Equal Temperament Research Articles

Explorations of Equal-Tempered Microtonality. Ivan Wyschnegradsky, Manual of Quarter-Tone Harmony, Edited by Noah Kaplan and translated by Rosalie Kaplan, Brooklyn, Underwolf, 2017. 28 pages with online audio supplement Julia Werntz, Steps to the Sea: Ear Training and Composing in a Minute Equal Temperament, Frog Peak Music, 2014. 47 pages with online audio supplement

Explorations of Equal-Tempered Microtonality / Ivan Wyschnegradsky, Manual of Quarter-Tone Harmony, Edited by Noah Kaplan and translated by Rosalie Kaplan, Brooklyn, Underwolf, 2017. 28 pages with online audio supplement / Julia Werntz, Steps to the Sea: Ear Training and Composing in a Minute Equal Temperament, Frog Peak Music, 2014. 47 pages with online audio supplement. An article from journal Circuit (Les voies de la pansonorité), on Érudit.

Playing with Shadows: Reinjection Loops and Historical Allusion in Georg Friedrich Haas’s Live Electronic Music

Recent scholarship suggests that the music of Austrian composer Georg Friedrich Haas can be understood as a dramatic confrontation between “clashing harmonic systems” (Hasegawa, 2015). Building on this observation, the present article focuses on Haas’s recent endeavors in the genre of live electronic music, showing how the composer deploys a relatively straightforward technical procedure—the reinjection loop,orthe delayed playback of recorded sound at various speeds—to juxtapose different modes of pitch organization, including twelve-tone equal temperament, ultrachromatic microtonality, and just intonation. After surveying the composer’s relationship to these historical idioms, the article presents detailed analyses of two pieces (Ein Schattenspieland String Quartet No. 4) to illustrate how the reinjection loop operates at multiple registers simultaneously, bringing the performers into contact with their immediate past, while also bringing Haas into dialogue with the shadows of his own compositional predecessors.

White-light Viewed from the Musical Perspective as Art Technology

When white light electromagnetic radiation passes an inclined transparent material such as a prism, it reveals 7-colors of a rainbow caused by the different refractive indices of the colors in white light. Basically, Isaac Newton found this in 1704 and it has been recognized under the name of color-music. The colors of white light have some similarity to the patterns of music. The frequency range of a rainbow from white light was analyzed in terms of bandwidth to compare with a music scale that had different notes of numbers in double frequencies (oneoctave) from the lower note to the upper note (one-octave). Then, the ranges of colors were compared with the existing music scale. The formula of equal temperament, which multiplies a lower note by 1.0594635, makes a half-step upper note (semitone) and was used for all 12 notes in an octave. We found that with 12-tone equal temperament, each tone has a corresponding color and a scale resembling the pattern of white light is a composite of a 6-note scale that was named ‘the rainbow scale’. Violet (C+) is not included in the numbers of the rainbow scale. The tones and light frequency pairs are C = 760 nm, E = 603 nm, F = 569.36 nm, G = 507.24 nm, A = 451.90 nm, A# = 426.53 nm, and higher C = 380 nm.

A Longitudinal Study of Intonation in an a cappella Singing Quintet

The skill to control pitch accurately is an important feature of performance in singing ensembles as it boosts musical excellence. Previous studies analyzing single performance sessions provide inconclusive and contrasting results on whether singers in ensembles tend to use a tuning system which deviates from equal temperament for their intonation. The present study observes the evolution of intonation in a newly formed student singing quintet during their first term of study. A semiprofessional singing quintet was recorded using head-worn microphones and electrolaryngograph electrodes to allow fundamental frequency (fo) evaluation of the individual voices. In addition, a camcorder was used to record verbal interactions between singers. The ensemble rehearsed a homophonic piece arranged for the study during five rehearsal sessions over four months. Singers practiced the piece for 10 minutes in each rehearsal, and performed three repetitions of the same pieces pre-rehearsal and post-rehearsal. Audio and electrolaryngograph data of the repeated performances, and video recordings of the rehearsals were analyzed. Aspects of intonation were then measured by extracting the fo values from the electrolaryngograph and acoustic signal, and compared within rehearsals (pre and post) and between rehearsals (rehearsals 1 to 5), and across repetitions (take 1 to 3). Time-stamped transcriptions of rehearsal discussions were used to identify verbal interactions related to tuning, the tuning strategies adopted, and their location (bar or chord) within the piece. Tuning of each singer was closer to equal temperament than just intonation, but the size of major thirds was slightly closer to just intonation, and minor thirds closer to equal temperament. These findings were consistent within and between rehearsals, and across repetitions. Tuning was highlighted as an important feature of rehearsal during the study term, and a range of strategies were adopted to solve tuning related issues. This study provides a novel holistic assessment of tuning strategies within a singing ensemble, furthering understanding of performance practices as well as revealing the complex approach needed for future research in this area. These findings are particularly important for directors and singers to tailor rehearsal strategies that address tuning in singing ensembles, showing that approaches need to be context driven rather than based on theoretical ideal.

Alternatives to Semitones and Quartertones: Music-Theoretical Suggestions

Thanks to free software synthesisers that support custom tuning files, it is now possible for a massive number of people to experiment with alternatives to semitones and quartertones. Indeed, contemporary musicians are not forced to choose 12-tone equal temperament or 24-tone equal temperament. I begin this article with a critique of 24-tone equal temperament. I then present a variation on the well-known whole-tone scale plus new symmetrical modes inspired by the French composer Olivier Messiaen. Finally, I show how one can map a little-known, non-octave scale by the German physicist Heinz Bohlen to a conventional, Halberstadt keyboard. Since one can only make symmetrical modes (modes a transpositions limitees) with an equal temperament, most of the musical intervals discussed in this article are approximations of frequency-ratios; they are not pure frequency-ratios.

Eight Artist-Level Violinists Performing Unaccompanied Bach: Are There Consistent Tuning Patterns?

This study was designed to explore the intonation performances of highly regarded concert violinists in an unaccompanied context. I chose eight solo artists representing different generations of performers (Heifetz, Grumiaux, Milstein, Perlman, Hahn, Midori, Barton Pine, and Shaham). Pitch performances were analyzed in commercial recordings spanning 65 years of two unaccompanied pieces composed by Bach, one in d minor and one in E major. I investigated whether similarities in intonation tendencies would occur among these respected artists, and whether the displayed patterns would approximate the theoretical tuning systems that have been studied previously: Just, Pythagorean, or equal temperament. The artists appear to have their own individual tendencies, and none of them conformed consistently to any theoretical tuning system. Although there are average tendencies on isolated intervals that approximate one tuning or the other, multiple repetitions show a wide range both between and within individual performers. However, performances of the thirds and sixths in the major key and less so in the minor key (intervals that illustrate the largest differences in tuning between systems) did display a propensity toward Pythagorean tuning of these scale degrees. Deviations away from all three tuning systems are common, and these artists never play exactly in one tuning or another.

Composer and Theorist Joel Mandelbaum on Microtonality

Mr. Mandelbaum (b. 1932) is a highly educated, skillful composer who writes in a clear, tuneful, and accessible tonal style to which the use of microtones add an original and fascinating flavor. Unfortunately, his music, such as his important, much-acclaimed operas on Jewish subjects, is all but unavailable on recordings. Of his microtonal music, only a short electronic work in 31- tone equal temperament is available on CD (“Andante cantabile” on the compilation “Electronical,” American Festival of Microtonal Music – Pitch P-200208).Among microtonal composers and scholars of tuning systems, Joel Mandelbaum is widely known for his seminal dissertation “Multiple Division of the Octave and the Tonal Resources of the 19-tone Equal Temperament” (Ph.D. thesis, University of Indiana, 1961; available online at http://anaphoria.com/mandelbaum.html). Publishing this 1994 interview now makes available a fine short introduction to a remarkable microtonal composer and theorist, and two of the most important and popular microtonal tuning systems: 19 and 31 equal divisions of the octave. After his most illuminating talk, Mr. Mandelbaum demonstrated various intervals and tunings on his famous Motorola Scalatron, an early digitally retunable electronic organ, which at the time was still in working condition. Keywords: microtonality, Joel Mandelbaum, microtonal composers.

Designing inharmonic strings

Uniform strings have a harmonic sound; non-uniform strings have an inharmonic sound. Given a precise description of a non-uniform string, its inharmonic spectrum can be calculated using standard techniques. This paper addresses the inverse problem: given a desired/specified spectrum, how can string parameters be chosen so as to achieve that specification? The design method casts the inverse problem in an optimization framework that can be solved using iterative techniques, and experiments show that viable solutions are often possible despite the multi-modal character of the cost function. Three properties of inharmonic strings are studied: their behavior under changes in tension, changes in density, and changes in length. These are important to the use of the inharmonic strings in musical instruments where it is often desirable that a set of strings has consistent timbre (spectrum) over many different pitches. Several different strings are designed: one that has partials derived from the golden ratio φ, one with overtones that beat with each other, and a family of strings designed for performance in n-tone equal temperament. An Online Supplement for this article contains details of these string designs for all n from 7 to 20 and can be accessed at http://dx.doi.org/10.1080/17459737.2018.1491649. Several of the strings have been constructed, and the predicted frequencies of the overtones are verified by direct measurement.

Quirinus van Blankenburg's Transporteur

Thirty-one tone equal temperament was studied by Christiaan Huygens, a prominent Dutch mathematician and scientist who lived in The Netherlands from 1629 to 1695. This temperament was rediscovered ...

A Psychocultural Theory of Musical Interval

The Pythagoreans linked musical intervals with integer ratios, cosmic order, and the human soul. The empirical approach of Aristoxenus, based on real musicians making real music, was neglected. Today, many music scholars and researchers still conceptualize intervals as ratios. We argue that this idea is fundamentally incorrect and present convergent evidence against it. There is no internally consistent “Just” scale: a 6th scale degree that is 5:3 above the 1st is not a perfect 5th (3:2) above the 2nd (9:8). Pythagorean tuning solves this problem, but creates another: ratios of psychologically implausible large numbers. Performers do not switch between two ratios of one interval (e.g., 5:4 and 81:64 for the major third), modern studies of performance intonation show no consistent preferences for specific ratios, and no known brain mechanism is sensitive to ratios in musical contexts. Moreover, physical frequency and perceived pitch are not the same. Rameau and Helmholtz derived musical intervals from the harmonic series, which is audible in everyday sounds including voiced speech; but those intervals, like musical intervals, are perceived categorically. Musical intervals and scales, although they depend in part on acoustic factors, are primarily psychocultural entities—not mathematical or physical. Intervals are historically and culturally variable distances that are learned from oral traditions. There is no perfect tuning for any interval; even octaves are stretched relative to 2:1. Twelve-tone equal temperament is not intrinsically better or worse than Just or Pythagorean. Ratio theory is an important chapter in the history Western musical thought, but it is inconsistent with a modern evidence-based understanding of musical structure, perception and cognition.

‘OF THE TEMPERAMENT OF THOSE MUSICAL INSTRUMENTS’: CONSIDERING TIBERIUS CAVALLO AND THE SCIENTIFIC OBSERVATION OF MUSICAL SOUNDS IN LATE EIGHTEENTH-CENTURY LONDON

The connection of music to scientific exploration in late Enlightenment London can be considered from various perspectives, perhaps most evidently through the binary of amateur–professional. These two realms intersected within natural philosophical observation, a practice that often served concurrently as entertainment and as study. The development of scientific instruments for the observation of various phenomena appeared in both professional and amateur contexts, contributing to technological growth and research. Natural philosopher Tiberius Cavallo (1749–1809) and his 1788 article on musical temperament (‘Of the Temperament of Those Musical Instruments, in Which the Tones, Keys, or Frets, are Fixed, as in the Harpsichord, Organ, Guitar, &c’) provide a captivating example of amateur interest overlapping effectively with the professional domain; as an amateur musician and professional scientist, Cavallo observed equal temperament in both mathematical and aesthetic terms. Consideration of his work promotes a more nuanced view of London as a place where scientific and musical ideas could meet and be ‘instrumentalized’, emphasizing the city's status as a vibrant arena for the interaction of scientific exploration, artistic endeavour and professional identities. In this sense, Cavallo's work on temperament was not merely a scientific activity; it reflected technological change during a stimulating period of scientific and musical progress in late eighteenth-century London. For example, instrument builders were actively developing ways to improve pitch control and tuning stability, as witnessed by numerous British patents for harp mechanisms, the addition of flute keys and keyboard construction.

Towards Song

This paper combines the separate but overlapping interests of its authors – one a composer and sound-artist, who often works with voice and word; the other a poet with an interest in how page-based poetry is actually read – in an enquiry into the performance of lyric poetry and its relations to song. The enquiry owes at least as much to the studio, to sound-editing software, to close listening and close reading as it does to existing literature on connected topics. The main precedent is Douglas Oliver’s 1989 publication, Poetry and Narrative in Performance, in which the author adopts the word performance to refer to any act of reading, whether aloud or silent. Like Oliver, the authors rely on the visual representations of the sound-shapes of specific sounded readings and also rely on access to sound files of some of the readings and their manipulations. In addition to the analytic value of having access to recordings and wave-forms and spectrograms, the authors experimented with using software (Audacity and Melodyne) to make subtle changes in pitch, pulling particular vowel sounds from their original positions to the nearest notes in a tempered scale. These experiments test, perhaps, a border-zone between lyric poetry and sung lyric. They also suggest a possible value for poets in using such software as support for their technologically unaided ears in testing out their own compositions.

The Effect of the Tuning System and Instrument Variables on Modal Dictation Performance

Ear training and musical literacy (ETML) education is one of the main dimensions of the bachelor degree program of music teacher education departments, which provides professional music education. In ETML education, hearing, sight-reading and dictation studies for Turkish music makams have an important place. In this study, it was aimed to investigate the effects of different instrument sources and tuning systems; namely, equal tempered or traditionally tempered, on the modal dictation performances of students in dictation exercises in the scope of ETML education. This research was conducted with 56 bachelor degree music education students who were taking ETML course. While equal tempered and traditional instruments (piano and kanun) were used as the sound source in the research, examples of equal difficulty level in Huseyni makam scale, which is one of the main makams of Turkish music, were used as dictation material. Both instruments were tuned in accordance with both the equal temperament and the traditional makam system so that four different dictation types were created. As a result of the research, it has been determined that the students are more successful when piano is used in dictation than kanun-dictated trials and in terms of tuning system, the students are found to be more successful with equal tempered system when compared with traditional tuning system. The instrument and tuning interaction was not statistically significant.

Equal Temperament: Autonomy and Identity in Chinese Public Speaking Clubs

AbstractYoung professionals in China are eagerly studying what are called communication skills, particularly public speaking. This article reads technologies of self‐presentation in the context of tentatively liberal social imaginaries, and as therapeutic resources for connecting with others. Entering Toastmasters public speaking clubs in Beijing, we see psychosocial techniques and institutional forms that anchor people otherwise floating in a fragmented urban space. Club members actively seek to be objectified on stage, to become self‐aware under the gaze of an audience. They practice locating themselves within an imagined public, as individuals. However, members also repurpose their clubs into comprehensive social resources, and they construct a critical historical consciousness. While members of public speaking clubs pursue self‐definition, they do so in ways that challenge liberal understandings of autonomy and identity. We can hear these tensions by listening to how Chinese therapists use psychology to articulate social critiques. While both psychotherapists and members of public speaking clubs participate in a modernist cultural politics, club members do not regard personal relationships as antagonistic. Young adults in urban China are getting on stage in order to connect with other people. Like their peers elsewhere, they are learning the power of self‐definition in a world of strangers.

Roughness of Two Simultaneous Harmonic Complex Tones On Just-Tempered and Equal-Tempered Scales

This study is focused on the perceived roughness of two simultaneous harmonic complex tones with ratios between their fundamental frequencies set to create intervals on just-tempered (JT) and equal-tempered (ET) scales. According to roughness theories, ET intervals should produce more roughness. However, previous studies have shown the opposite for intervals in which the lower fundamental frequency of the complex was equal to 261.6 Hz. The aim of this study is to verify and explain these results by using intervals composed of complexes whose spectral components were generated with either a sine starting phase or with a random starting phase. Results of the current study showed the same phenomenon as previous studies. To examine whether the explanation of the phenomenon lies in the function of the peripheral ear, three roughness models based upon this function were used: the Daniel and Weber (1997) model, the synchronization index (SI) model, and the model based on a hydrodynamic cochlear model. For most of the corresponding JT and ET intervals, only the Daniel and Weber (1997) model predicted less roughness in the ET intervals. In addition to this, the intervals were analyzed by a model simulating the auditory periphery. The results showed that a possible cause for the roughness differences may be in the frequencies of fluctuations of the signal in the peripheral ear. For JT intervals the fluctuations in the adjacent places on the simulated basilar membrane had either the same frequency or integer multiples of that frequency and were synchronized. Since a previous study showed that synchronized fluctuations in adjacent auditory filters lead to higher roughness than out of phase fluctuations (Terhardt, 1974), this may cause more roughness across JT and ET intervals.

Harmony outside the Iron Cage: Tanaka Shōhei’s Strategic Deconstruction of the Music-Theoretical Edifice

Tanaka Shōhei was a major figure in both Japanese and comparative musicology. Convinced by his tutor in Tokyo (Mendenhall) that Japanese music must embrace harmony in order to become “modern,” Tanaka was admonished by his advisor in Berlin (Helmholtz) that the Japanese must not adopt twelve-tone equal temperament because of its adverse effects on the ear—contradictory advice given that the latter was considered the simplest way of enabling the smooth functioning of the former. His solution involved a system of musical intonation whereby the octave was divided into fifty-three (as opposed to twelve) tones. In the context of his overall project, however, this theoretical innovation was only the first step. His goal was to reproduce a sort of harmonic “rationality” in Japanese music, while preserving its expressive melodic freedom: in Weberian terms, he wanted a different sort of modernity, one that escaped the confines of the iron cage.

Традиции Античности в музыкальном искусстве Возрождения и Классицизма

The urge to create an image of the ideal world, basing oneself on the capabilities of reason, brings together three significant periods in the history of art – Antiquity, the Renaissance and Classicism. The system of music theory first to have been developed for these aims was that of the Ancient Hellenes. In their aspiration to understand the laws of euphony they based their musical thinking upon the proportions of prime numbers, the highest objectivity of which made it possible to liken music to the perfection of the sounding Cosmos and its smaller copy – man, the microcosm. Thereby, the greatest achievement of Ancient Greek and Roman music (in its theory and practice) must be considered to be the assertion of perfect consonances and diatonicism as the model sample of Harmony – whether cosmic, human or musical. The formation of a qualitatively new musical theory during the period from the Renaissance to the 18th century was aroused by theoretical aspirations toward the creation of the image of nature and the natural, i.e. affective human being. Following the standards of imitation of Antiquity and nature established in the art and aesthetics of the Renaissance, the thinkers and masters of music developed a new theoretical foundation, most notably, the equally tempered scale and the teaching of the tertial structure of chords (the study of harmony). Similarly to the thinkers of Antiquity, they also based themselves on mathematical natural laws; however, of the utmost significance in this case was their connection to the natural sciences – namely, experiments in acoustics. As a result, the chief criterion for euphony for the music of the Early Modern Times was expressed not only by numerical proportions, but, first of all, by the laws of acoustics and the capabilities of human aural perception. Keywords: music theory, Cosmos, consonance, interval of a third, Antiquity, Renaissance, Classicism.

Aspectos históricos del temperamento de los instrumentos de teclado en España a finales del siglo XVII: de los proyectos toledanos a las realizaciones mediterráneas

This article is, primarily, a synthesis of decades of research into organ building in Spain. But it also deals with a topic rarely considered: the temperament of keyboard instruments at the end of the seventeenth and early eighteenth centuries in Spain. The recent discovery of a 1696 contract to build an organ at the Toledo Cathedral, signed by Liborna Echevarria, provides a new perspective on the way organs were built at the time. Moreover, this contract can be compared with the one signed by Domingo Mendoza in 1695 for the construction of an instrument in this very cathedral, which allows us to compare and study the differences between the two projects. Indeed, these differences are not only mechanical but represent the two main standard bearers of the art of Spanish organ building at its height. A new interpretation of both documents will also give us a better understanding of the two main approaches to keyboard temperament: the one tending towards equal temperament, with a “tonal” orientation, from classical harmony (characteristic of Liborna Echevarria and Domingo Aguirre), which dominated the Western part of the Peninsula; and the one tending towards a “pretonal” or “modal” temperament, whose origins are to be found in the Renaissance period and which disappeared during the eighteenth century. The latter was widely used in South-Eastern Spain, developed and revived around the city of Valencia and inspired by Juan de Andueza and his pupil Domingo Mendoza. Both approaches used friar Jose de Echevarria as their model.

Ладовые основы классицизма в контексте мировоззрения XVIII века

Classicism presents one of the most significant artistic achievements of humanity. The basis of Classicism formed in the art of music in the 18th century was comprised by the equal tempered scale and the major-minor modal system, which succeeded the Pythagorean tuning and the assemblage of the medieval church modes. The aforementioned innovations were the result of a lengthy goal-seeking scholarly search for thinkers and musicians for the end of creating an ideal musical temperament. This corresponded fully to the overall mentality of the epoch of the Enlightenment with its veneration of reason, both Divine and human. The aforementioned innovations comprised an artistic analogy to the images of the infinite homogenous space and a heliocentric system of the world, which sidelined to the second place the medieval perceptions of a hierarchical world order with the Earth as the absolute center. The socially-conceived analogue to the centralization of the mode and the tonal organization of sound may be conceived to be the formation of French absolutism, as well as the teaching created by the Enlighteners of an ideal concisely structured community of people equal before God ruled by an enlightened monarch. The cosmological and social analogies presented in the article testify to the fact that the modal foundations of musical Classicism corresponded in full to the central world outlook paradigm of the epoch, namely, the conviction of the Rational and Divine foundations of art, nature and society. Keywords: Classicism, the equal-tempered scale, major-minor, the heliocentric system, social structure.

Relative Nature of Thai Traditional Music through its Tuning System

Thai traditional music exhibits number of relative characteristics, though many of them are not expressed explicitly but only demonstrated orally. Its tuning system, despite having been put through many scientific studies for decades, cannot be given the absolute pitch frequencies. It is therefore possible that the tuning system does not have any absolute pitch frequencies but possesses relative nature. This papers aims to present the historic background, characteristics, past, current practices of Thai traditional music’s tuning system and to put forward the theoretical idea of non-existence of absolute tuning frequency and that of relative nature of Thai traditional music’s tuning system. The data was collected from previous researches on Thai traditional music’s tuning system and from sample instruments. The latter was then compared to each other and analyzed with respect to the former. The preliminary results were that tuning system of Thai traditional music was in the form of seven-tone equal temperament without any sharps or flats. But practically, it was not exactly equal as musicians still prefer the traditionalbiased tunings which is believed to be more tuneful. Regarding the tuning practices, the tuning of Fine Arts Department are most dominating among several others. Even though, the tuning of Fine Arts Department measure from various sources are slightly different in terms of frequencies and intervals. It can be concluded that the tuning system of Thai traditional music does not rely on specific pitch frequency, but is relative to intervals and personal preferences and its absolute pitch frequency is yet to be established.