Microtonal music

Microtonal music or microtonality is the use in music of microtones—intervals smaller than a semitone, also called "microintervals". It may also be extended to include any music using intervals not found in the customary Western tuning of twelve equal intervals per octave. In other words, a microtone may be thought of as a note that falls between the keys of a piano tuned in equal temperament. In Revising the musical equal temperament, Haye Hinrichsen defines equal temperament as “the frequency ratios of all intervals are invariant under transposition (translational shifts along the keyboard), i.e., to be constant. The standard twelve-tone equal temperament (ET), which was originally invented in ancient China and rediscovered in Europe in the 18th century, is determined by two additional conditions. Firstly the octave is divided into twelve semitones. Secondly the octave, the most fundamental of all intervals, is postulated to be pure (beatless), as described by the frequency ratio 2:1.”[2]

A music staff with sharps
Composer Charles Ives chose the chord above as a good candidate for a "fundamental" chord in the quarter tone scale, akin not to the tonic but to the major chord of traditional tonality[1]
Two examples of an Ives fundamental chord with quarter tones

TerminologyEdit

MicrotoneEdit

 
Quarter tone accidentals residing outside the Western semitone:
quarter tone sharp, sharp, three quarter tones sharp;
quarter tone flat, flat, (two variants of) three quarter tones flat

Microtonal music can refer to any music containing microtones. The words "microtone" and "microtonal" were coined before 1912 by Maud MacCarthy Mann in order to avoid the misnomer "quarter tone" when speaking of the srutis of Indian music.[3] Prior to this time the term "quarter tone" was used, confusingly, not only for an interval actually half the size of a semitone, but also for all intervals (considerably) smaller than a semitone.[4][5] It may have been even slightly earlier, perhaps as early as 1895, that the Mexican composer Julián Carrillo, writing in Spanish or French, coined the terms microtono/micro-ton and microtonalismo/micro-tonalité.[6]

In French, the usual term is the somewhat more self-explanatory micro-intervalle, and French sources give the equivalent German and English terms as Mikrointervall (or Kleinintervall) and micro interval (or microtone), respectively.[7][8][9][10] "Microinterval" is a frequent alternative in English, especially in translations of writings by French authors and in discussion of music by French composers.[11][12][13] In English, the two terms "microtone" and "microinterval" are synonymous.[14] The English analogue of the related French term, micro-intervalité, however, is rare or nonexistent, normally being translated as "microtonality"; in French, the terms micro-ton, microtonal (or micro-tonal), and microtonalité are also sometimes used, occasionally mixed in the same passage with micro-intervale and micro-intervalité.[6][15][16][17]

Ezra Sims, in the article "Microtone" in the second edition of the Harvard Dictionary of Music defines "microtone" as "an interval smaller than a semitone",[18] which corresponds with Aristoxenus's use of the term diesis.[19] However, the unsigned article "Comma, Schisma" in the same reference source calls comma, schisma and diaschisma "microintervals" but not "microtones",[20] and in the fourth edition of the same reference (which retains Sims's article on "Microtone") a new "Comma, Schisma" article by André Barbera calls them simply "intervals".[21] In the second edition of The New Grove Dictionary of Music and Musicians, Paul Griffiths, Mark Lindley, and Ioannis Zannos define "microtone" as a musical rather than an acoustical entity: "any musical interval or difference of pitch distinctly smaller than a semitone", including "the tiny enharmonic melodic intervals of ancient Greece, the several divisions of the octave into more than 12 parts, and various discrepancies among the intervals of just intonation or between a sharp and its enharmonically paired flat in various forms of mean-tone temperament", as well as the Indian sruti, and small intervals used in Byzantine chant, Arabic music theory from the 10th century onward, and similarly for Persian traditional music and Turkish music and various other Near Eastern musical traditions,[22] but do not actually name the "mathematical" terms schisma, comma, and diaschisma.

"Microtone" is also sometimes used to refer to individual notes, "microtonal pitches" added to and distinct from the familiar twelve notes of the chromatic scale,[23] as "enharmonic microtones",[24] for example.

In English the word "microtonality" is mentioned in 1946 by Rudi Blesh who related it to microtonal inflexions of the so-called "blues scales".[25] In Court B. Cutting's 2019 Microtonal Analysis of “Blues Notes” and the Blues Scale, he states that academic studies of the early blues concur that its pitch scale has within it three microtonal “blue notes” not found in 12 tone equal temperament intonation.[26] It was used still earlier by W. McNaught with reference to developments in "modernism" in a 1939 record review of the Columbia History of Music, Vol. 5.[27] In German the term Mikrotonalität came into use at least by 1958,[28][29] though "Mikrointervall" is still common today in contexts where very small intervals of early European tradition (diesis, comma, etc.) are described, as e.g. in the new Geschichte der Musiktheorie[30] while "Mikroton" seems to prevail in discussions of the avant-garde music and music of Eastern traditions.[citation needed] The term "microinterval" is used alongside "microtone" by American musicologist Margo Schulter in her articles on medieval music.[31][32]

MicrotonalEdit

The term "microtonal music" usually refers to music containing very small intervals but can include any tuning that differs from Western twelve-tone equal temperament. Traditional Indian systems of 22 śruti; Indonesian gamelan music; Thai, Burmese, and African music, and music using just intonation, meantone temperament or other alternative tunings may be considered microtonal.[33][22] Microtonal variation of intervals is standard practice in the African-American musical forms of spirituals, blues and jazz.[34]

Many microtonal equal divisions of the octave have been proposed, usually (but not always) in order to achieve approximation to the intervals of just intonation.[33][22]

Terminology other than "microtonal" has been used or proposed by some theorists and composers. In 1914, A. H. Fox Strangways objected that "'heterotone' would be a better name for śruti than the usual translation 'microtone'".[35] Modern Indian researchers yet write: "microtonal intervals called shrutis".[36] In Germany, Austria, and Czechoslovakia in the 1910s and 1920s the usual term continued to be Viertelton-Musik (quarter tone music[37][page needed]), and the type of intervallic structure found in such music was called the Vierteltonsystem,[38][39] which was (in the mentioned region) regarded as the main term for referring to music with microintervals, though as early as 1908 Georg Capellan had qualified his use of "quarter tone" with the alternative term "Bruchtonstufen (Viertel- und Dritteltöne)" (fractional degrees (quarter and third tones)).[40] Despite the inclusion of other fractions of a whole tone, this music continued to be described under the heading "Vierteltonmusik" until at least the 1990s, for example in the twelfth edition of the Riemann Musiklexikon,[41] and in the second edition of the popular Brockhaus Riemann Musiklexikon.[42]

Ivan Wyschnegradsky used the term ultra-chromatic for intervals smaller than the semitone and infra-chromatic for intervals larger than the semitone;[43] this same term has been used since 1934 by ethnomusicologist Victor Belaiev (Belyaev) in his studies of Azerbaijan and Turkish traditional music.[44][45][46] A similar term, subchromatic, has been used by theorist Marek Žabka.[47] Ivor Darreg proposed[when?][citation needed] the term xenharmonic; see xenharmonic music. The Austrian composer Franz Richter Herf and the music theorist Rolf Maedel, Herf's colleague at the Salzburg Mozarteum, preferred using the Greek word ekmelic when referring to "all the pitches lying outside the traditional twelve-tone system".[48] Some authors in Russia[49][50][51][52][53][54] and some musicology dissertations[55][56][57][58][59][60] disseminate the term микрохроматика (microchromatics), coined in the 1970s by Yuri Kholopov,[61] to describe a kind of 'intervallic genus' (интервальный род) for all possible microtonal structures, both ancient (as enharmonic genus—γένος ἐναρμόνιον—of Greeks) and modern (as quarter tone scales of Alois Haba); this generalization term allowed also to avoid derivatives such as микротональность (microtonality, which could be understood in Russian as a sub-tonality, which is subordinate to the dominating tonality, especially in the context of European music of the 19th century) and микротоника (microtonic, "a barely perceptible tonic"; see a clarification in Kholopov [2000][62]). Another Russian authors use more international adjective 'microtonal' and rendered it in Russian as 'микротоновый', but not 'microtonality' ('микротональность').[63][64][65][66] However, the terms 'микротональность'[67] and 'микротоника'[68] are also used. Some authors writing in French have adopted the term "micro-intervallique" to describe such music.[69][70] Italian musicologist Luca Conti dedicated two his monographs to microtonalismo,[71][72] which is the usual term in Italian, and also in Spanish (e.g., as found in the title of Rué [2000][73]). The analogous English form, "microtonalism", is also found occasionally instead of "microtonality", e.g., "At the time when serialism and neoclassicism were still incipient a third movement emerged: microtonalism".[74]

The term "macrotonal" has been used for intervals wider than twelve-tone equal temperament,[75][permanent dead link][better source needed] or where there are "fewer than twelve notes per octave", though "this term is not very satisfactory and is used only because there seems to be no other".[76] The term "macrotonal" has also been used for musical form.[77]

Examples of this can be found in various places, ranging from Claude Debussy's impressionistic harmonies to Aaron Copland's chords of stacked fifths, to John Luther Adams' Clouds of Forgetting, Clouds of Unknowing (1995), which gradually expands stacked-interval chords ranging from minor 2nds to major 7thsm. Louis Andriessen's De Staat (1972–1976) contains a number of "augmented" modes that are based on Greek scales but are asymmetrical to the octave.[78]

HistoryEdit

Greek Dorian mode (enharmonic genus) on E, divided into two tetrachords.

The Hellenic civilizations of ancient Greece left fragmentary records of their music, such as the Delphic Hymns. The ancient Greeks approached the creation of different musical intervals and modes by dividing and combining tetrachords, recognizing three genera of tetrachords: the enharmonic, the chromatic, and the diatonic. Ancient Greek intervals were of many different sizes, including microtones. The enharmonic genus in particular featured intervals of a distinctly "microtonal" nature, which were sometimes smaller than 50 cents, less than half of the contemporary Western semitone of 100 cents. In the ancient Greek enharmonic genus, the tetrachord contained a semitone of varying sizes (approximately 100 cents) divided into two equal intervals called dieses (single "diesis", δίεσις); in conjunction with a larger interval of roughly 400 cents, these intervals comprised the perfect fourth (approximately 498 cents, or the ratio of 4/3 in just intonation).[79] Theoretics usually described several diatonic and chromatic genera (some as chroai, "coloration" of one specific intervallic type), but the enarmonic genus was always the only one (argumented as one with the smallest intervals possible).

 
Vicentino's archicembalo in cents

Guillaume Costeley's "Chromatic Chanson", "Seigneur Dieu ta pitié" of 1558 used 1/3 comma meantone and explored the full compass of 19 pitches in the octave.[80]

The Italian Renaissance composer and theorist Nicola Vicentino (1511–1576) worked with microtonal intervals and built a keyboard with 36 keys to the octave known as the archicembalo. While theoretically an interpretation of ancient Greek tetrachordal theory, in effect Vicentino presented a circulating system of quarter-comma meantone, maintaining major thirds tuned in just intonation in all keys.[81]

In 1760 the French flautist Charles de Lusse [de] published a treatise, L'Art de la flute traversiere, all surviving copies of which conclude with a composition (possibly added a year or two after the actual publication of the volume) incorporating several quarter tones, titled Air à la grecque, accompanied by explanatory notes tying it to the realization of the Greek enharmonic genus and a chart of quarter tone fingerings for the entire range of the one-keyed flute. Shortly afterward, in a letter published in the Mercure de France in September 1764, the celebrated flautist Pierre-Gabriel Buffardin mentioned this piece and expressed an interest in quarter tones for the flute.[82][83]

Jacques Fromental Halévy composed a cantata "Prométhée enchaîné" for a solo voice, choir and orchestra (premiered in 1849), where in one movement (Choeur des Océanides) he used quarter tones, to imitate the enharmonic genus of Greeks.

In the 1910s and 1920s, quarter tones (24 equal pitches per octave) received attention from such composers as Charles Ives, Julián Carrillo, Alois Hába, Ivan Wyschnegradsky, and Mildred Couper.

Alexander John Ellis, who in the 1880s produced a translation of Hermann Helmholtz's On the Sensations of Tone, proposed an elaborate set of exotic just intonation tunings and non-harmonic tunings.[84] Ellis also studied the tunings of non-Western cultures and, in a report to the Royal Society, stated that they used neither equal divisions of the octave nor just intonation intervals.[85] Ellis inspired Harry Partch immensely.[86]

During the Exposition Universelle of 1889, Claude Debussy heard a Balinese gamelan performance and was exposed to non-Western tunings and rhythms. Some scholars have ascribed Debussy's subsequent innovative use of the whole-tone (six equal pitches per octave) tuning in such compositions as the Fantaisie for piano and orchestra and the Toccata from the suite Pour le piano to his exposure to the Balinese gamelan at the Paris exposition,[87] and have asserted his rebellion at this time "against the rule of equal temperament" and that the gamelan gave him "the confidence to embark (after the 1900 world exhibition) on his fully characteristic mature piano works, with their many bell- and gong-like sonorities and brilliant exploitation of the piano's natural resonance".[88] Still others have argued that Debussy's works like L'isle joyeuse, La cathédrale engloutie, Prélude à l'après-midi d'un faune, La mer, Pagodes, Danseuses de Delphes, and Cloches à travers les feuilles are marked by a more basic interest in the microtonal intervals found between the higher members of the overtone series, under the influence of Helmholtz's writings.[89] Emil Berliner's introduction of the phonograph in the 1890s allowed much non-Western music to be recorded and heard by Western composers, further spurring the use of non-12-equal tunings.[citation needed]

Major microtonal composers of the 1920s and 1930s include Alois Hába (quarter tones, or 24 equal pitches per octave, and sixth tones), Julián Carrillo (24 equal, 36, 48, 60, 72, and 96 equal pitches to the octave embodied in a series of specially custom-built pianos), Ivan Wyschnegradsky (third tones, quarter tones, sixth tones and twelfth tones, non octaving scales) and the early works of Harry Partch (just intonation using frequencies at ratios of prime integers 3, 5, 7, and 11, their powers, and products of those numbers, from a central frequency of G-196).[90]

Prominent microtonal composers or researchers of the 1940s and 1950s include Adriaan Daniel Fokker (31 equal tones per octave), Partch (continuing to build his handcrafted orchestra of microtonal just intonation instruments), and Eivind Groven.

Digital synthesizers from the Yamaha TX81Z (1987) on and inexpensive software synthesizers have contributed to the ease and popularity of exploring microtonal music.

Microtonality in electronic musicEdit

Electronic music facilitates the use of any kind of microtonal tuning, and sidesteps the need to develop new notational systems.[22] In 1954, Karlheinz Stockhausen built his electronic Studie II on an 81-step scale starting from 100 Hz with the interval of 51/25 between steps,[91] and in Gesang der Jünglinge (1955–56) he used various scales, ranging from seven up to sixty equal divisions of the octave.[92] In 1955, Ernst Krenek used 13 equal-tempered intervals per octave in his Whitsun oratorio, Spiritus intelligentiae, sanctus.[22]

In 1979–80 Easley Blackwood composed a set of Twelve Microtonal Etudes for Electronic Music Media, a cycle that explores all of the equal temperaments from 13 notes to the octave through 24 notes to the octave, including 15-ET and 19-ET.[93][full citation needed][page needed] "The project," he wrote, "was to explore the tonal and modal behavior of all [of these] equal tunings..., devise a notation for each tuning, and write a composition in each tuning to illustrate good chord progressions and the practical application of the notation".[94][full citation needed]

In 1986, Wendy Carlos experimented with many microtonal systems including just intonation, using alternate tuning scales she invented for the album Beauty In the Beast. "This whole formal discovery came a few weeks after I had completed the album, Beauty in the Beast, which is wholly in new tunings and timbres".[95]

In 2016, electronic music composed with arbitrary microtonal scales was explored on the album Radionics Radio: An Album of Musical Radionic Thought Frequencies by British composer Daniel Wilson, who derived his compositions' tunings from frequency-runs submitted by users of a custom-built web application replicating radionics-based electronic soundmaking equipment used by Oxford's De La Warr Laboratories in the late 1940s, thereby supposedly embodying thoughts and concepts within the tunings.[96]

Limitations of some synthesizersEdit

The General MIDI Specification does not directly support microtonal music, because each note-on and note-off message only represents one chromatic tone. However, microtonal scales can be emulated using pitch bending, such as in LilyPond's implementation.[97]

Although some synthesizers allow the creation of customized microtonal scales, this solution does not allow compositions to be transposed. For example, if each B note is raised one quarter tone, then the "raised 7th" would only affect a C major scale.

Microtonality in rock musicEdit

A form of microtone known as the blue note is an integral part of rock music and one of its predecessors, the blues. The blue notes, located on the third, fifth, and seventh notes of a diatonic major scale, are flattened by a variable microtone.[98] Joe Monzo has made a microtonal analysis of the song "Drunken Hearted Man",[99] written and recorded by the delta blues musician Robert Johnson.[100]

Musicians such as Jon Catler have incorporated microtonal guitars like 31-tone equal tempered guitar and a 62-tone just intonation guitar in blues and jazz rock music.[101]

English rock band Radiohead has used microtonal string arrangements in its music, such as on "How to Disappear Completely" from the album Kid A.[100]

American band Secret Chiefs 3 has been making its own custom "microtonal" instruments since the mid 1990s. The proprietary tuning system they use in their Ishraqiyun aspect is ratio-based, not equal temperament. The band's leader Trey Spruance, also of Mr. Bungle challenges the terminology of "microtonality" as a development that instead of liberating tonal sensibility to a universe of diverse possibilities, both new and historical, instead mainly serves to reinforce the idea that the universal standard for "tone" is the (western) semitone.[102]

Australian band King Gizzard and the Lizard Wizard utilises microtonal instruments, including custom microtonal guitars modified to play in 24-TET tuning. Tracks with these instruments appear on their 2017 albums Flying Microtonal Banana,[103] & Gumboot Soup, their 2020 album K.G, and their 2021 album L.W.[104]

American band Dollshot used quarter tones and other microtonal intervals in their album Lalande.[105]

American instrumental trio Consider the Source employs microtonal instruments in their music.[citation needed]

In the WestEdit

Eastern microtonal pioneersEdit

* Naseeruddin Saami (Karachi, Pakistan, 1950, microtone practitioner)

Western microtonal pioneersEdit

Modern Western microtonal composersEdit

Western microtonal researchersEdit

See alsoEdit

ReferencesEdit

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Further readingEdit

  • Adèr, Lidiâ Olegovna [Адэр, Лидия Олеговна]. 2011a. "Микротоновая идея: Истоки и предпосылки" [The Concept of Microtonality: Its Origin and Background]. Научный журнал Санкт-Петербургской консерватории [Opera musicologica: Naučnyj žurnal Sankt-Peterburgskoj konservatorii] 3–4, nos. 8–9:114–134.
  • Adèr, Lidiâ Olegovna [Адэр, Лидия Олеговна]. 2011b. "Микротоновый инструментарий—первые шаги от утопии к практике" [Microtonal Instruments: The First Steps from Utopia to Practice]. In Временник Зубовского института: Инструментализм в истории культуры [Instrumentalism in the history of culture], edited by Evgenia Vladimirovna Hazdan, 52–65. Vremennik Zubovskogo instituta 7. St. Petersburg: Rossijskij Institut Istorii Iskusstv.
  • Aron, Pietro. 1523. Thoscanello de la musica. Venice: Bernardino et Mattheo de Vitali. Facsimile edition, Monuments of music and music literature in facsimile: Second series, Music literature 69. New York: Broude Brothers, 1969. Second edition, as Toscanello in musica... nuovamente stampato con laggiunta da lui fatta et con diligentia corretto, Venice: Bernardino & Matheo de Vitali, 1529. Facsimile reprint, Bibliotheca musica Bononiensis, sezione 2., n. 10. Bologna: Forni Editori, 1969. Online edition of the 1529 text (in Italian). Third edition, as Toscanello in musica, Venice: Marchio Stessa, 1539. Facsimile edition, edited by Georg Frey. Kassel: Bärenreiter, 1970. Fourth edition, Venice, 1562. English edition, as Toscanello in music, translated by Peter Bergquist. 3 vols. Colorado College Music Press Translations, no. 4. Colorado Springs: Colorado College Music Press, 1970.
  • Barbieri, Patrizio. 1989. "An Unknown 15th-Century French Manuscript on Organ Building and Tuning". The Organ Yearbook: A Journal for the Players & Historians of Keyboard Instruments 20.
  • Barbieri, Patrizio. 2002. "The Evolution of Open-Chain Enharmonic Keyboards c1480–1650". In Chromatische und enharmonische Musik und Musikinstrumente des 16. und 17. Jahrhunderts/Chromatic and Enharmonic Music and Musical Instruments in the 16th and 17th Centuries. Schweizer Jahrbuch für Musikwissenschaft/Annales suisses de musicologie/Annuario svizzero di musicologia 22, edited by Joseph Willimann. Bern: Verlag Peter Lang AG. ISBN 3-03910-088-2.
  • Barbieri, Patrizio. 2003. "Temperaments, Historical". In Piano: An Encyclopedia, second edition, edited by Robert Palmieri and Margaret W. Palmieri,[page needed]. New York: Routledge.
  • Barbieri, Patrizio, Alessandro Barca, and conte Giordano Riccati. 1987. Acustica accordatura e temperamento nell'illuminismo Veneto. Pubblicazioni del Corso superiore di paleografia e semiografia musicale dall'umanesimo al barocco, Serie I: Studi e testi 5; Pubblicazioni del Corso superiore di paleografia e semiografia musicale dall'umanesimo al barocco, Documenti 2. Rome: Edizioni Torre d'Orfeo.
  • Barbieri, Patrizio, and Lindoro Massimo del Duca. 2001. "Late-Renaissance Quarter-tone Compositions (1555–1618): The Performance of the ETS-31 with a DSP System". In Musical Sounds from Past Millennia: Proceedings of the International Symposium on Musical Acoustics 2001, edited by Diego L. González, Domenico Stanzial, and Davide Bonsi. 2 vols. Venice: Fondazione Giorgio Cini.
  • Barlow, Clarence (ed.). 2001. "The Ratio Book." (Documentation of the Ratio Symposium Royal Conservatory The Hague 14–16 December 1992). Feedback Papers 43.
  • Blackwood, Easley. 1985. The Structure of Recognizable Diatonic Tunings. Princeton: Princeton University Press. ISBN 0-691-09129-3.
  • Blackwood, Easley. 1991. "Modes and Chord Progressions in Equal Tunings". Perspectives of New Music 29, no. 2 (Summer): 166–200.
  • Burns, Edward M. 1999. "Intervals, Scales, and Tuning." In The Psychology of Music, second edition, ed. Diana Deutsch. 215–264. San Diego: Academic Press. ISBN 0-12-213564-4.
  • Carr, Vanessa. 2008. "These Are Ghost Punks". Vanessa Carr's website (29 February). (Accessed 2 April 2009)
  • Colonna, Fabio. 1618. La sambuca lincea, overo Dell'istromento musico perfetto. Naples: C. Vitale. Facsimile reprint of a copy containing manuscript critical annotations by Scipione Stella (1618–1624), with an introduction by Patrizio Barbieri. Musurgiana 24. Lucca, Italy: Libreria Musicale Italiana, 1991.
  • Daniels, Arthur Michael. 1965. "Microtonality and Mean-Tone Temperament in the Harmonic System of Francisco Salinas". Journal of Music Theory 9, no. 1 (Spring): 2–51.
  • Dumbrill, Richard J. 2000. The Musicology and Organology of the Ancient Near East, second edition. London: Tadema Press. ISBN 0-9533633-0-9.
  • Fink, Robert. 1988. "The Oldest Song in the World". Archaeologia Musicalis 2, no. 2:98–100.
  • Fritsch, Johannes G. 2007. "Allgemeine Harmonik, Tonsysteme, Mikrotonalität: Ein geschichtlicher Überblick". In Orientierungen: Wege im Pluralismus der Gegenwartsmusik, edited by Jörn Peter Hiekel, 107–122. Veröffentlichungen des Instituts für Neue Musik und Musikerziehung Darmstadt 47. Mainz: Schott Musik International. ISBN 978-3-7957-1837-4.
  • Gilmore, Bob. 1998. Harry Partch: A Biography. New Haven: Yale University Press. ISBN 0-300-06521-3.
  • Haas, Georg Friedrich. 2007. "Mikrotonalität und spektrale Musik seit 1980". In Orientierungen: Wege im Pluralismus der Gegenwartsmusik, edited by Jörn Peter Hiekel, 123–129. Veröffentlichungen des Instituts für Neue Musik und Musikerziehung Darmstadt 47. Mainz: Schott Musik International. ISBN 978-3-7957-1837-4.
  • Hába, Alois. 1927. Neue Harmonielehre des diatonischen, chromatischen Viertel-, Drittel-, Sechstel- und Zwölftel-tonsystems. Leipzig: Kistner & Siegel.
  • Johnston, Ben. 2006. 'Maximum Clarity' and other writings on music, ed. B. Gilmore. Urbana and Chicago: University of Illinois Press.
  • Kotschy, Johannes. 2008. "Mikrotonalität: Eine Zeiterscheinung?" Österreichische Musikzeitschrift 63, no. 7 (July): 8–15.
  • Landman, Yuri. [2008]. "Third Bridge Helix: From Experimental Punk to Ancient Chinese Music and the Universal Physical Laws of Consonance". Perfect Sound Forever (online music magazine) (accessed 6 December 2008).
  • Landman, Yuri. n.d. "Yuichi Onoue's Kaisatsuko" on Hypercustom.com (accessed 31 March 2009).
  • Leedy, Douglas. 1991. "A Venerable Temperament Rediscovered". Perspectives of New Music 29, no. 2 (Summer): 202–211.
  • Lindley, Mark. 2001b. "Temperaments". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan Publishers.
  • Mandelbaum, M. Joel. 1961. "Multiple Division Of the Octave and the Tonal Resources of the 19 Tone Temperament". Ph.D. thesis. Bloomington: Indiana University.
  • Mosch, Ulrich. 2008. "Ultrachromatik und Mikrotonalität: Hans Zenders Grundlegung einer neuen Harmonik". In Hans Zender: Vielstimmig in sich, edited by Werner Grünzweig, Anouk Jeschke, and Jörn Peter Hiekel, 61–76. Archive zur Musik des 20. Jahrhunderts, No. 12. Hofheim:Wolke. ISBN 978-3-936000-25-2.
  • Noyze, Dave, and Richard D. James.. 2014. "Aphex Twin Syrobonkers! Interview: Part 2". Noyzelab Blogspot.com.au (Monday, 10 November)
  • Stahnke, Manfred. 2010. "About Backyards and Limbos: Microtonality Revisited". In Concepts, Experiments and Fieldwork: Studies in Systematic Musicology and Ethnomusicology, edited by Rolf Bader, Christiane Neuhaus, and Ulrich Morgenstern, with a prefaceby Achim Reichel, 297–314. Frankfurt am Main and New York: Peter Lang. ISBN 978-3-631-58902-1.
  • Vitale, Raoul. 1982. "La Musique suméro-accadienne: gamme et notation musicale". Ugarit-Forschungen 14: 241–263.
  • Werntz, Julia. 2001. "Adding Pitches: Some New Thoughts, Ten Years after Perspectives of New Music's 'Forum: Microtonality Today'". Perspectives of New Music 39, no. 2 (Summer): 159–210.
  • Wood, James. 1986. "Microtonality: Aesthetics and Practicality". The Musical Times 127, no. 1719 (June): 328–330.
  • Wyschnegradsky, Ivan. 1937. "La musique à quarts de ton et sa réalisation pratique". La Revue Musicale no. 171:26–33.
  • Zweifel, Paul. 1996. "Generalized Diatonic and Pentatonic Scales: A Group-Theoretic Approach". Perspectives of New Music 34, no. 1 (Winter): 140–161.

External linksEdit