Use in music of microtones (intervals smaller than a semitone)
For sounds on the time scale shorter than musical notes, see microsound.
"Microtone" redirects here. For the slicing tool, see Microtome.
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
Two examples of an Ives fundamental chord with quarter tones
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 16th 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.”
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. 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.
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. 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.
"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, as "enharmonic microtones", 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". 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. 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. 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 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.
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.
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'". Modern Indian researchers yet write: "microtonal intervals called shrutis". In Germany, Austria, and Czechoslovakia in the 1910s and 1920s the usual term continued to be Viertelton-Musik (quarter tone music[pageneeded]), 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)). 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, and in the second edition of the popular Brockhaus Riemann Musiklexikon.
Ivan Wyschnegradsky used the term ultra-chromatic for intervals smaller than the semitone and infra-chromatic for intervals larger than the semitone; 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 by theorist Marek Žabka.Ivor Darreg proposed[when?] 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". Some authors in Russia and some musicology dissertations disseminate the term микрохроматика (microchromatics), coined in the 1970s by Yuri Kholopov, 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 ). Another Russian authors use more international adjective 'microtonal' and rendered it in Russian as 'микротоновый', but not 'microtonality' ('микротональность'). However, the terms 'микротональность' and 'микротоника' are also used. Some authors writing in French have adopted the term "micro-intervallique" to describe such music. 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 Rué ). 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".
The term "macrotonal" has been used for intervals wider than twelve-tone equal temperament,[bettersourceneeded] 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". The term "macrotonal" has also been used for musical form.
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.
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). 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.
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.
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.
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.
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. 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. Ellis inspired Harry Partch immensely.
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, 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". 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.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).
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. In 1954, Karlheinz Stockhausen built his electronic Studie II on an 81-step scale starting from 100Hz with the interval of 51/25 between steps, and in Gesang der Jünglinge (1955–56) he used various scales, ranging from seven up to sixty equal divisions of the octave. In 1955, Ernst Krenek used 13 equal-tempered intervals per octave in his Whitsun oratorio, Spiritus intelligentiae, sanctus.
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".
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.
Limitations of some synthesizers
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.
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 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. Joe Monzo has made a microtonal analysis of the song "Drunken Hearted Man", written and recorded by the delta blues musician Robert Johnson.
English rock band Radiohead has used microtonal string arrangements in its music, such as on "How to Disappear Completely" from the album Kid A.
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.
Karlheinz Stockhausen (Germany, 1928–2007, in his electronic works many microtonal concepts, non-octaving scales in Studie II, just intonation in Gruppen and Stimmung, occasional microtonal instrumental and vocal writing throughout Licht)
An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, which gives an equal perceived step size as pitch is perceived roughly as the logarithm of frequency.
In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals consist of tones from a single harmonic series of an implied fundamental. For example, in the diagram, if the notes G3 and C4 are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just fourth.
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is ≈702 cents wide.
Meantone temperament is a musical temperament, that is a tuning system, obtained by narrowing the fifths so that their ratio is slightly less than 3:2, in order to push the thirds closer to pure. Meantone temperaments are constructed the same way as Pythagorean tuning, as a stack of equal fifths, but it is a temperament in that the fifths are not pure.
In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.
The chromatic scale is a set of twelve pitches used in tonal music, with notes separated by the interval of a semitone. Almost all western musical instruments, such as the piano, are made to produce the chromatic scale, while other instruments capable of continuously variable pitch, such as the trombone and violin, can also produce microtones, or notes between those available on a piano.
In music theory, the tritone is defined as a musical interval composed of three adjacent whole tones. For instance, the interval from F up to the B above it is a tritone as it can be decomposed into the three adjacent whole tones F–G, G–A, and A–B.
In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a written note, interval, or chord is an alternative way to write that note, interval, or chord. The term is derived from Latin enharmonicus, from Late Latin enarmonius, from Ancient Greek ἐναρμόνιος (enarmónios), from ἐν (en) and ἁρμονία (harmonía).
In jazz and blues, a blue note is a note that—for expressive purposes—is sung or played at a slightly different pitch from standard. Typically the alteration is between a quartertone and a semitone, but this varies depending on the musical context.
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale. For example, C is adjacent to C♯; the interval between them is a semitone.
In Western classical music, a minor sixth is a musical interval encompassing six staff positions, and is one of two commonly occurring sixths. It is qualified as minor because it is the smaller of the two: the minor sixth spans eight semitones, the major sixth nine. For example, the interval from A to F is a minor sixth, as the note F lies eight semitones above A, and there are six staff positions from A to F. Diminished and augmented sixths span the same number of staff positions, but consist of a different number of semitones.
Regular temperament is any tempered system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. For instance, in 12-TET, the system of music most commonly used in the Western world, the generator is a tempered fifth, which is the basis behind the circle of fifths.
A quarter tone is a pitch halfway between the usual notes of a chromatic scale or an interval about half as wide as a semitone, which itself is half a whole tone. Quarter tones divide the octave by 50 cents each, and have 24 different pitches.
In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways. Strictly speaking, there are only two kinds of comma, the syntonic comma, "the difference between a just major 3rd and four just perfect 5ths less two octaves", and the Pythagorean comma, "the difference between twelve 5ths and seven octaves". The word comma used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F♯ tuned using the D-based Pythagorean tuning system, and another F♯ tuned using the D-based quarter-comma meantone tuning system. Intervals separated by the ratio 81:80 are considered the same note because the 12-note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B♭ and A♯ are both approximated by the same interval although they are a septimal kleisma apart.
In music, 72 equal temperament, called twelfth-tone, 72-TET, 72-EDO, or 72-ET, is the tempered scale derived by dividing the octave into twelfth-tones, or in other words 72 equal steps. Play (help·info) Each step represents a frequency ratio of 72√2, or 16+2⁄3 cents, which divides the 100 cent "halftone" into 6 equal parts and is thus a "twelfth-tone". Since 72 is divisible by 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72, 72-EDO includes all those equal temperaments. Since it contains so many temperaments, 72-EDO contains at the same time tempered semitones, third-tones, quartertones and sixth-tones, which makes it a very versatile temperament.
Twelve-tone equal temperament is the musical system that divides the octave into 12 parts, all of which are equally tempered on a logarithmic scale, with a ratio equal to the 12th root of 2. That resulting smallest interval, 1⁄12 the width of an octave, is called a semitone or half step.
In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET or 31-EDO, also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equal-sized steps. Play (help·info) Each step represents a frequency ratio of 31√2, or 38.71 cents.
In music, 19 Tone Equal Temperament, called 19 TET, 19 EDO, or 19 ET, is the tempered scale derived by dividing the octave into 19 equal steps. Each step represents a frequency ratio of 19√2, or 63.16 cents.
A neutral third is a musical interval wider than a minor third play (help·info) but narrower than a major third play (help·info), named by Jan Pieter Land in 1880. Land makes reference to the neutral third attributed to Zalzal, described by Al-Farabi as corresponding to a ratio of 27:22 and by Avicenna as 39:32. The Zalzalian third may have been a mobile interval.
↑ Boatwright, Howard (1971). Boretz, Benjamin; Cone, Edward T. (eds.). Ives' Quarter-Tone Impressions. Perspectives on American Composers. Princeton: Princeton University Press. pp.8–9.
↑ Hinrichsen, H. (2016). "Revising the musical equal temperament". Caderno Brasileiro de Ensino de Física. 38 (1): 1310–1314. arXiv:1508.02292.
↑ Mann, Maud (MacCarthy) (16 January 1912). "Some Indian Conceptions of Music". Proceedings of the Musical Association, 38th Session (1911–1912): 44.
↑ Whitfield, Charles (1989). L'anglais musicologique: l'anglais des musiciens. Guides Musicologiques. Paris: Editions Beauchesne. p.13. ISBN2-7010-1181-7.
↑ Battier, Marc; Lacino, Thierry (1984). "Simulation and Extrapolation of Instrumental Sounds Using Direct Synthesis at IRCAM (A Propos of Resonance)". Contemporary Music Review. 1 (Musical Thought at IRCAM, edited by Tod Machover): 77–82. doi:10.1080/07494468400640081. hdl:2027/spo.bbp2372.1982.033.
↑ Boulez, Pierre (1958). Translated by Goehr, Alexander. "At the Ends of Fruitful Land ...". Die Reihe (Englished.). 1, Electronic Music: 22–23.
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↑ Maclagan, Susan J. (2009). A Dictionary for the Modern Flutist. Lanham, MS, and Plymouth: Scarecrow Press, Inc. p.109. ISBN978-0-8108-6711-6.
↑ Rigoni, Michel (1998). Karlheinz Stockhausen: ... un vaisseau lancé vers le ciel. Musique de notre temps: compositeurs (in French) (Second, revised, corrected, and enlargeded.). Lillebonne: Millénaire III Éditions. p.314. ISBN978-2-911906-02-2.
↑ Apel, Will (1974). The Harvard Dictionary of Music (Seconded.). Cambridge, Massachusetts, USA: Harvard University Press. p.527.
↑ Barbera, André (2003). "Comma, Schisma". In Randel, Don Michael (ed.). Harvard Dictionary of Music (Fourthed.). Cambridge, Massachusetts, USA: The Belknap Press of Harvard University Press. p.193. ISBN978-0-674-01163-2.
↑ Dahlhaus, Carl; Eggebrecht, Hans Heinrich; Oehl, Kurt, eds. (1995). Brockhaus Riemann Musiklexikon in vier Bänden und einem Ergänzungsband, Volume 4[Brockhaus Riemann Music Lexicon in Four Volumes and a Supplementary Volume] (Seconded.). Mainz: Atlantis-Schott Musikbuch-Verlag. p.304.
↑ Wyschnegradsky, Ivan (1972). "L'Ultrachromatisme et les espaces non octaviants". La Revue musicale (290–91): 84–87.
↑ Belyaev, Victor M. [Беляев, В. М.] (1971). "Азербайджанская народная песня (1960)" [Azerbaijan Folk Songs]. In Belyaev, Victor (ed.). О музыкальном фольклоре и древней письменности[On Musical Folklore and Ancient Literature]. Moscow: Sov'etskii Kompozitor. pp.108–156.
↑ Belyaev, Victor M. [Беляев, В. М.] (1971). "Турецкая музыка (1934)" [Turkish Music]. In Belyaev, Victor (ed.). О музыкальном фольклоре и древней письменности[On Musical Folklore and Ancient Literature]. Moscow: Sov'etskii Kompozitor. pp.163–176.
↑ "Микрохроматика" [Mikrochromatika / Microchromatics]. Музыкальный энциклопедический словарь[Musical Encyclopedic Dictionary] (in Russian). Moscow: Sov'etskaya Entsiklopediya. 1990. p.344.
↑ "Микрохроматические интервалы" [Mikrochromaticheskie Intervali / Microchromatic Intervals]. Музыкальный словарь Гроува[Grove Music Dictionary] (in Russian) (Seconded.). Moscow: Praktika. 2007. p.563.
↑ Akopyan, Levon. Музыка ХХ века. Энциклопедический словарь[Music of the Twentieth Century: An Academic Dictionary] (in Russian). Moscow: Praktika. pp.353–354.
↑ Tsenova, V. S. [Ценова, В. С.], ed. (2007). Теория современной композиции[The Theory of Modern Composition] (in Russian). Moscow: Muzyka. pp.65, 123, 152 etc.
↑ Kholopov, Y. [Ю. ХОЛОПОВ]; Kirillina, L. [л. КИРИЛЛИНА]; Kyuregyan, T. [Т. КЮРЕГЯН]; Lyzhov, G. [г. ЛЫЖОВ]; Pospelova, R. [Р. ПОСПЕЛОВА]; Tsenova, V. [В. ЦЕНОВА] (2006), Музыкально-теоретические системы[Musical-Theoretical Systems](PDF) (in Russian), Moscow: Kompozitor, pp.86 etc.
↑ Klishin, A. G. (2010). Проблемы музыкального строя в начале Нового времени[Problems of Musical Structure in the Early Modern Era] (PhD diss.). Moscow: Moscow Conservatory.
↑ Gurenko, N.A. [Гуренко Н.А] (2010). Микрохроматика И. Вышнеградского: История, теория, практика освоения[The Microchomatics of I. Wyschnegradsky: History, Theory, Practice, Development] (PhD diss.). Urals Mussorgsky State Conservatoire.
↑ Polunina, E. N. [Полунина Е.Н.] (2010). Микрохроматика в музыкальном искусстве позднего Возрождения[Microchromatics in Music of the Late Renaissance] (PhD diss.). Vladivostok: Far East State Academy of Arts.
↑ Rovner, A. A. (2010). Сергей Протопопов: композиторское творчество и теоретические работы[Sergey Protopopov: composer's output and theoretical works] (PhD diss.). Moscow: Moscow Conservatory.
↑ Nikoltsev, I. D. (2013). Микрохроматика в системе современного музыкального мышления[Microchomatics in Contemporary Musical Thought] (PhD diss.). Moscow: Moscow Conservatory.
↑ More references can be located on the disserCat website, at "Поиск диссертаций"[Search for dissertations].
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