Quarter tone

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File:Quarter tone on C.mid
Quarter tone on C

A quarter tone is a pitch halfway between the usual notes of a chromatic scale or an interval about half as wide (orally, or logarithmically) as a semitone, which itself is half a whole tone. Quarter tones divide the octave by 50 cents each, and have 24 different pitches.

File:Heckel ViertelTonTRP C.jpg
Trumpet with 3 normal valves and a quartering on the extension valve (right)

Quarter tones have their roots in the music of the Middle East and more specifically in Persian traditional music.<ref>Hormoz Farhat (2004). The Dastgah Concept in Persian Music. Cambridge University Press. Template:ISBN</ref> However, the first evidenced proposal of the equally-tempered quarter tone scale, or 24 equal temperament, was made by 19th-century music theorists Heinrich Richter in 1823<ref name=JR>Julian Rushton, "Quarter-Tone", The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell (London: Macmillan, 2001).</ref> and Mikhail Mishaqa about 1840.<ref>Touma, Habib Hassan (1996). The Music of the Arabs, p. 16. Translator: Laurie Schwartz. Portland, Oregon: Amadeus Press. Template:ISBN.</ref> Composers who have written music using this scale include: Pierre Boulez, Julián Carrillo, Mildred Couper, George Enescu, Alberto Ginastera, Gérard Grisey, Alois Hába, Ljubica Marić, Charles Ives, Tristan Murail, Krzysztof Penderecki, Giacinto Scelsi, Ammar El Sherei, Karlheinz Stockhausen, Tui St. George Tucker, Ivan Wyschnegradsky, Iannis Xenakis, and Seppe Gebruers (See List of quarter tone pieces.)

TypesTemplate:AnchorTemplate:AnchorEdit

Equal-tempered tuning systemsEdit

File:Ives quarter tone fundamental chord.png
Composer Charles Ives chose the four-note chord above (C–DTemplate:Music–G–ATemplate:Music) as good possibility for a "fundamental" chord in the quarter-tone scale, akin not to the tonic but to the major chord of traditional tonality.<ref name="Boatwright">Boatwright, Howard (1965). "Ives' Quarter-Tone Impressions", Perspectives of New Music 3, no. 2 (Spring–Summer): pp. 22–31; citations on pp. 27–28; reprinted in Perspectives on American Composers, edited by Benjamin Boretz and Edward T. Cone, pp. 3–12, New York: W. W. Norton, 1971, citation on pp. 8–9. "These two chords outlined above might be termed major and minor."</ref>File:Ives fundamental chord (quarter tones).oggFile:Ives quarter tone fundamental chord arp.mid

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File:Subminor seventh on C.png
The "subminor seventh": Template:Nowrap=Template:Nowrap, 19 quarter tones. It approximates the harmonic seventh, Template:Nowrap. Maneri-Sims notation: Template:Nowrap

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., DTemplate:Music–ETemplate:Music).<ref name=JR /> In the quarter-tone scale, also called 24-tone equal temperament (24-TET), the quarter tone is 50 cents, or a frequency ratio of Template:Radic or approximately 1.0293, and divides the octave into 24 equal steps (equal temperament). 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. 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 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 (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 systemsEdit

In just intonation 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 minor third (6:5) and septimal minor third (7:6).

Composer Ben Johnston, to accommodate the just septimal quarter tone, uses a small "7" (Template:Music) as an accidental to indicate a note is lowered 49 cents, or an upside down "7" (Template:Music) to indicate a note is raised 49 cents,<ref>Douglas Keislar; Easley Blackwood; John Eaton; Lou Harrison; Ben Johnston; Joel Mandelbaum; William Schottstaedt. p.193. "Six American Composers on Nonstandard Tunnings", Perspectives of New Music, vol. 29, no. 1. (Winter 1991), pp. 176–211.</ref> or a ratio of 36:35.<ref name="Fonville">Fonville, John (Summer, 1991). "Ben Johnston's Extended Just Intonation: A Guide for Interpreters", p. 114, Perspectives of New Music, vol. 29, no. 2, pp. 106–137.</ref> Johnston uses an upward and downward arrow to indicate a note is raised or lowered by a ratio of 33:32, or 53 cents.<ref name="Fonville" /> The Maneri-Sims notation system designed for 72-et uses the accidentals Template:Music and Template:Music for a quarter tone (36:35 or 48.77 cents) up and down.

Playing quarter tonesEdit

File:QuartertoneClarinet.jpg
Quarter tone clarinet by Fritz Schüller viewed from four sides.

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". Because 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 pianos, organs, and accordions
  • Fretted string instruments such as guitars, bass guitars, and ukuleles (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 that use keys or valves
    • Woodwind instruments, 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, to play at least some quarter tones, if not a whole scale)
    • Valved brass instruments (trumpet, tuba) (though, as with woodwinds, embouchure manipulation, as well as harmonic 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 (although note bending is a common technique)

Conventional musical instruments that can play quarter tones include

Other instruments can be used to play quarter tones when using audio signal processing 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.Template:Citation needed

Music of the Middle EastEdit

Template:See also

Many Persian dastgah and Arabic maqamat contain intervals of three-quarter tone size; a short list of these follows.<ref name=Spector>Template:Cite journal</ref>

  1. Bayati ({{#invoke:Lang|lang}}): D ETemplate:Music F G A BTemplate:Music C D
    <score override_ogg="Arabic-scale bayati.ogg">

\relative c' { \time 8/4 \omit Staff.TimeSignature d4 eeh f g a bes c d \bar "|" } </score>

  1. Rast ({{#invoke:Lang|lang}}):
    C D ETemplate:Music F G A BTemplate:Music C (ascending)
    C BTemplate:Music A G F ETemplate:Music D C (descending)
    <score override_ogg="Arabic-scale rast.ogg">

\relative c' { \time 8/4 \omit Staff.TimeSignature c4 d eeh f g a beh c \bar "|" } </score>

  1. Saba ({{#invoke:Lang|lang}}): D ETemplate:Music F GTemplate:Music A BTemplate:Music C D
    <score override_ogg="Arabic-scale saba.ogg">

\relative c' { \time 8/4 \omit Staff.TimeSignature d4 eeh f ges a bes c d \bar "|" } </score>

  1. Segah ({{#invoke:Lang|lang}}): ETemplate:Music F G A BTemplate:Music C D ETemplate:Music
    <score sound="1">

\relative c' { \time 8/4 \omit Staff.TimeSignature eeh f g a beh c d eeh \bar "|" } </score>

  1. ‘Ajam ({{#invoke:Lang|lang}})
  2. Hoseyni

The Islamic philosopher and scientist Al-Farabi described a number of intervals in his work in music, including a number of quarter tones.

Assyrian/Syriac Church Music Scale:<ref>Asaad, Gabriel (1990). Syria's Music Throughout History</ref>

  1. Qadmoyo (Bayati)
  2. Trayono (Hussayni)
  3. Tlithoyo (Segah)
  4. Rbiʿoyo (Rast)
  5. Hmishoyo
  6. Shtithoyo (ʿAjam)
  7. Shbiʿoyo
  8. Tminoyo

Quarter-tone scaleEdit

Known as gadwal in Arabic,<ref name=Spector /> 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.<ref name=Marcus> Template:Cite journal </ref> The invention of the scale is attributed to Mishaqa who wrote a book devoted to the topic<ref name=Mishaqa> Template:Cite book </ref> but made clear that his teacher, Sheikh Muhammad al-Attar (1764–1828), was one among many already familiar with the concept.<ref name=Maalouf> Template:Cite journal </ref>

<score sound="1"> \relative c' { \cadenzaOn \omit Staff.TimeSignature \tempo 1 = 90 \set Score.tempoHideNote = ##t c1 cih cis cisih d dih dis disih e eih f fih fis fisih g gih gis gisih a aih ais aisih b bih \bar "|" c \bar "|." \break c1 ceh b beh bes beseh a aeh aes aeseh g geh ges geseh f feh e eeh ees eeseh d deh des deseh \bar "|" c \bar "|." } </score>

The quarter tone scale may be primarily a theoretical 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.<ref name=Marcus/>

Previously, pitches of a mode were chosen from a scale consisting of seventeen tones, developed by Safi al-Din al-Urmawi in the thirteenth century.<ref name=Maalouf/>

File:24ed2.svg
19-Limit just intonation intervals approximated in 24 Template:Sc

Composer Charles Ives chose the chord C–DTemplate:Music–F–GTemplate:Music–BTemplate:Music 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<ref name=Boatwright/> Here is the secondary "minor" and its "first inversion": <score> \relative c' { \omit Staff.TimeSignature \set Score.tempoHideNote = ##t <c dih f gih bes>1 \bar "|" <c f gih bes dih>1 \bar "|." } </score>

In popular Western musicEdit

The bass descent of Nancy Sinatra's version of "These Boots Are Made for Walkin'Template:-" includes quarter tone descents.<ref> Template:Cite book </ref> Several quarter-tone albums have been recorded by Jute Gyte, a one-man avantgarde black metal band from Missouri, US.<ref> Template:Cite news </ref><ref> Template:Cite AV media </ref> Another quartertone metal album was issued by the Swedish band Massive Audio Nerve.<ref> Template:Cite news </ref> Australian psychedelic rock band King Gizzard & the Lizard Wizard's albums Flying Microtonal Banana, K.G., and L.W. heavily emphasize quarter-tones and used a custom-built guitar in 24 Template:Sc tuning.<ref> Template:Cite news </ref> Jazz violinist / violist Mat Maneri, in conjunction with his father Joe Maneri, made a crossover fusion album, Pentagon (2005),<ref> Template:Cite AV media </ref> that featured experiments in hip hop with quarter tone pianos, as well as electric organ and mellotron textures, along with distorted trombone, in a post-Bitches Brew type of mixed jazz / rock.<ref> Template:Cite magazine </ref>

Later, Seppe Gebruers started playing and improvising with two pianos tuned a quarter-tone apart. In 2019 he started a research project at the Royal Conservatory of Ghent, titled 'Unexplored possibilities of contemporary improvisation and the influence of microtonality in the creation process'.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> With two pianos tuned a quarter tone apart Gebruers recorded 'The Room: Time & Space' (2018) in a trio formation with drummer Paul Lovens and bassist Hugo Anthunes. In his solo project 'Playing with standards' (album release January 2023), Gebruers plays with famous songs including jazz standards. With Paul Lytton and Nils Vermeulen he forms a 'Playing with standards' trio.

Ancient Greek tetrachordsEdit

The enharmonic genus of the Greek tetrachord consisted of a ditone or an approximate major third, and a semitone, which was divided into two microtones. Aristoxenos, 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).<ref name=Chalmers-Divisions-5-49> Template:Cite book </ref><ref name=West1992> Template:Cite book </ref>

File:Greek Dorian enharmonic genus.png
Greek Dorian enharmonic genus: two disjunct tetrachords each of a quarter tone, quarter tone, and major third.File:Greek Dorian mode on E, enharmonic genus.mid

Interval size in equal temperamentEdit

Here are the sizes of some common intervals in a 24-note equally tempered scale, with the interval names proposed by Alois Hába (neutral third, etc.) and Ivan Wyschnegradsky (major fourth, etc.):

Interval name Size
(steps)
Size
(cents)
MIDI Just ratio Just
(cents)
MIDI Error
(cents)
octave 24 1200 File:Perfect octave on C.mid 2:1 1200.00 File:Perfect octave on C.mid Template:00.00
semidiminished octave 23 1150 File:Supermajor seventh on C.mid 35:18 1151.23 File:Septimal supermajor seventh on C.mid Template:01.23
supermajor seventh 23 1150 File:Supermajor seventh on C.mid 27:14 1137.04 File:Septimal major seventh on C.mid +12.96
major seventh 22 1100 File:Major seventh on C.mid 15:8Template:0 1088.27 File:Just major seventh on C.mid +11.73
neutral seventh, major tone 21 1050 File:Neutral seventh on C.mid 11:6Template:0 1049.36 File:Undecimal neutral seventh on C.mid +Template:00.64
neutral seventh, minor tone 21 1050 File:Neutral seventh on C.mid 20:11 1035.00 File:Lesser undecimal neutral seventh on C.mid +15.00
large just minor seventh 20 1000 File:Minor seventh on C.mid 9:5 1017.60 File:Greater just minor seventh on C.mid −17.60
small just minor seventh 20 1000 File:Minor seventh on C.mid 16:9Template:0 Template:0996.09 File:Lesser just minor seventh on C.mid +Template:03.91
supermajor sixth/subminor seventh 19 Template:0950 File:Subminor seventh on C.mid 7:4 Template:0968.83 File:Harmonic seventh on C.mid −18.83
major sixth 18 Template:0900 File:Major sixth on C.mid 5:3 Template:0884.36 File:Just major sixth on C.mid +15.64
neutral sixth 17 Template:0850 File:Neutral sixth on C.mid 18:11 Template:0852.59 File:Undecimal neutral sixth on C.mid Template:02.59
minor sixth 16 Template:0800 File:Minor sixth on C.mid 8:5 Template:0813.69 File:Just minor sixth on C.mid −13.69
subminor sixth 15 Template:0750 File:Subminor sixth on C.mid 14:9Template:0 Template:0764.92 File:Septimal minor sixth on C.mid −14.92
perfect fifth 14 Template:0700 File:Perfect fifth on C.mid 3:2 Template:0701.96 File:Just perfect fifth on C.mid Template:01.96
minor fifth 13 Template:0650 File:Thirteen quarter tones on C.mid 16:11 Template:0648.68 File:Eleventh harmonic inverse on C.mid +Template:01.32
lesser septimal tritone 12 Template:0600 File:Tritone on C.mid 7:5 Template:0582.51 File:Lesser septimal tritone on C.mid +17.49
major fourth 11 Template:0550 File:Eleven quarter tones on C.mid 11:8Template:0 Template:0551.32 File:Eleventh harmonic on C.mid Template:01.32
perfect fourth 10 Template:0500 File:Perfect fourth on C.mid 4:3 Template:0498.04 File:Just perfect fourth on C.mid +Template:01.96
tridecimal major third Template:09 Template:0450 File:Nine quarter tones on C.mid 13:10 Template:0454.21 File:Tridecimal major third on C.mid Template:04.21
septimal major third Template:09 Template:0450 File:Nine quarter tones on C.mid 9:7 Template:0435.08 File:Septimal major third on C.mid +14.92
major third Template:08 Template:0400 File:Major third on C.mid 5:4 Template:0386.31 File:Just major third on C.mid +13.69
undecimal neutral third Template:07 Template:0350 File:Neutral third on C.mid 11:9Template:0 Template:0347.41 File:Undecimal neutral third on C.mid +Template:02.59
minor third Template:06 Template:0300 File:Minor third on C.mid 6:5 Template:0315.64 File:Just minor third on C.mid −15.64
septimal minor third Template:05 Template:0250 File:Five quarter tones on C.mid 7:6 Template:0266.87 File:Septimal minor third on C.mid −16.87
tridecimal five-quarter tone Template:05 Template:0250 File:Five quarter tones on C.mid 15:13 Template:0247.74 File:Tridecimal five-quarter tone on C.mid +Template:02.26
septimal whole tone Template:05 Template:0250 File:Five quarter tones on C.mid 8:7 Template:0231.17 File:Septimal major second on C.mid +18.83
major second, major tone Template:04 Template:0200 File:Major second on C.mid 9:8 Template:0203.91 File:Major tone on C.mid Template:03.91
major second, minor tone Template:04 Template:0200 File:Major second on C.mid 10:9Template:0 Template:0182.40 File:Minor tone on C.mid +17.60
neutral second, greater undecimal Template:03 Template:0150 File:Neutral second on C.mid 11:10 Template:0165.00 File:Greater undecimal neutral second on C.mid −15.00
neutral second, lesser undecimal Template:03 Template:0150 File:Neutral second on C.mid 12:11 Template:0150.64 File:Lesser undecimal neutral second on C.mid Template:00.64
15:14 semitone Template:02 Template:0100 File:Minor second on C.mid 15:14 Template:0119.44 File:Septimal diatonic semitone on C.mid −19.44
diatonic semitone, just Template:02 Template:0100 File:Minor second on C.mid 16:15 Template:0111.73 File:Just diatonic semitone on C.mid −11.73
21:20 semitone Template:02 Template:0100 File:Minor second on C.mid 21:20 Template:084.47 File:Septimal chromatic semitone on C.mid +15.53
28:27 semitone Template:01 Template:050 File:Quarter tone on C.mid 28:27 Template:062.96 File:Septimal minor second on C.mid −12.96
33:32 semitone Template:01 Template:050 File:Quarter tone on C.mid 33:32 Template:053.27 File:Thirty-third harmonic on C.mid −3.27
unison Template:00 Template:00 File:Middle C.mid 1:1 Template:00.00 File:Middle C.mid Template:00.00

Moving from 12-TET 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 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.

See alsoEdit

ReferencesEdit

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Further readingEdit

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External linksEdit

Template:Intervals Template:Microtonal music Template:Musical tuning