Tuning and Temperament

Gegen ein immer noch wirkmächtiges Bild der Aufklärung als einer Form des Rationalismus will dieser Sammelband " die ästhetische Welterschlie-ßung und Selbstreflexion in den Mittelpunkt stellen " , weil, so die etwas über-raschende Behauptung der Herausgeber, " das Denken der Aufklärung erst auf einer solchen Basis ihre politischen Forderungen richtig stellen " könne (S. 9). Nun könnte man natürlich sofort einigermaßen gespannt sein, wie diese richtig gestellten politischen Forderungen aussehen, aber zunächst ist man doch skeptisch, wenn " das " Denken " der " Aufklärung irgendwelche po-litischen Forderungen stellen soll, denn wie sich schon zwanglos aus Jo-nathan Israels Forschungen ergibt, existiert ein solches Denken nicht als Einheit, so daß es auch nicht " ihre " politischen Forderungen an sich geben kann. Es geht aber in dem Band 1 auch nicht vorrangig um sozialpolitische Fragen, sondern zentral um Musikgeschichte und Musikwissenschaft. Den-noch soll der ganze Band auf eine " Aktualisierung der musikalischen Para-digmen und auf deren sozialpolitische Dimension selbst " orientieren (siehe dazu auch den letzten Beitrag des Bandes aus der Feder von Boris Pre-višić). Was dann dazu gesagt wird, wirkt eher unbestimmt und vage, ohne daß recht greifbar wäre, worauf das nun konkret hinauslaufen soll: so seien " Stimmung und Vielstimmigkeit als Paradigmen der heutigen Kultur " in ih-rem Profil aus der Sicht der Aufklärung zu schärfen " , und zwar auf der Grundlage von ästhetischen Modellen und Kunstwerken. Weiter heißt es: " Der Zusammenhang zwischen Stimmungen und Vielstimmigkeit ist leider noch viel weniger als gesellschaftspolitisches Desiderat in der Gegenwart angekommen. Ein verbindlicher Pluralismus tut not, der in der Aufklärung angedacht worden war, bevor er dann im Getöse des 19. und 20. Jahrhun-derts wieder verschüttet wurde " (S. 16). Offenbar soll die in der Musik wal-tende Vielstimmigkeit auf andere Bereiche bezogen werden, politisch also 1 Inhaltsverzeichnis: https://d-nb.info/1131375556/04

An approach towards outlining a practical understanding of ancient music. Particularly dealing with aspects of ancient Greek music, as recounted by the translated fragments of Aristoxenus, Ptolemy and others. In brief, a summary guide defining a method for practically realising harmonic relations, rhythmic poetics, the varieties of the tonal genera, and tetrachord theory using standard techniques of rational string division.
(Please note: This is an incomplete review copy and as such does not contain the full text of this manuscript. If you would like the full picture, please do get in touch with me.)

Fifths and major thirds beat rate characteristics of famous historical temperaments are analysed. It appears that beat rate characteristics might be the actual determining factors for Baroque temperaments, mainly because beat rates are of main importance to interpreting musicians regarding harmony and possible musical affects, and to auditory tuners because of quality and ease of tuning. It is, on the other hand, not always clear whether published ratios, cents or comma's are deduced from theoretic calculations or from concrete results on monochord measurements or settings. The revealed reality and importance of beat rate characteristics of temperaments raises additional arguments for acceptability of the Jobin proposal concerning a probable Bach temperament, or for almost identical beat rate alternatives. A novel hypothesis is proposed concerning the spirals drawn on top of the title page of "Das wohltemperirte Clavier" of Johan Sebastian Bach. 1 Preamble The commonly published and dominating factors with discussions on musical temperaments are probably the investigations on purity deviations of musical intervals, measured in ratios, cents or commas. And still, musical interval beats and their beating rates are probably more affecting musical factors to interpreting musicians and auditory tuners of keyboard musical instruments. More attention might therefore have to be paid to those characteristics : beats are undesired and directly observable. Approximate auditory beat rate evaluations do not require any tool nor calculation. Impurity measurements in ratios, cents or commas on the other hand, are often nothing more but rather abstract concepts to many musicians, not of direct use or interest when playing music and also not for auditory tuning. This paper is an attempt to confirm and elucidate the importance and practical applicability of beat rate evaluations in the determination of musical temperaments, especially some Baroque ones. 2 The auditory music keyboard tuning The elementary basic concepts of musical temperaments, seen from the point of view of the interpreting musician and the auditory music keyboard tuner are discussed in this paragraph. There is of course much more that can be written on this subject, see for example : "Le Clavier Bien Obtempéré", A. Calvet, 2020.

A short introduction to some connections between music and math.  Polyrhythms may be analyzed using basic number theory concepts such as GCD and LCM.  Tuning using only rational multiples of a fundamental frequency (just intonation) leads to ideas in abstract algebra, including abelian groups, lattices, and factor groups.

The Bible, Homer and other myths were largely written down long after their inception, from memorable oral narratives composed by a type of author who was not a writer. Plato saved the number science implicit in oral works from extinction through including it in his own literary output. A number science can develop without a formal arithmetic through using only geometric procedures and these are to be found in the terminal Stone Age, famous for its enduring megalithic monuments. This paper illustrates the structure of this harmonic science as it is found within historical texts and monuments, whilst suggesting that the harmonic ratios between astronomical periods would most simply explain how our oldest myths were orally composed and performed cosmographies, skillfully woven around the invariant structure of numerical harmony and the sky, these being taken as responsible for creating and maintaining the world.

The evolution from meantone into the 18th century French circular "tempérament ordinaire" is described. The latter is accurately reconstructed, based on the analysis of sources by Rameau, d'Alembert and Rousseau.

More recently this work has been thoroughly updated, with a fully revised Rameau temperament, a detailed assessment of the differences between d’Alembert and Rousseau, and an analysis of the diverging tuning trends between pipe organs and other instruments in 18th century France. See the sections 8.3 pp. 108-112 and 21.3  pp. 418-429 of the eBook "Unequal Temperaments: Theory, History and Practice", 3rd. ed., 2013. See http://temper.braybaroque.ie/

Mit einem Minimum an mathematischen Formeln und in leicht verständlicher Sprache bietet das vorliegende Buch auf dem aktuellen Stand der Forschung eine grundlegende Einfüh-rung in die einzelnen Fachgebiete der Musikal ischen Akustik (Instrumenten-, Raum-und Psychoakustik, Auditorische Szenenanalyse, Musikalische Wahrnehmung und Klangsyn-these) sowie in die Geschichte, Perspektiven und Methoden dieses vielschichtigen Bereichs der Musikwissenschaft. Am Beispiel einer Konzertsituation wird der Weg von der Schallerzeugung am Musikinstru-ment bzw. der Singstimme über die Schallausbreitung im Raum bis hin zum Ohr des Zuhörers und der dahinter stattfindenden kognitiven und alltagsrelevanten Schallverarbeitung erläu-tert. Hierbei werden sowohl die aktuellen Messmethoden und akustischen Ana lyseverfahren berücksichtigt als auch die Funktionsweise und Geschichte der elektronischen Musikinstru-mente und der modernen Audiotechnik.

In the following essay, I review a series of my own works composed for guitar(s). I argue that these works, which were composed discontinuously over a period of fifteen years, expose a process oriented towards the construction —or the discovery— of a personal guitar (and a personal music). The works take as a given certain initial conditions of the guitar, which are the result of 1) The medium is characterized by an endogamy of guitar composer-performers and/or 2) The inertia of historical repertoire within the genre isolates the guitar from experimental processes necessary for further development. To transcend this perceived barrier, I argue there must be a reflexive work leaning towards expanding the musical language of the guitar. In the first part of this paper, I narrate a progression from what could be called a conventional guitar environment to a more personalized relationship with the instrument and the task of composing for it within the scope of contemporary classical music. In the second part, I list a series of technical resources that are associated with the search of a singular compositional language for the guitar, establishing links between certain techniques and the desired expressive result. Finally, I offer some preliminary conclusions about my path composing for this instrument, and I sketch some possible ways to continue developing my creative work for guitar(s).

Augusto Novaro's book, Sistema natural de la música, can be considered one of the most important synthesis for understanding the research of acoustic and musical theory, of microtonal tuning systems and of the rectification of the well tempered system. Novaro's experiments led him to make diverse instruments with the purpose of applying his theories, but also to write several papers in which he sought to explain the basis of his speculations. His theoretical labor did not result in musical compositions, but was limited to conferences and private concerts. Modern new instruments and technological resources finally outmoded his research methods. Not-withstanding, we can now situate him as a true pioneer of "just intonation" and as one of the researchers that delved into the tuning problems of the well tempered system.

Una spiegazione riassuntiva del temperamento musicale: principi acustici, caratteristiche musicali ed evoluzione storica.
[A concise exposition of musical temperament: acoustic principles, musical features and historical evolution. 17pp.]

Traduzione dalle versioni originali in Spagnolo e in Inglese (non pubblicata) effettuata dalla cattedra di clavicembalo del Conservatorio G. Verdi di Torino, poi revisata dall'autore.
[Translation from the original Spanish and English (unpublished) versions, by the chair of harpsichord of the Conservatorio G. Verdi of Turin, with final revision by the author.]

Organ tuning and temperament are essential elements of the soundscape of an organ.
It is commonly assumed today that organ temperament practice shifted away from the well-documented standard of meantone temperament toward circulating temperaments (well-tempered or equal) around 1700. Written records and evidence from extant instruments, however, strongly suggest that unmodified meantone temperament remained common practice in North German organs until the 1740s. Even advocates of the new temperament designs such as Andreas Werckmeister had to admit that organ builders did not follow their suggestions. There is no evidence that retunings resulted in well-tempered systems; only meantone and equal temperaments are mentioned in the sources for retunings.
The reason for suggesting new temperament systems was the requirements of ensemble accompaniment. The simultaneous use of instruments tuned at different pitches (Chorton, Kammerton) required extensive transposition, a practice that, due to the development of major-minor tonality, became increasingly difficult if not impossible with organs tuned in meantone temperament. Modifications of this temperament were suggested and sometimes carried out, but were generally not successful.
Re-tempering an existing organ was a time-consuming task: in a large organ it could take months of work even for an experienced builder. The circumstances under which the organs had to be tuned constituted an important impediment to the introduction of new temperaments. An analysis of payments to bellows-treaders as recorded in church account books shows that the organs of St. Marien, Lübeck, were not retuned during the tenures of Franz Tunder and Dieterich Buxtehude. Thus, some of their organ works could not have been played on the organs available to them during their lifetimes. At this time, however, playing from a score was not acceptable for a professional organist, who was expected to extemporize even complex contrapuntal music. Organ music seems to have been written down mainly for study purposes.

Proposition d'un algorithme d'évaluation global, des « bons tempéraments », et ceux « de Bach » Broekaert Johan ; broekaert.devriendt(at)telenet.be Résumé Une formule de mesure globale des impuretés de tempérament musical est élaborée. Elle semble conduire à des résultats significatifs concernant l'évaluation de tempéraments. En plus, les résultats de classement s'accordent très bien avec les recherches de pointe, historiques et musicologiques, du professeur H. Kelletat, qui démontre quelles caractéristiques essentielles devraient être propres aux prétendus « tempéraments de Bach ».

Die Festlegung der richtigen musikalischen 'Temperatur' und damit der richtigen Stimmung löst um 1700 heftige Kontroversen aus. Es geht dabei um alles: um die Verteidigung von alten und neuen Weltbildern. Es geht um die Frage, was Musik eigentlich ist, aber auch, in welcher Beziehung sie zum Menschen, zu seiner sozialen Organisation und zur Natur steht. Mit Andreas Werckmeister, Johann Mattheson, Barthold Heinrich Brockes und weiteren Zeitzeugen beleuchtet Silvan Moosmüller musiktheoretische wie allegorische Argumentationsstrategien und setzt sie ins Verhältnis zur zeitgenönisschen Dichtung und Wissenschaft. Im Spannungsfeld zwischen Sphärenharmonie und physiologischer Ästhetik kommt ein dichtes Geflecht in den Blick, in dem sich musikalische, kosmologische, mathematisch-physikalische und ästhetische Semantiken der Stimmung aufs Engste miteinander verzahnen.

The long-necked lute bağlama has been predicated the main folk instrument of Turkey, originated from Ancient Mesopotamia. In Anatolia the first instruments of this kind appeared in early Hittite time, approximately XVII. century B.C. Since then, in Anatolia nearly all civilizations have used this type of instrument in order to accompany singing voices, as a melody instrument, for rhythmical accompaniment, to work out homophonic structures and for theoretical guidance by means of fret distribution. These lutes have always represented a great variety in structure, stringing and tuning systems as well as performance styles. Bağlamas can be played both fingerstyles as well as by using a plectrum. Extraordinary right-hand techniques have been developed to perform very efficient rhythmic patterns. Different tuning systems support the efficiency of developing artfully melodies. In a bağlama performance, each of the player's hands holds specific functions. In coordination with the right hand, the left hand is dedicated to the melodic dimension. Apart from the rhythmic and melodic functions, a bağlama also serves as tool for working out homophonic or sometimes polyphonic accompaniments. This paper will present tuning systems of the bağlama most common in Turkey and will investigate their variations.

"Una presentación concisa sobre el temperamento musical: principios acústicos, características musicales y evolución histórica. 17pp.
[A concise exposition of musical temperament: acoustic principles, musical features and historical evolution. 17pp.]"

Pourquoi les interrogations sur la définition du ton et des intervalles constitutifs des gammes ont-elles préoccupé tant de compositeurs et de musicologues depuis l'antiquité grecque? Le ton, la tierce, la quinte, l'octave sont-ils des données absolues? Non, bien au contraire. Au cours du temps, et cela au moins depuis l'antiquité grecque, les intervalles ont été déterminés sur des bases mathématiques et techniques, en faisant appel à la science des nombres, tout en cherchant à s'adapter aux instruments existants. La gamme tempérée a succédé à d'autres gammes, telles que la gamme pythagoricienne, la gamme de Zarlino, et divers systèmes de tempéraments. Elle est une convention passagère, correspondant à la mentalité d'une époque. L'essentiel réside dans l'effet sonore qui résulte de la gamme, et dans la façon dont elle résonne et agit sur le corps et sur l'âme. Des recherches récentes tentent de trouver une intonation qui soit juste pour le corps et pour l'Être. Mais est-ce possible de façon définitive et universelle?

La seguente tesi di ricerca si occupa di illustrare sufficientemente quali possono essere o non essere i limiti topici attraverso cui il compositore di musica elettronica opererebbe all’occorrenza specialmente nel campo dell’armonia, della melodia e del ritmo.

Öz Akort sistemlerinin Batı müzik dünyasındaki evrimi sürekli daha mükemmeli hedefleyen bir arayış sürecini ifade eder. Pisagor dizisi ile başlayan matematiksel ilkelere dayalı ideal bir sistem kurma uğraşı yüzyıllar içinde birbirinden farklı sistemlerin ortaya çıkması ile sürmüştür. Her sistem, kendisinden önceki sistemlerin yanıt veremediği estetik ve pratik beklentilere yönelik bir çözüm olarak tasarlanmıştır. Bununla birlikte her sistemin kendi döneminin estetik anlayışının bir sonucu olması, herhangi bir sistemin sınırının da bu anlayıştaki değişimlere bağlı olduğunu göstermektedir. Bu bakımdan akort sistemlerinin mükemmelliği tasarım idealleri açısından görecelidir. Her biri kendi zamanının perspektifi açısından avantajlar sunan akort sistemleri başka bir zamana göre yetersiz kalabilmiştir. Bu durum herhangi bir sistemin diğerinden üstün olduğunu değil, her sistemin kendi döneminin estetik çerçevesi içerisinde değerlendirilmesi gerektiğini göstermektedir. Mükemmel bir sonuç olarak ileri sürülen eşit tampere sistemin diğer sistemler aleyhine yükselişi de üstünlüğünün değil, belirli bir dönemde ortaya çıkan ve sonuçları sürmekte olan bir anlayışın sonucudur. Bu anlayış, sistemin önceki sistemlerden farklı bir biçimde bir kültürel ideal olarak kabul edilmesinden kaynaklıdır. Böylece diğer sistemler ortadan kalkarken eşit tampere sistem bir değerlendirme ölçütü haline gelmiştir. Bu bağlamda, bu yazıda Pisagor dizisinden itibaren akort ve tampereman yöntemleri incelenerek, bunları ortaya çıkartan koşullar, sistemlerin temel ilkeleri, avantaj ve dezavantajları ile eşit tampere sistemin hâkimiyetinin sebep ve sonuçları üzerinde durulacaktır.
Abstract The evolution of tuning systems in the Western music world is an example of a continuous process of seeking perfection. Efforts to produce an ideal system based on mathematical principles started with the Pythagorean scale and continued with the emergence of different systems over centuries. Each system was designed as a solution for the aesthetic and practical expectations that the previous systems could not meet. On the other hand, the fact that each system resulted from the sense of aesthetics of its own time indicates that the limits of any system depend on the changes in the sense of aesthetics. In this regard, the excellence of a tuning system is relative in terms of the ideals of design. Although the tuning systems provide certain advantages from the perspective of their own time, they may be insufficient for another period. This does not show that any system is superior to another system; rather, it indicates that each system should be evaluated within the aesthetic framework of its own time. The rise of the equal temperament system, which is suggested as an excellent result, against the other systems is the result of an understanding that emerged at a certain time and the results of which continue, rather than its superiority to the other systems. This understanding arises from the fact that this system is accepted as a cultural ideal, unlike the previous systems. Thus, while the other systems have disappeared, the equal temperament system has become an assessment criterion. In this regard, this paper analyzes the tuning and temperament methods from the Pythagorean scale onwards and addresses the conditions that created them, the main principles of the systems, their advantages and disadvantages, and the reasons for and results of the dominance of the equal temperament system.

Il temperamento di Vallotti/Young è un sistema di accordatura ideale dal punto di vista sia storico che pratico per la musica dell'alto barocco. Indipendentemente dall'ambito storico e geografico di origine, Vallotti è assai vicino alla "media" dei temperamenti circolanti descritti dai principali scrittori barocchi tedeschi: Werckmeister e Neidhardt.

È questa una traduzione in italiano di un articolo pubblicato soltanto in lingua inglese.

Per un analisi acustico, storico e pratico del temperamento di Vallotti vedi le sezioni 9.5, 9.7, 9.8, 11.10, 13.16, 13.17 and 21.17 dell'eBook
"Unequal Temperaments: Theory, History and Practice", 3ª ed., 2013.
Vedi http://temper.braybaroque.ie/

This is a response to ‘Unequal Temperaments’, a review by Bradley Lehman of Claudio Di Veroli's book "Unequal Temperaments: Theory, History and Practice", 2nd ed., Bray Baroque, Bray, Ireland 2009. http://temper.braybaroque.ie/.
Lehman's review had been published in The Viola da Gamba Society Journal, vol. 3, part 2 (2009), pp. 137-163.

This is my PhD dissertation. The third chapter might be interesting since it shows that the natural scale was the primary scale used in European antiquity (the so-called Folk Music Scale). It combined with the early diatonic scale (stacking thirds on the diatonic scale that those instruments couldn’t play) to create our modern scale. The Highland bagpipe is an ossified hybrid between the two.

The fourth chapter might be of interest to composers because it explains the rhythm of languages (especially English) and also the pitch-patterning of the spoken word which is used as the basis of melody.

The link brings you to the University of Otago library which is open and non-private. So you have to force your web browser to continue to the site.

ABSTRACT:
The purpose of this study was to investigate through archival and musicological analysis the audio recordings of Fenian lays made in the middle of the last century. These recordings were made from informants who learned the material orally; they contain cultural elements that assist in comprehending the musical mechanics of Fenian lays at a time when their performance practices were being extirpated by foreign musical influences. These elements include Indo-European (IE) thematic material, poetics, language register, pitch structuring, rhythm, and vocal techniques. Audio recordings of Fenian lays from Ireland, Scotland, and Nova Scotia, Canada were analysed in terms of their linguistic-musical material. Results show that the rhythm of the lays did not display a repetitive musical metre but the more complex structure of speech. However, rhythmic patterns did alter with volume. Also, resonance tuning was apparent. Many characteristics associated with volume in lay recordings exist in declaimed speech as well; both may be seen to act as a bridge between speech and metered song. Lay poetry appeared to be syllabic, which is unusual for a stress-timed language; this reflects an Indo-European genesis that is supported by the presence of oral formulaic language. Both stress and accent shifted pitch by poetic line to match spoken characteristics. A high language register was present, which does not indicate composition by the intelligentsia for use at court, but rather a fear-induced protective linguistic device apparent in all social classes. Moreover, the addition of delineated pitch to spoken declamation may be seen as an attempt to further increase the communicative register. The pitch structure was seen to be anachronistic, matching the linear scales played by pastoral instruments, particularly that of wooden shepherd trumpets used since at least the beginning in the Early Neolithic Age.

Vallotti/Young temperament is an ideal historical and practical tuning system for High Baroque music making. Regardless of its historical formulation and area coverage, it matches very closely the “average” of the circular temperaments described by the main German Baroque writers on temperament: Werckmeister and Neidhardt.

For a full acoustical, historical and practical treatment of Vallotti's temperament see the sections 9.5, 9.7, 9.8, 11.10, 13.16, 13.17 and 21.17 of the eBook "Unequal Temperaments: Theory, History and Practice", 3rd. ed., 2013. See http://temper.braybaroque.ie/

This volume is based on the conviction that microtonality is a fundamental change-indicating concept in Western music history. The book focuses on the development of microtonal music in Eastern and Central Europe from World War I to the present. The authors examine how diverse concepts of microtonality have given way to new composition theories and practices in the region, which has long been marginalized in general histories of avant-garde and post-avant-garde music. These scholars hold the view that even between WWI and WWII, microtonal music and its theoretical reflection were outstanding contributions of Eastern and East-Central European composers to the contemporary discourse of avant-garde music. That provoked radical changes in the composition and performance practice of new music and affected several generations, sustaining and transforming early avant-garde insights.

During Bach’s lifetime, the history of temperament in Western music took new turns, but his relationship to this development is far from well understood. The article explores a number of relevant questions to help define what we actually can know about this matter:

• Which sources about musical temperament are relevant in regard to Bach’s practice and what can be learned from them?

• How detailed, plausible or credible is the relevant information?

• Do we know whether Bach had any influence on the choice of temperament in organ projects? If so, to what extent and what was his advice?

• To what degree do extant contemporary reports, expert opinions and historical instruments allow us to reconstruct temperaments in the organ building of Bach’s age?

• What role did the temperament of instruments with fixed pitches (like keyboards) play in interaction with free-intonating instruments and singers?


It appears that conclusive evidence for answering these question does not exist, which might be liberating:
In the end, it remains a matter for the individual to decide which temperament seems to be musically apt for the performance of any of Bach’s compositions.

"Did Bach indeed leave a record of how he tuned his keyboard instruments as Bradley Lehman's 2005 article, 'Bach's extraordinary temperament: our Rosetta Stone' would have us believe? With a bit of cloak and dagger work, can we get to the bottom of this Da Vinci Code of tuning secrets? Has a Bach temperament solution miraculously appeared by way of a tantalising hint Bach may have left for us at the top of the title page of The Well-Tempered Clavier  in the form of a cursive calligraphic flourish, a secret codified message, a memory jogging reminder; or was it even just a doodle?

Sadly, Lehman's "Bach temperament" is not only incomplete, but misleading, suggesting something which is neither probable, nor representative of the tuning traditions of Bach's time and likely vastly different from that which Bach knew, or used, or would have found acceptable. That the calligraphic scrolling glyph on the title page of The Well-Tempered Clavier is indeed a diagram of a temperament can be supported in that it can be broken down into structural elements which compare in structure and placement with intrinsic values of temperament intervals found elsewhere in the literature of the period. So how did he really tune his harpsichord?

This paper seeks to provide an overview of the Bach temperament issues raised by Bradley Lehman and attempts to provide an actual working historical temperament possibility as a Bach temperament. This paper is an abridged version and under review pending its full publication. In writing this paper I have carefully and intentionally avoided modern scholastic writings as much as possible and chosen to keep focus on historical work in order that subjective modern ideas not cloud the issues discussed. "