Quarter tone

"24 equal temperament" redirects here. For other uses, see Arab tone system.

A quarter tone <phonos file="Quarter tone on C.mid">play</phonos>, is a pitch halfway between the usual notes of a chromatic scale or an interval about half as wide (aurally, or logarithmically) as a semitone, which is half a whole tone.

File:Heckel ViertelTonTRP C.jpg

Trumpet with 3 normal valves and a quartering on the extension valve (right).

Many composers are known for having written music including quarter tones or the quarter-tone scale (24 equal temperament), first proposed by 19th-century music theorist Mikha'il Mishaqah, and in 1823 by the German theorist Heinrich Richter, including: 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 Alexandrovich Wyschnegradsky, and Iannis Xenakis (see List of quarter tone pieces).

Types of quarter tones

Composer Charles Ives chose the four-note chord above 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. <phonos file="Ives fundamental chord (quarter tones).ogg">Play</phonos> or <phonos file="Ives quarter tone fundamental chord arp.mid">play</phonos>

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 2^1/24 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 <phonos file="Neutral second on C.mid">play</phonos> or neutral second, half of a minor third.

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, 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).

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 3 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.

Composer Ben Johnston, to accommodate the just septimal quarter tone, uses a small "7" () as an accidental to indicate a note is lowered 49 cents, or an upside down "∠" () to indicate a note is raised 49 cents, or a ratio of 36/35. Johnston uses an upward and downward arrow to indicate a note is raised or lowered by a ratio of 33/32, or 53 cents.

Playing quarter tones on musical instruments

A quarter tone clarinet by Fritz Schüller.

Because many musical instruments manufactured today 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 or with special tunings)
• 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, 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)

Conventional musical instruments that can play quarter tones include

• Electronic instruments:
  • Synthesizers, using either special keyboard controllers or continuous-pitch controllers such as fingerboard controllers, or when controlled by a sequencer capable of outputting quarter-tone control signals
  • Theremins and other continuously pitched instruments
• Fretless string instruments, such as fretless guitars, fretless electric basses, ouds, members of the huqin family of instruments, and members of the violin family
• String instruments with movable frets (such as the sitar)
• Specially fretted string instruments
• Pedal steel guitar
• Wind instruments whose main means of tone-control is a slide, such as trombones, the tromboon, slide trumpet and slide whistles
• Specially keyed woodwind instruments
• Valved brass instruments with extra, quarter-tone valves
• Kazoo
• Pitched percussion instruments, when tuning permits (e.g., timpani), or using special techniques

Experimental instruments have been built to play in quarter tones, for example a quarter tone clarinet by Fritz Schüller (1883–1977) of Markneukirchen.

Other instruments can be used to play quarter tones when using audio signal processing effects such as pitch shifting.

Pairs of conventional instruments tuned a quarter tone apart can be used to play some quarter tone music. Indeed, quarter-tone pianos have been built, which consist essentially of two pianos stacked one above the other in a single case, one tuned a quarter tone higher than the other.

Music of the Middle East

See also: Persian traditional music, Arabic music, and Arab tone system

While the use of quarter tones in modern Western music is a more recent and experimental phenomenon, these and other microtonal intervals have been an important part of the music of Iran (Persia), the Arab world, Armenia, Turkey, Assyria, Kurdistan, and neighboring lands and areas for many centuries.

File:QuartertoneMaqams.png

300px

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

1. Shoor (Bayati)

   شور (بیاتی)

   D E F G A B♭ C D

2. Rast

   راست

   C D E F G A B C

   with a B♭ replacing the B in the descending scale

3. Sabba

   صبا

   D E F G♭ A B♭ C D

4. Siga

   سه گاه

   E F G A B C D E

5. ‘Ajam
6. Hussayni

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:

• 1 - Qadmoyo (Bayati)
• 2 - Trayono (Hussayni)
• 3 - Tlithoyo (Segah)
• 4 - Rbi‘oyo (Rast)
• 5 - Hmishoyo
• 6 - Shtithoyo (‘Ajam)
• 7 - Shbi‘oyo
• 8 - Tminoyo

Quarter tone scale

Quarter tone scale on C ascending and descending. <phonos file="Quarter tone scale on C.mid">Play</phonos>

File:Ives quarter tone secondary chord.png

Composer Charles Ives chose the chord above 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.

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. The invention of the scale is attributed to Mikhail Mishaqa whose work Essay on the Art of Music for the Emir Shihāb (al-Risāla al-shihābiyya fi 'l-ṣinā‘a al-mūsīqiyya) is devoted to the topic but also makes clear his teacher Sheikh Muhammad al-‘Attār (1764-1828) was one of many already familiar with the concept.

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.

Previously, pitches of a mode were chosen from a scale consisting of seventeen tones, developed by Safi 'I-Din al-Urmawi in the thirteenth century.

In popular music

The Japanese multi-instrumentalist and experimental musical instrument builder Yuichi Onoue developed a 24-TET quarter tone tuning on his guitar. Norwegian guitarist Ronni Le Tekrø of the band TNT used a quarter-step guitar on the band's third studio album, Intuition.

Ancient Greek tetrachords

Greek Dorian enharmonic genus: two disjunct tetrachords each of a quarter tone, quarter tone, and major third. <phonos file="Greek Dorian mode on E, enharmonic genus.mid">Play</phonos>

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).

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 Alois Hába (neutral third, etc.) and Ivan Wyschnegradsky (major fourth, etc.):

interval name	size (steps)	size (cents)	midi	just ratio	just (cents)	midi	error
octave	24	1200	<phonos file="Perfect octave on C.mid">play</phonos>	2:1	1200.00	<phonos file="Perfect octave on C.mid">play</phonos>	0.00
semidiminished octave	23	1150		35:18	1151.23	<phonos file="Perfect octave on C.mid">play</phonos>	−1.23
supermajor seventh	23	1150		27:14	1137.03	<phonos file="Septimal major seventh on C.mid">play</phonos>	+12.96
major seventh	22	1100	<phonos file="Major seventh on C.mid">play</phonos>	15:8	1088.27	<phonos file="Just major seventh on C.mid">play</phonos>	+11.73
neutral seventh	21	1050	<phonos file="Neutral seventh on C.mid">play</phonos>	11:6	1049.36		+0.64
large just minor seventh	20	1000	<phonos file="Minor seventh on C.mid">play</phonos>	9:5	1017.60	<phonos file="Greater just minor seventh on C.mid">play</phonos>	−17.60
small just minor seventh	20	1000	<phonos file="Minor seventh on C.mid">play</phonos>	16:9	996.09	<phonos file="Lesser just minor seventh on C.mid">play</phonos>	+3.91
supermajor sixth/subminor seventh	19	950		7:4	968.83	<phonos file="Harmonic seventh on C.mid">play</phonos>	−18.83
major sixth	18	900	<phonos file="Major sixth on C.mid">play</phonos>	5:3	884.36	<phonos file="Just major sixth on C.mid">play</phonos>	+15.64
neutral sixth	17	850	<phonos file="Neutral sixth on C.mid">play</phonos>	18:11	852.59	<phonos file="Undecimal neutral sixth on C.mid">play</phonos>	−2.59
minor sixth	16	800	<phonos file="Minor sixth on C.mid">play</phonos>	8:5	813.69	<phonos file="Just minor sixth on C.mid">play</phonos>	−13.69
subminor sixth	15	750		14:9	764.92	<phonos file="Septimal minor sixth on C.mid">play</phonos>	−14.92
perfect fifth	14	700	<phonos file="Perfect fifth on C.mid">play</phonos>	3:2	701.95	<phonos file="Just perfect fifth on C.mid">play</phonos>	−1.95
minor fifth	13	650	<phonos file="Thirteen quarter tones on C.mid">play</phonos>	16:11	648.68		+1.32
lesser septimal tritone	12	600	<phonos file="Tritone on C.mid">play</phonos>	7:5	582.51	<phonos file="Lesser septimal tritone on C.mid">play</phonos>	+17.49
major fourth	11	550	<phonos file="Eleven quarter tones on C.mid">play</phonos>	11:8	551.32	<phonos file="Eleventh harmonic on C.mid">play</phonos>	−1.32
perfect fourth	10	500	<phonos file="Perfect fourth on C.mid">play</phonos>	4:3	498.05	<phonos file="Just perfect fourth on C.mid">play</phonos>	+1.95
tridecimal major third	9	450	<phonos file="Nine quarter tones on C.mid">play</phonos>	13:10	454.21	<phonos file="Tridecimal major third on C.mid">play</phonos>	−4.21
septimal major third	9	450	<phonos file="Nine quarter tones on C.mid">play</phonos>	9:7	435.08	<phonos file="Septimal major third on C.mid">play</phonos>	+14.92
major third	8	400	<phonos file="Major third on C.mid">play</phonos>	5:4	386.31	<phonos file="Just major third on C.mid">play</phonos>	+13.69
undecimal neutral third	7	350	<phonos file="Neutral third on C.mid">play</phonos>	11:9	347.41	<phonos file="Undecimal neutral third on C.mid">play</phonos>	+2.59
minor third	6	300	<phonos file="Minor third on C.mid">play</phonos>	6:5	315.64	<phonos file="Just minor third on C.mid">play</phonos>	−15.64
septimal minor third	5	250	<phonos file="Five quarter tones on C.mid">play</phonos>	7:6	266.88	<phonos file="Septimal minor third on C.mid">play</phonos>	−16.88
tridecimal minor third	5	250	<phonos file="Five quarter tones on C.mid">play</phonos>	15:13	247.74	<phonos file="Tridecimal minor third on C.mid">play</phonos>	+2.26
septimal whole tone	5	250	<phonos file="Five quarter tones on C.mid">play</phonos>	8:7	231.17	<phonos file="Septimal major second on C.mid">play</phonos>	+18.83
whole tone, major tone	4	200	<phonos file="Major second on C.mid">play</phonos>	9:8	203.91	<phonos file="Major tone on C.mid">play</phonos>	−3.91
whole tone, minor tone	4	200	<phonos file="Major second on C.mid">play</phonos>	10:9	182.40	<phonos file="Minor tone on C.mid">play</phonos>	+17.60
neutral second, greater undecimal	3	150	<phonos file="Neutral second on C.mid">play</phonos>	11:10	165.00	<phonos file="Greater undecimal neutral second on C.mid">play</phonos>	−15.00
neutral second, lesser undecimal	3	150	<phonos file="Neutral second on C.mid">play</phonos>	12:11	150.64	<phonos file="Lesser undecimal neutral second on C.mid">play</phonos>	−0.64
15:14 semitone	2	100	<phonos file="Minor second on C.mid">play</phonos>	15:14	119.44	<phonos file="Septimal diatonic semitone on C.mid">play</phonos>	−19.44
diatonic semitone, just	2	100	<phonos file="Minor second on C.mid">play</phonos>	16:15	111.73	<phonos file="Just diatonic semitone on C.mid">play</phonos>	−11.73
21:20 semitone	2	100	<phonos file="Minor second on C.mid">play</phonos>	21:20	84.47	<phonos file="Just diatonic semitone on C.mid">play</phonos>	+15.53
28:27 semitone	1	50	<phonos file="Quarter tone on C.mid">play</phonos>	28:27	62.96	<phonos file="Just diatonic semitone on C.mid">play</phonos>	−12.96
septimal quarter tone	1	50	<phonos file="Quarter tone on C.mid">play</phonos>	36:35	48.77	<phonos file="Septimal quarter tone on C.mid">play</phonos>	+1.23

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 also

• Musical temperament
• Microtonal music
• List of quarter tone pieces
• List of meantone intervals

Further reading

• Bartolozzi, Bruno (1967). New Sounds for Woodwind. London, New York: Oxford University Press.
• Bousted, Donald (2002). "Microtonality, the Recorder and the Quarter-Tone Recorder Manual". The Recorder Magazine 22, no. 3 (Fall): 99–102.
• Bousted, Donald (2005). "Next Step Quarter-Tone Resources: Melody". The Recorder Magazine 25, no. 3 (Fall): 88–91.
• Caravan, Ronald R. (1979). Preliminary Exercises and Etudes in Contemporary Techniques for Clarinet: Introductory Material for the Study of Multiphonics, Quarter Tones, and Timbre Variation. [Oswego, N.Y.]: Ethos Publications.
• Ellis, Don (1975). Quarter Tones: A Text with Musical Examples, Exercises and Etudes. Plainview, N.Y.: Harold Branch Pub. Co.
• MacDonald, John (1822). A Treatise on the Harmonic System Arising from the Vibrations of the Aliquot Divisions of Strings According to the Gradual Progress of the Notes from the Middle, to the Remote Extremes: Explaining Simply, by Curved Delineations, the Manner in Which the Harmonic Tones, Half and Quarter Notes, Are Generated and Produced on Every Corresponding Part of the String; and under a Copious Explanatory Description Illustrated by Musical and Appropriate Plates, Giving an Easy and Familiar Adaptation of the Whole to the Purposes of Composition and Instrumental Music, and More Particularly, to the Practice of the Violin, Tenor, Violoncello and Double Bass, on All the Strings, and in Every Compass of These Instruments, by Every Practical Mode of Execution; with Some Musical Animadversions Introductory of the General Subject, Briefly Alluding to the Rise and Progress of Music, and to the Corrections of Temperament: and Stating Various Improvements of Instruments, Experimentally Ascertained: Concluding with an Application or Two of the Principle of Musical Notes, to Purposes of Utility, and a Reference to Terms Less Generally Noticed. London: Printed for the Author, and Sold by T. Preston.
• Möllendorff, Willi, and Joe Monzo (2001). Music with Quarter-Tones: Experiences at the Bichromatic Harmonium. [United States]: J. Monzo.
• Rees, Carla (2007). "Eva Kingma and the Quarter-Tone Flute". Pan: The Flute Magazine 26, no. 4:23-29.
• Rewoldt, Todd (2000). "Altissimo Quarter-Tones for the Alto Saxophone". Saxophone Symposium 25:56–69.

External links

• "quarter-tone / 24-edo", TonalSoft.com.