A quarter tone is a pitch halfway between the usual notes of a

chromatic scale The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce th ...

or an interval about half as wide (aurally, or logarithmically) as a

semitone 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 no ...

, which itself is half a

whole tone In Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a musical interval encompassing two adjacent staff positions (see Interval number for more det ...

. Quarter tones divide the octave by 50 cents each, and have 24 different pitches. Heckel ViertelTonTRP C

Quarter tone has its roots in the music of the Middle East and more specifically in

Persian traditional music Persian traditional music or Iranian traditional music, also known as Persian classical music or Iranian classical music, refers to the classical music of Iran (also known as ''Persia''). It consists of characteristics developed through the coun ...

. However, the first evidenced proposal of quarter tones, or the quarter-tone scale (24 equal temperament), was made by 19th-century music theorists Heinrich Richter in 1823 Julian Rushton, "Quarter-Tone", ''

The New Grove Dictionary of Music and Musicians ''The New Grove Dictionary of Music and Musicians'' is an encyclopedic dictionary of music and musicians. Along with the German-language '' Die Musik in Geschichte und Gegenwart'', it is one of the largest reference works on the history and t ...

'', second edition, edited by

Stanley Sadie Stanley John Sadie (; 30 October 1930 – 21 March 2005) was an influential and prolific British musicologist, music critic, and editor. He was editor of the sixth edition of the '' Grove Dictionary of Music and Musicians'' (1980), which was pub ...

and John Tyrrell (London: Macmillan, 2001). and

Mikhail Mishaqa Mikhail Mishaqa or Michael Mishaka (March 20, 1800 – July 19, 1888; ar, ميخائيل مشاقة, gr, Μιχαήλ Μισάκα), also known as Doctor Mishaqa, was born in Rashmayyā, Lebanon, and is reputed to be "the first historian of mo ...

about 1840. Composers who have written music using this scale include:

Pierre Boulez Pierre Louis Joseph Boulez (; 26 March 1925 – 5 January 2016) was a French composer, conductor and writer, and the founder of several musical institutions. He was one of the dominant figures of post-war Western classical music. Born in Mo ...

, Julián Carrillo,

Mildred Couper Mildred Couper (December 10, 1887 in Buenos Aires, Argentina – August 9, 1974 in Santa Barbara, United States) was a prominent composer and pianist, and one of the first American musicians to experiment with quarter-tone music. She was b ...

George Enescu George Enescu (; – 4 May 1955), known in France as Georges Enesco, was a Romanian composer, violinist, conductor and teacher. Regarded as one of the greatest musicians in Romanian history, Enescu is featured on the Romanian five lei. Biogr ...

Alberto Ginastera Alberto Evaristo Ginastera (; April 11, 1916June 25, 1983) was an Argentinian composer of classical music. He is considered to be one of the most important 20th-century classical composers of the Americas. Biography Ginastera was born in Buenos ...

, Gérard Grisey,

Alois Hába Alois Hába (21 June 1893 – 18 November 1973) was a Czech composer, music theorist and teacher. He belongs to the important discoverers in modern classical music, and major composers of microtonal music, especially using the quarter-tone scal ...

Ljubica Marić Ljubica Marić (Љубица Марић , 18 March 1909 – 17 September 2003) was a composer from Yugoslavia. She was a pupil of Josip Štolcer-Slavenski. She was known for being inspired by Byzantine Empire, Byzantine Eastern Orthodox Church, O ...

Charles Ives Charles Edward Ives (; October 20, 1874May 19, 1954) was an American modernist composer, one of the first American composers of international renown. His music was largely ignored during his early career, and many of his works went unperformed ...

Tristan Murail Tristan Murail (born 11 March 1947) is a French composer associated with the "spectral" technique of composition. Among his compositions is the large orchestral work ''Gondwana''. Early life and studies Murail was born in Le Havre, France. His fa ...

Krzysztof Penderecki Krzysztof Eugeniusz Penderecki (; 23 November 1933 – 29 March 2020) was a Polish composer and conductor. His best known works include ''Threnody to the Victims of Hiroshima'', Symphony No. 3, his '' St Luke Passion'', '' Polish Requiem'', ' ...

Giacinto Scelsi Giacinto Francesco Maria Scelsi (; 8 January 1905 – 9 August 1988, sometimes cited as 8 August 1988) was an Italian composer who also wrote surrealist poetry in French. He is best known for having composed music based around only one pitch, ...

Ammar El Sherei Ammar Ali Mohamed Ibrahim Ali Al Sherei ( ar, عمار علي محمد إبراهيم علي الشريعي) or more commonly known as Ammar El Sherei (16 April 1948 – 7 December 2012) was an Egyptian music icon, performer and composer. Ea ...

Karlheinz Stockhausen Karlheinz Stockhausen (; 22 August 1928 – 5 December 2007) was a German composer, widely acknowledged by critics as one of the most important but also controversial composers of the 20th and early 21st centuries. He is known for his groundb ...

, Tui St. George Tucker, Ivan Wyschnegradsky, and

Iannis Xenakis Giannis Klearchou Xenakis (also spelled for professional purposes as Yannis or Iannis Xenakis; el, Γιάννης "Ιωάννης" Κλέαρχου Ξενάκης, ; 29 May 1922 – 4 February 2001) was a Romanian-born Greek-French avant-garde c ...

. (See

List of quarter tone pieces A selection of compositions using quarter tones: A * Thomas Adès **''Asyla'' calls for an upright piano tuned a quarter-tone flat. B *Jan Bach **Concert Variations for solo euphonium; "each variation is based on different performance techniqu ...

Types

Equal-tempered tuning systems

The term ''quarter tone'' can refer to a number of different intervals, all very close in size. For example, some 17th- and 18th-century theorists used the term to describe the distance between a sharp and enharmonically distinct flat in mean-tone temperaments (e.g., D–E). In the quarter-tone scale, also called 24-tone equal temperament (24-TET), the quarter tone is 50 cents, or a frequency ratio of or approximately 1.0293, and divides the octave into 24 equal steps (

equal temperament An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, ...

). In this scale the quarter tone is the smallest step. A semitone is thus made of two steps, and three steps make a three-quarter tone or neutral second, half of a

minor third In music theory, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions (see: interval number). The minor third is one of two com ...

. The 8-TET scale is composed of three-quarter tones. Four steps make a whole tone. Quarter tones and intervals close to them also occur in a number of other equally tempered tuning systems.

22-TET In music, 22 equal temperament, called 22-TET, 22-EDO, or 22-ET, is the tempered scale derived by dividing the octave into 22 equal steps (equal frequency ratios). Each step represents a frequency ratio of , or 54.55 cents (). When composing wit ...

contains an interval of 54.55 cents, slightly wider than a quarter-tone, whereas 53-TET has an interval of 45.28 cents, slightly smaller. 72-TET also has equally tempered quarter-tones, and indeed contains three quarter-tone scales, since 72 is divisible by 24. The smallest interval in

31 equal temperament In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET (31 tone ET) or 31-EDO (equal division of the octave), also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equal-sized steps (equa ...

(the "diesis" of 38.71 cents) is half a chromatic semitone, one-third of a diatonic semitone and one-fifth of a whole tone, so it may function as a quarter tone, a fifth-tone ''or'' a sixth-tone.

Just intonation tuning systems

just intonation In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios (such as 3:2 or 4:3) of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals (and ...

the quarter tone can be represented by the septimal quarter tone, 36:35 (48.77 cents), or by the undecimal quarter tone (i.e. the thirty-third harmonic), 33:32 (53.27 cents), approximately half the semitone of 16:15 or 25:24. The ratio of 36:35 is only 1.23 cents narrower than a 24-TET quarter tone. This just ratio is also the difference between a

(6:5) and septimal minor third (7:6). Composer

Ben Johnston Ben Johnston may refer to: * Ben Johnston (rugby union) (born 1978), British rugby player * Ben Johnston (composer) (1926–2019), American contemporary composer of concert music * Bennett Johnston, Jr. (born 1932), Washington, D.C.-based lobbyist ...

, to accommodate the

just Just or JUST may refer to: __NOTOC__ People * Just (surname) * Just (given name) Arts and entertainment * ''Just'', a 1998 album by Dave Lindholm * "Just" (song), a song by Radiohead * "Just", a song from the album '' Lost and Found'' by Mudvayn ...

septimal quarter tone, uses a small "7" () as an accidental to indicate a note is lowered 49 cents, or an upside down "7" () to indicate a note is raised 49 cents, or a ratio of 36:35. Fonville, John (Summer, 1991). "Ben Johnston's Extended Just Intonation: A Guide for Interpreters", p. 114, ''

Perspectives of New Music ''Perspectives of New Music'' (PNM) is a peer-reviewed academic journal specializing in music theory and analysis. It was established in 1962 by Arthur Berger and Benjamin Boretz (who were its initial editors-in-chief). ''Perspectives'' was firs ...

'', vol. 29, no. 2, pp. 106–137. Johnston uses an upward and downward arrow to indicate a note is raised or lowered by a ratio of 33:32, or 53 cents. The Maneri-Sims notation system designed for 72-et uses the accidentals and for a quarter tone (36:35 or 48.77 cents) up and down.

Playing quarter tones

Any tunable musical instrument can be used to perform quarter tones, if two players and two identical instruments, with one tuned a quarter tone higher, are used. As this requires neither a special instrument nor special techniques, much quarter toned music is written for pairs of pianos, violins, harps, etc. The retuning of the instrument, and then returning it to its former pitch, is easy for violins, harder for harps, and slow and relatively expensive for pianos. The following deals with the ability of single instruments to produce quarter tones. In Western instruments, this means "in addition to the usual 12-tone system". This man of because of the many musical instruments manufactured today (2018) are designed for the 12-tone scale, not all are usable for playing quarter tones. Sometimes special playing techniques must be used. Conventional musical instruments that ''cannot'' play quarter tones (except by using special techniques—see below) include: * Most standard or unmodified non-electronic keyboard instruments, such as

piano The piano is a stringed keyboard instrument in which the strings are struck by wooden hammers that are coated with a softer material (modern hammers are covered with dense wool felt; some early pianos used leather). It is played using a keyboa ...

s, organs, and

accordion Accordions (from 19th-century German ''Akkordeon'', from ''Akkord''—"musical chord, concord of sounds") are a family of box-shaped musical instruments of the bellows-driven free-reed aerophone type (producing sound as air flows past a ree ...

s *

Fret A fret is any of the thin strips of material, usually metal wire, inserted laterally at specific positions along the neck or fretboard of a stringed instrument. Frets usually extend across the full width of the neck. On some historical instru ...

ted

string instruments String instruments, stringed instruments, or chordophones are musical instruments that produce sound from vibrating strings when a performer plays or sounds the strings in some manner. Musicians play some string instruments by plucking the st ...

such as

guitars The guitar is a fretted musical instrument that typically has six strings. It is usually held flat against the player's body and played by strumming or plucking the strings with the dominant hand, while simultaneously pressing selected strin ...

bass guitars The bass guitar, electric bass or simply bass (), is the lowest-pitched member of the string family. It is a plucked string instrument similar in appearance and construction to an electric or an acoustic guitar, but with a longer neck and sc ...

, and

ukulele The ukulele ( ; from haw, ukulele , approximately ), also called Uke, is a member of the lute family of instruments of Portuguese origin and popularized in Hawaii. It generally employs four nylon strings. The tone and volume of the instrumen ...

s (though on these it is possible to play quarter tones by pitch-bending, with special tunings, or with customized necks) * Pitched percussion instruments, if standard techniques are used, and if the instruments are not tunable * Western

wind instruments A wind instrument is a musical instrument that contains some type of resonator (usually a tube) in which a column of air is set into vibration by the player blowing into (or over) a mouthpiece set at or near the end of the resonator. The pit ...

that use keys or valves **

Woodwind instruments Woodwind instruments are a family of musical instruments within the greater category of wind instruments. Common examples include flute, clarinet, oboe, bassoon, and saxophone. There are two main types of woodwind instruments: flutes and re ...

, such as clarinets, saxophones, flutes, and oboes (though with many of these, it is still possible using non-standard techniques such as special fingerings or by the player manipulating their

embouchure Embouchure () or lipping is the use of the lips, facial muscles, tongue, and teeth in playing a wind instrument. This includes shaping the lips to the mouthpiece of a woodwind instrument or the mouthpiece of a brass instrument. The word is o ...

, to play at least ''some'' quarter tones, if not a whole scale) ** Valved brass instruments (

trumpet The trumpet is a brass instrument commonly used in classical and jazz ensembles. The trumpet group ranges from the piccolo trumpet—with the highest register in the brass family—to the bass trumpet, pitched one octave below the standard ...

tuba The tuba (; ) is the lowest-pitched musical instrument in the brass family. As with all brass instruments, the sound is produced by lip vibrationa buzzinto a mouthpiece. It first appeared in the mid-19th century, making it one of the ne ...

) (though, as with woodwinds, embouchure manipulation, as well as

harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', t ...

tones that fall closer to quarter-tones than half-tones, make quarter-tone scales possible; the horn technique of adjusting pitch with the right hand in the bell makes this instrument an exception) *

Harmonica The harmonica, also known as a French harp or mouth organ, is a free reed wind instrument used worldwide in many musical genres, notably in blues, American folk music, classical music, jazz, country, and rock. The many types of harmonica in ...

(although note bending is a common technique) Conventional musical instruments that ''can'' play quarter tones include * Electronic instruments: **

Synthesizer A synthesizer (also spelled synthesiser) is an electronic musical instrument that generates audio signals. Synthesizers typically create sounds by generating waveforms through methods including subtractive synthesis, additive synthesis a ...

s, using either special keyboard controllers or continuous-pitch controllers such as fingerboard controllers, or when controlled by a

sequencer Sequencer may refer to: Technology * Drum sequencer (controller), an electromechanical system for controlling a sequence of events automatically * DNA sequencer, a machine used to automatically produce a sequence readout from a biological DNA sam ...

capable of outputting quarter-tone control signals. **

Theremin The theremin (; originally known as the ætherphone/etherphone, thereminophone or termenvox/thereminvox) is an electronic musical instrument controlled without physical contact by the performer (who is known as a thereminist). It is named afte ...

s and other continuously pitched instruments * Fretless string instruments, such as the

violin family The violin family of musical instruments was developed in Italy in the 16th century. At the time the name of this family of instruments was viole da braccio which was used to distinguish them from the viol family (viole ''da gamba''). The stan ...

fretless guitar A fretless guitar is a guitar with a fingerboard without frets, typically a standard instrument that has had the frets removed, though some custom-built and commercial fretless guitars are occasionally made. The classic fretless guitar was first ...

s, fretless electric basses, ouds, and members of the

huqin ''Huqin'' () is a family of bowed string instruments, more specifically, a spike fiddle popularly used in Chinese music. The instruments consist of a round, hexagonal, or octagonal sound box at the bottom with a neck attached that protrudes u ...

family of instruments. * String instruments with movable frets (such as the

sitar The sitar ( or ; ) is a plucked stringed instrument, originating from the Indian subcontinent, used in Hindustani classical music. The instrument was invented in medieval India, flourished in the 18th century, and arrived at its present form ...

) * Specially fretted string instruments (such as the Turkish

bağlama The ''bağlama'' or ''saz'' is a family of plucked string instruments, long-necked lutes used in Ottoman classical music, Turkish folk music, Turkish Arabesque music, Azerbaijani music, Kurdish music, Armenian music and in parts of Syria, I ...

). * Fretted string instruments specially tuned to quarter tones *

Pedal steel guitar The pedal steel guitar is a console-type of steel guitar with pedals and knee levers that change the pitch of certain strings to enable playing more varied and complex music than any previous steel guitar design. Like all steel guitars, it can ...

* Wind instruments whose main means of tone-control is a slide, such as

trombone The trombone (german: Posaune, Italian, French: ''trombone'') is a musical instrument in the brass family. As with all brass instruments, sound is produced when the player's vibrating lips cause the air column inside the instrument to vibrate ...

s, the tromboon invented by

P. D. Q. Bach P. D. Q. Bach is a fictional composer invented by the American musical satirist Peter Schickele, who developed a five-decade-long career performing the "discovered" works of the "only forgotten son" of the Bach family. Schickele's music combines ...

, the slide trumpet and the

slide whistle A slide whistle (variously known as a swanee or swannee whistle, lotos flute piston flute, or jazz flute) is a wind instrument consisting of a fipple like a recorder's and a tube with a piston in it. Thus it has an air reed like some woodw ...

* Specially keyed woodwind instruments. A quarter tone clarinet was built by Fritz Schüller (1883–1977) of

Markneukirchen Markneukirchen () is a town in the Vogtlandkreis district, in Saxony, Germany, close to the Czech border. It lies in between the Erzgebirge and the Fichtelgebirge in the Elstergebirge, southeast of Plauen, and northeast of Aš (Czech Republic ...

, an
a quarter tone mechanism for flutes by Eva Kingma
* Valved brass instruments with extra, quarter-tone valves, and natural brass instruments that play through the 11th and 13th partials of the harmonic series *

Voice The human voice consists of sound made by a human being using the vocal tract, including talking, singing, laughing, crying, screaming, shouting, humming or yelling. The human voice frequency is specifically a part of human sound producti ...

Kazoo The kazoo is an American musical instrument that adds a "buzzing" timbral quality to a player's voice when the player vocalizes into it. It is a type of '' mirliton'' (which itself is a membranophone), one of a class of instruments which modifie ...

* Pitched percussion instruments, when tuning permits (e.g., timpani), or using special techniques Other instruments can be used to play quarter tones when using

audio signal processing Audio signal processing is a subfield of signal processing that is concerned with the electronic manipulation of audio signals. Audio signals are electronic representations of sound waves— longitudinal waves which travel through air, consist ...

effects such as pitch shifting. Quarter-tone pianos have been built, which consist essentially of two pianos with two keyboards stacked one above the other in a single case, one tuned a quarter tone higher than the other.

Music of the Middle East

Many Persian

dastgah Dastgāh ( fa, دستگاه) is the standard musical system in Persian art music, standardised in the 19th century following the transition of Persian music from the Maqam modal system. A consists of a collection of musical melodies, . In a son ...

and Arabic ''maqamat'' contain intervals of three-quarter tone size; a short list of these follows. #

Bayati {{About, a form of Azerbaijani folk poetry, other uses, Bayat (disambiguation) Bayati ( az, Bayatı) is one of the oldest forms of Azerbaijani folk poetry. A bayati consists of four lines, each of which has seven syllables. The rhyme scheme is ...

(): D E F G A B C D #: \relative c' # Rast (): #:C D E F G A B C (ascending) #:C B A G F E D C (descending) #: \relative c' #Saba (): D E F G A B C D #: \relative c' # Sigah (): E F G A B C D E #: \relative c' # ‘Ajam () #Hoseyni The Islamic philosopher and scientist

Al-Farabi Abu Nasr Muhammad Al-Farabi ( fa, ابونصر محمد فارابی), ( ar, أبو نصر محمد الفارابي), known in the West as Alpharabius; (c. 872 – between 14 December, 950 and 12 January, 951)PDF version was a renowned early Isl ...

described a number of intervals in his work in music, including a number of quarter tones. Assyrian/Syriac Church Music Scale: # Qadmoyo (Bayati) # Trayono (Hussayni) # Tlithoyo (Segah) # Rbiʿoyo (Rast) # Hmishoyo # Shtithoyo (ʿAjam) # Shbiʿoyo # Tminoyo

Quarter-tone scale

Known as ''gadwal'' in Arabic, the quarter-tone scale was developed in the Middle East in the eighteenth century and many of the first detailed writings in the nineteenth century Syria describe the scale as being of 24 equal tones.Marcus, Scott (1993)."The Interface between Theory and Practice: Intonation in Arab Music", '' Asian Music'', vol. 24, no. 2. (Spring–Summer, 1993), pp. 39–58. The invention of the scale is attributed to

whose work ''Essay on the Art of Music for the Emir Shihāb (al-Risāla al-shihābiyya fi 'l-ṣināʿa al-mūsīqiyya)'' is devoted to the topic but also makes clear his teacher Sheikh Muhammad al-Attar (1764–1828) was one of many already familiar with the concept.Maalouf, Shireen (2003). "Mikhiiʾil Mishiiqa: Virtual Founder of the Twenty-Four Equal Quartertone Scale", ''

Journal of the American Oriental Society The ''Journal of the American Oriental Society'' is a quarterly academic journal published by the American Oriental Society since 1843. \relative c' The quarter tone scale may be primarily a

theoretical A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be ...

construct in Arabic music. The quarter tone gives musicians a "conceptual map" they can use to discuss and compare intervals by number of quarter tones, and this may be one of the reasons it accompanies a renewed interest in theory, with instruction in music theory a mainstream requirement since that period. Previously, pitches of a mode were chosen from a scale consisting of seventeen tones, developed by

Safi al-Din al-Urmawi Safi al-Din al-Urmawi al-Baghdadi ( fa, صفی الدین اورموی) or Safi al-Din Abd al-Mu'min ibn Yusuf ibn al-Fakhir al-Urmawi al-Baghdadi (born c. 1216 AD in Urmia, died in 1294 AD in Baghdad) was a renowned musician and writer on the ...

in the thirteenth century. Composer

chose the chord C–D–F–G–B as good possibility for a "secondary" chord in the quarter-tone scale, akin to the minor chord of traditional tonality. He considered that it may be built upon any degree of the quarter tone scale Here is the secondary "minor" and its "first inversion": \relative c'

In popular music

Several quarter-tone albums have been recorded by Jute Gyte, a one-man avantgarde black metal band from Missouri, USA. Another quartertone metal album was issued by the Swedish band Massive Audio Nerve. Australian

psychedelic rock Psychedelic rock is a rock music genre that is inspired, influenced, or representative of psychedelic culture, which is centered on perception-altering hallucinogenic drugs. The music incorporated new electronic sound effects and recording te ...

band King Gizzard & the Lizard Wizard's album '' Flying Microtonal Banana'' heavily emphasizes quarter-tones and used a custom-built guitar in 24-TET tuning. Jazz violinist/violist Mat Maneri, in conjunction with his father Joe Maneri, made a crossover fusion album, ''Pentagon'' (2005), that featured experiments in hip hop with quarter tone pianos, as well as electric organ and

mellotron The Mellotron is an electro-mechanical musical instrument developed in Birmingham, England, in 1963. It is played by pressing its keys, each of which pushes a length of magnetic tape against a capstan, which pulls it across a playback head. ...

textures, along with distorted trombone, in a post-Bitches Brew type of mixed Jazz/rock."Mat Maneri: ''Pentagon''"
by Will Layman, 1 December 2005, ''

PopMatters ''PopMatters'' is an international online magazine of cultural criticism that covers aspects of popular culture. ''PopMatters'' publishes reviews, interviews, and essays on cultural products and expressions in areas such as music, television ...

Ancient Greek tetrachords

The

enharmonic genus In the musical system of ancient Greece, genus (Greek: γένος 'genos'' pl. γένη 'genē'' Latin: ''genus'', pl. ''genera'' "type, kind") is a term used to describe certain classes of intonations of the two movable notes within a tetrach ...

of the

Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...

tetrachord In music theory, a tetrachord ( el, τετράχορδoν; lat, tetrachordum) is a series of four notes separated by three intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency pr ...

consisted of a ditone or an approximate

major third In classical music, a third is a Interval (music), musical interval encompassing three staff positions (see Interval (music)#Number, Interval number for more details), and the major third () is a third spanning four semitones.Allen Forte, ...

, and a

, which was divided into two

microtones 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 t ...

Aristoxenos Aristoxenus of Tarentum ( el, Ἀριστόξενος ὁ Ταραντῖνος; born 375, fl. 335 BC) was a Greek Peripatetic philosopher, and a pupil of Aristotle. Most of his writings, which dealt with philosophy, ethics and music, have been ...

, Didymos and others presented the semitone as being divided into two approximate quarter tone intervals of about the same size, while other ancient Greek theorists described the microtones resulting from dividing the semitone of the enharmonic genus as unequal in size (i.e., one smaller than a quarter tone and one larger).Chalmers, John H. Jr. (1993). ''Divisions of the Tetrachord''. Hanover, New Hampshire: Frog Peak Music. Chapter 5, Page 49

Interval size in equal temperament

Here are the sizes of some common intervals in a 24-note equally tempered scale, with the interval names proposed by

(neutral third, etc.) and Ivan Wyschnegradsky (major fourth, etc.): Moving from

12-TET Twelve-tone equal temperament (12-TET) is the musical system that divides the octave into 12 parts, all of which are equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the 12th root of 2 ( ≈ 1.05946). That resulti ...

to 24-TET allows the better approximation of a number of intervals. Intervals matched particularly closely include the neutral second, neutral third, and (11:8) ratio, or the 11th harmonic. The septimal minor third and

septimal major third In music, the septimal major third , also called the supermajor third (by Hermann von Helmholtz among others Hermann L. F. von Helmholtz (2007). '' Sensations of Tone'', p. 187. .) and sometimes '' Bohlen–Pierce third'' is the musical interval ...

are approximated rather poorly; the (13:10) and (15:13) ratios, involving the 13th harmonic, are matched very closely. Overall, 24-TET can be viewed as matching the 11th and 13th harmonics more closely than the 7th.

References

External links

quarter-tone / 24-edo
, ''TonalSoft.com''. {{Musical tuning Musical scales