi.”
These symbols can be converted into Roman numerals, as shown in Example 3.2. There
are three “passing” sonorities in the continuo reduction: [4, [1, 5, 8]], [4, [1, 5, 10]], and [7,
[2, 6, 9]]. All of these result from the bass note moving up a step from a previous chord.
These are left in the dataset for each song, but each computational process has some kind of
12
The data specifying the key and final remain the same for reasons that will become apparent later.
There are no inversions in “Felici gl’Animi,” and they are in general not very common in this literature.
This will be covered in more detail in later chapters.
13
46
( [ b a s s note ,
[0 ,[0 ,4 ,7]] ,
[0 ,[0 ,4 ,7]] ,
[5 ,[0 ,4 ,7]] ,
[5 ,[0 ,4 ,7]] ,
[10 ,[0 ,4 ,7]]
[7 ,[0 ,4 ,7]] ,
[7 ,[0 ,3 ,7]] ,
[ pitch clas ses of alfabeto
[7 ,[0 ,4 ,7]] , [5 ,[0 ,4 ,7]] ,
[10 ,[0 ,4 ,7]] , [9 ,[0 ,3 ,7]] ,
[4 ,[1 ,5 ,8]] , [2 ,[0 ,3 ,7]] ,
[7 ,[2 ,6 ,9]] , [9 ,[0 ,4 ,7]] ,
, [9 ,[0 ,3 ,7]] , [7 ,[0 ,3 ,7]] ,
[0 ,[0 ,4 ,7]] , [10 ,[0 ,4 ,7]] ,
[5 ,[0 ,4 ,7]] , [7 ,[0 ,4 ,7]] ,
symbol ] ] )
[7 ,[0 ,4 ,7]] ,
[7 ,[0 ,4 ,7]]
[4 ,[1 ,5 ,10]]
[2 ,[0 ,4 ,7]] ,
[5 ,[0 ,4 ,7]]
[9 ,[0 ,3 ,7]]
[0 ,[0 ,4 ,7]]
,
,
,
,
Figure 3.2: A continuo representation of Kapsperger’s “Felici gl’Animi” represented in
Python syntax
threshold that chords must pass. If a chord does not meet the threshold (for example, if [4,
[1, 5, 8]] does not meet 1% or .1% of the entire corpus, or whatever the threshold may be),
it will be omitted. The details of this conversion process will be explored in greater detail
in later chapters, but it is effective in removing harmonies like the passing harmonies above
as well as possible errors in the original notation or in the data input process. This is why
Example 3.2 does not contain the “passing” harmonies.
3.3
Conclusion
Some extant alfabeto songbooks were not included in this study due to availability and
readability. A list of other alfabeto songbooks can be found in Boye’s database (Boye 2013e)
and Gavito’s dissertation (Gavito 2006). However, the total number of songs in the alfabeto
corpus is 529, which is quite large considering the focus of genre, time period, geography,
and instrumentation of this corpus study. My intention for encoding the alfabeto corpus
as I did was to be faithful to the spirit of the alfabeto system. Alfabeto chords are not
concerned with inversion or voice leading, so it seems that representing those chords as sets
of pitch-class integers reflects the alfabeto system. The methodology of identifying alfabeto
chords explained in this chapter will be the basis for the analyses in the following chapters.
47
CHAPTER 4
QUESTIONS OF MODE AND KEY
“The change from ‘modal’
nomenclature, be it octenary or
dodecachordal, to ‘tonal’ nomenclature
is a function of the concomitant
developments in the organization and
control of pitch relationships....”
Harold Powers, “From Psalmody to
Tonality” (Powers 1998, p. 275)
With the rise of monody and vertically-oriented harmony, the seventeenth century ushered in significant changes to harmonic practice. The modal framework was also changing
from a multi-mode system to a major-minor modality, which is reflected in the change of
notation. It is often assumed that European music before common-practice tonality was
built on a system of six, eight, or twelve modes and that the music from the eighteenth
century and onward was built on two: major and minor. Historical notation supports this
with the number of signatures and final cadences possible in any given system. It is possible
that accidentals and musica ficta could have bent modes towards each other. For example,
7̂ in a piece in what we would today call Mixolydian mode could be consistently raised while
perhaps 4̂ in a Lydian mode piece could be consistently lowered.
4.1
Historical Notation
The alfabeto corpus was notated in the multi-mode system, yet its triadic focus and
IV→V→I cadences point to later tonal practices in the major-minor system. Richard Hudson, in his discussion of guitar schema, asserts that the guitar “gave to art music...a concept
of mode that led eventually to fully developed major-minor tonality”(Hudson 1970, p. 163).1
1
This is dismissed as “simplistic” by Dean (Dean 2009, p. 15).
48
Table 4.1: Mode possibilities in the durus/mollis system
Signature
♮ (durus)
♭ (mollis)
Final Cadence
C, D, E, F, G, A
C, D, F, G, A B♭
This chapter presents a corpus study that measures the number of statistically distinct modes
in the alfabeto corpus and compares the results to corpora of J.S. Bach, Palestrina, and the
Franco-Flemish school. The results give an insight into the modal framework of these corpora and shed light on differences between time periods and genres. Before proceeding, I
want to address a few assumptions that are often made when conducting corpus studies on
harmony:
1. Notation indicates mode in early music.
2. There is a clear division between modal practice and common practice tonality.
3. Major and minor keys can be extrapolated regardless of time period.
4. All notated music from a given historical period operated under the same modal framework.
Looking at the first assumption, notation does seem to indicate modes rather clearly
in early music. While common practice tonality operated under a system of several key
signatures that were each associated with a major and minor key (Figure 4.1), early music
often operated under two signatures—no flats (durus) and B flat (mollis)—with several
possible final cadences for each signature (Table 4.1). There were instances of two flats in
music before the seventeenth century as well as conflicting signatures where one voice may
be in mollis while another in durus.
For now, let’s consider two possible modal frameworks for the durus/mollis system from a
statistical point of view. The first could be that each modal pair is a distinct mode, which the
notation would suggest (Table 4.2). Or, perhaps the influence of accidentals and/or musica
ficta causes the multi-mode system to coalesce into only two that are distinguishable only
by the quality of 3̂, in which case the modal framework would be closer to the major-minor
system than the multi-mode system (Table 4.3).
49
Figure 4.1: Heinichen’s musical circle (1711) in the common-practice system
Table 4.2: Notated modes
Mode
Ionian
Dorian
Phrygian
Lydian
Mixolydian
Aeolian
Signature:Final
♮:C, ♭:F
♮:D, ♭:G
♮:E, ♭:A
♮:F, ♭:B♭
♮:G, ♭:C
♮:A, ♭:D
Table 4.3: Possible coalesced modes due to accidentals and/or musica ficta
Mode
Major
Minor
Signature:Final
♮:C, ♮:F, ♮:G,
♭:F, ♭:B♭, ♭:C
♮:D, ♮:E, ♮:A,
♭:D, ♭:G, ♭:A
50
Much has been written about modality and its theories in early music. From a historyof-theory perspective, there were competing theories of mode during the time in which many
of the pieces analyzed were composed including theories of six church modes, eight modes,
or twelve modes. Furthermore, mode labels were originally an organizational system to label
existing church chants rather than a guide to composition.
The second assumption has been thoroughly debunked by Gregory Barnett (Barnett
2002), Harold Powers (Powers 1998), and others.2 The seventeenth century was indeed
a period when styles, theories, and notation were changing, but the change depended on
geographical location and genre. There was no strict division between modality and common
practice tonality nor was it a gradual evolution from one practice to another (Barnett 2002,
pp. 407–8).
Albrecht and Huron’s recent article on modality and chronology discusses the third assumption. Their research traces pitch class distribution of major and minor keys through
fifty-year epochs and find that while there is a distinction between major and minor, other
modes can be found through their algorithm before the seventeenth century (Albrecht and
Huron 2014; Albrecht and Huron 2012). This is to say that one should consider time of
composition when using key-finding algorithms.
Perhaps because of their limited data size for each epoch, Albrecht and Huron make the
fourth assumption, and that is the primary focus of this chapter. Thinking of today, I think
it is fair to say that not all music composed operates under the same harmonic or modal
framework. Music of Thomas Adès, John Luther Adams, and Kaija Saariaho do not all use
the same harmonic framework even among their own compositions. Include Taylor Swift
and Kendrick Lamar, and we can really see the diversity of musical practices today. I am
not suggesting that there was such an extreme diversity of musical styles in the seventeenth
century, but it is worth investigating the modal differences between secular and sacred genres.
One important question about the early seventeenth century is whether musical practice
preceded the change in notation. Christensen rightfully warns against letting our desire to
2
Alexander Dean provides a thorough overview of differences between English “keys” and Italian “chiave”
in the seventeenth century (Dean 2009, pp. 282–7).
51
find “tonality” neglect compositional practice of the time (Christensen 1992), but how often
are scale degrees altered towards what would become major and minor, and is it frequent
and consistent enough to change the modal framework of an entire corpus? Barnett writes
that the change in system to major-minor tonality was a result of the need to describe music
practice:
Theorists began speaking in terms of major-minor tonality when their purpose
changed from handing down and refining an inherited theoretical tradition to
creating a practical construct that most simply and accurately fit the music
around them. (Barnett 2002, p. 408)
Every song in the alfabeto corpus is written in the mollis/durus system, such as in
Table 4.1, but this may not mean that the songs are written in several different modes from
the perspective of pitch and chord content. Songs in a flat key with a G final (♭:G) will
often lower the sixth degree from E♮ to E♭, and songs in (♭:B♭) often include E♭ rather than
E♮. This chapter will answer whether these altered pitches are significant enough to coalesce
what looks to be many modes into an early major-minor paradigm.
4.2
Previous Computational Approaches
Clustering algorithms can provide meaningful answers to how many modes there were
in the alfabeto corpus and in other corpora. Rather than looking at the final cadence and
key signature, clustering algorithms can focus on the pitch classes used and can therefore
determine whether or not the use of accidentals and alfabeto symbols are present enough to
change the underlying modal framework that is implied by the notation.
Modal analyses are commonly employed in seventeenth-century music, which is especially
appropriate for sacred music.3 Finding modes typically involves locating cadences and hexachords. Such methods are locally-focused and of course involve a theoretical assumption of
what modes are, how many there are, and how they behave. Computational approaches are
in many ways the opposite; the approach is more global and does not try to fit a selection of
3
One example is in Chafe’s analyses which look at Monteverdi’s output through a modal context (Chafe
1992).
52
Key Profiles from Temperley Marvin 2008
25
Frequency of Use
20
major
minor
15
10
5
0
1̂
1̂/ 2̂
2̂
2̂/ 3̂
3̂
4̂
4̂/ 5̂
Scale Degrees
5̂
5̂/ 6̂
6̂
6̂/ 7̂
7̂
Figure 4.2: Major and minor key profiles from (Temperley and Marvin 2008); the graph
shows the frequency (in percent) that each scale degree occurs in the Haydn/Mozart string
quartet corpus
music into a historic modal theory. Rather than making assumptions of a theoretical framework for composition, the clustering algorithms produce results based on given parameters
such as pitch-class distribution or chord frequency.
David Temperley and Betsy Marvin created major and minor key profiles based on a
corpus study of the string quartets by Mozart and Haydn (Temperley and Marvin 2008).
A key profile is the frequency each pitch class is used with pitch-class 0 being tonic. The
key profile of Temperley and Marvin’s study can be seen in Figure 4.2. A key for a given
excerpt of music can be found by counting the frequency of each pitch class and determining
which key profile to which it is most similar. Key profiles are related to cognitive studies
by Krumhansl and Kessler, which found that frequency of pitch classes corresponds with
the perceived key (Krumhansl and Kessler 1982). Temperley also conducted a similar study
of European folk melodies and created two key profiles (major and minor) that reflect the
corpus, which can be seen in Table 4.4.
Joshua Albrecht and David Huron used a clustering algorithm called Ward’s method to
53
Table 4.4: Temperley’s key profiles, ordered from 1̂–7̂ (pitch classes 0–11)
pitch class
Major (M )
Minor (m)
0
.18
.19
1
2
0 .16
.01 .15
3
4
0 .19
.18 0
5
6
.11 .01
.14 0
7
.21
.20
8
9
0 .08
.04 .01
10 11
0 .06
.05 .02
determine how many modes are present in a corpus (Albrecht and Huron 2014; Albrecht and
Huron 2012). Ward’s method is a hierarchical clustering algorithm that shows the degree to
which data samples are similar or different. The results are plotted on a dendrogram (tree
diagram) that shows the nested clustering. They used the final ten measures of each piece of
music and found the frequency that each pitch-class occurs, which results in a twelve-member
vector like Temperley’s key profiles. The k-means cluster algorithm measures the Euclidean
distance between each vector then processes the data (Albrecht and Huron 2014, pp. 225-6).
Albrecht and Huron found that the major-minor system (two modes) grew out of a multimodal system around the beginning of the seventeenth century. Interestingly, they find that
there is still a significant split between “major” and “minor” throughout the centuries studied
but that sub-clusters of major and minor become present before the seventeenth century
(Albrecht and Huron 2014, p. 241).
In the discussion of their results, they suggest that further research could show differences
between genre, region, and smaller epochs. The sample size of each epoch was also relatively
small, and a larger one could further reinforce their claims. The study of the alfabeto corpus
provides a larger study that takes genre and location into consideration.
4.3
Scale Degree Frequency Analysis
To determine the number of modes in a given corpus, I will use a clustering algorithm
that measures pitch class data and clusters similar songs together. A key profile, like those
of Temperley and Marvin (Temperley 2007; Temperley and Marvin 2008), is created for
each song or piece in every corpus by tallying the number of times each scale degree is
used, which is represented as pitch class integers with 1̂ being pitch class 0. Key profiles
are 12-dimensional vectors with one dimension for each pitch class. The number for each
54
dimension is the frequency (in percent) that that a given class occurs in a given song.
You may perhaps imagine each song being plotted on a 12-dimensional graph based on the
coordinates of its key profile, although imagining such a thing may prove difficult. To find
the similarity of each song, the Euclidean distance is found between every song in the corpus:
d(p, q) =
qP
n
i=1 (qi
− pi )2 . For example, the Euclidean distance of Temperley and Marvin’s
major and minor key profiles is just over 22 units. Songs with similar key profiles will have
smaller distances such as 2 or 3 units while songs with different key profiles will be higher.
Similar songs will therefore be grouped together in clusters in the 12-dimensional space, and
the number of statistically distinct modes will reflect the number of clusters in the data.
To find the optimal number of modes, k-means clustering is used, which is an unsupervised machine learning algorithm that clusters data points into a given number of clusters
(k). K-means clustering is called an unsupervised machine learning algorithm because the
algorithm decides which songs belong in which clusters based on what it learns from the
data. For example, given the Euclidean distances between songs, the algorithm partitions
the songs into k number of clusters. The partitioning is scored by measuring the inertia (within-cluster sum-of-squares), which the algorithm tries to minimize (Pedregosa et al.
2011):
Pn
i=0
minµj ∈C (||xj − µi ||2 ). The partitioning is then adjusted and scored again. The
readjustment continues until the change in inertia score reaches a desired minimum, which
is called convergence.
While k-means clustering gives a way to reveal clusters in a corpus, we still do not know
the optimal number of clusters, or k. To achieve this, the k-means clustering algorithm is
performed for 2–12 clusters or modes. For each mode number, two scoring metrics are used:
the silhouette coefficient and the completeness score. The silhouette coefficient measures
the distinctiveness of each cluster—how compact data points are within the cluster and how
far apart cluster centroids are from each other. A score of 1 means perfect separation, 0
means no separation, and -1 means incorrect clustering. The completeness score measures
the success that all songs of a given notated key or mode land within the same cluster. A
score of 1 means that all pieces in the same notated mode or key ended up in the same
55
clusters, while a score of 0 means that no pieces in the same notated mode or key ended up
in the same cluster.
The number of modes that have the highest silhouette and completeness scores will be
selected for the number of distinguishable modes. The silhouette coefficient score is what
determines the number of statistically distinct modes, while the completeness score shows
whether notated modes are in agreement with the statistically distinct modes. If a corpus
in the multi-mode system has a high completeness score for six modes, the notated modes
reflect the statistically distinct modes. It is possible to have more than one well-scored
cluster number as well as no good options, which would imply that there is no clear modal
framework for the given corpus.
To visualize the data, the twelve dimensions are reduced to two using the principal
component analysis (PCA) algorithm. Each song is then plotted and labeled with its key
signature. The distance between the songs plotted on the graph approximates the Euclidean
distance of their key profiles, and the graphs are partitioned into k clusters based on their
silhouette and completeness scores. Cluster centroids are numbered, and the mode of all
songs that belong to that cluster can be found by finding the top seven key profile dimensions
with the highest frequency. For the sake of convenience, I will be using the Greek names
of the modes as we use today but with the understanding that these names were not the
naming convention in the seventeenth century.
Another method is also used to scale the dimensions, which is the NEATO algorithm
developed by Graphviz (S. C. North 2004). The NEATO graphs will be used to show the
distances between notated keys where the data for each key is the average of all songs with
that key rather than looking at each individual song. The NEATO and PCA graphs do not
have an x or y axis but instead show distance between points on the graph as differences in
data.
4.3.1
Chronological Comparison
Before looking at the alfabeto corpus, I will first analyze corpora from adjacent centuries
to the alfabeto corpus. This will help give a frame of reference for the alfabeto corpus.
56
Music21 has a large number of scores for Palestrina and J. S. Bach, and I used both for these
analyses. I created a Python script that counted each pitch class4 and converted the result
to a key profile. Each piece is labeled with its respective key—either a major or minor key
(Bach) or a notated mode (Palestrina).
Figure 4.3 shows the results of a cluster analysis of the Palestrina corpus from music21.
Each point on the graph is a song and is labeled with its notated mode. Figure 4.3b shows
the silhouette and completeness scores, and it can be easily seen that the songs cluster into
as many as five clusters. The centroids for five clusters are:
1. Minor (Aeolian)
2. Major (Ionian)
3. Dorian
4. Phrygian
5. Mixolydian
The “Lydian” mode is the only notated mode that is not statistically distinguishable.
Its ♯4̂ is consistently lowered and is mixed in with the “Ionian” cluster. The other modes,
however, are statistically distinguishable, which shows that the Palestrina corpus is indeed
built upon the multi-modal framework in which it was notated.
The NEATO graph in Figure 4.4 takes the averages (arithmetic means) of all songs that
share the same notated mode and compares them using Graphviz’s NEATO multidimensional
scaling: the Euclidean distance between each notated mode arrays is found, and the NEATO
algorithm scales the twelve dimensions down to two. The result is that each mode pairs with
its transposed kind, if it exists in the corpus. This can be seen in the NEATO Figure as well
as in the full k-means graph. It should also be noted that many of the mode averages are
very distant from either of Temperley’s key profiles.
As a contrast, Figure 4.5a shows a cluster analysis of the Bach corpus from music21.
Reflecting the major-minor system in which the Bach corpus is notated, each point on the
4
Repeated pitches are retained, although I tested the algorithm with repeated pitches removed and found
that the change in results was insignificant.
57
(a) PCA-reduced graph of the Palestrina corpus; numbers indicate cluster
centroids, which reflects the statistically-distinct modes
completeness score
silhouette score
score
1.0
0.8
0.6
0.4
0.2
0.0 2
4
6
8
10
12
number of clusters (modes)
(b) Completeness and silhouette scores of the Palestrina corpus
Figure 4.3: K-means clustering of pitch frequency in the Palestrina corpus
58
♭:C
♮:G
♭:F
♮:F
♮:C
♭:G
M
♮:D
m
♮:A
♭:D
♮:E
Figure 4.4: NEATO reduction of notated mode averages for the Palestrina corpus; distance
between nodes is proportional to average key profile difference (Euclidean distance); M and
m are Temperley and Marvin’s major and minor key profiles, respectively
59
(a) PCA-reduced graph of the Bach corpus; numbers indicate cluster centroids, which reflects the statistically-distinct modes
completeness score
silhouette score
score
1.0
0.8
0.6
0.4
0.2
0.0 2
4
6
8
10
12
number of clusters (modes)
(b) Completeness and silhouette scores for the Bach corpus
Figure 4.5: K-means clustering of pitch frequency in the Bach Corpus
60
em
dm
fm
bm
am
gm
f♯m
m
cm
M
GM FM
AMB♭M
DM
EM
CM
E♭M
A♭M
Figure 4.6: NEATO reduction of notated mode averages for the Bach corpus; distance
between nodes is proportional to average key profile difference (Euclidean distance); M and
m are Temperley and Marvin’s major and minor key profiles, respectively
graph is labeled as a major or minor key rather than a notated mode. Despite a couple of
outliers, the Bach corpus clusters into two modes—major and minor. The completeness score
for two modes is a perfect 1.0, which means that all songs with the same key were clustered
into the same mode. Likewise the silhouette score of just over 0.5 shows that there is a clear
distinction between the two modes. No other number of modes have a high silhouette or
completeness score. The cluster centroids also reflect the major-minor system:
1. Major (Ionian)
2. Minor (Aeolian)
61
The NEATO graph in Figure 4.6 shows the distances between the key averages. This
again shows a clear major-minor system as well as a close connection to Temperley’s major
and minor key profiles. The minor cluster is less compact than the major one, and this is
likely due to the variance of scale degrees 6̂ and 7̂. The A♭ major average is further away
from the other major keys because there are fewer pieces in A♭ major in the Bach corpus
and therefore not a very large data set.
The clustering results of the alfabeto corpus can be found in Figure 4.7. There is a clear
separation into “major” and “minor” clusters—all of the modes with minor 3̂ are grouped
together, and those with major 3̂ are in the other group. There are more outliers than in the
Bach corpus, but there are clearly not more than two modes that have a high score; there is
a dramatic drop in score after two modes. The cluster centroids also show major and minor:
1. Major (Ionian)
2. Minor (Aeolian)
The NEATO graph in Figure 4.8 gives an interesting perspective. Despite a clear clustering into only two modes, the average of each key still preserves the modal system and
drifts further away from Temperley’s key profiles. The ♭:C node is an outlier because only
two alfabeto songs are in that notated mode. These modal pairs, however, are vestigial and
are not distinguishable as a whole because of the results of the k-means scores.
Lastly, Figure 4.9 shows the keys and modes from Palestrina, Bach, and the alfabeto
corpora in a NEATO graph. Each key is labeled with a P, B, or A, which refers to the
composer and/or corpus. All related modes are connected by a black line, and they are
more or less close to each other. However, it is important to stress that these are key
averages; the Palestrina song clusters from Figure 4.3 each contain only one mode (itself
and its transposition), while the alfabeto song clusters from Figure 4.8 are separated into
major and minor. The modal separation only happens when all songs of a similar mode are
averaged. As is to be expected, the Bach keys are closest to Temperley’s major and minor
key profiles.
62
(a) PCA-reduced graph of the alfabeto corpus; numbers indicate cluster
centroids, which reflects the statistically-distinct modes
completeness score
silhouette score
score
1.0
0.8
0.6
0.4
0.2
0.0 2
4
6
8
10
12
number of clusters (modes)
(b) Completeness and silhouette Scores for the alfabeto corpus
Figure 4.7: K-means clustering of pitch frequency in the alfabeto corpus
63
♮:F
M
♭:B♭
♮:C
♭:F
♮:G
♭:C
♮:D
♭:G
♮:A
♭:D
♮:E
m
Figure 4.8: NEATO reduction of notated mode averages for the alfabeto corpus; distance
between nodes is proportional to average key profile difference (Euclidean distance); M and
m are Temperley and Marvin’s major and minor key profiles, respectively
64
P ♭:D
P ♮:A
P ♮:E
A ♮:E
A ♭:D
P ♭:G
A ♮:A
A ♭:G
B f♯m
B cm
B dm
B am
B gmB bm B em
P ♮:D
A ♮:D
P ♭:C
B fm
A ♭:C
m
P ♮:G
A ♮:G
A ♭:F
P ♮:F
B AM
B B♭M B EM
A ♮:C
B GM
B FM B DMB E♭M
B CM
P ♭:F
M
A ♮:F
A ♭:B♭
P ♮:C
B A♭M
Figure 4.9: NEATO graph of key averages for Palestrina, Bach, and alfabeto corpora, labeled
with “P,” “B,” “A,” respectively
65
4.3.2
Genre Comparison
As described in Chapter 2, most alfabeto songs were considered low-brow secular music
that were associated with taverns rather than high courts (even though the music was composed by the same court composers as “high-brow” music). This “low-brow” and dismissive
status of alfabeto songs highlights a divide between secular and sacred genres. Perhaps secular and sacred genres were not composed with the same theoretical frameworks, and that
the number of modes may depend on genre rather than chronology. I do not have another
seventeenth-century corpus of a very large size, but there is another corpus that is divided
by genre. The Josquin Research Project (The Josquin Research Project 2016) provides several hundred scores from the Franco-Flemish School that are grouped into Mass movements,
motets, and secular songs.
The vast majority of the Franco-Flemish corpus is notated in the durus/mollis system,
which implies multiple modes like the Palestrina corpus. The question is whether the different
genres operate under the same modal framework even though the music was published in
the same time and location and by the same composers.
The mass movements cluster well in two to six modes (Figure 4.10). None of the clusters
are as distinct as the Palestrina corpus, but the completeness score shows that the corpus
can be divided into six modes, and the notated modes still mostly cluster together. The
cluster centroids of the five modes shown in Figure 4.10a shows that the notated modes are
in agreement with the statistically distinct modes:
1. Dorian
2. Major (Ionian)
3. Phrygian
4. Mixolydian
5. Minor (Aeolian)
The only mode that does not become a distinct cluster is again the Lydian mode. The
same is true for the motet corpus, although there is a stronger peak at five modes in the
66
(a) Masses k-means
completeness score
silhouette score
score
1.0
0.8
0.6
0.4
0.2
0.0 2
4
6
8
10
12
number of clusters (modes)
(b) Masses score
Figure 4.10: K-means clustering of key profiles in the Franco-Flemish School corpus (Masses)
67
(a) Motets k-means
completeness score
silhouette score
score
1.0
0.8
0.6
0.4
0.2
0.0 2
4
6
8
10
12
number of clusters (modes)
(b) Motets score
Figure 4.11: K-means clustering of key profiles in the Franco-Flemish School corpus (motets)
68
completeness score (Figure 4.11). Once again, the Lydian mode is not distinguishable, which
shows that accidentals and/or ficta consistently lower the ♯4̂. The cluster centroids of the
motets are:
1. Mixolydian
2. Phrygian
3. Dorian
4. Major (Ionian)
5. Minor (Aeolian)
The secular song corpus, however, clusters best in two modes with a slight secondary
peak at four modes (Figure 4.11). The clusters are not as distinct as later corpora, but it
is clear that two modes better represents this secular song corpus than any other number
of modes. Like the Bach and alfabeto corpora, the centroids of the Franco-Flemish secular
songs are again major and minor:
1. Minor (Aeolian)
2. Major (Ionian)
The results of this study show that different genres have different modal frameworks even
if composed within the same time period, country, and even by the same composer. Secular
genres cluster into only two modes (despite the music’s notation) long before sacred genres.
This leaves open speculation of the modal framework for vernacular genres that were not
notated. Perhaps the alfabeto corpus is the closest we can come to vernacular music. Further
digitization of early music can extend this study, especially regarding seventeenth-century
repertoire. Given enough music, comparisons could also be made geographically.
Of course, understanding the harmonic practices of a corpus includes more than pitch
class frequency, including how those pitches are grouped and move from one to another,
but this study is an important first step. It gives a view of the modal framework in early
69
(a) Secular k-means
completeness score
silhouette score
score
1.0
0.8
0.6
0.4
0.2
0.0 2
4
6
8
10
12
number of clusters (modes)
(b) Secular score
Figure 4.12: K-means clustering of key profiles in the Franco-Flemish School corpus (secular
songs)
70
music that provides a foundation for other ways of investigating harmonic practice—a foundation that recognizes the different harmonic practices of secular and sacred genres and that
sometimes compositional practices were not always in line with notational or theoretical
conventions of their time.
4.3.3
Bass Harmonization Analysis
While key profile clustering clearly shows that the alfabeto corpus is built upon a majorminor framework, another measurement reflects some of the writing from the songbooks.
The scalla di musica, which was discussed in Chapter 2, appears to present only two modes,
durus and mollis (see Figure 4.13). Each of the scales have different chords that harmonize
the ascending bass notes. In the many books that include the scalla, only two scales are
ever shown: ♮:G and ♭:F. Perhaps the scalla was intended as a guide for amateurs to know
how to harmonize a bass note within a particular key, without regard to the final cadence.
There are also typographical errors and unusual harmonizations, which are detailed in full
by Alexander Dean (Dean 2009, pp. 293–6).
Despite the problems with the scalla di musica, it does show that there was a focus on harmonizing bass notes—a relatively new practice at the time. To test the scalla, I investigated
pairing of bass notes and alfabeto symbols5 to compare the actual notated harmonization
with the scalla. Figure 4.14 shows the scales with percentages of each harmonization. There
are seven rows of percentages below the scales. The first three pairs of rows are for root
position, first inversion, and second inversion. Each of those row pairs are broken into major
and minor triads. The seventh row is for other harmonizations that do not make a major or
minor triad with the bass. These are mostly instances where the bass changes but the chord
does not.
There are no instructions, apart from the title “Scalla di Musica per B. Quadro,” that
suggest whether these harmonizations are for ♮:G and ♭:F specifically or if these are the typical
harmonizations of the given bass notes given the key (Quadro/Durus or Molle/Mollis).6
Figure 4.15 is a k-means clustering analysis of chord-frequency use for every song in the
5
6
For a review of how the bass notes and chords were paired, see Chapter 3.
The latter option is argued by Eisenhardt (Eisenhardt 2015, p. 20).
71
(a) Original from (Obizzi 1627)
G
Am
5
3
Scalla di Musica per B. Quadro.
5
3
F
C
D
E
F
G
6
3
5
3
5
3
5
3
5
3
5
3
5
3
Scalla di Musica per B. Molle.
Gm
5
3
G
5
3
Am
B♭
C
Dm
E
5
3
5
3
5
3
5
3
5
3
5
3
5
3
(b) In modern notation
Figure 4.13: Scalla di musica
72
F
Am
Gm
(a) Harmonization percentages for ♮:G
(b) Harmonization percentages for ♮:G
Figure 4.14: Percent of bass harmonizations in the alfabeto corpus: all numbers sum to 100,
rows indicate inversion and mode (root, first, second; major and minor, respectively; and
“other”)
73
alfabeto corpus. Rather than transposing all of the songs, the algorithm instead records the
frequency that each alfabeto symbol is used for every song. The results show that the songs
cluster into two “modes.” Furthermore, the clusters mostly reflect durus and mollis, shown
by the data points in Figure 4.15a.
The results in Tables 4.5 show the bass harmonization frequencies for each possible bass
note (untransposed). Table 4.4c compares the cluster centroid data with the durus and
mollis harmonizations from the scalla di musica. The actual data matches the scalla di
musica surprisingly well. However, it is clear by looking at Figures 4.4a and 4.4b that
the music does not always harmonize a particular bass note with a particular chord. The
following analyses will take a closer look at the bass note harmonization and its interaction
with the notated modes.
For this study, each song is once again transposed, and the bass note harmonization is
measured over each scale degree with 1̂ assigned as final bass note of each song. K-means
clustering is again used to determine the number of modes based on bass note harmonization
frequency. The same algorithms that were used for clustering key profiles are used again.
The difference is that the input data is an 84-dimensional vector, which is based on the
eighty-four bass note-alfabeto pairs in Figure 4.14. It is important to note that no seventh
chords, diminished triads, or other types of harmonies are specifically accounted for; all such
harmonies are included in the “other” category. This decision was made to reflect the triadic
structure of the alfabeto system, even though it may have an effect on the other corpora.
The results can be found in Figure 4.16. Again, the songs are neatly clustered into two
clusters, and both the silhouette and completion scores clearly favor two clusters. The results
mostly match those of the key profile clustering analyses. The centroids for each cluster are
shown in Table 4.6. This provides a more complete picture of the differences between major
and minor than the key profiles alone. There are also some interesting differences between
the major and minor cluster centroids such as the preference of major keys to harmonize 3̂
with 63 while the minor third degree (♭3̂) of the minor centroid is almost always harmonized in
root position ( 53 ). Like the key profile clusters, the NEATO key average graph in Figure 4.17
shows that the modal pairs are still mostly closer to each other than to others.
74
(a) PCA-reduced graph of alfabeto letter usage
completeness score
silhouette score
score
1.0
0.8
0.6
0.4
0.2
0.0 2
4
6
8
10
12
number of clusters (modes)
(b) Silhouette and Completeness Score for alfabeto letter usage
Figure 4.15: K-means clustering of alfabeto letter frequency (untransposed); breaks into
mollis (centroid 1) and durus (centroid 2)
75
Table 4.5: Frequency of bass note harmonization centroids of Figure 4.15 from the alfabeto
corpus (untransposed); all numbers sum to 100
(a) Percentage of bass note harmonization of centroid 1
M
m
M
6
3 m
6 M
4 m
other
5
3
C
10.7
3.3
0.0
0.1
0.2
0.0
1.7
♯/♭
0.0
0.0
0.2
0.0
0.0
0.0
1.1
D
7.0
6.7
0.8
0.0
0.4
0.5
1.6
♯/♭
4.1
0.0
0.0
0.7
0.1
0.0
0.6
E
0.5
0.7
1.6
0.0
0.0
0.1
1.4
F
14.9
0.0
0.0
0.2
0.3
0.0
1.1
♯/♭
0.0
0.0
0.5
0.0
0.0
0.0
0.1
G
5.7
9.8
0.2
0.0
0.1
0.2
1.3
♯/♭
0.0
0.0
0.0
0.0
0.0
0.0
0.0
A
2.6
3.6
1.6
0.0
0.1
0.1
1.5
♯/♭
11.0
0.0
0.0
0.4
0.1
0.0
0.9
B
0.0
0.0
0.5
0.0
0.0
0.0
0.1
♯/♭
0.9
0.0
0.0
0.0
0.0
0.0
0.3
B
0.4
0.0
5.3
0.0
0.1
0.0
1.5
(b) Percentage of bass note harmonization of centroid 2
M
m
M
6
3 m
6 M
4 m
other
5
3
C
13.0
0.1
0.0
0.2
0.2
0.0
1.4
♯/♭
0.0
0.0
0.7
0.0
0.0
0.0
1.9
D
10.2
5.3
0.1
0.0
0.6
0.0
1.9
♯/♭
0.1
0.0
0.0
0.0
0.0
0.0
0.0
E
4.8
3.4
2.9
0.0
0.3
0.3
1.7
F
6.6
0.0
0.0
0.3
0.0
0.0
1.5
♯/♭
0.0
0.0
2.1
0.0
0.0
0.0
0.3
G
16.1
1.2
0.0
0.1
0.1
0.0
1.2
♯/♭
0.0
0.0
0.3
0.0
0.0
0.0
0.0
A
4.6
7.2
0.4
0.0
0.4
0.1
1.3
(c) Comparison of scalla harmonization and most frequent harmonization in the
alfabeto corpus (data)
C
♮
♭
scalla
data
scalla
data
5
3
5
3
5
3
5
3
♯/♭
-
D
5
♯3
5
♯3
5
3
5
3
♯/♭
5
3
E
F
5
♯3
5
♯3
♮5
♯3
6
3
5
3
5
3
5
3
5
3
76
♯/♭
6
3
-
G
5
3
5
3
5
3
5
3
♯/♭
-
A
5
3
5
♭3
5
3
5
3
♯/♭
5
3
5
3
B
6
3
6
3
-
(a) Clustering of bass harmonization data: alfabeto corpus
completeness score
silhouette score
score
1.0
0.8
0.6
0.4
0.2
0.0 2
4
6
8
10
12
number of clusters (modes)
(b) Silhouette and completeness score for bass harmonization: alfabeto corpus
Figure 4.16: Clustering of bass harmonization in the alfabeto corpus
77
Table 4.6: Frequency of bass note harmonization centroids from Figure 4.16, alfabeto corpus;
all numbers sum to 100
(a) Percentage of bass note harmonization of centroid 1 (Major)
M
m
M
6
3 m
6 M
4 m
other
5
3
1̂
18.7
0.5
0.0
0.1
0.3
0.0
1.0
♯1̂/♭2̂
0.0
0.0
0.2
0.0
0.0
0.0
1.6
2̂
4.5
6.6
0.2
0.0
0.4
0.1
1.5
♯2̂/♭3̂
0.3
0.0
0.0
0.0
0.0
0.0
0.1
3̂
1.1
2.3
4.5
0.0
0.1
0.1
1.6
4̂
12.5
0.1
0.0
0.3
0.0
0.0
1.2
♯4̂/♭5̂
0.0
0.0
0.6
0.0
0.0
0.0
0.2
5̂
14.1
2.7
0.0
0.0
0.4
0.0
1.6
♯5̂/♭6̂
0.0
0.0
0.1
0.0
0.0
0.0
0.0
6̂
3.6
5.2
1.9
0.0
0.3
0.3
1.5
♯6̂/♭7̂
3.8
0.0
0.0
0.3
0.0
0.0
0.7
7̂
0.1
0.5
2.5
0.0
0.0
0.0
0.9
(b) Percentage of bass note harmonization of centroid 2 (Minor)
M
m
6 M
3 m
6 M
4 m
other
5
3
1̂
7.3
11.1
0.3
0.0
0.2
0.2
1.4
♯1̂/♭2̂
0.4
0.0
0.0
0.0
0.0
0.0
1.6
2̂
1.9
2.0
1.7
0.0
0.1
0.1
1.7
♯2̂/♭3̂
10.9
0.0
0.0
0.4
0.1
0.0
1.0
3̂
0.0
0.0
0.6
0.0
0.0
0.0
0.2
4̂
7.1
6.4
0.0
0.1
0.2
0.0
2.1
♯4̂/♭5̂
0.0
0.0
0.1
0.0
0.0
0.0
0.0
5̂
9.1
4.7
1.2
0.0
0.3
0.4
1.7
♯5̂/♭6̂
5.9
0.0
0.0
0.9
0.1
0.0
0.9
6̂
0.0
0.2
1.2
0.0
0.0
0.0
1.0
♯6̂/♭7̂
11.2
0.6
0.0
0.2
0.3
0.0
1.3
7̂
0.0
0.0
0.7
0.0
0.0
0.0
0.2
♮:F
♭:G
♭:B♭
♮:D
♭:D
♮:C
♮:A
♭:F
♭:C
♮:G
♮:E
Figure 4.17: NEATO graph of continuo harmonization in the alfabeto corpus
78
To place the alfabeto corpus in context with the surrounding centuries, as was done
with the key profile analyses, the same algorithm was applied to Bach, Palestrina, and the
Franco-Flemish corpora. The data is represented in a way that is stylistically appropriate for
Baroque music since it is measuring bass harmonization. The earlier corpora, Palestrina and
the Franco-Flemish genres, are unlikely to cluster well because all of the music was composed
before the practice of basso continuo. The highly contrapuntal textures also contribute to
the low cluster score. Since these corpora do not have convenient chord symbols such as
the alfabeto corpus, “chords” are identified in a pseudo-continuo notation (as described in
Chapter 2):
1. chordify7 each musical score;
2. record the original key signature and final cadence;
3. convert each pitch to a pitch-class integer with pitch-class 0 assigned to the final bass
note of the score;
4. create a pseudo-continuo notation with the bass note of each “chord” and the set of
the entire “chord” (a I6 chord would be notated as [4, [0, 3, 8]], and a V chord would
be notated as [7, [0, 4, 7]]);
5. remove any harmonies that have a cardinality less than three;
6. remove adjacent subset harmonies (ex. if V leads to V7 , only input V7 ), adjacent
identical harmonies, and arpeggiated harmonies (ex. I→I6 ); and
7. remove any harmonies that do not occur above a certain threshold in the entire corpus
(ex. remove all harmonies that do not account for at least 1% of the total harmonies
in the corpus); the remaining harmonies tend to be more salient and prioritize more
chordal areas such as cadences.
This method is also used in Chapter 5, and the reasons for some of the decisions above are
explained in more detail. In general, these steps represent a balance of chord-label accuracy
and computational simplicity. Any philosophical problems with the methodology further
reinforces the usefulness of the alfabeto system where chords are clearly labeled.
7
“Chordify” is a Python method from music21. Each vertical sonority is isolated and treated as a “chord.”
This method is sometimes called “salami-slicing.”
79
Table 4.7: Frequency of bass note harmonization centroids from Figure 4.18, Bach corpus;
all numbers sum to 100
(a) Percentage of bass note harmonization of centroid 1 (Major)
M
m
6 M
3 m
6 M
4 m
other
5
3
1̂
11.7
0.0
0.0
0.9
0.3
0.0
7.2
♯1̂/♭2̂
0.0
0.0
0.1
0.0
0.0
0.0
6.2
2̂
1.2
2.2
0.1
0.2
0.4
0.0
8.4
♯2̂/♭3̂
0.0
0.0
0.0
0.0
0.0
0.0
0.2
3̂
0.5
1.3
4.9
0.0
0.0
0.2
4.1
4̂
2.8
0.1
0.0
1.0
0.0
0.0
6.9
♯4̂/♭5̂
0.0
0.0
0.4
0.0
0.0
0.0
1.6
5̂
7.4
0.2
0.0
0.7
1.0
0.0
10.1
♯5̂/♭6̂
0.0
0.0
0.5
0.1
0.0
0.0
0.7
6̂
0.4
4.4
2.5
0.0
0.0
0.2
5.3
♯6̂/♭7̂
0.2
0.0
0.0
0.1
0.0
0.0
0.7
7̂
0.0
0.0
3.1
0.0
0.0
0.2
4.8
♯6̂/♭7̂
3.5
0.1
0.0
0.9
0.6
0.0
5.6
7̂
0.0
0.0
1.3
0.0
0.0
0.0
2.2
(b) Percentage of bass note harmonization of centroid 2 (Minor)
M
m
M
6
3 m
6 M
4 m
other
5
3
1̂
2.0
10.0
1.0
0.0
0.0
0.2
6.1
♯1̂/♭2̂
0.2
0.0
0.0
0.1
0.0
0.0
8.1
2̂
0.3
0.2
1.7
0.0
0.1
0.1
6.5
♯2̂/♭3̂
4.4
0.0
0.0
2.9
0.0
0.0
3.3
3̂
0.0
0.0
0.5
0.0
0.0
0.0
0.7
4̂
0.8
2.1
0.0
0.2
0.2
0.0
9.2
♯4̂/♭5̂
0.0
0.0
0.2
0.0
0.0
0.0
1.2
5̂
4.1
1.7
2.5
0.0
0.1
1.4
8.8
♯5̂/♭6̂
2.5
0.0
0.0
2.4
0.0
0.0
5.0
6̂
0.0
0.0
0.7
0.0
0.0
0.0
2.1
The Bach corpus has the highest scores for two clusters (Figure 4.18), and the centroid
bass harmonizations can be found in Table 4.7. As is to be expected, the “other” category is
quite high because all non-triadic harmonies fall under that category. Nevertheless, the bass
note harmonization closely resembles that of the alfabeto corpus. Even though non-triadic
harmonies are sent to the “other” column, the Bach data still has more distinct clusters than
the alfabeto corpus, which shows a greater consistency of bass note harmonization. The
NEATO key average shows that the keys again mostly cluster together and do not break off
into pairs (Figure 4.19).
The Palestrina corpus does not have very distinct clustering; it scores best with two
clusters, but not as neatly as the Bach or alfabeto corpora. The “other” columns in Table 4.8
highlight the non-triadic nature of the Palestrina corpus. For example, the harmonization
of 1̂ is primarily “other” rather than a triad. Even with these issues, the clusters still
resemble those of the key profile clustering analyses. Furthermore, the NEATO graph in
80
(a) PCA-reduced graph of bass harmonization data: Bach corpus
completeness score
silhouette score
score
1.0
0.8
0.6
0.4
0.2
0.0 2
4
6
8
10
12
number of clusters (modes)
(b) Silhouette and completeness score for bass harmonization: Bach corpus
Figure 4.18: Clustering of bass harmonization: Bach corpus
81
cm
am
bm
E♭M
FM
gm
B♭M
GM
AM
f♯m
em
DM
CM
dm
EM
fm
A♭M
Figure 4.19: NEATO graph of continuo harmonization in the Bach corpus
82
Table 4.8: Frequency of bass note harmonization centroids from Figure 4.20, Palestrina
corpus; all numbers sum to 100
(a) Percentage of bass note harmonization of centroid 1 (“Minor”)
M
m
6 M
3 m
6 M
4 m
other
5
3
1̂
2.6
8.0
1.3
0.0
0.1
0.5
9.2
♯1̂/♭2̂
0.8
0.0
0.0
0.8
0.0
0.0
6.2
2̂
1.1
1.4
1.2
0.0
0.0
0.4
4.1
♯2̂/♭3̂
4.2
0.0
0.0
2.4
0.2
0.0
3.9
3̂
0.0
0.0
0.4
0.0
0.0
0.0
0.1
4̂
2.4
4.8
0.4
0.6
0.2
0.1
7.2
♯4̂/♭5̂
0.0
0.0
0.1
0.0
0.0
0.0
0.1
5̂
2.9
5.1
1.8
0.0
0.0
0.8
8.0
♯5̂/♭6̂
2.6
0.0
0.0
2.0
0.0
0.0
3.0
6̂
0.0
0.0
0.6
0.0
0.0
0.0
1.3
♯6̂/♭7̂
3.6
1.1
0.0
1.6
0.3
0.0
4.9
7̂
0.0
0.0
0.4
0.0
0.0
0.0
0.1
♯6̂/♭7̂
2.0
0.0
0.0
1.2
0.0
0.0
2.3
7̂
0.0
0.0
1.2
0.0
0.0
0.1
1.2
(b) Percentage of bass note harmonization of centroid 2 (“Major”)
M
m
M
6
3 m
6 M
4 m
other
5
3
1̂
10.8
0.3
0.0
1.5
0.5
0.0
9.5
♯1̂/♭2̂
0.0
0.0
0.0
0.0
0.0
0.0
3.5
2̂
1.5
4.2
0.9
0.0
0.2
0.2
6.9
♯2̂/♭3̂
0.2
0.0
0.0
0.2
0.0
0.0
0.3
3̂
0.2
1.2
2.5
0.0
0.0
0.2
2.5
4̂
7.0
0.0
0.0
1.8
0.1
0.0
6.5
♯4̂/♭5̂
0.0
0.0
0.2
0.0
0.0
0.0
0.0
5̂
7.0
2.6
0.1
0.8
0.7
0.0
9.9
♯5̂/♭6̂
0.0
0.0
0.0
0.0
0.0
0.0
0.0
6̂
0.3
3.6
2.5
0.0
0.0
0.3
4.7
Figure 4.21 shows clear modal pairing. The cluster results of the Franco-Flemish genres are
given (Figures 4.23–4.25), and their clusters also resemble those of the key profile clustering.
4.3.4
Measuring Idiomatic Chord Changes
One area that has drawn considerable interest when I have presented research on modality and clustering is how idiomatic chord choices may have influenced the modal clustering
in the alfabeto corpus. Since the fingering of each chord is already in the dataset, it is
easy to measure the average distances of chord movement. Each average will be paired
with its notated mode so the average distances can be compared. To measure chord-change
distance, the taxicab distance between the fingering vector of each chord is measured. Taxicab distance is essentially the absolute distance between two vectors, or more formally:
83
(a) PCA-reduced graph of bass harmonization data: Palestrina corpus
completeness score
silhouette score
score
1.0
0.8
0.6
0.4
0.2
0.0 2
4
6
8
10
12
number of clusters (modes)
(b) Silhouette and completeness score for bass harmonization: Palestrina
corpus
Figure 4.20: Clustering of bass harmonization: Palestrina corpus
84
♭:G
♮:D
♮:A
♮:F
♭:D
♮:C
♭:F
♮:E
♭:C
♮:G
Figure 4.21: NEATO graph of continuo harmonization in the Palestrina corpus
d1 (p, q) =
Pn
i=1
|pi − qi |.8 For example, consider a G chord (alfabeto chord A) moving to a D
chord (alfabeto chord C). The taxicab distance between the two fingering vectors is 5 units,
indicating minimal movement.
Chord
A
C
absolute distance
total distance
Fingering Vector
[2 0 0 3 3]
[0 0 2 3 2]
20201
5 units
All such distances are calculated between each pair of adjacent chords (chords 1 and
2, chords 2 and 3, etc.). The average distance is calculated as the mean of all adjacent
distances. The results are quite similar, such that key could not be determined by adjacent
chord distances. However, it is possible that the clear preference for close movement might
play a role in restricting which chord should follow which. For example, going back to the
earlier example of the third degree, ♮:G and ♭:F, the taxicab method may illustrate the
8
For an example of taxicab distance applied to twentieth-century classical guitar hand shapes, see (Tompkins 2015).
85
A ♭:B♭
A ♮:F
A ♭:F
A ♮:G
A ♮:C
A ♭:C
B GM
B B♭M
B E♭M
B FM
B AM
P ♮:F
B DM
B CM
P ♮:C
A ♮:D
B EM
P ♭:F
A ♭:G
A ♮:A
B A♭M
A ♭:D
P ♮:G
A ♮:E
B gm
P ♭:C
B am
B dm
B bm
B fm
B f♯m
B em
P ♮:D
P ♭:G
B cm
P ♮:A
P ♭:D
P ♮:E
Figure 4.22: NEATO graph of continuo harmonization in all corpora; black lines connect
modal pairs; Bach, Palestrina, and alfabeto corpora are labeled with “B,” “P,” and “A,”
respectively
86
(a) PCA-reduced graph of bass harmonization data: Franco-Flemish Mass
corpus
completeness score
silhouette score
score
1.0
0.8
0.6
0.4
0.2
0.0 2
4
6
8
10
12
number of clusters (modes)
(b) Silhouette and completeness score for bass harmonization: FrancoFlemish Mass corpus
Figure 4.23: Clustering of bass harmonization: Franco-Flemish Mass corpus
87
(a) PCA-reduced graph of bass harmonization data: Franco-Flemish motet
corpus
completeness score
silhouette score
score
1.0
0.8
0.6
0.4
0.2
0.0 2
4
6
8
10
12
number of clusters (modes)
(b) Silhouette and completeness score for bass harmonization: FrancoFlemish motet corpus
Figure 4.24: Clustering of bass harmonization: Franco-Flemish motet corpus
88
(a) PCA-reduced graph of bass harmonization data: Franco-Flemish secular
song corpus
completeness score
silhouette score
score
1.0
0.8
0.6
0.4
0.2
0.0 2
4
6
8
10
12
number of clusters (modes)
(b) Silhouette and completeness score for bass harmonization: FrancoFlemish secular song Corpus
Figure 4.25: Clustering of bass harmonization: Franco-Flemish secular song corpus
89
Table 4.9: Taxicab distances of common chord movements
Key
♮:G
♭:F
I
[2 0 0 3 3]
[3 3 2 1 1]
iii
[2 4 4 3 2]
[0 2 2 1 0]
I–iii
9 units
5 units
I–ii–iii
21 units
21 units
I–ii–I
22 units
22 units
different choices. As has already been described, a G chord is a desirable chord because of
its ease of fingering, whereas an F chord is not. If there is harmonic movement over a bass
movement of 1̂ − 2̂ − 3̂, the first two chords are almost always root-position triads (I, ii).
The harmony over 3̂ for ♮:G is typically I6 , but for ♭:F, it is typically iii.
Table 4.9 shows the measurements of common movements. Surprisingly, they are almost
all the same. Furthermore, an analysis of every alfabeto song shows that the average chord to
chord taxicab distance is nearly identical regardless of the key. Perhaps string-fret distance
does not play a significant role in alfabeto symbol choice. Of course, taxicab distance does
not completely capture the difficulty of certain chords and changing between them. For
example, some small changes require a difficult finger shift while some large changes do not.
It is possible that idiomatic movement still plays a role in determining chord choice, but the
algorithm for measuring such changes needs to account for more than taxicab distance. More
study in this area could provide an interesting insight into the role idiomatic movements play
in determining musical choices—or perhaps not.
4.4
Conclusion
The clustering agreement of the first two studies shows very strong evidence of the alfabeto corpus operating in a major-minor paradigm, despite its multi-modal notation. While
the corpus clusters into two groups, the bass harmonization and pitch class choice is not
exactly the same as that of later functional tonality, as can be seen when comparing pitch
frequency with Temperley’s key profiles and the bass harmonization with Bach’s. Nevertheless, the major-minor split in modes is very clear.
The studies on idiomatic chord motion show the pragmatic side of performance practice:
do what is easier. If another instrument were not harmonizing the bass notes, it might not
have mattered how 3̂ was harmonized.
90
Perhaps the most important results from this study are the differences between genres.
Results of this chapter show that the modal framework of a given corpus depends on the
genre, at least in early music. Finding the modal framework of a corpus is only the beginning
of understanding its harmonic practice, but it provides an important foundation. In the
following chapter, the results from these studies are critical for dividing the corpus into
major and minor pieces to investigate harmonic function, which provides a more complete
picture of the harmonic practices of the alfabeto corpus.
91
CHAPTER 5
HARMONIC MOTION
“A tone taken by itself is an acoustical
datum, not a musical phenomenon. It
becomes a musical phenomenon only
in association with other tones.”
Carl Dahlhaus, Studies on the Origin
of Harmonic Tonality, (Dahlhaus 1990,
p. 162)
“Meaning gives that order their but words, just not is it.” If the previous sentence did not
make sense, try reading it backwards. While the syntax in music is not the same as that of
language, order still matters. The chord progression IV→V→I carries a different meaning
than V→IV→I: the first is the most common cadential model of the alfabeto corpus; the
second would be much more likely in a 12-bar blues progression, and would not indicate a
cadence in the early seventeenth century. If a book were analyzed by a computer and every
word cataloged, the computer could probably infer the language the book was written in and
perhaps a relative time period given the spelling of the words. However, the meaning of the
book, or even whether fiction or nonfiction, is unlikely to be understood by word counting.
That can only be done by reading the words in order.
Chapter 4 found that the alfabeto corpus is built on a major-minor framework despite
being notated in a durus-mollis system. However, much more than pitch-class frequency
and even bass-note harmonization frequency must be taken into account to understand the
harmonic practices of the alfabeto corpus. In common-practice tonal music, chords are
typically expected to progress in a particular way. Rameau theorized that the fundamental
bass of chords should progress by fifths or thirds (Lester 1994, p. 115). In the nineteenth
century, Riemann developed the theory of harmonic function that further described the
possibilities of chord progressions.
92
The focus of this chapter is on harmonic function, but in a more general sense: a harmonic
function is defined as a group of chords that move in similar ways. For example, if ii and IV
generally progress to V, they will be included in the same function. Given this definition,
there can be different numbers of functions, and chords are not presupposed to belong to any
given function. Like the previous chapter on modality, this chapter uses machine learning to
let the music determine the theoretical generalizations rather than contemporaneous theories.
Given that alfabeto tablature is in its essence a chordal practice, it is worthwhile to
investigate whether the chords have some kind of function. This chapter analyzes how
individual chords function and how many functions best represent the harmonic practices of
the alfabeto corpus. The alfabeto corpus will again be compared to the Palestrina and Bach
corpora from Chapter 4.
5.1
Harmonic Schema in Seventeenth-Century Guitar
Music
In his 1970 article “The Concept of Mode in Italian Guitar Music During the First Half
of the 17th Century,” Richard Hudson presents the argument that much of the alfabeto
repertoire is an expansion of a simple I→IV→V→I progression. This schema came from
the ritornellos of instrumental guitar music, which could be expanded by inserting other
chords in the middle and by repeating a section (such as I→IV) (Hudson 1970). Hudson,
Dean, and Christensen suggest that the I→IV→V→I and its variants came from the guitar’s
oral tradition. While Hudson focuses on instrumental guitar music, the songs in the alfabeto
corpus are heavily centered around I, IV, and V. Almost every final cadence is IV→V→I,
and the internal phrases often also conclude with IV→V→I, although the minor-mode songs
may use iv and i.1 Some internal cadences seem jarring and end abruptly away from tonic,
such as the cadence on II in Kapsperger’s “Felici gl’Animi” in Example 3.2.
1
It is not uncommon for a final cadence to end with a minor triad. Giulia Nuti expressed surprise at such
a case in Montesardo (Montesardo 1612), as well as other harmonic choices (Nuti 2007, pp. 31–2). But for
instrumental and vocal alfabeto music alike, it is a common occurrence.
93
5.2
Machine-Learning Approaches to Finding Chord
Functions
Finding harmonic functions in music through computational methods has been the subject of recent scholarship. The general goal of such methods is to group chords together
that typically move to members of some other group of chords. For example, IV and ii can
be considered a subdominant group (S), I and vi can be a tonic group (T), while V and
vii◦ can be a dominant group (D). In functional harmony, T→S→D→T, and members of
the same function can be played adjacently, such as from IV to ii or V to V7 . This is the
way in which harmonic function is treated in recent computational literature (White 2013b;
White 2014; Jacoby, Tishby, and Tymoczko 2015). Although the three-function model is
widely used in music theory, computational methods can find other numbers of functions
that may prove interesting, especially when dealing with corpora that were composed before
common-practice tonality.
5.2.1
Some Philosophical Issues
As discussed in Chapter 1, several issues arise when using anachronistic concepts such
as Roman numerals, harmonic function, and even “chords” to early music, which can be
summed up in the following two questions:
1. Is there harmonic function in music that was written prior to the common practice?
2. What are “chords” in early music, and how can a computer find them?
With regard to the first question, compositional practice often precedes theories that explain it. This was the case in Chapter 4 where entire corpora of secular music was composed
in a major-minor system despite the notational and theoretical dominance of the multi-mode
system. Going back to the discussion of analytical philosophy in Chapter 1, the first question is also somewhat analogous to the issue of “authenticity” in the performance practice of
“early” music—and the reality of being a modern listener and performer. Richard Taruskin
discussed this at length regarding performing, listening, and musicological study (Taruskin
94
1982), and Gjerdingen discussed the differences between “presentist” and “historicist” approaches to computational corpus studies (Gjerdingen 2014).
Being a “modern” analyst does not prove that there is harmonic function within a corpus
of music. As discussed previously, this chapter uses a more generalized definition of harmonic
function that a computer algorithm can search for: a harmonic function is a group of chords
that behave in a similar way. In other words, chords that come from and move to similar
chords will belong to the same function. If no such grouping is possible due to equal likelihood
of any chord moving to another, the computer will determine that there is no harmonic
function in that corpus of music. Like the mode-finding algorithm of Chapter 4, the number
of chord clusters will reflect the number of functions in a corpus. Determining that number
of clusters is a primary focus of this chapter.
The second question is a more basic, yet important one. Contrapuntal music from Bach,
Palestrina, or the secular genres of the Franco-Flemish school are not particularly chordal,
and it may possibly be disingenuous to attempt to find chords. Fortunately, the alfabeto
corpus is clearly built upon a system of triads, to the extent that voice leading on the
guitar does not adhere to any rules of counterpoint. Still, there are two ways in which
chords are found in the alfabeto corpus: the first is to take only the alfabeto chords as root
position triads, and the second is to pair each alfabeto chord with its associated bass note,
which allows for possibilities of inversions. The chordal framework of the alfabeto corpus
greatly reduces the complexity of finding chords, and provides a simple answer to the second
question.
For the other corpora such as Palestrina, Bach, and the Franco-Flemish school, finding
“chords” is a more difficult process that necessitates much nuance and caution. Music21
includes a method called “chordify,” which isolates every vertical sonority in a piece of
music.2 My methodology for isolating harmonies in the non-alfabeto corpora is listed in the
previous chapter.
The results should provide a computerized continuo reduction that can substituted for
Roman numerals, which will be used in the graphs of this chapter. It is important to note
2
This is often referred to as “salami-slicing.”
95
that the Roman numerals in this chapter are not an analysis but are only showing the
relationship of each chord to the final chord of the piece, which is labeled I or occasionally
i. It should be noted that, as addressed in Chapter 4, mono-tonal Roman numeral labels do
not work as well with corpora after 1750 because of the practice of long-term modulation.3
If the focus of this chapter were on contrapuntal genres rather than the chordal alfabeto
corpus, the method described above would likely be replaced by a scale-degree function that
has a linear approach to harmony.
As mentioned previously, the Roman numerals are not intended to be an analysis in
themselves but simply show the relationship of a chord’s root to the final cadence of a song.
Furthermore, Roman numerals are labeled as if they are derived from a diatonic major scale
for the sake of computational and visual simplicity. For example, a major chord on the third
scale degree of a minor key will be written as ♭III regardless of the notated mode of the
song.
5.2.2
Some Statistical Data of Alfabeto Chords
The first step to determine chord function is to find patterns in chordal motion. If there
is harmonic function within the alfabeto corpus, the likelihood of chord motion will not be
equal among all chords. Table 5.1 is a pairwise histogram that shows the number of times
each chord in the left column moves to a chord in the upper row. Each Roman numeral
is labeled based on the relationship of the alfabeto chord to the final chord of the song.
Inversions are labeled when the bass note associated with the alfabeto chord is different
from the root. A more detailed explanation of how I extrapolate data from the alfabeto
corpus can be found in Chapter 3. The modes chosen for all of the studies in this chapter
(major and minor in the alfabeto corpus) are based on the modes found in Chapter 4.
From the data in Table 5.1 I created a bigram analysis of chord movement, which is
visualized in Figure 5.1. For clarity, only chord motion that accounts for at least two percent
of total motion is shown. The nodes on the graph are chords, and the arrows indicate motion
between those nodes. The thickness of the arrow lines is directly proportional to the number
3
One example is the prolonged modulated passages in typical sonatas and rondos. To analyze such
corpora, one must run key-finding algorithms over the sections to find the modulations.
96
Table 5.1: Pairwise histogram of the alfabeto corpus’s major-mode songs: left column is
antecedent chords, upper row is consequent chords
II
ii
vi
IV6
IV
V
I
iii
V6
III
♭VII
I6
v
VI
II
ii
vi IV6
0
21
9
0
1
0
74
7
16 30
0
1
5
2
0
0
9
87 45
35
30 75 138 28
142 217 92
41
11 63 18
0
4
2
27
4
3
0
24
0
1
8
62
28
10 72
8
4
21 31 17
13
151 127 7
4
IV
9
31
95
27
0
138
452
132
0
6
22
284
75
22
V
246
71
138
28
511
0
230
17
15
0
25
45
0
16
I
44
170
26
19
197
773
0
21
189
8
148
31
21
12
iii V6 III
17 17 15
66 13 36
32 50 24
2 26
1
68 0
9
39 16
3
15 105 4
0
5
0
2
0
0
1
0
0
0
1
0
0
1
0
0
0
0
6
8
4
♭VII
8
38
39
67
106
11
99
0
0
0
0
0
8
10
I6
10
107
26
5
140
60
69
0
2
1
1
0
1
7
v VI
17 8
25 80
47 0
9
0
56 8
13 82
30 19
0
1
0
3
0 64
58 12
4
3
0 92
9
0
of times the connected chords move from one to another. Several things can be seen from
Figure 5.1 such as how the major songs center around I→IV→V→I while the minor songs
include significant interplay with the relative major.
There is clearly some function with regard to the likelihood of chord motion: V is much
more likely go move to I than IV or ii. This bigram analysis only shows motion from one
chord to another, which results in a local view of harmonic function. Larger ngrams can
show longer sections of music and give a bigger picture of how chords are linked together.
A trigram analysis, for example, will show the likelihood of two chords in a particular order
moving to another chord. This can be expanded to 4-grams and higher, but the data becomes
less useful because there are soon too many unique n-grams with none of them coming to
the fore.
5.2.3
Chord Clusters and Harmonic Function
North American music theory teaches three harmonic functions—tonic (T), subdominant
(S), and dominant (D)—where T→S→D→T. Each of these functions are clusters of chords.
For example, IV and ii6 are both members of S, while V and vii◦ are both members of D.
97
i
bVII
bIII
I6
IV
iv
IV
V
I
ii
II
I
V
(a) Major songs
(b) Minor songs
Figure 5.1: Bigram graph of chord motion in the alfabeto corpus
98
Some chords can have a fuzzy membership, belonging to more than one cluster. For example,
vi can sometimes be a member of S or T depending on context. Chord clusters are created
because of the function of their members, and the chord clusters are named after their
functions.
Like the other computational elements in the previous chapter, the clustering algorithms
do not try to fit the results into a music-theoretic paradigm. However, the interpretation
of the data will use familiar function and chord labels when appropriate. For example, if I,
IV, and V cluster into separate functions, those clusters will be called T, S, D, respectively.
The labels do not necessarily imply anything except that the other members of the cluster
behave similarly to I, IV, and V. Given that these three chords are so prevalent compared to
others and are the basic form of Hudson’s ritornello schema, I will use them as prototypes of
a cluster. These three chords also influence my decision to prioritize three clusters, although
more and fewer will be explored.
5.2.4
Hierarchical Clustering
One fairly simple way to determine how chords are related to each other based on their
motion is with a dendrogram (tree diagram), which displays hierarchical clustering, as seen in
Figure 5.2. The data for each Roman numeral come from Table 5.1. The pairwise histogram
is normalized by percent, and numbers are represented so that each chord is labeled with
the percent that it will be preceded by and followed with each chord. The resulting array
therefore has a length that is double the number of chords. To rank similarity between chords,
the Euclidean distance of each array is measured. The dendrograms use the Ward algorithm
to display the data. Similar chords have a smaller Euclidean distance than different ones.
For example, I6 and iii are very similar while I and V are very different.
Just as the number of clusters in the analyses from Chapter 4 indicated the number of
modes in a corpus, the number of chord clusters in this analysis indicate the number of
functions. Being a hierarchical clustering algorithm, any number of clusters between 1 and
the number of chords can be found by placing a threshold on the y-axis of the dendrogram.
For example, in Figure 5.2a a threshold of 50 yields three clusters, which can be seen in
99
Table 5.2: Three clusters from the dendrogram analysis
Function
T
S
D
Chord
I,iii,VI,i,vi
II,ii,IV
V,♭VII,III,v
Table 5.2. As was expected, I, IV, and V became members of different clusters, so these
clusters can therefore be labeled T, S, and D, respectively.
5.2.5
Information Bottleneck Method
Another method of finding chord function comes from a recent article by Nori Jacoby,
Naftali Tishby, and Dmitri Tymoczko (Jacoby, Tishby, and Tymoczko 2015) that uses the
information bottleneck method.4 The goal of Information Bottleneck is to group similarly
functioning chords together, and the results are quite similar to the hierarchical clustering
above. The information bottleneck approach takes a pairwise histogram, which can be
seen in Table 5.1 for the alfabeto corpus’s major-mode songs. Chord motion from the
pairwise histogram (Table 5.1) is processed through the bottleneck algorithm, which tries to
group similarly-moving chords into the same function by squeezing the chords through an
“information bottleneck.” The data for 2–8 functions in the alfabeto corpus can be found in
Table 5.3.
Jacoby’s web applet, written in MATLAB, iterates over the histogram data and clusters
the chords above based on similar motion.5 The information bottleneck method prioritizes
maximal accuracy to minimal complexity; in other words, it tries to find the clustering that
best describes the musical surface (accuracy) and that is minimally complex by Occam’s razor.6 Jacoby’s bottleneck program finds an optimal curve of accuracy vs. complexity given
the data where accuracy is a measurement of how well the clusters describe the musical
surface and complexity is how complicated the clustering information is, based on Occam’s
4
This method builds on previous work by Tishby, Pereira, and Bailek (Tishby, Pereira, and Bialek 1999).
cluster.norijacoby.com
6
This is formalized in (Jacoby, Tishby, and Tymoczko 2015).
5
100
Chord Clustering
60
Difference
50
40
30
20
10
0
I
iii
VI
i
vi
II
ii
IV
V
♭VII
III
v
Chords
(a) A dendrogram of the major-mode alfabeto corpus (chords only)
Chord Clustering
70
60
Difference
50
40
30
20
10
0
III
v
I
6
iii
VI
IV
6
vi
I
II
ii
IV
V
6
V
♭VII
Chords
(b) A dendrogram of the major-mode alfabeto corpus (bass and chords)
Figure 5.2: dendrograms of major-mode alfabeto songs: x-axis shows chords, y-axis shows
Euclidean distance between chord motion data
101
Table 5.3: Harmonic functions in the alfabeto corpus (major songs only) using the information bottleneck method
Number of Functions
2
3
4
5
6
7
8
Function Number
1
2
1
2
3
1
2
3
4
1
2
3
4
5
1
2
3
4
5
6
1
2
3
4
5
6
7
1
2
3
4
5
6
7
8
102
Chords
II, VI, ♭VII, v, III
I, V, IV, ii, vi, I6 , iii, V6 , IV6
I, vi, I6 , VI, iii, IV6
V, ♭VII, v, V6 , III
IV, ii, II
I, vi, VI, IV6
V, ♭VII, V6
IV, ii, II
6
I , v, iii, III
I6 , v, iii
I, vi, VI, IV6
IV, II
ii, III
V, ♭VII, V6
I6 , v, iii
I, VI
vi, IV6
ii, III
V, ♭VII, V6
IV, II
IV, II
V, ♭VII, V6
vi, IV6
I6, iii
v, III
ii
I, VI
ii
VI
I
IV, II
vi, IV6
V, ♭VII, V6
I6 , iii
v, III
Table 5.4: Author-labeled functions; TSD: Tonic, Subdominant, Dominant, Other; Mm:
Major, Minor; DC: Diatonic, Chromatic
II
ii
vi
IV6
IV
V
I
iii
V6
III
♭VII
I6
v
VI
TSD Mm
O
M
S
m
S
m
S
M
S
M
D
M
T
M
S
m
D
M
O
M
O
M
T
M
O
m
O
M
DC
C
D
D
D
D
D
D
D
D
C
C
D
C
C
Table 5.5: Pair-wise histogram of sample progression functions
T
S
D
T
S
D
1
6
1
6
0
1
6
1
6
1
3
0
0
0
Razor. The goal of the information bottleneck approach is to maximize accuracy and minimize complexity.
Figure 5.3 shows the optimal curve for the data in Figure 5.1. The function numbers
(squares) represent the number of functions column from Table 5.3. Jacoby’s program also
tests user-determined functions, such as in Table 5.4 where I tried three function categories:
TSD (Tonic, Subdominant, Dominant); Mm (major, minor); DC (diatonic, chromatic).
The optimal curve was calculated after 200 iterations, while the chord clusters were calculated
after 2,000 clusters.7 The user-made functions (triangles in Figure 5.3) fall short of the
optimal curve, but the computed clusters are much more successful.
7
The optimal curve does not need as many iterations to be accurate. This many iterations are on the
cautious side; fewer would have also been sufficient and cluster the chords the same way.
103
Figure 5.3: Information bottleneck graph: alfabeto corpus major; the black line is the optimized score of complexity and accuracy; squares indicate number of functions (as labeled in
Table 5.3); triangles are author-labeled functions (as seen below); function labels that best
represent the music are closest to the line of complexity/accuracy
104
Any number of functions from two through one minus the total number of chords in the
histogram can be found through the algorithm. This can be compared with the dendrogram
pairings in Figure 5.2b. There is no exact metric for determining the number of clusters that
best fits the corpus, although one can find which number of functions has an ideal balance
of simplicity and accuracy. One of the limitations of the information bottleneck method is
that it is a concrete clustering method rather than a fuzzy one. This means that each chord,
and every instance of it, is forced into one concrete function. Fuzzy clustering could allow
for some chords to belong to different functions based on their context or to belong partly to
more than one function. For example, a vi chord may sometimes be tonic or subdominant
in function or even exist nebulously between the two. It may also act as a chord that brings
a progression from tonic to subdominant.
5.2.6
Hidden Markov Model
Another way of clustering chords is through dynamic processes, which takes a chord’s
placement within each song into consideration. One such method that has been used for
various corpus analyses is the hidden Markov model (HMM). A Markov chain uses n-grams to
determine the probability of a chord (for example) given n-1 number of chords. Consider the
following progression I→vii◦ →I→vi→IV→V→I. A first order Markov chain (bigram) will
calculate the probability of each chord given one previous chord. It starts at the beginning
and “walks” through the progression. A hidden Markov model moves beyond the surface
chords (called “tokens” in machine learning).
The basis of HMMs is that a series of surface tokens (chords) are emissions from “hidden”
states (functions). In other words, the series of surface chords are bi-products of some
number of functions. The input is a dataset of chord progressions, which in the case of the
alfabeto corpus is the chord progression for each song with adjacent duplicates removed.
The algorithm iterates over the progressions and finds which chords are emitted from which
functions. A given number of states (functions) are given for testing (training), and a
starting probability of the transition between those states and their emission is given (random
probabilities in this case). With each iteration of the data, the probability numbers are
105
corrected to better represent the musical surface. This is why it is “machine learning” rather
than “brute force;” the computer does not find the “answer” by simple calculation but rather
from going over the data until the changes in probability are minimized. This approach was
used by Ian Quinn and Christopher White in several different scenarios and music genres
from Bach (Quinn and White 2013) to popular music (Quinn and White 2015). A review of
Quinn and White’s work can be found in Chapter 1.
There are several important differences between HMMs and information bottleneck. The
first is that HMMs produces fuzzy clustering, which means that some chords may belong to
more than one function, while information bottleneck produces discrete clustering. Once an
HMM is trained on a corpus, it can be given a chord progression to “decode” the functions
within based on the trained data. This allows for some chords to change functions based
on context. For example, a vi chord could be closer to a tonic function if it follows V or
a subdominant function if it follows I. The other difference is in some ways philosophical;
information bottleneck considers functions to be a generalized clustering of chords based on
how they move. In other words, chord movement creates functions. HMMs, on the other
hand, considers chords to be “emissions” from “hidden” functions. In other words, harmonic
functions create chords, even though only the chords are known. Jacoby et al. consider their
information bottleneck method to be “analytical” while the HMM method is more from a
composer’s standpoint (Jacoby, Tishby, and Tymoczko 2015, p. 17). They also claim that
information bottleneck has better success clustering models with a higher function number
than HMMs because it contains fewer analytical “degrees of freedom” (Jacoby, Tishby, and
Tymoczko 2015, p. 18).
For the analyses in this chapter and in Chapter 6, I will use the hidden Markov model
approach because of its fuzzy clustering as well as its predictive ability. I used the Python
module hmmlearn to create multinomial8 HMMs based on the chord extractions from the
alfabeto, Bach, and Palestrina corpora that were used in the previous chapter.
8
Multinomial HMMs use discrete data, such as chords. This is compared to Gaussian HMMs that use nondiscrete data. The input for multinomial HMMs assigns an integer for each chord in the corpus. Gaussian
HMMs also take non-integers and would be useful to assess any aspect in music that deals with non-discrete
data.
106
Table 5.6: HMM results in three clusters: major-mode alfabeto songs
(a) Chords only
T
S
D
I
1
0
0
i II
.91 0
0
1
0
0
ii ♭III
0
.69
.85 .12
.15 .19
III
0
0
.94
iii IV
.77 0
0
1
.77 0
iv V
0
0
.72 0
.28 1
v ♭VI VI
.21 .54
1
0
.46
0
.72
0
0
vi ♭VII
.56 .12
.38 .19
0
.7
vii
0
.62
.38
(b) Chords with bass (continuo)
6
I
1
0
0
T
S
D
I
II
.8 0
.12 1
0
0
ii III
0
0
.77 0
.22 1
iii IV
.88 0
0
1
.12 0
IV6
.46
.38
.16
V V6 v VI
0
0
.4
1
0 .47 0
0
.99 .51 .58 0
vi ♭VII
.55
0
.4
.22
0
.72
Table 5.7: HMM results in three clusters: minor-mode alfabeto songs
(a) Chords only
T
S
D
I
1
0
0
i II
.8 0
.19 1
0
0
ii ♭III
0
.99
.33
0
.67
0
IV
0
1
0
iv V
0
0
.95 0
0
1
v ♭VI ♭VII ♭vii
.37 .24
0
0
0
.75
0
0
.57
0
.99
.97
(b) Chords with bass (continuo)
T
S
D
I
1
0
0
i II
.64 0
.24 1
.11 0
ii ♭III ♭III6
0
.99
.13
.48
0
.39
.52
0
.49
IV
0
1
0
IV6
.22
.77
0
iv V
0
0
.94 0
0
1
v ♭VI ♭VII ♭VII6
.6 .28
0
0
0 .72
0
.71
.4
0
.97
.26
The probability of each chord belonging to a function was calculated, as was the probability that a given function will emit a given chord.9 The result is a fuzzy clustering of
the numerals into a given number of functions. Once again, I, IV, and V are members of
different clusters and their probability of being in that membership is nearly 100%. Other
chords, such as ii are members of more than one function. The results from the chord and
continuo data of the major-mode alfabeto corpus can be seen in Table 5.6.
9
This algorithm is a slightly modified version of one created by Nori Jacoby and used in (Jacoby, Tishby,
and Tymoczko 2015).
107
5.3
5.3.1
A Hidden Markov Model Analysis of Harmonic
Function
Determining the Number of Functions
In Chapter 4, the number of clusters were found by measuring the silhouette coefficient
and completeness score for each cluster number (k). This gets more complicated when
determining the number of functions that best represents a particular corpus of music. The
primary point of White’s and/or Quinn’s work is to find the optimal number of functions
in a corpus. This is an interesting question, because it can potentially test theoretical
constructions such as the three-function tonic-subdominant-dominant paradigm. There is
not a widely-applied method for determining the number of functions (states) when they
are not known. For this section, I will be referring to methodology by Quinn and White
(Quinn and White 2013; Quinn and White 2015; White 2013b), and Deese, who attempted
to duplicate White’s study (Deese 2016). I will simply call these examples the “Quinn and
White method.”
Quinn and White attempt to find the number of states that produces the most consistent results. To achieve this, they create several HMMs using a modified Monte Carlo
method, which involves dividing the corpus into fifths. Eighty percent of the corpus (4/5) is
used as training data, which results in probability information for chord transition, function
transition, and chord emission (from the possible functions). The remaining twenty percent
(1/5) is used as testing data, which uses the trained data to label (decode) each chord with
a function.10 This process is repeated 300 times.11 After that is completed, the fifths are
shifted and the process is repeated so that eventually every part of the corpus has been a
part of the training and testing portion. The result is that every chord in each training set
is associated with a 300-member vector that contains function labels. They then create a
10
It also provides the likelihood of that choice, which can be interesting for chords that do not cleanly fit
into a single function such as vi.
11
This is true in (White 2013b), while (Quinn and White 2015) only repeats the process 100 times.
108
distance matrix based on the number of differences the vectors for each chord. For example,
consider the following three possibilities of test results for the following chords:12
Chord:
I:
V:
IV:
ii:
vi:
T
D
S
S
T
Function
T T T
D D D
S S S
S D S
S S T
T
D
S
S
S
The chords I, V, and IV are all maximally different. IV and ii are similar 80% of the
time, and so on.13 Quinn and White use the squared number of differences to create their
difference matrices. For example, the difference between I and V would be 25, while the
difference between IV and ii would be 1.
The difference matrix is clustered using k-means clustering where k is equal to the number
of functions. The silhouette score finds the degree to which the clusters are separate and
distinct. A high silhouette score would indicate that chords were consistently labeled the
same function. Optimal function numbers are determined by peaks in the silhouette score,
with a peak being defined as a score that is higher than the number of clusters that precedes
and follows it (White 2013b, pp. 194-8).
In White’s Bach analysis,14 he shows peaks for 3 and 13 functions, although it should be
noted that the silhouette score for 13 functions is much lower than for 3.15 Deese attempted
to replicate this experiment and found the optimal number of functions to be 3 and 11. My
own study, although on the entire music21 Bach corpus rather than just chorales, showed
peaks at 3, 5, and 14 functions, with a slight peak at 10, which can be seen in Figure 5.4.
Deese suggests that the inconsistency of the higher numbers of functions is the problem with
the limitations of HMMs (Deese 2016). It should also be noted that Deese only tested 50
HMMs, while White tested 300. My analyses tested 500 HMMs but randomized the Markov
chain Monte Carlo process rather than dividing the corpus into fifths.
12
The actual labels are numbers, and those numbers change from test to test, but I have provided familiar
function labels to better illustrate the idea.
13
Note that these represent single chords in a particular song, not a generalization of such chords.
14
This work was in collaboration with Ian Quinn and has been published in (White 2013b) and presented
in (Quinn and White 2013)
15
White notes that half of their tests showed the latter peak to be for 14 functions rather than 14.
109
1.0
Palestrina major
Bach major
Alfabeto major
Silhouette Score
0.9
0.8
0.7
0.6
0.5
2
3
4
5
6
7 8 9 10 11 12 13 14 15
Number of Functions
(a) Silhouette scores for three corpora using the Hamming metric.
1.0
Palestrina major
Bach major
Alfabeto major
Silhouette Score
0.9
0.8
0.7
0.6
0.5
2
3
4
5
6
7 8 9 10 11 12 13 14 15
Number of Functions
(b) Silhouette scores for three corpora using Quinn and White’s metric.
Figure 5.4: A comparison of silhouette scores for 2–15 clusters using different distance metric.
110
Quinn and White’s purpose for squaring the number of differences is to exaggerate the
different chords. However, the number could potentially change significantly if the number
of tested HMMs is increased. I have found that simply using the percent difference16 gives
very similar results and is not subjected to as much change when the number of HMMs
is increased or decreased. A comparison of my method and Quinn/White’s can be seen in
Figure 5.4.
The primary issue is that unlike previous difference matrices, this uses nominal variables:
the data points are labels rather than intervals. Kaufman and Rousseeuw write that the
Hamming score is the most common for such variables (Kaufman and Rousseeuw 2009,
pp. 28-9). Other methods are suggested for weighting the data, but none of the suggestions
correspond to Quinn and White’s approach.
Quinn and White’s method aims to show the most predictively consistent number of
functions. While I think this is a good starting point, I do not think it necessarily diminishes
the usefulness of other functions. Rather, I think it is indicative of how many chords are
transitional, or have a fuzzy membership to more than one function. I have experimented
with a hierarchical method of looking at HMM functions that shows the likelihood of chords
being present in any of the possible functions.
Finding the “correct” number of states when the number of states is unknown seems
to be a current issue with HMM research. Frequently used methods include the Bayesian
information criterion (BIC) and Akaike information criterion (AIC). Both approaches score
the likelihood of the HMM with a penalty for higher numbers of states. These approaches
may be problematic for some solutions, and several others have been proposed (Gassiat,
Rousseau, et al. 2014; Nam, Aston, and Johansen 2014; Celeux and Durand 2008; Robert,
Ryden, and Titterington 2000). Another promising approach is the reversible jump Markov
chain Monte Carlo method, which is an algorithm that can determine the number of states
based on the distribution of data (Green 1995, pp. 711–2). However, its implementation is
beyond the scope of this study. It is also common for the number of states to be known,
or at least assumed. For example, HMMs are used in speech recognition, and the number
16
This is also called the Hamming difference.
111
of states is sometimes three, based on the beginning, middle, and ending sounds of words.
The alfabeto corpus has many instances of I→IV→V→I, and there are stylistic reasons to
consider other progressions as an elaboration of this basic progression.17
I will use my modified version of Quinn and White’s method for the following analyses
because of its focus on consistency, the results of which can be seen in Figure 5.5. However,
I will not limit the analyses to only three functions; other numbers of functions can parse
out important behaviors of some harmonies. Beyond determining the appropriate number of
functions in a corpus, I think Quinn and White’s silhouette score method provides additional
opportunity to ask whether a given corpus has harmonic function. For example, the Bach
and alfabeto corpora both score high for two and three functions and then steeply drop off.
The Palestrina is mostly flat and does not score high for any number of functions, which
indicates that the corpus is not built on a system of functional harmony.18
5.3.2
Visualizing HMM Analyses
Like the k-means graphs of the previous chapter, my intention is to have the graphs
communicate the necessary information while remaining easy to read. Three things need to
be visualized at once: the number of functions, the emission probability of those functions
(percent likelihood that a particular chord is “emitted” from a function), and the transition
probability between functions (percent likelihood that a function will be followed by the
next).
Each function will be represented by a pie chart that is divided into sections that are
proportional to the emission probability. Any harmony that makes up less than 3% of
the total emission probability becomes a member of the “other” category. The center of
each function pie chart has a different number. These numbers are arbitrary and exist to
distinguish between functions. I also wanted to resist using traditional functional labels even
when a model reflects common-practice harmonic function. Each pie chart was created in
Python using matplotlib (Hunter 2007).
17
This has been explored in (Hudson 1970) where he finds basic passacaglia patterns to be I→IV→V→I
that are then varied throughout the piece. While he only studies instrumental alfabeto scores, it is likely
that this tradition of expanding upon the basic passacaglia model extended into the vocal repertoire.
18
This is, of course, to be expected.
112
1.0
Alfabeto (major)
Alfabeto (minor)
Bach (major)
Palestrina (major)
Silhouette Score
0.9
0.8
0.7
0.6
0.5
2
3
4
5
6
7
8
Number of Functions
9
10
11
Figure 5.5: Function fitness scores of HMMs using the modified Quinn/White method
113
The transition probability is visualized using arrows that point from function to function, or sometimes back onto the same function. The width of these arrows are directly
proportional to the transition frequency. In other words, thick arrows indicate common and
frequent movement while thin arrows indicate less common motion. Any transition that had
a probability less that 1% does not have an arrow for the sake of clarity.
The entire graph of nodes (pie charts) and arrows was constructed in Python using NetworkX (Daniel A Schult and Swart 2008) and pygraphviz (Hagberg, D. Schult, and Renieris
nodate), which processed the image through a Graphviz (Ellson et al. 2001) algorithm. The
images are similar to—and inspired by—those of Quinn and White (Quinn and White 2013;
Quinn and White 2015).
5.3.3
Alfabeto Corpus
Figure 5.6 shows the HMM analysis of the major and minor-mode songs of the alfabeto
corpus. It is striking how well both models reflect a common-practice T→S→D format.
This supports Hudson’s argument that the I→IV→V→I passacaglia schema of the alfabeto
guitar’s vernacular and instrumental practices brought a kind of functional harmony to the
alfabeto corpus (Hudson 1970).
Figure 5.6 should be compared with the bigram graphs in Figure 5.1; the HMM analysis
accurately depicts the relative major and its dominant (♭III, ♭VII) in the minor-mode songs
as belonging to the minor tonic and its dominant (i, V), respectively. It should be noted
that while the HMM graphs reflect common-practice functions, there is some significant
“retrogressive” motion. It is not uncommon for V to move directly to IV, which can be
seen in Kapsperger’s “Felici gl’Animi” (Example 3.2 and 5.1). This shows that while the
functions in the alfabeto corpus behave in a way consistent with common-practice tonality,
the corpus has more freedom of chord movement (the alfabeto system being limited to triads
notwithstanding).
As can be seen in Figure 5.5, the major and minor songs in the alfabeto corpus are
best represented by three functions. More specifically, HMMs with three functions most
consistently labels chords the same way under three functions. This is perhaps due to the
114
heavy presence of IV, V, I and their distinct movements, although perhaps not. Other
function graphs can be found in Section 7.1 of this dissertation and indeed show interesting
properties of some chords.
5.3.4
Bach Corpus
The three functions in the Bach corpus, major and minor, are shown in Figure 5.7. The
major-mode pieces show a clear T→S→D (function nodes 1→3→2, respectively). Notice
also how the arrows are weighted stronger than in the alfabeto corpus. This shows that
retrogressive progressions are very uncommon. The function transitions in the minor-mode
pieces is somewhat less clear but still shows a strong T→S→D motion. It is likely that
some of the ambiguities, if not all, are due to issues with using the chordify method on
contrapuntal textures.
In both major and minor-mode graphs, there is significant self-mapping in the subdominant function (node 3). This can be viewed as a kind of prolongation where a chord progression may move to a different subdominant chord before moving to the dominant (node 2). It
is also significant that many more chords make up the Bach corpus, which is to be expected
due to the constraints of the alfabeto system. The subdominant node in the Bach majormode functions is mostly divided evenly while it is dominated by IV in the alfabeto corpus.
Despite the differences, however, the alfabeto and Bach corpora have striking similarities.
5.3.5
Palestrina Corpus
As previously stated, the Palestrina corpus does not score well in any number of functions using the modified Quinn/White method. Many of the “chords” are likely to have
been incidental merging of contrapuntal lines that were parsed with the chordify function.
Nevertheless, there are some strong transitions. The harmony V4 clearly resolves to V or v,
which then may resolve to I. In other words, the HMM functions seem to have identified two
things: there are some common suspensions that resolve, and there is some V→I motion.
All other progressions seem a bit capricious and inconsequential.
115
1
1
2
2
3
3
(a) Major-mode songs
(b) Minor-mode songs
Figure 5.6: Three functions in the alfabeto corpus using hidden Markov model; nodes indicate
emission frequency of each function; line width is proportional to transition frequency
116
1
1
2
2
3
3
(a) Major-mode pieces
(b) Minor-mode pieces
Figure 5.7: Three functions in the Bach corpus using hidden Markov model; nodes indicate
emission frequency of each function; line width is proportional to transition frequency
117
It should be noted that the I 64 in Bach and Palestrina corpora should often be notated
as V 64 when the chord acts as a cadential six-four, but are labeled the former way for
computational reasons. The functions clearly show that the six-four is a double-suspension,
but the computer input needed a way to distinguish between a dominant with a double
suspension and a second-inversion dominant chord.
5.3.6
Harmonic Analyses from the Alfabeto Corpus
Given the data above, I had the trained HMM alfabeto corpus analyze the chords of
Kapsperger’s “Felici gl’Animi” from his fourth book of villanellas (Kapsperger [Kapsberger]
1623). Chords are labeled with Roman numerals (related to the final cadence, not a functional analysis), function numbers from Figure 5.6, and the probability that the chord belongs
to each of those functions.
The HMM analysis of “Felici gl’Animi” shows that every I, IV, and V belongs completely to separate functions. Other numerals, such as vi have a fuzzy membership, which
is appropriate for its function.
5.4
Conclusion
The results from this chapter should not be interpreted to suggest that composers were
thinking of functional harmony over a century before it was codified. Instead, this chapter
shows that chord movements in the alfabeto corpus progress in predictable and generalizable
ways. This shows that alfabeto symbols were not carelessly sprinkled onto publications
for the sake of popularity (Nuti 2007, p. 31) but progressed in such a way that had clear
harmonic function similar to later codified theories. On a local level, there are certainly
harmonic movements that may seem jarring, but the corpus overall is built upon a system
of regular harmonic function.
118
1
1
2
2
3
3
(a) Major-mode pieces
(b) Minor-mode pieces
Figure 5.8: Three functions in the Palestrina corpus using hidden Markov model; nodes indicate emission frequency of each function; line width is proportional to transition frequency
119
Felici gl'Animi
G
G
Fe li
ci
gl'an
1(S)
2(D)
3(T)
9
I
li,
1(S)
2(D)
3(T)
14
i mi Che
F
Em
I
gior
ni
D
V
mean
an
O
pia
cer
dil
IV
ii
.99
0
.01
.14
.86
0
F
o
pia
bVII
o
IV
sta
bi
le
O
Dm
C
pur
er
vi
cer
sta
bi
ta
no,
E
le
Em
v
0
.75
.25
Dm
o
pou
er
vi
.27
.73
0
.25
0
.75
ses pia ce
l'
core
se
ren
an
II
0
0
1
1
0
0
D
d'
am
a
bi
IV
V
1
0
0
C
0
1
0
D
G
ta
d'a
ma
v
IV
V
I
0
.75
.25
1
0
0
bi
0
1
0
Example 5.1: “Felici gl’Animi” with function emission probabilities (0–1)
120
o
VI
uo
A
Di
1
0
0
0
0
1
li
Em
.25
0
.75
G
1(S) 0
2(D) 0
3(T) 1
E
1(S) .27
2(D) .73
3(T) 0
teu
bVII
I
et
da
0
1
0
C
I
V
1
0
0
ro anni
Am
G
0
1
0
o
pet
G
Nel
le
i mi
IV
C
.22
0
.78
fan
0
1
0
D
no Ne cure e
V
bVII vi
.27
.73
0
da
C
F
18
greg gie gui
0
0
1
D
G
G. G. Kapsperger [Ger. Kapsberger]
transcribed by Daniel Tompkins
le
0
0
1
CHAPTER 6
A COMPARATIVE ANALYSIS WITHIN
THE ALFABETO CORPUS
Felici gl’animi
Che greggie guida no
G.G. Kapsperger, “Felici gl’Animi”
(Kapsperger [Kapsberger] 1623)1
At the beginning of this dissertation, I wrote that large-scale harmonic features are representative of the alfabeto corpus as a whole but that there is some wiggle room in the smaller
details. This chapter will primarily focus on Kapsperger’s seven books of villanellas, including their detailed harmonic practice and how it changed over the thirty years of publication.
This chapter will also look at differences between Kapsperger’s realization of figured bass
on the guitar and theorbo, which will speak to the larger issue of the role instrumentation
played in determining continuo realization. After analyzing Kapsperger’s villanellas, the
comparative analysis will be expanded to include the entire alfabeto corpus.
6.1
Kapsperger’s Harmonic Practices
Kapsperger’s publication show a versatility in obeying contrapuntal norms. His sacred
music follows conventional voice leading, as does the theorbo tablature accompaniment in his
first and third books of villanellas. His other secular vocal music (without alfabeto tablature)
follows the style of monody and include written-out vocal embellishments. While he clearly
has a command for writing conventional voice leading on the theorbo—which is difficult
given the re-entrant tuning of the instrument—his solo theorbo publications show his ability
to deviate from the norm. One of the most strikingly deviant pieces is “Colascione”—named
after the instrument of the same name—from his fourth book of tablature for solo theorbo
1
“Happy are the souls who lead the flock”
121
Example 6.1: “Colascione” from Kapsperger’s Libro Quarto di Intavolatura d’Chitarrone,
built almost entirely from open parallel fifths
(see Example 6.1 (Kapsperger [Kapsberger] 1640a)). The piece is almost entirely built from
parallel open fifths, which today sounds more similar to a power-chord metal song than a
seventeenth-century composition.
Kapsperger did not include a scalla di musica with any of his seven books of villanellas, but he did provide a comprehensive continuo harmonization guide in his third book of
tablature for solo theorbo (see Figure 6.2) (Kapsperger [Kapsberger] 1626, p. 46). Like the
scalla published with some Italian alfabeto songbooks, Kapsperger divides his into durus
and mollis. It seems clear that Kapsperger is showing typical harmonizations over a given
bass and signature without regard to any particular final. He also includes inversions and
triads with a major third. Perhaps unsurprisingly, Kapsperger realizes all sharp (raised) bass
notes as first inversion triads, even though there is no notated figure.
122
Example 6.2: “Colascione” from Kapsperger’s Libro Quarto di Intavolatura d’Chitarrone
(Transcription), built almost entirely from open parallel fifths
123
(a) Bass Harmonization from (Kapsperger [Kapsberger] 1626) (durus)
(b) Modern Transcription (durus)
Figure 6.1: Bass Harmonization from (Kapsperger [Kapsberger] 1626) in the durus system
124
(a) Bass Harmonization from (Kapsperger [Kapsberger] 1626) (mollis)
(b) Modern Transcription (mollis)
Figure 6.2: Bass Harmonization from (Kapsperger [Kapsberger] 1626) in the mollis system
125
6.1.1
Differences in Guitar and Theorbo Continuo Tablatures
The differences between the guitar and theorbo tablatures in Kapsperger’s first and third
books of villanellas have been addressed several times. As discussed in Chapter 3, such
differences lead to questions regarding who placed the alfabeto tablature in the score, its
reliability, or perhaps implies that the theorbo and guitar tablatures were two options that
were not intended to be played together. Nevertheless, this deserves a closer look.
The theorbo is an instrument that has the capability of playing a bass line and harmonizing it. A skilled theorboist, as Kapsperger was, could harmonize a bass line using correct
voice leading. This, however, was in spite of the theorbo’s tuning topology; the two strings
on the bottom of the instrument are lower than the two above it, which creates re-entrant
tuning. In other words, arpeggiating a chord from the bass note to the bottom of the instrument would not create a contour of low to high but one that rises, falls, and rises again.2
This is the reason Kapsperger chose the chord voicings and fingerings he did in Figures 6.1
and 6.2.
Not being bound by the same constraints as the alfabeto system, the theorbo can play
diminished triads, seventh chords, and suspensions. These differences can be seen in the
opening measures of Kapsperger’s “Fiorite Valli” from his first book of villanellas (Kapsperger
[Kapsberger] 1610), transcribed in Figure 6.3. Consider the first three bass notes of the
second measure: the alfabeto chords harmonize them as C–Dm–Gm, while the theorbo
harmonizes them as C–B♭(first inversion)–Em. The theorbo part is essentially playing all of
the vocal parts, but the alfabeto seems to be uncertain which part to follow. The Dm in
the following measure is following the outer voices, but the theorbo, considering each voice,
plays a G chord in second inversion.3
The opening measures of “Fiorite Valli” presents a more extreme case of theorbo-guitar
disagreement than most of Kapsperger’s other songs that include accompaniment for theorbo
and guitar, but it is clear that the instruments have some accompanimental disagreements.
2
For an extensive overview of theorbo types and tunings, see (N. North 1987).
The B♮ in the theorbo part is left unresolved due to issues with the re-entrant tuning. Including the C
would have required an awkward and very difficult fingering.
3
126
These differences highlight the practical reality that continuo harmonization varied from
instrument to instrument, especially in the early seventeenth century.
6.2
Comparative Analysis of Kapsperger’s Villanellas
(1610–1640)
Kapsperger’s seven books of villanellas provide an excellent starting point for a closer
analysis within the alfabeto corpus. The books span from the first publication to one of
the last, the genre and place of publication is the same for all books, and Kapsperger was a
guitarist himself. This section will show a detailed view of Kapsperger’s harmonic practices
in his seven books of villanellas based on the alfabeto tablature. Only major-mode songs
will be analyzed because they provide the largest dataset.
6.2.1
Chord Frequency Analysis
Perhaps the simplest way to compare each book in the corpus is to look at the frequency
that each chord is used. The alfabeto system allows for twenty-four chords, one major and
minor for each chromatic note, even though most of the chords used are the first few in the
alfabeto decoder (Eisenhardt 2015).
I created a program in Python that counts the how often each triad is used for all majormode songs in each book. The result is a 24-dimensional vector that is a “chord profile”
of each book. The books can be measured for similarity by finding the Euclidean distance
between each book. There is no need (and too few data points) for a cluster algorithm, so
the results of the Euclidean data will be represented two ways. A distance matrix shows the
Euclidean distance between the book on the left column to the book on the upper row, seen
in Table 6.1.
This data can be visualized graphically through multidimensional scaling using the NEATO
algorithm (S. C. North 2004). Each node is placed a distance away from all others in a way
that is mostly proportional to the distances found in Table 6.1. The results can be seen
in Figure 6.4, which shows that the book similarity mostly reflects the date of publication.
127
(a) Original
B♭
F Gm
86
Fio ri te val
86
Fio ri te val
6
8
Fio ri te val
Theorbo
li
C Dm Gm F
Cm
Gm F Dm
cam pag nea me
ne
vi
vi cri stall
i
Dm E♭ Cm B♭
fe
li cia re
ne
li
cam pag nea me
ne
vi
vi cri stall
i
fe
li cia re
ne
li
cam pag nea me
ne
vi
vi cri stall
i
fe
li cia re
ne
6
8
6
8
2015
(b) Transcription
%
%
%
Figure 6.3: “Fiorite Valli (Flowering Valleys)” from (Kapsperger [Kapsberger] 1610), first
four measures
128
Table 6.1: Distance matrix of all seven books of villanellas by Kapsperger
1:
2:
3:
4:
5:
6:
7:
1610
1619
1619
1626
1630
1632
1640
1: 1610
0
12.44
12.29
12.42
15.32
9.15
19.24
2: 1619
3: 1619
4: 1626
0
3.64
3.74
5.91
10.45
10.52
0
6.01
8.43
11.37
11.50
0
5.21
9.39
10.61
5: 1630 6: 1632
0
12.62
7.69
0
16.96
7: 1640
0
Book 6 is a notable exception, but it is also the only book that has more minor-mode songs
than major-mode songs and therefore has a weaker sample size.
6.2.2
N-Gram Analysis
Moving beyond chord frequency, it is important to also look at how those chords interact
with each other. An n-gram is a way of measuring the frequency of succession of various
things—chords in this case. It is the likelihood of a given chord given n − 1 chords. In
common practice harmony, a bigram analysis of chords would show that a V chord is likely
given IV, but a IV chord is unlikely given V. This is a simpler method than the hidden
Markov model analysis in Chapter 5, but it is ideal for focusing on isolated harmonies.
The computational methods in Chapters 4 and 5 assume a kind of chord egalitarianism—
that is that no chord or function is more important than another. In common practice
tonality, some chordal movements are arguably more important than others such as cadences.
In the following analysis, I have primarily focused on n-grams that are most likely to involve
cadences. This provides an interesting look into “pre-dominant” harmonies and will give
more context to the harmonic function analyses in Chapter 5.
Bigram graphs in Figure 6.6 highlight the differences between Kapsperger’s first book of
villanellas (1610) and the last (1640). These graphs show all motion, with weighted arrows
that reflect movement frequency, that accounts for at least 2% of all bigram motion. The
passacaglia progression (IV→V→I) is central to book 1, with a significant input from a
secondary dominant (II). The seventh book is less focused on the passacaglia progression
and includes a greater interaction with iii and ii.
129
Figure 6.4: MDS graph of Table 6.1. Each node represents a book, and distance corresponds
to difference.
130
II
V
V
I
IV
iii
IV
I
ii
(a) Book 1 (1610)
(b) Book 7 (1640)
Figure 6.5: Bigram motion that makes up at least 2% of total motion (major keys only)
131
Table 6.2: Book similarity measuring bigram frequency
1:
2:
3:
4:
5:
6:
7:
1610
1619
1619
1626
1630
1632
1640
1: 1610
0
10.96
11.42
12.71
13.08
10.28
16.54
2: 1619
3: 1619
4: 1626
0
5.03
6.92
7.12
11.13
11.06
0
7.27
8.02
11.60
11.52
0
7.63
12.56
10.76
5: 1630 6: 1632
0
12.76
8.82
0
15.55
7: 1640
0
Table 6.3: Book similarity measuring bigram frequency
1:
2:
3:
4:
5:
6:
7:
1610
1619
1619
1626
1630
1632
1640
1: 1610
0
9.73
9.12
11.06
10.98
10.47
12.10
2: 1619
3: 1619
4: 1626
0
7.02
7.79
7.01
10.39
8.80
0
7.69
8.25
10.69
9.79
0
8.13
11.57
8.97
5: 1630 6: 1632
0
11.39
7.56
0
12.55
7: 1640
0
The books can be compared in the same way they were in the previous section except
the number of dimensions will be the one hundred most frequent bigrams. The Euclidean
distance between each 100-dimensional vector is measured between each book, the results of
which can be seen in table 6.2. This can also be visualized graphically, which can be seen in
Figure 6.6.
A trigram analysis takes the bigram a step further by looking at frequency of three
chords that occur in a particular order than just two. Table 6.3 shows a distance matrix of
the books’ trigram data, and it is visualized in a NEATO graph in Figure 6.7. The differences
between the book become less pronounced, a trend that continues with higher n-grams. This
is because there are fewer cases of similar three-chord patterns that show up rather than
bigrams. In a sample size as small as the Kapsperger books, it is possible that only a few
instances of a trigram in one book over another could create difference when it is not really
very meaningful.
132
Figure 6.6: NEATO graph of Table 6.2. Each node represents a book, and distance corresponds to difference of bigram frequency.
133
Figure 6.7: NEATO graph of Table 6.3. Each node represents a book, and distance corresponds to difference of trigram frequency.
134
6.2.3
Harmonic Function
These observations are supported when analyzing harmonic function as well. As in the
previous chapter, I created a hidden Markov model analysis for each Kapsperger book.
Figure 6.8 shows HMM results for 4 functions in books 1 and 7, respectively. Book 1 relies
very heavily on IV→V→I motion while book 7 uses many more chords. This is shown
when contrasting function 2 of book 1 and function 4 of book 7. Keep in mind that the
function numbers are irrelevant. These functions both “function” in the same manner, and
the numbers are assigned randomly at initiation of the HMMs.
Possible ways of measuring HMM similarity include measuring the emission probabilities,
but the size of the data does not create reliable data.
One thing that remains clear through each type of comparison is Kapsperger’s increase
in use of iii over the course of the seven books. One reason for this may be that Kapsperger
began to try to harmonize every bass note rather than allowing for several to go by underneath a single alfabeto chord. However, looking through the songs this appears not to be the
case. Rather, Kapsperger seems to prefer conjunct bass lines near cadences (1̂–2̂–3̂–4̂–5̂–1̂)
in his later books.
It should be noted, however, that these differences highlighted are relatively small and
that the overall focus on I→IV→V→I is apparent in all of his books—and others of the
alfabeto corpus. Figure 6.10 compares all of the books of the alfabeto corpus.
6.2.4
Kapsperger with the Alfabeto Corpus
Using the chord frequency analysis, I measured every book in the alfabeto corpus to see
how Kapsperger fits in with the other books. The results of the comparison can be seen
in Figure 6.9. Each node is labeled with the author and year, and connecting lines were
omitted to decrease visual clutter. It is interesting to note that composition year does not
seem to have an effect on similarity, but several authors’ books are quite similar. These
include Milanuzzi 1628 and 1635, Kapsperger’s 1619a,b, 1623, and 1630, and Milanuzzi’s
1622 and 1625. This is not to say that the “composer” parameter always shows similarity;
Rontani 1623 and 1620 are very different.
135
1
2
4
1
2
4
3
3
(a) Book 1 (1610)
(b) Book 7 (1640)
Figure 6.8: Four functions in the major-key songs
136
Kapsperger
1640
Giaccio
1618
Kapsperger
1630
Kapsperger
1623
Kapsperger
1619b
Landi
1620
Marini
1622
Milanuzzi
1622
Milanuzzi
1625
Vitali
1622
Giaccio
1620
Stefani
1622
Sabbatini
1652
Kapsperger
1632
Stefani
1623
Stefani
1621
Rontani
1620
Kapsperger
1619a
Obizzi
1627
DIndia
1621
Ghizzolo
1623
Kapsperger
1610
DIndia
1623
Milanuzzi
1635 Milanuzzi
1628
Milanuzzi
1630
Rontani
1619
Montesardo
1612
Rontani
1623
Figure 6.9: NEATO graph of all books from the alfabeto corpus (major-key songs only),
chord frequency
137
Given the number of sources, I also processed the data using a dendrogram (tree diagram)
that uses Ward’s method. The result is a hierarchical clustering of book similarity, with the
difference on the x-axis (Figure 6.10). While there is some obvious clustering of books, there
is no clear trend of composer, year, publisher, or city of publication.
6.3
Conclusion
Kapsperger was a composer who could follow conventions of voice leading in some contexts
and completely go against them in others. His ability as a theorboist, lutenist, and guitarist
make his alfabeto song publications worth studying. Further research into his books of
villanellas that include alfabeto and theorbo tablature will lead to better understanding of
the role instrument choice made in realizing a continuo score—especially in an era where
such a thing was a mostly untheorized performance practice.
138
Book Clustering
Vitali 1622
DIndia 1621
DIndia 1623
Marini 1622
Stefani 1622
Giaccio 1620
Kapsperger 1632
Stefani 1623
Stefani 1621
Sabbatini 1652
Kapsperger 1623
Kapsperger 1619a
Book
Kapsperger 1619b
Obizzi 1627
Kapsperger 1630
Giaccio 1618
Kapsperger 1640
Milanuzzi 1622
Milanuzzi 1625
Landi 1620
Rontani 1620
Rontani 1623
Montesardo 1612
Kapsperger 1610
Milanuzzi 1630
Rontani 1619
Milanuzzi 1628
Milanuzzi 1635
Ghizzolo 1623
0
5
10
15
20
25
30
35
Difference
Figure 6.10: A dendrogram comparing all of the books from the alfabeto corpus, calculated
using the Ward method
139
CHAPTER 7
CONCLUSION: THE BAROQUE GUITAR,
ITS NOTATION, AND ITS PLACE IN
MUSIC THEORY, REVISITED
“...the guitar is no more than a cowbell,
so easy to play, especially in rasgueado,
that there is not a stable boy who is
not a musician on the guitar.”
Nicola Matteis, Sebastián de
Covarrubias, Tesoro del lengua
Castellana a Española, 16111
As I mentioned in the beginning of this dissertation, the seventeenth century was a time
when harmonic practices and theories were changing. It saw the rise of basso continuo, the
eventual notation change to a major-minor system, and a triadic-based harmonic system that
was bass-focused. The alfabeto corpus is important to understand these changes because it
existed in the first half of the seventeenth century.
The alfabeto corpus is also important because it gives a notated continuo realization—
however imperfect or limited it may have been—that lends itself to computational methods.
Although it was seen as low-brow and unsophisticated music, the alfabeto system itself was
quite forward thinking; the system was entirely triadic with no regard to counterpoint, and
the scalla di musica was perhaps an early version of what would become François Campion’s
règle de l’octave. The alfabeto letters provide a clear understanding of what a “chord” was,
which bypasses many of the issues in studying harmony with computational methods.
The results of the analyses in Chapter 4 found that the underlying modal framework
for the alfabeto corpus is major-minor rather than the multi-mode system in which it is
1
“...la guitarra no es más que un cencerro, tan fácil de nañer, especialmente en lo rasgado, que no ay
moco de cavallos que no sea música de guitarra.” Sebastián de Covarrubias, Tesoro del lengua Castellana a
Española, (Madrid, 1611); translated by Lux Eisenhardt (Eisenhardt 2015, p. 15).
140
notated. This gives important insight into the way in which the songs were composed and
how one may go about analyzing their harmony. The modal framework of the alfabeto
corpus in addition to its triadic focus shows how forward-thinking the seemingly low-brow,
unsophisticated music turned out to be. The alfabeto corpus also has harmonic function
that is strikingly similar to what would become standard in the common practice. This
was perhaps an outgrowth of the vernacular Spanish dances such as the passacaglia, which
centered around I→IV→V→I.
The alfabeto corpus is only a slice of the published guitar music in the seventeenth
century. The instrumental rasgueado style lasted throughout the seventeenth century and
was practiced well beyond Italy. It also features many more advanced and virtuosic chord
changes than the alfabeto song repertoire. The same studies of mode and harmonic function
could apply to that corpus, although the songs were rarely notated with a key signature.
Several guitarists, such as Gaspar Sanz, included schema for several of the vernacular dances,
although Sanz’s book was published after the rise and fall of the alfabeto corpus. These
schemas could be compared to the harmonic progressions in the alfabeto corpus. However,
it is possible that the Italian alfabeto songs had an impact on Sanz’s schema because he
traveled to Italy to perfect his craft on the “Spanish” guitar.
The detailed study of harmony in Chapter 6 showed Kapsperger’s sometimes peculiar
compositional style as well as some of the interesting ways of continuo realization. One such
example is the use of parallel triads, such as iii→IV→V rather than I6 →IV→V, although
the latter was certainly also used. The Kapsperger analyses also showed the limits of the
alfabeto system; the lack of seventh chords, diminished triads, and suspensions occasionally
created issues with harmonizing 7̂ and caused some disagreement with the occasional figured
bass. These small details did not affect the overall data of the larger corpus studies, but
they are nonetheless interesting and important.
From a computational standpoint, this dissertation offered a corpus with clearly defined
“chords,” which conveniently takes away some of the analyst’s often heavy footprint. The
application of k-means clustering to find modal frameworks through the silhouette coefficient
and completeness score offers a compelling way to look for modal frameworks in any cor141
pus and clarifies previous studies of modality and machine learning. The harmonic function
analyses offer a useful comparison of various methods and discuss the problem of finding
the correct number of functions within a corpus. This dissertation also offers methods for
visualizing results of computational studies such as the hidden Markov model visualizations,
NEATO graphs, and principal component analysis graphs. Furthermore, I hope this dissertation offers a corpus study that is first and foremost built upon knowledge of the alfabeto
system and its use in Italy and beyond.
Back to the guitar, it is fascinating to ponder how an instrument came from the taverns
to the hands of Louis XIV and François Campion—both of whom had a significant effect
on musical composition and theory for centuries. The guitar became a common continuo
instrument, and perhaps the alfabeto tradition helped to shape the practice of continuo
itself. As a guitarist, I find it fascinating how similar the alfabeto songbooks are to what one
finds in music stores today—popular songs that everyone hears with guitar chords (likely
printed by the publisher) and a melody for people to play and sing at home. Furthermore,
the existing copies of poetry with alfabeto chords and no notation is strikingly similar to
what one finds on the internet today. Guitar websites also often include finger placement
guides to help amateur guitarists who do not read music play their favorite songs.
Perhaps the alfabeto corpus was indeed a mix of vernacular dance music and songs
associated with taverns that even a stable boy could play. But this “stable boy” music was
forward thinking with harmonic elements that would later become standard practices and
theories.
7.1
Further Research
Beyond the scope of the Baroque guitar, this dissertation offers many opportunities for
further research. The methodology for determining mode from Chapter 4 can be applied to
many different corpora. Current robust digital collections include the Liber Usualis,2 and
the thousands of popular chord charts from iRealPro.3 As more music becomes digitized,
2
3
http://kern.humdrum.org/cgi-bin/browse?l=/mcgill/liber
http://irealpro.com
142
there will be many opportunities to gain a better understanding of the modal context of
many different kinds of music.
The key profiles used to determine mode are based on cognitive studies by Krumhansl
and Kessler, which focuses on only major and minor. Given that the cluster analysis implies
several modes in sacred music, a revised cognitive study could include more modes than
major and minor. This also leads to interesting questions about historical music cognition
and its change over time.
The methodologies for determining harmonic function from Chapter 5 can also be expanded to include other digitized music. Beyond expanding the application of the methodology, there is more work to be done on the methodology itself. As described in Chapter 5,
there may be more efficient and accurate ways of determining the number of functions that
best represents a corpus of music. Several methods have been applied in other fields but have
yet to be applied to music. In addition to finding the best number of functions, it would also
be useful to understand if a corpus exhibits harmonic function to begin with—in a similar
way to which I measured the scores of mode clustering for different numbers of modes.
Of course, machine learning is a rapidly growing field of study. Complex machine learning
methods such as deep learning and artificial neural networks (ANN) offer new possibilities for
understanding many aspects of music. As computers and algorithms become more capable,
there is an ever expanding horizon of opportunities on which we can use computational
methods to test our own assumptions and better understand music.
143
APPENDIX A
ADDITIONAL HMM FUNCTION GRAPHS
Chapter 5 found that three functions best represents the alfabeto corpus, and those three
functions were described in detail. However, other numbers of functions can elucidate some
interesting properties of the alfabeto corpus. The function numbers (in the middle of each pie
chart) of each graph are arbitrarily numbered by the HMM algorithm and are not intended
to correspond with other graphs. Only transitions that account for at least two percent
of all function transitions are included to reduce arrow clutter. Perhaps one of the more
interesting discoveries when comparing different models is to observe the dual function a
single chord may have. Larger graphs can be produced by executing the code found on
GitHub (https://github.com/tompkinsguitar/) or by contacting the author.
1
1
2
2
(a) Major-mode songs
(b) Minor-mode songs
Figure A.1: Two functions in the alfabeto corpus using hidden Markov model; nodes indicate
emission frequency of each function; line width is proportional to transition frequency
144
1
1
2
2
4
4
3
3
(a) Major-mode songs
(b) Minor-mode songs
Figure A.2: Four functions in the alfabeto corpus using hidden Markov model; nodes indicate
emission frequency of each function; line width is proportional to transition frequency
145
1
2
1
5
3
4
4
3
5
(a) Major-mode songs
2
(b) Minor-mode songs
Figure A.3: Five functions in the alfabeto corpus using hidden Markov model; nodes indicate
emission frequency of each function; line width is proportional to transition frequency
146
1
1
2
4
3
3
5
6
4
5
6
2
(a) Major-mode songs
(b) Minor-mode songs
Figure A.4: Six functions in the alfabeto corpus using hidden Markov model; nodes indicate
emission frequency of each function; line width is proportional to transition frequency
147
4
1
6
2
3
1
4
2
7
3
5
7
6
5
(a) Major-mode songs
(b) Minor-mode songs
Figure A.5: Seven functions in the alfabeto corpus using hidden Markov model; nodes indicate emission frequency of each function; line width is proportional to transition frequency
148
1
1
2
4
2
5
3
5
6
8
7
7
8
6
3
(a) Major-mode songs
4
(b) Minor-mode songs
Figure A.6: Eight functions in the alfabeto corpus using hidden Markov model; nodes indicate emission frequency of each function; line width is proportional to transition frequency
149
1
1
3
5
4
7
5
8
8
6
2
9
3
2
7
4
9
6
(a) Major-mode songs
(b) Minor-mode songs
Figure A.7: Nine functions in the alfabeto corpus using hidden Markov model; nodes indicate
emission frequency of each function; line width is proportional to transition frequency
150
1
1
6
4
3
3
8
10
6
2
8
10
7
9
4
7
2
5
5
9
(a) Major-mode songs
(b) Minor-mode songs
Figure A.8: Ten functions in the alfabeto corpus using hidden Markov model; nodes indicate
emission frequency of each function; line width is proportional to transition frequency
151
1
1
3
3
7
5
8
5
2
2
10
6
11
4
11
6
8
10
4
9
9
7
(a) Major-mode songs
(b) Minor-mode songs
Figure A.9: Eleven functions in the alfabeto corpus using hidden Markov model; nodes indicate emission frequency of each function; line width is proportional to transition frequency
152
1
2
6
1
7
6
12
3
3
8
8
10
2
5
5
9
10
12
4
11
9
4
11
(a) Major-mode songs
7
(b) Minor-mode songs
Figure A.10: Twelve functions in the alfabeto corpus using hidden Markov model; nodes indicate emission frequency of each function; line width is proportional to transition frequency
153
6
2
1
7
1
12
2
9
12
11
7
9
4
3
13
8
3
6
11
5
4
8
13
10
5
10
(a) Major-mode songs
(b) Minor-mode songs
Figure A.11: Thirteen functions in the alfabeto corpus using hidden Markov model; nodes
indicate emission frequency of each function; line width is proportional to transition frequency
154
1
1
4
5
11
8
2
7
2
13
10
11
5
3
10
7
12
9
12
9
3
6
14
13
4
6
14
8
(a) Major-mode songs
(b) Minor-mode songs
Figure A.12: Fourteen functions in the alfabeto corpus using hidden Markov model; nodes
indicate emission frequency of each function; line width is proportional to transition frequency
155
1
7
11
14
8
13
2
1
5
4
3
3
12
12
7
10
9
8
4
15
15
13
2
6
11
14
6
10
(a) Major-mode songs
9
5
(b) Minor-mode songs
Figure A.13: Fifteen functions in the alfabeto corpus using hidden Markov model; nodes indicate emission frequency of each function; line width is proportional to transition frequency
156
APPENDIX B
LIST OF TERMS
bigram a type of n-gram where n = 2
chordify a music21 Python method that isolates every possible vertical sonority in a musical
score; colloquially referred to as "salami slicing"
completeness score an algorithm that scores the results of k-means clustering; measures
the success that all songs of a given notated key or mode land within the same cluster
from 0 (no pieces in the same notated mode or key ended up in the same cluster) and
1 (all pieces in the same notated mode or key ended up in the same clusters)
course a pair of strings on a lute or guitar that are played as a single string; may be tuned
in unisons or octaves (bourdon)
dendrogram a tree-diagram that is the result of a hierarchical clustering algorithm; the
length of the branches is proportional to the difference (typically Euclidean distance)
between data points
Euclidean distance measures the shortest distance between two points; on two-dimensional
graph:
q
(y2 − y1 )2 + (x2 − x1 )2
formally:
d(p, q) =
v
u n
uX
t (q
i
− pi )2
i=1
French tablature a system of horizontal lines that represent courses of a lute, guitar, or
theorbo where the top line is the highest-sounding course; letters are placed on the
lines that indicate which on which fret a string or course should be pressed
harmonic function chords that are grouped together based on their likelihood of movement
157
hidden Markov model a Bayesian statistical algorithm that consists of unobserved (hidden) states and emission tokens of those states; the transition probabilities between
states and emissions as well as the emission probability from states to tokens can be
found by training and fitting data; applied to harmonic function, chords are known
(observed) emissions that are emitted from hidden functions; the algorithm iterates
over the chords in the order in which they occur and determine the likelihood that a
chord is emitted from a particular state
information bottleneck an algorithm created by Jacoby, Tishby, and Tymoczko 2015,
based on Tishby, Pereira, and Bialek 1999, to group similarly-moving chords into
functions; models that maximize accuracy and minimize complexity are selected for
functions
Italian tablature a system of horizontal lines that represent courses of a lute, guitar, or
theorbo where the bottom line is the highest-sounding course; numbers are placed on
the lines that indicate which on which fret a string or course should be pressed
k-means clustering an unsupervised machine learning algorithm that partitions data into
k number of partitions or clusters
Monte Carlo method repeated random sampling to determine a certain statistic result or
measurement, such as drawing a square with a circle inside it, scattering dots over the
surface (pseudo-randomly), counting the number of dots inside and outside the circle,
and repeating; the result is a useful way to approximate pi (Kalos and Whitlock 2008)
n-gram the likelihood of something given n − 1 things
Neapolitan tablature an early variation of Italian tablature but without the number 0
(first fret would be labeled with a 2); horizontal lines were occasionally flipped like in
French tablature
NEATO a graphing algorithm where the distance between nodes on a graph reflects the calculated difference (typically Euclidean distance of the data associated with the nodes)
between the nodes
principal component analysis an algorithm that compresses multi-dimensional data into
fewer dimensions
158
rasgueado Spanish; refers to the "strummed" style of accompaniment on the five-course
guitar; also see battuto
scalla di musica two scales, one in durus, on in mollis, that show alfabeto symbols and
Italian tablature to harmonize each bass note; found at the beginning of many alfabeto
songbooks; possible precursor to François Campion’s règle de l’octave
silhouette coefficient an algorithm that scores the results of k-means clustering; measures
the distance between clusters from -1 (incorrectly labeled clusters) to 1 (very distinct
clusters); higher score indicates distinguishable modes
taxicab distance a non-Euclidean measurement between two points by measuring the sum
of the absolute difference, or formally:
d1 (p, q) =
n
X
|pi − qi |
i=1
on a two-dimensional plane, the measurement can be visualized as navigating city
blocks; also called Manhattan distance, city block distance, etc.
trigram a type of n-gram where n = 3
Ward’s method an algorithm, similar to k-means clustering, that clusters data hierarchically
159
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BIOGRAPHICAL SKETCH
Daniel Tompkins is a native of Brevard, North Carolina where he began guitar studies at
age seven. He graduated from Appalachian State University in 2010 with a Bachelor of
Music in classical guitar performance and in 2013 with a Master of Music in classical guitar
performance where he was named a Fellow of the Cratis Williams Society of Outstanding
Graduates of the Graduate School. While studying at Appalachian State University, he
began studying the lute and theorbo and has since performed with many early music
ensembles. His primary guitar teachers include Douglas James, Robert Teixeira, and Marc
Yaxley. Daniel received a Masters of Music in music theory from Florida State University
in 2015 and promptly began his doctoral studies. Daniel has presented at several
conferences including the Society for Music Theory (SMT), Music Theory Southeast
(MTSE), Texas Society for Music Theory (TSMT), Graduate Association of Musicologists
und Theorists (GAMuT), College Music Society Mid-Atlantic, and Florida State Music
Theory Forum. He has won four awards for conference papers: the Student paper award
(MTSE, 2017), Colvin Student Presentation Award (TSMT, 2017), Student Paper Award
(GAMuT, 2015), and the Ruskin Cooper Best Student Paper Award (College Music
Society Mid-Atlantic, 2013). As a teacher, Daniel has served as an adjunct professor of
guitar at Emory and Henry College, where he taught applied guitar, class guitar, and
guitar ensemble. At Appalachian State University, he taught applied guitar, class guitar,
and directed the guitar orchestra as a graduate assistant. At Florida State University, he
has taught the core undergraduate and aural skills curriculum as an instructor of record.
Daniel’s current research interests include historical performance practice, computational
analysis and composition, mathematics and music theory, and meter in post-progressive
rock. In addition to his research and teaching activities, Daniel is active as a performer on
the lute and theorbo.
175
Early Seventeenth-Century Harmonic Practice: A Corpus Study of Tonality, Modality, and Harmonic Function in Italian Secular Song with Baroque Guitar Accompaniment in Alfabeto Tablature
This dissertation investigates harmonic properties of Italian secular songs with Baroque guitar accompaniment in alfabeto tablature, which was a system of letters and other symbols that represented a chord shape to be strummed using all five courses. All of the resulting chords form major or minor triads. These letters were printed in songbooks for guitarists to accompany secular songs. This triadic accompaniment has significant implications for understanding harmonic practices during the seventeenth century.
The purpose of this dissertation is to use computational methods to better understand the harmonic features of the alfabeto song repertoire, which can perhaps be a window through which we can understand the broader harmonic practices of secular genres in the early seventeenth century. Machine learning algorithms are employed to understand the modal framework and harmonic function of the alfabeto corpus. K-means clustering is used to determine how many modes are present within the corpus, building upon the recent work of Albrecht and Huron. To understand the harmonic function within the alfabeto corpus, hidden Markov models analyze the number of functions and which chords belong to each of those functions using a similar methodology to Quinn and White.
The analyses also compare the alfabeto corpus to corpora of other genres and eras such as J.S. Bach, Palestrina, and Franco-Flemish composers. The alfabeto corpus is also compared with itself to better understand its style and any changes over time. This dissertation provides a fresh look at harmonic practice in the seventeenth century by using statistical models that are informed by historical performance practice....Read more
