Just Intonation

Between 1971 and 2006, James Tenney created more than fifty significant works of what is now commonly called ‘spectral music’. In their materials and procedures, his compositions frequently paralleled, sometimes anticipated, and in some instances strikingly contrasted with spectralist developments in Europe. This article provides an analytical introduction to the spectralist component of Tenney’s large and varied output, and explores its emergence and place within a North American tradition of spectral music composition. Among the compositional concerns addressed are the duality of timbre and harmony, the harmonic series as a structural resource, the instrumental synthesis of speech, rhythmic analogs of spectral structures, the expansion of traditional harmonic means, and practical performance considerations.

This study examines the deviation in the intonation of simultaneously sounding tones under the condition of an embedded melody task. Two professional musicians (trumpet players) were chosen as subjects to play the missing upper voice of a four-part audio example, while listening via headphones to the remaining three parts in adaptive five-limit just intonation and equal temperament. The experimental paradigm was that of a controlled varied condition with a 2 (tuning systems) ´ 5 (interval categories) ´ 5 (renditions) ´ 2 (players) factorial design. An analysis of variance showed a nonsignificant difference between the average deviation of harmonic intonation in the two systems used. Mean deviations of 4.9 cents (SD = 6.5 cents) in the equal-temperament condition and of 6.7 cents (SD = 8.1 cents) in the just-intonation condition were found. Thus, we assume that the musicians employed the same intonation for equaltemperament and just-intonation versions (an unconscious "always the same" strategy) and could not successfully adapt their performances to the just-intonation tuning system. Fewer deviations could be observed in the equal-temperament condition. This overall tendency can be interpreted as a "burn in" effect and is probably the consequence of longterm intonation practice with equal-temperament. Finally, a theoretical model of intonation is developed by use of factor analysis. Four factors that determine intonation patterns were revealed: the "major third factor," the "minor third and partials factor," the "instrumental tuning factor," and the "octave-minor seventh factor." To summarize, even in expert musicians, intonation is not determined by abstract tuning systems but is the result of an interaction among compositional features, the acoustics of the particular musical instrument, and deviation patterns in specific intervals.

Technology, Aesthetics, Communication AN INTERNATIONAL JOURNAL 2 2005 ESTRATTO I PISA ROMA ISTITUTI EDITORIALI E POLIGRAFICI INTERNAZIONALI MMV e LUCA CONTI ARTEFATTI E TEORIA NEL SISTEMA NATURAL DE LA MÙSICA (1951) DI AUGUSTO NOVARO S ONO scarsi i documenti che aiutano a ricostruire la vicenda di Augusto Carlos Novaro (Tacubaya, Città del Messico, 3 gennaio 1891-ivi, si novembre 1960).

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.

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.

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.

La teoría musical moderna, basada en la escala temperada de doce tonos, fue una respuesta a las necesidades impuestas por la Entonación Justa en la que no se podía modular de manera satisfactoria. Este documento demuestra los detalles teóricos de ese proceso, con el cuidado de clarificar conceptos que deberían ser conocidos como parte de la teoría musical moderna, a la vez que los expande según las ideas de diversos autores.
Palabras Clave: Temperamento, Entonación Justa, Identidad, Límite, Coma.

This paper introduces polytempic polymicrotonality as a new musical aesthetic. Microtonality is the basis for its inception, from which the discussion proposes music with more than one microtonal tuning system. Examples from the literature are discussed to give an historic framework showing that this tendency has been present throughout human musical history. Polytempo is a tool for which polymicrotonal structures can function in relief from its background. Polytempo acts as a frame, or ground structure, that is multi-dimensional, akin to the advancement of perspective in Renaissance art. Examples of music literature are displayed for musical precedence in this area, focusing on Charles Ives’s Universe Symphony, unfinished since 1925, and realized recently by Johnny Reinhard in 1996.

Philosopher/mathematician/musician Pythagoras of Samos (ca. 570—ca. 490 BCE) is credited with the historic discovery of the relationship between musical harmony and mathematics, and his teachings about the divine nature of music and numbers live through to today in the concept of musica universalis or “the music of the spheres.” Although Pythagoras left no writings behind, his ideas found expression in the works of Plato, Iamblichus, Nicomachus, Boethius, and others. In the 20th century these ideas inspired the works of American microtonal composer/theorist Harry Partch, along with German “harmonicists” such as Hans Kayser and Rudolf Haase. By tracing a line between the Lambdoma of Pythagoras and the Tonality Diamond of Harry Partch, and through comprehensive review of Pythagorean harmonic concepts embedded in various ancient philosophical texts along with modern works on microtonal music theory, we examine the connection between the two disciplines, and ponder whether a study of either one is truly complete without an understanding of the other.

This dissertation examines how late nineteenth-century discourse on tuning and temperament was co-produced with the global politics of nationalism, transnationalism, colonialism, and imperialism.  I concentrate on the history of “just intonation”—a tuning system whose intervallic proportions are predicated on the logic of simple whole-integer ratios—as it was theorized by a network of musical scholars spanning Asia, Europe, and North America.  My focus on this history reveals how, on the one hand, political forces shaped the development of comparative musicology and methods of score-based analysis.  I also demonstrate, on the other, how the increasing entanglements between “Western” and “non-Western” scholarship produced new expressions of cultural hybridity that fundamentally reconfigured national politics, through the reciprocal exchange of musical ideas and instruments.  Based on my findings, I challenge persistent myths of theoretical autonomy and cultural exceptionalism, and assess both the merits and pitfalls of attempting to write a global history of music theory.
In four chapters, I examine three branches of just-intonation discourse and evaluate their connections to the political relations between indigenous—settler populations (Native American communities/United States), colonizer—colonized groups (England/India), and nation-states pursuing diplomatic relations (Germany/Japan).  Drawing on the methods of Science, Technology, and Society Studies (STS), I lay out how the epistemological frameworks of scholarship within these contexts drew upon statistical techniques, myths of cultural origins, and theories of nature that were integral to the formation of nineteenth-century nation-states and imperial regimes.  I argue that the construction of just-intonation keyboards with as many as fifty-three tones per octave played a particularly important role within this context, as tools for scientific investigation as well as for normalizing and standardizing national musical practices.  I then build on the insights of transnational historiography and post-colonial theory in examining the subaltern response to these approaches, identifying two destabilizing strategies: the provincialization of “Western” musical discourse, and the defamiliarization of “Western music.”  I conclude with reflections on my own experiences playing just-intonation instruments, suggesting how the global history of theory might incorporate insights from music cognition, critical organology, and performance studies so as to expand its purview from the history of ideas.

Pythagoras developed an eight-note musical scale in which ratios of small whole numbers were used to define the length of the strings. While Pythagoras committed nothing to writing, later Pythagoreans wrote a good deal about these matters, and these principles were used to tune musical instruments until fairly modern times. In the seventeenth century, equal-tempered chromatic scales were introduced in which the frequencies of successive notes differ by a common ratio. In the early days of Rosicrucian activity in America, H. Spencer Lewis published a system of " natural harmonics " in which the notes of the chromatic scale were correlated with the colors of visible light and certain vowel sounds. Dr. Lewis used a value of middle C (C4), 273 Hz, that was based on a specific wavelength of green light in the middle of the visible spectrum. Modern scales, on the other hand, use values such as 256 Hz and 262 Hz for middle C. As the physiological and psychic effects attributed to the musical notes may be a function of the scale used, this paper examines and compares different musical scales. Based on this examination, it is believed that no one frequency for middle C is best for all occasions. Résumé Pythagore a développé une gamme musicale de huit notes (accord pythagoricien) qui utilisait les rapports de petits nombres entiers pour définir l'accord. Tandis que Pythagore ne cocha guère ses théories par écrit, des pythagoriciens ont ultérieurement écrit de nombreux traités sur cet argument et ses principes ont été utilisés pour accorder les instruments de musique jusqu'à la fin du Moyen-Age. Le XVIIe siècle vit l'introduction des gammes tempérées (tempérament égal), dans lesquelles les fréquences des notes successives sont reliées par une relation commune (intervalles égaux). Dans les premiers temps d'activité de la Rose-Croix en Amérique, le Docteur H. Spencer Lewis publia un système d' « harmoniques naturels », dans lequel les notes de la gamme chromatique sont en corrélation avec les couleurs de la lumière visible et avec certaines voyelles. Le Dr. Lewis utilisa à cet effet une valeur pour le do central de 273 Hz, qui correspond à une longueur d'onde spécifique de la lumière verte du milieu du spectre visible. Les gammes actuelles, en revanche, utilisent des valeurs d'environ 262 Hz. Du moment que les effets physiologiques et psychiques attribués aux notes musicales peuvent différer selon les fréquences utilisées, cet article examine et compare les différentes gammes musicales. On en déduit qu'aucune fréquence unique pour le do central n'est idoine pour toutes les circonstances.

It has been an abiding opinion in many nations and thinkers that music has great power to affect the morals of listeners for good and ill. This paper, written to celebrate the 25th anniversary of Paul Rahe's Republics Ancient and Modern, analyses how modern Western music in all its forms is, because of the tempered scale, profoundly corrupt and corrupting, and a powerful tool of tyranny.

Lafkioui, M. (2009). L'intonation et sa fonction de structurateur hiérarchique des syntagmes propositionnels sans indice morphématique. Le cas du tarifit. In S. Chaker, A. Mettouchi, & G. Phillipson (a cura di), Etudes de phonétique et linguistique berbères. Hommage à Naïma Louali ...

Corso in Tradizioni musicali extraeuropee a indirizzo indologico
TESI DI DIPLOMA ACCADEMICO DI I LIVELLO
Il concetto di śruti: dall'indagine storica all'ipotesi sperimentale
- una rielaborazione attualizzata dell'analisi di Jairazbhoy-Stone (1963) Relatore: Chiar.mo Prof. Roberto PERINU
Candidato: Paolo FALLILONE (matricola n° 3264)

In 1852, the theorist Johanna Kinkel urged musicians to “emancipate the quartertone [for] a new world of sound!” Her call to arms was quickly countered by Wolfgang Heinrich Riehl, who denigrated such “enharmonic” sounds as “effeminate” in contrast to traditional German diatonicism. This debate occurred during the contemporaneous development of experimental keyboards that could enable Kinkel’s musical “emancipation,” featuring up to fifty-three equally tempered divisions of the octave in approximation of the pure intervals of just intonation. This article describes two such instruments as case studies: T. P. Thompson’s enharmonic organ for the “abolition” of temperament, and Tanaka Shōhei’s enharmonium for the restoration of music’s “natural purity.” I show how the language these theorists used reveals the larger political and cultural forces shaping the late nineteenth-century development of music theory and comparative musicology and trace how their instruments were construed as tools for transnational exchange and the global promotion of European civilization.

A new theory of musical consonance discovered by Jacques Dudon through the creation of photosonic disks.
J.Dudon found that all traditional musical scales of the world show a high degree of differential coherence, that can be experienced in their acoustic qualities. Differential coherence completes the usual consonance theories, showing the importance of specific small variations for certain notes within musical scales, often considered as single notes. Differential coherence helped J.Dudon to create also many new scales and systems as well as new "fractal" sounds optimising tuning and timbre adequations.

Published in “The New Grove Dictionary of Music and Musicians” (1980 and later editions)

Although this article covers neither 21st- nor even 20th-century compositional experiments, it does show why “just intonation,” though described (more or less vaguely) by quite a few Western music theorists from the 16th century through the 19th, had no bearing on the development of  mainstream keyboard music. So, the text is given here.

We find a fifth approximation of the Just Intonation which generalizes Equal Temperament. The intervals causing a dilemma are the second and the minor seventh and the tritone because they are unambiguous in Just Intonation (the relative frequencies 10/9, 9/8, 8/7 and 7/4, 16/9, 18/10 and 45/32, 64/45, respectively). If we do not consider the second and seventh with the relative frequencies 8/7 and 7/4, respectively, all the music intervals in this approximation either coincide with he Just Intonation interval values (the octave, fifth, fourth, second (9/8) and the minor seventh (16/9)) or are exactly the one comma distant from the corresponding Just Intonation intervals. This comma is 32 805/32 768 ≈ 1.00112915, which is less than the ratio of frequencies of the perfect and the equal tempered fifths (≈ 1.00112989).

During the 1980s, Julien Jalâl Ed-Dine Weiss, founder of the Al-Kindi ensemble of Aleppo, invented a qānūn in just intonation with which he attempted to solve a major discrepancy between the theory and practice of maqām-scales. Weiss objected to the introduction of Western standards, observing that they distort the significance of interval ratios and prevent a comparative understanding of the modal system as a transnational phenomenon. In the twentieth century, the implementation of equal-semitone temperament emerged simultaneously with a notable invasion of sociological criteria into musical inquiry. The polarity observed between westernisation and tradition can be seen most visibly in the present search for identity amongst Middle-and Near-Eastern musicians, but this schismogenic process can also be observed in the history of the Western avant-garde, where microtonal explorations have been halted in favour of extra-musical conceptuality. While cross-cultural musicians are faced with a new climate of distrust, it seems most likely that the principles that draw us apart may originate in the very patterns of thought in which our notion of culture operates. Weiss's tuning system may serve as a helpful tool to foster a new and universal epistemology of tone, bridging and transcending the apparent contradictions between the two spheres.

Robert Wannamaker's monumental two-volume study explores the influential music and ideas of American composer, theorist, writer, performer, and educator James Tenney. Delving into the whole of Tenney's far-ranging oeuvre, Wannamaker provides in-depth, aurally grounded analyses of works linked to the artist's revolutionary theories of musical form, timbre, and harmonic perception.

Volume 1, Contexts and Paradigms, chronologically surveys Tenney's creative development and output. Wannamaker begins each section with biographical, aesthetic, and technical context that focuses on a distinct period in Tenney's career. From there, he analyzes a small number of pieces that illuminate the concerns, characteristics, and techniques that emerged in Tenney's music during that time. Wannamaker supplements the text with musical examples, graphs, and diagrams while also drawing on unpublished material and newly available primary sources to flesh out each work and the ideas that shaped it.

A landmark in experimental music scholarship, The Music of James Tenney is a first-of-its-kind consideration of the music titan and his work.

(Click the cover image to read the Kindle sample.)

This is a set of data for use by composers and music theorists working in extended just intonation.  The table contains approximately 20,000 frequency ratios - the total set available using four or fewer pitches each related to one another by one of the tunable (i.e., tunable by ear on acoustic instruments) frequency ratios, a complete list of which was compiled and tested by composers Marc Sabat and Wolfgang von Schweinitz.  Each ratio is accompanied by its distance from a reference pitch (1/1) in cents, its harmonic distance (a linear measurement of relative consonance with respect to a reference pitch), the minimum number of tunable intervals from the reference pitch required to arrive at the given ratio, and a prime factorization of the numerator and denominators of the ratio.

During the 1980s, Julien Jalâl Ed-Dine Weiss, founder of the Al-Kindi ensemble of Aleppo, invented a qānūn in just intonation with which he attempted to solve a major discrepancy between the theory and practice of maqām-scales. Weiss objected to the introduction of Western standards, observing that they distort the significance of interval ratios and prevent a comparative understanding of the modal system as a transnational phenomenon. In the twentieth century, the implementation of equal-semitone temperament emerged simultaneously with a notable invasion of sociological criteria into musical inquiry. The polarity observed between westernisation and tradition can be seen most visibly in the present search for identity amongst Middle- and Near-Eastern musicians, but this schismogenic process can also be observed in the history of the Western avant-garde, where microtonal explorations have been halted in favour of extra-musical conceptuality. While cross-cultural musicians are faced w...

This essay documents some initial speculations regarding how harmonies (might) evolve in extended just intonation, connecting back to various practices from two perspectives that have been influential to my work. The first perspective, which is the primary investigation, concerns itself with an intervallic conception of just intonation, centring around Harry Partch's technique of Otonalities and Utonalities interacting through Tonality Flux: close contrapuntal proximities bridging microtonal chordal structures. An analysis of Partch's 1943 composition "Dark Brother," one of his earliest compositions to use this technique extensively, is proposed, contextualised within his 43-tone "Monophonic" system and greater aesthetic interests. This is followed by further approaches to just intonation composition from the perspective of the extended harmonic series and spectral interaction in acoustic sounds. Recent works and practices from composers La Monte Young, Éliane Radigue, Ellen Fullman, and Catherine Lamb are considered, with a focus on the shifting modalities and neighbouring partials in Lamb's string quartet "divisio spiralis" (2019). Finally, I connect this discussion to my current compositional interests, which have been exploring a method of microtonal modulation through arbitrarily near enharmonic connections in Harmonic Space called "enharmonic proximities".

This is a systemic study of atonal music and our broken relationship with nature in view of the global catastrophe.

Drawing on Gregory Bateson's ecology of mind and Claude Lévi-Strauss's structural mythology, as well as recent observations in neurobiology, I show how deeply the abstraction of modernist art is intertwined with capitalism and a parasitic understanding of cultural history.

The "Black Hole" defines the threshold between sound  and modern abstraction, exploiting - not unlike a monetary currency - its source material in order to transform it into aesthetic meaning as a form of digital value. My counter-proposal is an utopic reconsideration of sound's mathematical property, just intonation and equilibrium in a world of turmoil. 

"Healing requires systemic knowledge. I observe a) that music reflects the structure of our destructive economy; b) that an epistemology of tone can correct our ecology towards sustainability. I conclude that our dilemma originates in a disturbed relationship with nature which Modernism consolidates in a virtual (and entirely 'worldly') form of 'transcendence'. My proposal - a kind of "Green New Deal" - neither relies on 'recycling' nor on 'waste avoidance', but on a process similar to 'composting'."   

written in German in 2019.

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.

During the 1980s, Julien Jalâl Ed-Dine Weiss, founder of the Al-Kindi ensemble of Aleppo, invented a qānūn in just intonation with which he attempted to solve a major discrepancy between the theory and practice of maqām-scales. Weiss objected to the introduction of Western standards, observing that they distort the significance of interval ratios and prevent a comparative understanding of the modal system as a transnational phenomenon. In the twentieth century, the implementation of equal-semitone temperament emerged simultaneously with a notable invasion of sociological criteria into musical inquiry. The polarity observed between westernisation and tradition can be seen most visibly in the present search for identity amongst Middle- and Near-Eastern musicians, but this schismogenic process can also be observed in the history of the Western avant-garde, where microtonal explorations have been halted in favour of extra-musical conceptuality. While cross-cultural musicians are faced w...

I really enjoyed reading this book more than 500 pages long, even though, not being an English native speaker, it took me a while to finish it. It is a very detailed portrait of a great human being besides an excellent musician. I do not mean to diminish his musical value but what I really love about him is his voracious appetite for the many aspects of his life that this book is capable to convey to the reader.