Tuning and Temperament

This paper deals with the tuning question as it is discussed in Gioseffo Zarlino’s principal musical-theoretical treatise, Istitutioni harmoniche (1558). First, Zarlino’s tuning system in vocal music (the syntonic diatonic) is presented; then, its shortcomings are discussed; and finally, its adjustment (temperament) for the use in instrumental music is explained.

The creation of the Harrison-Slye tuning for reso-phonic guitar was inspired by Lou Harrison for the composition of his piece "Scenes from Nek Chand" (2001-02). Harrison’s primary scale resource for his composition was the scale consisting of partials 6–12 in the harmonic series, producing the 6-tone scale 1/1, 7/6, 4/3, 3/2, 5/3, 11/6. In turn, Wiliam Slye and Bill Alves were called upon to suggest the intervals to be used to fill out the remaining six available pitches in the twelve pitch system. This paper constitutes a catalog of scale resources available in this tuning based on the use of trichords and tetrachords as building blocks. I began this catalog as a pre-compositional exercise for composing for the Harrison-Slye tuning. I make it public in this form because it may be of some use to other composers who wish to explore the possibilities of this tuning. The current form of the document may be amended as I discover new insights into the tuning. The paper is also available on my website at jarradpowell.com.

Part two of three, practical instructions for listening to musical intervals in preparation for tuning, performing limited harp maintenance with an ear towards successful tuning no matter the system chosen, and a review and comparison of six specific tuning systems that are frequently used on diatonic harps.

Resumo O fenômeno dos sons resultantes foi abordado inicialmente pelo organista Sorge (1703–1778) e pelo violinista Tartini (1692–1770), baseado em suas experiências práticas. Helmholtz fortaleceu o conhecimento teórico e tornou-se a referência nas áreas de acústica e psicoacústica a partir dos seus estudos sobre sons combinatórios. Howard e Angus (2009) consideram que relação musical entre sons resultantes e notas musicais depende do sistema de afinação em uso e este artigo teve o objetivo de identificar as relações de afinação entre sons resultantes gerados a partir dos intervalos produzidos entre a nota Lá4 e Lá3. As notas resultantes foram analisadas a partir dos valores das frequências nos sistemas de afinação Igual, Pitagórica e Justa. O estudo foi realizado a partir de uma metodologia quantitativa que investigou os valores das frequências tendo como referência o modelo matemático de Roederer (2008) e das razões para Afinação Pitagórica e Justa propostos por Gunther (2008). Resultados mostram que cada tabela de afinação apresenta intervalos com valores que podem ser exatos com um sistema de afinação ou diferir em até meio tom em outro.
Tunning relationships between combination tones and temperaments Abstract Combination tones have been identified by Sorge (1703–1778) and Tartini (1692–1770) based on their instrumental experiences. Helmholtz established the theoretical framework for further studies up to today and Howard & Angus (2009) consider the musical relationship of combination tones to notes played therefore depends on the tuning system in use. This article aimed to identify the combination tones present in intervals within the octave based on Equal, Pythagoric and Just temperaments. This study analyses combination tones frequencies based on Roederer mathematical model. The tuning tables were based on ratios indicated by Gunther (2008). Results show that combination notes can be exact in one temperament or differ up to a semitone in another temperament.

Az egyenlő temperálás a mai nyugati zenénk hangolási paradigmája. Jópár zenész és hallgató generáció nőtt fel, akik nem is igen hallhattak más harmóniai minőséget, mint ami az oktáv 12 egyenlő félhangra való felosztásából származik. Már egy ideje tudjuk, hogy ez nem mindig volt így, hogy Bach vagy Mozart zenéje máshogy hangzott saját korukban, mint ma. Az egyenlő temperálás rendszerének szélesebb elterjedése az 19oo-as századfordulóra tehető, de ennél sokkal korábbi források is vannak a megszerkesztéséről, hírdetéséről, javaslatáról. Tudjuk, hogy már rég ismert volt az egyenlő temperálás fogalma és megvalósításának módja, amikor mégis nagy buzgósággal más rendszereket kerestek és alkalmaztak az zeneteoretikusok, -szerzők és előadók. Tehát az egyenlő temperálás térhódítása nem egy egysíkú fejlődési folyamat, hanem egy sokkal összetettebb jelenség.
Ennek a jelenségnek egy szűkebb szegmensét fogom bemutatni előadásomban.  A mellet fogok érvelni, hogy Marin Mersenne 1636-os Harmonie Universelle-jében érte el az egyenlő temperálás elmélete a szükséges (és egy módon maximális) pontosságot matematikai, konstrukciós és fogalmi (és kommunikációs) szinten is. Ehhez elemzem saját írását és sorra veszem az őt megelőző egyenlő temperálást valamilyen módon hírdető zeneteoretikusokat (Lanfranco, Salinas, Vicenzo Galilei, Zarlino, Stevin, Faulhaber). Az ilyen módon egymás mellé állított elméleteket összehasonlítom s rendszerezem az egyenlő temperálás különböző aspektusaira vonatkozóan.

A hypothetical virtual reconstruction of a quest for a well tempered auditory musical keyboard tuning derived from the meantone is worked out, based on Werckmeister's concepts about well tempering, and musical insights and auditory music keyboard tuning practices at Baroque times, but also nowadays. The elaborated Well Tempered Meantone turns out to fit best with an optimum, based on presumed auditory tuning quality criteria. A number of characteristics that can be related to J. S. Bach, are discussed.

This paper (in the Giornale italiano di filologia) provides a close reading of Nicomachus’ ‘very tortuous’ account of Pythagoras’ alleged addition of an eighth string to the seven-stringed lyre. The primary focus is on Philolaus’ anomalous use of the string name τρίτη in fragment 6a. This well-known conundrum is reconsidered in light of the ‘epicentric’ arrangement of the Archaic lyre’s seven strings. It is argued that τρίτη and παραμέση were originally alternative names for the same string (as Nicomachus implies). This ambivalence was later bifurcated when lyres began to have eight strings around 480–460 BCE, a musical development that is plausibly reflected in a minority tradition about Simonides; the Pythagorean legend is a secondary development.

17.2.2016

Patrizio Barbieri, “Harps versus pianos: Parisian querelles on tuning 1770-1830”

ABSTRACT

This study aims at illustrating the scientific quarrels on the evolution of piano and harp tuning in the French capital, starting from the so-called tempérament ordinaire  an intermediate solution between the meantone and the equal, still widespread in the 1820s  and the final acceptance of equal temperament (c1830).
The controversies on the temperament of harps rose from the late eighteenth century, as a result of the increasing use of pedals on this instrument, then much in favour in Paris: unlike ordinary pianos and organs, with the innovation proposed by the Cousineaus in 1782 harps could actually play, in every octave of just seven strings, as many as 21 different notes, without causing any difficulty to the player as far as fingering is concerned. The theoretical aspect of the matter had been examined by the Parisian Académie des sciences, to which, according to current practice, makers applied in order to obtain the desired approbation for their innovations. These querelles on temperament and on the new instruments mainly involved scientists like de Prony, Biot, Vandermonde, Lacépède and Blein, besides representatives of the musical world like Fétis, Naderman, Dizi and Erard. They only died down toward 1830, when equal temperament was accepted by the Parisian musical world, almost a century after Rameau’s proposal (1737).
The features of Cousineau’s enharmonic harp had repercussions also on keyboard string instruments, like Johann Georg Roser’s “Piano-forté de la parfaite harmonie” (c1780), which offered the possibility of differentiating sharps and enharmonically equivalent flats; being, like the new harps, an isomorphic instrument, on its six keyboards the very same fingering could be used for many different tonalities. Also on this 31-note-per-octave piano new information has been found.


Author’s biography
PATRIZIO BARBIERI has taught at the University of Lecce (History of music theory, Musical acoustics, Applied acoustics) and in Rome (Università Gregoriana, and Rome Tor Vergata University). In 2008 he was awarded the Frances Densmore Prize by the American Musical Instrument Society for the best English-language article on musical instruments published in the two-year period 2007-08. He is the author of three books and over one hundred articles. He is currently engaged  as an independent researcher  in the history of musical acoustics, organology, music theory and printing.

Extended Abstract

The full text of this paper is written in Turkish.

Julien Jalâl Ed-Dine Weiss, originally of French-Swiss descent, belongs to the foremost performers of the Middle-Eastern maqām tradition on the inter-national stage of World Musics. In 1976, at the age of 23, he started to play the Arab lute and then the qānūn, and henceforth left his country for studying with different teachers in North Africa, the Hejaz, and Turkey. Unlike any other West-European musi-cian who converted to a foreign idiom, Weiss soon acquired notable fame as a qānūn player and foun-ded one of today’s most widely acknowledged formations for Middle-Eastern music: the ensemble “Al-Kindi”. Through the study of the theoretical tradition as well as through practical experience in many Near-Eastern countries, Weiss acquired an exceptionally broad perspective over the whole repertoire. Feeling unsatisfied with tuning and pitch supply of common qānūn models, he started to develop a number of novel prototypes that would provide him with justly tuned intervals in an extended range of microtones. Two instruments have so far been constructed, one in 1990 and specialized in the tuning customs of the Arab world, another in 2007 and more suitable for the interpretation of the Ottoman repertoire. Today, these models constitute the most advanced qānūns that have ever been built.  Their advanced structure reflects Weiss’ immense command over the microtonal language of his art and his confidence as a modern Cartesian thinker between two cultural spheres. In their just intonation, drawing on simple integer ratios and providing the decisive intervals of the medieval treatises, these instruments are able to please acousticians, theorists, and practical performers.     
The aim of this study is to offer a basic introduction into structure and tuning of Julien Weiss’ extended qānūn models. The primary sources were acquired in a manner comparable to field research, in con-tinuous oral exchange with the musician. As a traveling artist who regularly performs with musicians from a variety of local backgrounds, Weiss discovered early in his career that theory and practice of Middle-Eastern music often contradict each other. Whereas all commonly used qānūn models are tuned upon the tempered 12-tone scale, temperament proves to be useless in the context of Middle-Eastern music. Since  the earliest Arab treatises, theorists have distinguished major and minor semi- and whole-tones. The controversy over the note Segāh and other “neutral” scale steps of the fundamental scale can be traced back to Safīy al-Dīn’s neo-Pythagorean 17-note scale from the 13th century. Weiss has put enormous stress on the study of Safīy al-Dīn’s theoretical treatises, the “Šarafīy-yah” and the “Kitāb al-’Adwār” that created the earliest major disagreement to the older theories, such as those of Al-Farābī and ’Ibn Sīna. In shifting upwards, the important medius frets were reinter-preted upon a strictly Pythagorean system, com-posed uniquely of commas and limmas. Weiss, in pointing on a number of theoretical inconsistencies, suggests that the Systematist 17-note scale was ne-ver practiced in the way described by Safīy al-Dīn and that it was practically impossible to perform it with the lute fretting described in the Šarafyyīah. Today, the same theoretical discrepancy divides the Middle-Eastern tradition into two distinct spheres of influence: Performers in the Arab world understand the “neutral” degrees on the fundamental scale as “quarter-tones”,while the  Turkish theories of Raûf Yektâ, Hüseyin Sadettin Arel and Suphi Ezgi have resurrected the Systematist school within the extended 24-note pitch supply of the 20th century. Many different systems were proposed at the Congress of Cairo (1932). However, the participants departed without defining a general, compulsory system for the intonation of principal modal genres. As a mature tradition, the Middle-Eastern maqām system can no more rely on simplicist attempts to-wards one single fundamental scale. Similarly to Western polyphonic music – in which the intonation of apparently identical pitches changes upon harmonic context – the Middle-Eastern system is composed of a plurality of dimensions. Although many modal genres seem to share common note names and pitch content, their refined charac-teristics often draw on subtle nuances in the intonation of apparently the same intervals. In disposing of six different “quarter-tone” intervals on every course of strings while keeping Pytha-gorean and harmonic ratios in precise tuning, Weiss’ extended mandal systems finally enable exact tuning in the totality of local and historic contexts within the Middle-Eastern tradition. Comparably to Dimitrie Cantemir’s tanbūr scale from the 17th century, they may well be described as the closest reflection of a Middle-Eastern tuning system in the 21 st century.       
Keywords: Qānūn, Middle-Eastern Tuning System, Microtones, Fundamental Main Notes, Turkish & Arab Maqām Music, fieldwork.

A detailed analysis of the spectra and frequencies of tones produced on metallophones in a Balinese orchestra clarifies a number of issues that arise in studying timbre and tuning. Among these are the relative importance of accuracy and completeness of measurements, the relevance of acoustical description to musical perception and style, the appropriateness of statistical generalizations and numerical formulations to musical practice and theory, and the possibility of bi-musical responses to divergent tuning systems. Byproducts of the investigation include a proposed basis for inter-cultural ear training and a novel formulation of the unison, octave, and other intervals (including wide and narrow variants), as well as traditional scales and interval collections that have existed side-by-side in widely varying forms within individual cultures.

To determine whether there are any significant features shared by the best-documented Western and non-Western scales, the author compares the diatonic, pentatonic, equiheptatonic, sléndro, pélog, and in scales as well as referential collections of Arabian maqamat and South Asian ragas. All are found to be based on cycles of intervallic bisectors of the octave or sections thereof in which the sizes of intervals between scale degrees are coordinated with the cycles from which they are derived.

The two schools are situated in two kinds of environments-Kamalgazi, Kolkata is the place where most of the children come from the nuclear family set up-parents are professional-they leave their children at the school gate in the morning and take them after their office hour from their own age old parents’ place as their parents are instructed to take away their children from schools. Children are guided by different people and enjoy different family environment in their childhood.
In Behala, Sakherbazar most of the children live in joint family structure-parents are professional-they leave their children in school on the way to their working place.
So problems they face in daily life are not same –some differences are there in child’s life

Cristofori's instruments Giuhana Montanan has published a use-ful list of Cristofon instruments (EM, Aug 1991, p.383), but her reliance on Boalch (this, and other sources men-tioned here were cited by Montanan) or citing of its opinions leads to errors or confusion in some ...

The subject of the report is the technical and psychophysiological aspects of historical and modern intonation systems in European musical practice. The heterogeneous intonation approaches of various performing traditions and possible factors of their diversification are considered. Particular attention is paid to the intonation specificity of non-tempered instruments.
The presentation include a demo of the Sectio Canonis web application, designed for theoretical and practical study and development of non-trivial tunings and temperaments.

El presente método brinda un catálogo de escalas disponibles para el sistema de afinación 13EDO, mostrando las diferentes posiciones distribuidas en cada temperamento, buscando hacer claro el panorama de todo el diapasón y las secciones alcanzables por los dedos.

This paper explores a new and ambitious tuning system for the Middle Eastern qānūn, conceived and practiced by the French virtuoso Julien Jalâl Ed-Dine Weiss (b. 1953). Since the invention of mandal-s, movable bridges that enable microtonal alterations during performance, the qānūn has become a major source for tuning and fixation of the Middle Eastern pitch inventory. Whereas Arab qānūn-s are based on "quarter tones", Turkish instruments divide the octave into 72 notes. All commonly fabricated models, however, have in common that their tuning is based on 12-semitone equal temperament. This does not conform with the Pythagorean tuning on which the heptatonic framework of the qānūn is traditionally based. For that reason, common qānūn models only approximate the interval sizes that have been described throughout the theoretical tradition by means of arithmetic ratios. Equal temperament causes transformations in the tuning of commonly agreed scale degrees and complicates the qānūn's interaction with both justly tuned and fretless instruments to an audible extent. As the founder of the internationally acclaimed Al-Kindi ensemble, Weiss has repeatedly criticized this deficiency. Since 1990, he has constructed nine qānūn prototypes with which he performs in diverse regional contexts within the Middle-Eastern tradition. As the outcome of many years of practical experience, Weiss's qānūn models currently offer the widest range of microtones in strictly just intonation. At the same time, their tuning remains truthful to the theoretical approaches of the major historic treatises. This study explores Weiss's motivations in regard to the principal developments of the Middle Eastern fundamental scale. The discussion includes those tuning approaches from the recent past that influenced Weiss in the conception of his prototypes: the professional Arab qānūn prior to the 1970s, the frets on Necdet Yaşar's Turkish tanbūr, and the qānūn of Aleppo that vanished in the 1980s. Weiss's approach reveals that the tuning system of the transnational maqām phenomenon has spread to such complexity that its fundamental supply of pitches cannot be thought of except in terms of local variations in a plurality of traditions. His instrumental prototypes may reconcile the diverse scholarly approaches with both practical and theoretical objectives. Please download the full text from the AAWM website.

According to a widely accepted interpretation, the “unclear” (la-zakû) intervals of Mesopotamian tuning were tritones, which became “clear” (zakû) intervals—in modern terms, perfect 5ths or 4ths—by tightening or loosening a particular string on a harp or lyre. Nonetheless, Sam Mirelman has recently cautioned (2013) that we cannot be sure such an interpretation is correct. Meanwhile, Yitzhaq Feder’s (2014) survey of contexts for “zakû” in Mesopotamian texts has shown that it referred to things that were “pure” not only in the sense that they were “not impure,” but in the even more precise sense that they had been “purified” insofar as their impurity had been removed.
Such a construal of the contrast between “zakû” and “la-zakû” is consistent with the view that Mesopotamian tuning comprised a process in which the psychoacoustical impurity of a la-zakû interval, namely, its beating or roughness, was removed by tightening or loosening a string, resulting in a purified, zakû interval. Further, if one assumes that the modular interval of Mesopotamian tuning corresponded to the perfect octave (an assumption that has never been disputed), one can reduce the possible ways in which a harp or lyre was tuned from twelve to two. Further still, close reading of a recently published transliteration and photograph of a cuneiform tablet reduces these two ways to one. Finally, applying the tuning process to the well-known Hurrian notations results in music that is cognate with a musical framework Patricia Carpenter characterized as “discant structure.”

This dissertation concerns interval intonation, tuning systems, and temperaments and their relevance to music education. It considers a historically well-known tuning discrepancy, the syntonic comma, from three different perspectives: 1) recognition, 2) preferences of usage, and 3) utility in music education. The first objective was investigating to what extent students in music-intensive classes and university music students are able to recognize a mistuning of the syntonic comma in musical passages tuned in just intonation. This was investigated through an experiment comprising 40 chord progressions. The second objective was to determine the preferences of university music students and teachers for dealing with the syntonic comma by way of centrally recognized tuning alternatives through an experiment with 30 pairs of chord progressions. The third objective was to consider how the present results and other recent research about the syntonic comma could help with teaching intonation skills. The first experiment, recognition of mistuning, revealed a wide distribution among listeners (n = 168). The recognition of mistuning in vertical harmony decreased significantly according to the complexity of the chord progressions. Mistuning between successive notes of the melody line helped recognition. The recognition of mistuning was also influenced by singing experience: In one category of chord progressions, participants representing the bass voice in a choir detected mistuned triads better than participants representing the top voice. There was also a significant correlation of the mutually interrelated variables of instrumental experience and age with recognition of mistuning. In Experiment 2, intonation preferences, music university students and teachers (n = 93) were asked a preference from among four tuning strategies with pairs of differently tuned but otherwise identical chord sequences. Meantone tuning (85.9%) and equal temperament (85.2%) were overwhelmingly preferred, with local tempering falling behind, and there was overall rejection of pitch drift. The results motivate the development of new exercises for intonation skills, and the experiments shed light on the problem of the syntonic comma, proving potential for practical music pedagogy. Several suggestions for further research are presented.

In the usual seven-note diatonic scale, the maximally smooth cycle of triads contains a long section that uses only major and minor triads, the same triad that forms maximally smooth cycles within the twelve-note chromatic scale. Tonal music exploits this property of the scale to create sequences of similar chords. The goal of this study is to determine the extent to which such long chains containing inversionally related species exist in maximally smooth trichord cycles within microtonal scale systems that share certain properties with the diatonic. The study thus combines neo-Riemannian theory, especially Cohn’s concept of maximally smooth cycles, with the diatonic scale theory developed by Clough and other authors. The patterns of maximally smooth trichord cycles depend on the type of scale within which they occur, and on the cardinalities of the scales. Among all scales, the usual diatonic supports the longest possible chain.