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词条 96 equal temperament
释义

  1. History and use

  2. Notation

  3. Interval size

  4. Scale diagram

     Modes 

  5. See also

  6. References

  7. Further reading

In music, 96 equal temperament, called 96-TET, 96-EDO ("Equal Division of the Octave"), or 96-ET, is the tempered scale derived by dividing the octave into 96 equal steps (equal frequency ratios). Each step represents a frequency ratio of {{radic|2|96}}, or 12.5 cents. Since 96 factors into 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96, it contains all of those temperaments. Most humans can only hear differences of 6 cents on notes played sequentially, and this amount varies according the pitch, so the use of larger divisions of octave can be considered unnecessary. Smaller differences in pitch may be considered vibrato or stylistic devices.

History and use

96-EDO was first advocated by Julián Carrillo in 1924, with a 16th-tone piano. It was also advocated more recently by Pascale Criton and Vincent-Olivier Gagnon.[1]

Notation

Since 96 = 24 × 4, quarter-tone notation can be used, and split into four parts.

One can split it into four parts like this:

C, C{{music|up}}, C{{music|up}}{{music|up}}/C{{music|t}}{{music|down}}{{music|down}}, C{{music|t}}{{music|down}}, C{{music|t}}, ..., C{{music|down}}, C

Since it can get confusing with so many accidentals, Julián Carrillo proposed referring to notes by step number from C (e.g. 0, 1, 2, 3, 4, ..., 95, 0)

Interval size

Below are some intervals in 96-EDO and how well they approximate just intonation.

interval name size (steps) size (cents) midi just ratio just (cents) midi error (cents)
octave961200Perfect octave on C.mid|play}}2:11200.00Perfect octave on C.mid|play}}{{0|+0}}0.00
semidiminished octave921150Supermajor seventh on C.mid|play}}35:181151.23Septimal supermajor seventh on C.mid|play}}−{{0}}1.23
supermajor seventh911137.527:141137.04Septimal major seventh on C.mid|play}}+{{0}}0.46
major seventh871087.515:8{{0}}1088.27Just major seventh on C.mid|play}}−{{0}}0.77
neutral seventh, major tone841050Neutral seventh on C.mid|play}}11:6{{0}}1049.36Undecimal neutral seventh on C.mid|play}}+{{0}}0.64
neutral seventh, minor tone831037.520:111035.00Lesser undecimal neutral seventh on C.mid|play}}+{{0}}2.50
large just minor seventh811012.59:51017.60Greater just minor seventh on C.mid|play}}−{{0}}5.10
small just minor seventh801000Minor seventh on C.mid|play}}16:9{{0}}{{0}}996.09Lesser just minor seventh on C.mid|play}}+{{0}}3.91
supermajor sixth/subminor seventh78{{0}}9757:4{{0}}968.83Harmonic seventh on C.mid|play}}+{{0}}6.17
major sixth71{{0}}887.55:3{{0}}884.36Just major sixth on C.mid|play}}+{{0}}3.14
neutral sixth68{{0}}850Neutral sixth on C.mid|play}}18:11{{0}}852.59Undecimal neutral sixth on C.mid|play}}−{{0}}2.59
minor sixth65{{0}}812.58:5{{0}}813.69Just minor sixth on C.mid|play}}−{{0}}1.19
subminor sixth61{{0}}762.514:9{{0}}{{0}}764.92Septimal minor sixth on C.mid|play}}−{{0}}2.42
perfect fifth56{{0}}700Perfect fifth on C.mid|play}}3:2{{0}}701.96Just perfect fifth on C.mid|play}}−{{0}}1.96
minor fifth52{{0}}650Thirteen quarter tones on C.mid|play}}16:11{{0}}648.68Eleventh harmonic inverse on C.mid|play}}+{{0}}1.32
lesser septimal tritone47{{0}}587.57:5{{0}}582.51Lesser septimal tritone on C.mid|play}}+{{0}}4.99
major fourth44{{0}}550Eleven quarter tones on C.mid|play}}11:8{{0}}{{0}}551.32Eleventh harmonic on C.mid|play}}−{{0}}1.32
perfect fourth40{{0}}500Perfect fourth on C.mid|play}}4:3{{0}}498.04Just perfect fourth on C.mid|play}}+{{0}}1.96
tridecimal major third36{{0}}450Nine quarter tones on C.mid|play}}13:10{{0}}454.21Tridecimal major third on C.mid|play}}−{{0}}4.21
septimal major third35{{0}}437.59:7{{0}}435.08Septimal major third on C.mid|play}}+{{0}}2.42
major third31{{0}}387.55:4{{0}}386.31Just major third on C.mid|play}}+{{0}}1.19
undecimal neutral third28{{0}}350Neutral third on C.mid|play}}{{0}}11:9{{0}}347.41Undecimal neutral third on C.mid|play}}+{{0}}2.59
Superminor third27{{0}}337.5{{0}}17:14{{0}}336.13Superminor third on C.mid|play}}+{{0}}1.37
77th harmonic26{{0}}32513 steps in 48-et on C.mid|play}}{{0}}77:64{{0}}320.14Seventy-seventh harmonic on C.mid|play}}+{{0}}4.86
minor third25{{0}}312.56:5{{0}}315.64Just minor third on C.mid|play}}−{{0}}3.14
Second septimal minor third24{{0}}300Minor third on C.mid|play}}25:21{{0}}301.85Second septimal minor third on C.mid|play}}−{{0}}1.85
Tridecimal minor third23{{0}}287.513:11{{0}}289.21Tridecimal minor third on C.mid|play}}−{{0}}1.71
augmented second, just22{{0}}27511 steps in 48-et on C.mid|play}}75:64{{0}}274.58Just augmented second on C.mid|play}}+{{0}}0.42
septimal minor third21{{0}}262.57:6{{0}}266.87Septimal minor third on C.mid|play}}−{{0}}4.37
tridecimal five-quarter tone20{{0}}250Five quarter tones on C.mid|play}}15:13{{0}}247.74Tridecimal five-quarter tone on C.mid|play}}+{{0}}2.26
septimal whole tone18{{0}}2258:7{{0}}231.17Septimal major second on C.mid|play}}−{{0}}6.17
major second, major tone16{{0}}200Major second on C.mid|play}}9:8{{0}}203.91Major tone on C.mid|play}}−{{0}}3.91
major second, minor tone15{{0}}187.510:9{{0}}{{0}}182.40Minor tone on C.mid|play}}+{{0}}5.10
neutral second, greater undecimal13{{0}}162.511:10{{0}}165.00Greater undecimal neutral second on C.mid|play}}−{{0}}2.50
neutral second, lesser undecimal12{{0}}150Neutral second on C.mid|play}}12:11{{0}}150.64Lesser undecimal neutral second on C.mid|play}}−{{0}}0.64
Greater tridecimal ⅔-tone11{{0}}137.513:12{{0}}138.57Greater tridecimal two-third tone on C.mid|play}}−{{0}}1.07
Septimal diatonic semitone10{{0}}1255 steps in 48-et on C.mid|play}}15:14{{0}}119.44Septimal diatonic semitone on C.mid|play}}+{{0}}5.56
diatonic semitone, just{{0}}9{{0}}112.516:15{{0}}111.73Just diatonic semitone on C.mid|play}}+{{0}}0.77
Undecimal minor second (121st subharmonic){{0}}8{{0}}100Minor second on C.mid|play}}128:121{{0|00}}97.36121st subharmonic on C.mid|play}}−{{0}}2.64
Septimal chromatic semitone{{0}}7{{0}}87.521:20{{0|00}}84.47Septimal chromatic semitone on C.mid|play}}+{{0}}3.03
Just chromatic semitone{{0}}6{{0}}751 step in 16-et on C.mid|play}}25:24{{0|00}}70.67Just chromatic semitone on C.mid|play}}+{{0}}4.33
Septimal minor second{{0}}5{{0}}62.528:27{{0|00}}62.96Septimal minor second on C.mid|play}}−{{0}}0.46
Undecimal quarter-tone (33rd harmonic){{0}}4{{0|00}}50Quarter tone on C.mid|play}}33:32{{0|00}}53.27Thirty-third harmonic on C.mid|play}}−{{0}}3.27
Undecimal diesis{{0}}3{{0|00}}37.545:44{{0|00}}38.91Undecimal diesis on C.mid|play}}−{{0}}1.41
septimal comma{{0}}2{{0|00}}25Eighth-tone on C.mid|play}}64:63{{0|00}}27.26Septimal comma on C.mid|play}}−{{0}}2.26
septimal semicomma{{0}}1{{0|00}}12.5Sixteenth-tone on C.mid|play}}126:125{{0|00}}13.79Septimal semicomma on C.mid|play}}−{{0}}1.29
unison{{0}}0{{0|000}}0Middle C.mid|play}}1:1{{0|000}}0.00Middle C.mid|play}}{{0|+0}}0.00

Moving from 12-EDO to 96-EDO allows the better approximation of a number of intervals, such as the minor third and major sixth.

Scale diagram

{{Missing information|section|the scale diagram|date=February 2019}}

Modes

96-EDO contains all of the 12-EDO modes. However, it contains better approximations to some intervals (such as the minor third).

See also

  • Musical temperament
  • Equal temperament

References

1. ^{{cite web |last1=Monzo |first1=Joe |title=Equal-Temperament |url=http://tonalsoft.com/enc/e/equal-temperament.aspx#edo-table |website=Tonalsoft Encyclopedia of Microtonal Music Theory |publisher=Joe Monzo |accessdate=26 February 2019 |date=2005}}

Further reading

  • Sonido 13, Julián Carillo's theory of 96-EDO
{{Microtonal music}}{{Musical tuning}}

2 : Equal temperaments|Microtonality
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