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Microtonal music

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Microtonal music

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 (Boatwright 1971, 8–9).     or About this sound   

Microtonal music or microtonality is the use in music of microtones—intervals smaller than a semitone, which are 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.

Terminology

Microtone

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, therefore it is important to comprehend what a "microtone" is. 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 (Mann 1912, 44). 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 (
;
). 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 (Donval 2006, 119). 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 (
;
;
;
. "Microinterval" is a frequent alternative in English, especially in translations of writings by French authors and in discussion of music by French composers (
;
;
). In English, the two terms "microtone" and "microinterval" are synonymous (Maclagan 2009, 109). The English analogue of the related French term, micro-intervallité, 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-intervalle and micro-intervalité (
;
;
). Ezra Sims, in the article "Microtone" in the second edition of the Harvard Dictionary of Music (Apel 1974, 527) defines "microtone" as "an interval smaller than a semitone", which corresponds with Aristoxenus's use of the term diesis (Richter 2001). However, the unsigned article "Comma, Schisma" in the same reference source calls comma, schisma and diaschisma "microintervals" but not "microtones" (Apel 1974, 188), 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" (Barbera 2003). 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 various other Near Eastern musical traditions (
), 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 distict from the familiar twelve notes of the chromatic scale (Von Gunden 1986, 59).

In English the word "microtonality" is mentioned in 1946 by Rudi Blesh who related it to microtonal inflexions of the so called "blues scales" (Blesh 1946, 234). 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 (McNaught 1939, 102). In German the term Mikrotonalität came into use at least by 1958 (
;
), 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 (Zaminer 2006, 94) while "Mikroton" seems to prevail in discussions of the avant-garde music and music of Eastern traditions. The term "microinterval" is used alongside "microtone" by American musicologist Margo Schulter in her articles on medieval music (Schulter 1998; Schulter 2001).

Microtonal

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 (
;
). Microtonal variation of intervals is standard practice in the African-American musical forms of spirituals, blues and jazz (Cook and Pople 2004, 124–26). 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 (
;
). 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'" (Strangways 1914, 127n). In Germany, Austria, and Czechoslovakia in the 1910s and 1920s the usual term continued to be Viertelton-Musik (quarter-tone music;
), and the type of intervallic structure found in such music was called the Vierteltonsystem (
,
), 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)) (Capellen 1908, 184). 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 (Riemann 1967, 1032–33), and in the second edition of the popular Brockhaus Riemann Musiklexikon (Dahlhaus, Eggebrecht, and Oehl 1995, 304). Ivan Wyschnegradsky used the term ultra-chromatic for intervals smaller than the semitone and infra-chromatic for intervals larger than the semitone (Wyschnegradsky 1972, 84–87); this same term has been used since 1934 by ethnomusicologist Victor Belaiev (Belyaev) in his studies of Azerbaijan and Turkish traditional music (
;
). A similar term, subchromatic, has been used recently by theorist Marek Žabka (Žabka 2014). Ivor Darreg proposed (where and when?) the term xenharmonic. (See xenharmonic music). The Austrian composer Franz Richter Herf and his colleague at the Salzburg Mozarteum, the music theorist Rolf Maedel, preferred using the Greek word ekmelic when referring to "all the pitches lying outside the traditional twelve-tone system" (Hesse 1991, 216–17). Most modern Russian music dictionaries (
;
;
), the universal Great Russian encyclopedia (Lebedev 2012), textbooks (e.g.,
;
;
), and musicology dissertations (
;
;
;
;
, more references can be located here) use the term микрохроматика (microchromatics), coined in the 1970s by Yuri Kholopov (Kholopov 1976), 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)). A few Russian authors borrowed from English the adjective 'microtonal' and rendered it in Russian as 'микротоновый', but not 'microtonality' ('микротональность') (
;
). Some authors writing in French have adopted the term "micro-intervallique" to describe such music (e.g.,
;
). Italian musicologist Luca Conti dedicated two his monographs to microtonalismo (
;
), which is the usual term in Italian, and also in Spanish (e.g., as found in the title of
). 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" (Chou 1971, 218).

History

Greek Dorian mode (Play  

The Hellenic civilizations of ancient Greece left fragmentary records of their music—e.g. 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) (West 1992, 160–72). 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).

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 (Lindley 2001a).

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 (Barbour 1951, 117–18).

In 1760 the French flautist [Charles?] De Lusse 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 (
;
).

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 quartertones, 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 (Helmholtz 1885, 514–27). Ellis also studied the tunings of non-Western cultures and, in a report to the Royal Society, stated that they did not use either equal divisions of the octave or just intonation intervals (Ellis 1884). Ellis inspired Harry Partch immensely (Partch 1979, vii).

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 (Lesure 2001), 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" (Howat 2001). 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 (Don 1991, 69 et passim). 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.

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) (Partch 1979, chapt. 8, "Application of the 11 Limit", 119–37).

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 music

Electronic music facilitates the use of any kind of microtonal tuning, and sidesteps the need to develop new notational systems (Griffiths, Lindley, and Zannos 2001). 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 (Stockhausen 1964, 37), and in Gesang der Jünglinge (1955–56) he used various scales, ranging from seven up to sixty equal divisions of the octave (Decroupet and Ungeheuer 1998, 105, 116, 119–21). In 1955, Ernst Krenek used 13 equal-tempered intervals per octave in his Whitsun oratorio, Spiritus intelligentiae, sanctus (Griffiths, Lindley, and Zannos 2001).

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 (Blackwood and Kust 2005, ). "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" (Blackwood n.d.).

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" (Carlos 1989–96).

Aphex Twin has been making microtonal electronic music with software and various analog and digital synthesizers since his 1994 album, Selected Ambient Works Volume II. (Noyze and James 2014)

Microtonality in rock music

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 (Ferguson 1999, 20).

Musicians like 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 (Couture n.d.).

The band Radiohead have used microtonal string arrangements in their music, such as on "How to Disappear Completely" from their album Kid A (Wilson, Penderecki, and Greenwood 2012).

See also

Western microtonal pioneers

Pioneers of modern Western microtonal music include:

Recent microtonal composers

Microtonal researchers

References

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Further reading

  • 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 . 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,. New York: Routledge.
  • Barbieri, Patrizio. 2008. Enharmonic instruments and music, 1470-1900. Latina: Il Levante Libreria Editrice. ISBN 978-88-95203-14-0.
  • 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. 215–64. 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, 107–22. 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.
  • Orientierungen: Wege im Pluralismus der Gegenwartsmusik, edited by Jörn Peter Hiekel, 123–29. 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.
  • Jedrzejewski, Franck. 2003. Dictionnaire des musiques microtonales [Dictionary of Microtonal Musics]. Paris: L'Harmattan. ISBN 2-7475-5576-3.
  • Jedrzejewski, Franck. 2014. Dictionnaire des musiques microtonales. Nouvelle édition revue et augmentée (1892–2013) [Dictionary of Microtonal Musics]. Paris: L'Harmattan. ISBN 978-2343035406.
  • 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): 202–11.
  • 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.
  • 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: 241–63.
  • 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): 328–30.
  • Zweifel, Paul. 1996. "Generalized Diatonic and Pentatonic Scales: A Group-Theoretic Approach". Perspectives of New Music 34, no. 1 (Winter): 140–61.

External links

  • Microtonal music at DMOZ
  • Aikin, Jim. 2003. Jim Aikin's article on alternative tuning in electronic music
  • Anon. [n.d.]. "Nicola Vicentino (1511–1576)". IVO: Sacred Music in the Italian Cinquecento outside Venice and Rome, edited by Chris Whent. Here Of A Sunday Morning website. (Accessed 19 August 2008)
  • Chalmers, John. Dr. John Chalmers Divisions of the Tetrachord
  • Loli, Charles. 2008. " Microtonalismo". (Article on alternative tuning in Peruvian music)
  • Solís Winkler, Ernesto. 2004. "Julián Carrillo and the 13th Sound: A Microtonal Musical System". (Accessed 19 August 2008)
  • Wilson, Erv. "Wilson Archives of papers on microtonal theory"
  • Links to microtonal composers at Xenharmonic Wiki
  • Links to microtonal projects around the world at Xenharmonic Wiki
  • Offtonic Microtonal Synthesizer, a browser-based synth to explore microtonal tunings with a QWERTY keyboard



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