Twelve-tone Equal Temperament Research Articles

Classical guitar intonation and compensation: The well-tempered guitar.

An intuitive model of classical guitar intonation is presented that includes the effects of the resonant length of the fretted string, linear mass density, tension, and bending stiffness. An expression is derived for the vibration frequencies of a stiff string using asymmetric boundary conditions at the saddle and the fret. Based on logarithmic frequency differences ("cents") that decouple these physical effects, Taylor series expansions are introduced that exhibit clearly the origins of frequency deviations of fretted notes from the corresponding 12-tone equal temperament (12-TET) values. A simple in situ technique is demonstrated for measurement of the changes in frequency of open strings arising from small adjustments in length, and a simple procedure is proposed that any interested guitarist can use to estimate the corresponding frequency shifts due to tension and bending stiffness for their own guitars and string sets. Based on these results, a least-squares fit method is employed to select values of saddle and nut setbacks that map fretted frequencies-for a particular string set and guitar-almost perfectly onto their 12-TET targets. A general approach to tempering an "off-the-shelf" guitar is shown to further reduce the tonal errors inherent in any fretted musical instrument.

Navigating a Two-dimensional Field / across the plane

Composing for the piano is beset by constant compromise. As a composer whose musical language is exemplified by glissandi, microtones, timbral transformation, and states of destabilisation and distortion, the piano—at least as traditionally focused upon the keyboard—invokes feelings of frustration and constriction. With the instrument’s design predicated by performance upon the keyboard, twelve-tone equal temperament, a clarity of pitch and timbre (even at extreme dynamics),1 and a uniform timbral profile, I find myself restricted to a narrow band of exploratory possibilities. This is amplified by my classical training as a pianist in which my fingers default to learnt gestures, familiar pitch structures, typical voicings, and are always drawn to the keyboard as a primary location of activity. In an attempt to divorce these learnt behaviours, open up a more exploratory compositional space, and re-work the many existing notational models that still focus upon the five-line stave, this article outlines my development of a new notational model that serves as both a scoring tool and, most crucially, a lens in which to approach and compose for the piano. The aims for this notational model include the following: • decentralising the keyboard as only a subcomponent of the entire instrument • capturing the entire body of the piano and invite exploration • allowing for manipulations to the strings, preparations, object-oriented composition, and exploration of glissandi, distortion, and timbral transformation • appearing ultimately coherent and intuitive to read. These aims will be further illustrated through my recent work, Fourteen piano transcriptions from across the plane [plain] (2023), composed specifically to explore the possibilities and limits of this notation.

Fine-Tuning a Global History of Music Theory: Divergences, Zhu Zaiyu, and Music-Theoretical Instruments

AbstractThe issue of equal temperament offers an object lesson in the challenges of the new global history of music theory: Twelve-tone equal temperament was mathematically formulated at almost the same time in Ming-dynasty China and sixteenth-century Europe. While the old debate got stuck on questions of dates and cultural rivalries, recent work in comparative humanities, especially Kuriyama (2002), opens up new avenues. His concept of “divergence” is applied to the specific “music-theoretical instruments” in which Chinese and European theories of tuning manifested themselves in sound. Zhu Zaiyu’s pathbreaking 1584 theory is reexamined specifically from this angle: He credits the qin (zither) for holding knowledge that the 12 lü, the traditional Chinese pitch-pipes, could not convey. Zhu’s example—and the concept of “divergence”—offers ways forward for a new, materially oriented, global history of music theory.

Wavescapes: A visual hierarchical analysis of tonality using the discrete Fourier transform

Many structural aspects of music, such as tonality, can be expressed using hierarchical representations. In music analysis, so-called keyscapes can be used to map a key estimate (e.g., C major, F minor) to each subsection of a piece of music, thus providing an intuitive visual representation of its tonality, in particular of the hierarchical organization of local and global keys. However, that approach is limited in that the mapping relies on assumptions that are specific to common-practice tonality, such as the existence of 24 major and minor keys. This limitation can be circumvented by applying the discrete Fourier transform (DFT) to the tonal space. The DFT does not rely on style-specific theoretical assumptions but only presupposes an encoding of the music as pitch classes in 12-tone equal temperament. We introduce wavescapes, a novel visualization method for tonal hierarchies that combines the visual representation of keyscapes with music analysis based on the DFT. Since wavescapes produce visual analyses deterministically, a number of potential subjective biases are removed. By concentrating on one or more Fourier coefficients, the role of the analyst is thus focused on the interpretation and contextualization of the results. We illustrate the usefulness of this method for computational music theory by analyzing eight compositions from different historical epochs and composers (Josquin, Bach, Liszt, Chopin, Scriabin, Webern, Coltrane, Ligeti) in terms of the phase and magnitude of several Fourier coefficients. We also provide a Python library that allows such visualizations to be easily generated for any piece of music for which a symbolic score or audio recording is available.

Based on the Cross-strait Exchanges, The Yangqin Development and Gradual Change of Taiwanese Gezai Opera (Xiang Opera) in Fujian and Taiwan

Gezi Opera is the only kind of opera that uses Yangqin as the accompaniment instrument in the social and cultural inheritance and reproduction of immigrants from Fujian and Taiwan. In terms of cultural communication, Gezi Opera and Xiang opera Yangqin are regarded as sister flowers. Although they come down in one continuous line and share the same root, with the confrontation between the two sides in 1949 and the transfer of regional time and space, they interrupted each other's exchanges and took their own development in the historical context and social structure. From the perspective of culture and civilization, this paper examines the shape variation of Yangqin as an accompaniment instrument of Gezi Opera after the reform and opening up on both sides of the Strait in 1987. Thus, when tradition meets modernity, even if there is a century of culture, it is hard to escape the bad luck of abandoning it like shoes. Keywords: traditional dulcimer; Taiwanese opera; Xiang opera; twelvetone equal temperament DOI: 10.7176/RHSS/11-22-02 Publication date: November 30 th 2021

On the use of nuclear magnetic resonance spectroscopy in music composition- principles, practice and possibilities

Nuclear Magnetic Resonance (NMR) spectroscopy is an analytical technique commonly used across the natural sciences that works upon the detection of magnetically spin-aligned atomic nuclei resonating with an externally applied radio source. This work presents an overview of music created using NMR data and outlines principles for its use in composition. Methods for the sonification and musical interpretation of NMR spectra are demonstrated and opportunities for its development as a musical technique are identified. Abbreviations: 12-TET: Twelve-Tone Equal Temperament; ADC: Analogue to Digital Converter; COSY: Correlation Spectroscopy; FID: Free Induction Decay; FT: Fourier Transform; NMR: Nuclear Magnetic Resonance; PCM: Pulse Code Modulation; PPM: Parts per million; SNR: Signal-to-noise ratio; TMS: Tetramethylsilane

19-Tone Theory and Applications

Microtonal music is one of those subjects that has always been around, but few people have ever had the will to investigate it thoroughly. The main reason why more people have not dealt with microtonal music is that there are almost no instruments that allow composers to experiment with it. In spite of all this, the music of many cultures even at the present time employs non-equaltempered scales, and even Western music did until the eighteenth century, when mathematicians worked out the logarithmic basis of equal temperament. In this article, the author explains how he became interested in 19-tone equal temperament and how he explored the possible resources available in such a system. This involves creating a chord grammar based on similarity relationships, similar to what he has used in his music written in 12-tone equal temperament. Through these considerations he discovered a particular set of chords that have special properties in terms of their interval content, number of transpositions, and relationships to other chords. Finally, the author explains how he used these properties in the music he composed in this temperament.

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.

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.

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.

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.

Microtonal Tunings in Electronic Dance Music: A Survey of Precedent and Potential

While numerous contemporary composers have explored the endless possibilities afforded by microtonal tuning systems, Western popular music has hitherto rarely ventured from the harmonic comfort zone of 12-tone equal temperament. This article highlights an unlikely exception to this trend in the emergence of a community of microtonal electronic dance musicians, and examines how changes in technology, culture, and distribution might have led to its development.

The physics of musical scales: Theory and experiment

The theory of musical scales involves mathematical ratios, harmonic resonators, beats, and human perception and provides an interesting application of the physics of waves and sound. We first review the history and physics of musical scales, with an emphasis on four historically important scales: twelve-tone equal temperament, Pythagorean, quarter-comma meantone, and Ptolemaic just intonation. We then present an easy way for students and teachers to directly experience the qualities of different scales using MIDI synthesis.

Mathematical aspects of musical scales and series

Musical scales may be interpreted as specific increasing sequences of real numbers. Periodic scales, i.e. musical scales whose pattern is periodic, feature in this article. Of special interest are regular scales of which are given seven properties in four theorems which are new results, engaging the concepts of: the period of scale, the length of period of scale, the pattern, being homomorphous, a transposition and having the same key. This paper considers the idea of musical scales in general, hence it is possible to consider more complex musical scales not only in the twelve-tone equal temperament, and also proposes tools to investigate traditional musical scales as Messiaen’s Modes of limited transposition [4, p.58], the major and the minor scales as well as ancient Greek scales inherited by medieval Western music. The paper also goes beyond the notion of musical scale and contains new interesting result which gives way to determine when one can form a series from a shorter sequence of sounds by transposing the sequence.

A proposal of a color music notation system on a single melody for music beginners

Music teachers often encounter obstructions in teaching beginners in music reading. Conventional notational symbols require beginners to spend significant amount of time in memorizing, which discourages learning at early stage. This article proposes a newly-developed color music notation system that may improve the recognition of the staff and the numbered musical notation for music beginners. The focus is on single melodies that are relatively easy for music beginners to read and to play, which may increase students’ confidence at the first step of music learning. The color music notation system that associates color music scores with color and music synesthesia by human vision and audition is proposed. Pitch, duration, range, and intensity of music elements are coded as colors, lattices, graphs, and shape sizes, respectively. Based on the 12-tone equal temperament, we adopt the Itten color wheel and the natural color system with 12 primary colors to retain colors accurately when the notation is reproduced. Hopefully, beginners can use the novel colored music scores to read and to play music quickly, accurately, and confidently.

Exploring microtonal performance of “…Plainte…” by Klaus Huber for viola d’amore in third-tone tuning

Contemporary microtonal music poses considerable challenges to professional performers. Here, we investigate five renditions of Klaus Huber’s “…Plainte…” for viola d’amore in third-tone tuning with respect to pitch accuracy at the acoustic level. The analysis also includes examining tones on open strings, which were played on the composer’s instrument. Significant differences between intended and realized fundamental frequency values were observed. The tuning of instruments deviated 14 to 42.5 cents from theoretical values. Influences of sound production on fundamental frequencies in performances were analysed, supplemented by additional analyses using the composer’s instrument. In each performance, the range of fundamental frequency deviations of tones played on open strings was considerably narrower than when tones were played with fingerboard stopping. In latter case, the range of deviations exceeded a third-tone interval (67 cents) in all performances and was larger than reported on tonal music in twelve-tone equal temperament. These results suggest specific limitations in the realization of third-tone music as performed on a fretless string instrument by highly-trained professional performers. The fact that three of the analysed recordings are commercially available further suggests that rather large deviations of fundamental frequencies from values described by music theory are accepted aesthetically.

A System for Tuning Instruments Using Recorded Music Instead of Theory-Based Frequency Presets

Musical instrument tuners are devices that help musicians to adjust their instruments such that the played notes have the desired fundamental frequencies. In a conventional tuner, the reference tuning frequencies are preset, where the presets are obtained from tuning (musical scale) theory, such as twelve-tone equal temperament, or are user-specified temperaments. For many kinds of music in oral traditions, especially nonwestern music, widely accepted theoretical presets for tuning frequencies are not available because of the use of non-standard tunings. For such contexts, the “reference” is a master musician or a recording of a master musician. In this article, a tuning method and technology are presented that help the musician to tune the instrument according to a given (user-provided) recording. The method makes use of simultaneous audio and visual feedback during the tuning process, in which novel approaches are used for both modalities. For audio feedback, loopable stable frames, obtained automatically from the recording, are looped and played continuously. For visual feedback, a superimposed plot of the auto-difference functions is displayed instead of the conventional tuner's approach of detecting frequencies and displaying the amount of frequency difference between the input and the reference.

A MIDI Sequencer That Widens Access to the Compositional Possibilities of Novel Tunings

March 01 2012 A MIDI Sequencer That Widens Access to the Compositional Possibilities of Novel Tunings Anthony Prechtl, Anthony Prechtl *Music, Mind & Technology Department of Music PL 35(M) 40014 University of Jyväskylä, Finland aprechtl@gmail.com Search for other works by this author on: This Site Google Scholar Andrew J. Milne, Andrew J. Milne *Music, Mind & Technology Department of Music PL 35(M) 40014 University of Jyväskylä, Finland aprechtl@gmail.com Search for other works by this author on: This Site Google Scholar Simon Holland, Simon Holland *Music, Mind & Technology Department of Music PL 35(M) 40014 University of Jyväskylä, Finland aprechtl@gmail.com Search for other works by this author on: This Site Google Scholar Robin Laney, Robin Laney *Music, Mind & Technology Department of Music PL 35(M) 40014 University of Jyväskylä, Finland aprechtl@gmail.com Search for other works by this author on: This Site Google Scholar David B. Sharp David B. Sharp *Music, Mind & Technology Department of Music PL 35(M) 40014 University of Jyväskylä, Finland aprechtl@gmail.com†Computing Department The Open University Walton Hall, Milton Keynes Buckinghamshire, MK7 6AA UK andymilne@tonalcentre.orgs.holland@open.ac.ukr.c.laney@open.ac.uk**Department of Design, Development, Environment and Materials The Open University Walton Hall, Milton Keynes Buckinghamshire, MK7 6AA UK d.sharp@open.ac.uk Search for other works by this author on: This Site Google Scholar Author and Article Information Anthony Prechtl *Music, Mind & Technology Department of Music PL 35(M) 40014 University of Jyväskylä, Finland aprechtl@gmail.com Andrew J. Milne *Music, Mind & Technology Department of Music PL 35(M) 40014 University of Jyväskylä, Finland aprechtl@gmail.com Simon Holland *Music, Mind & Technology Department of Music PL 35(M) 40014 University of Jyväskylä, Finland aprechtl@gmail.com Robin Laney *Music, Mind & Technology Department of Music PL 35(M) 40014 University of Jyväskylä, Finland aprechtl@gmail.com David B. Sharp *Music, Mind & Technology Department of Music PL 35(M) 40014 University of Jyväskylä, Finland aprechtl@gmail.com†Computing Department The Open University Walton Hall, Milton Keynes Buckinghamshire, MK7 6AA UK andymilne@tonalcentre.orgs.holland@open.ac.ukr.c.laney@open.ac.uk**Department of Design, Development, Environment and Materials The Open University Walton Hall, Milton Keynes Buckinghamshire, MK7 6AA UK d.sharp@open.ac.uk Online ISSN: 1531-5169 Print ISSN: 0148-9267 © 2012 Massachusetts Institute of Technology.2012 Computer Music Journal (2012) 36 (1): 42–54. https://doi.org/10.1162/COMJ_a_00104 Cite Icon Cite Permissions Share Icon Share Facebook Twitter LinkedIn MailTo Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Search Site Citation Anthony Prechtl, Andrew J. Milne, Simon Holland, Robin Laney, David B. Sharp; A MIDI Sequencer That Widens Access to the Compositional Possibilities of Novel Tunings. Computer Music Journal 2012; 36 (1): 42–54. doi: https://doi.org/10.1162/COMJ_a_00104 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAll JournalsComputer Music Journal Search Advanced Search This content is only available as a PDF. © 2012 Massachusetts Institute of Technology.2012 Article PDF first page preview Close Modal You do not currently have access to this content.

편종의 음색 분석 연구

편종의 소리합성을 위한 연구의 첫 단계로 FFT 분석을 통하여 그 음색을 분석하였다. 현대 편종 음계의 음고는 서양의 12평균율의 음고와 큰 차이가 없었다. 편종의 음색은 시간의 흐름에 따라 변화한다. 타격시 모든 주파수 대역에 걸쳐 생성된 노이즈들은 단시간에 사라지고, 고유 배음만이 남는다. 그 후 높은 주파수 성분의 배음부터 차례로 소멸되고 여음부에는 두개의 배음만이 남는데, 이 때 남은 배음의 주파수에 의하여 맥놀이 현상인 발견된다. 본 연구에서 사용된 편종은 비교적 최근에 제작된 편종으로 추후 고대 편종의 음색에 관한 연구가 병행된다면 편종 원형의 소리 합성이 가능할 것이다. As the first step in research for sound synthesis of Pyungjong, timbre was analyzed by FFT analysis. There was no significant difference in the pitch of scale between modern Pyungjong and western twelve-tone equal temperament scale. The timbre of Pyunjong chagnes over time. When striking Pyunjong, the noises to be generated across all frequencies will disappear in a short period of time, and a unique partial only is left. Then it disappear from the partial of high frequency components in order, and two partials remain at release time. At this time, the beating is occurring by the frequency of the remaining partial. Pyungjong used in this study is relatively a recently manufactured Pyungjong, and for original sound synthesis of Pyungjong, a study of ancient Pyungjong's timbre should be additionally conducted later.