96 equal temperament
{{Short description|Musical scale with a 96-step octave}}
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 , 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 that are played sequentially, and this amount varies according to 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.{{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 |access-date=26 February 2019 |date=2005}}
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
As it can become 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)
Since the 16th-tone piano has a 97-key layout arranged in 8 conventional piano "octaves", music for it is usually notated according to the key the player has to strike. While the entire range of the instrument is only C4–C5, the notation ranges from C0 to C8. Thus, written D0 corresponds to sounding C{{music|up}}{{music|up}}4 or note 2, and written A♭/G♯2 corresponds to sounding E4 or note 32.
Interval size
Below are some intervals in 96-EDO and how well they approximate just intonation.
| class="wikitable"
! interval name ! size (steps) ! size (cents) ! midi ! just ratio ! just (cents) ! midi ! error (cents) |
| octave
|align=center| 96 |align=center| 1200 | {{Audio|Perfect octave on C.mid|play}} |align=center| 2:1 |align=center| 1200.00 | {{Audio|Perfect octave on C.mid|play}} |align=center| {{0|+0}}0.00 |
| semidiminished octave
|align=center| 92 |align=center| 1150 | {{Audio|Supermajor seventh on C.mid|play}} |align=center| 35:18 |align=center| 1151.23 | {{Audio|Septimal supermajor seventh on C.mid|play}} |align=center| −{{0}}1.23 |
| supermajor seventh
|align=center| 91 |align=center| 1137.5 | |align=center| 27:14 |align=center| 1137.04 | {{Audio|Septimal major seventh on C.mid|play}} |align=center| +{{0}}0.46 |
| major seventh
|align=center| 87 |align=center| 1087.5 | |align=center| 15:8{{0}} |align=center| 1088.27 | {{Audio|Just major seventh on C.mid|play}} |align=center| −{{0}}0.77 |
| neutral seventh, major tone
|align=center| 84 |align=center| 1050 | {{Audio|Neutral seventh on C.mid|play}} |align=center| 11:6{{0}} |align=center| 1049.36 | {{Audio|Undecimal neutral seventh on C.mid|play}} |align=center| +{{0}}0.64 |
| neutral seventh, minor tone
|align=center| 83 |align=center| 1037.5 | |align=center| 20:11 |align=center| 1035.00 | {{Audio|Lesser undecimal neutral seventh on C.mid|play}} |align=center| +{{0}}2.50 |
| large just minor seventh
|align=center| 81 |align=center| 1012.5 | |align=center| 9:5 |align=center| 1017.60 | {{Audio|Greater just minor seventh on C.mid|play}} |align=center| −{{0}}5.10 |
| small just minor seventh
|align=center| 80 |align=center| 1000 | {{Audio|Minor seventh on C.mid|play}} |align=center| 16:9{{0}} |align=center| {{0}}996.09 | {{Audio|Lesser just minor seventh on C.mid|play}} |align=center| +{{0}}3.91 |
| harmonic seventh
|align=center| 78 |align=center| {{0}}975 | |align=center| 7:4 |align=center| {{0}}968.83 | {{Audio|Harmonic seventh on C.mid|play}} |align=center| +{{0}}6.17 |
| supermajor sixth
|align=center| 75 |align=center| 937.5 | |align=center| 12:7 |align=center| 933.13 | {{Audio|Septimal major sixth on C.mid|play}} |align=center| + 4.17 |
| major sixth
|align=center| 71 |align=center| {{0}}887.5 | |align=center| 5:3 |align=center| {{0}}884.36 | {{Audio|Just major sixth on C.mid|play}} |align=center| +{{0}}3.14 |
| neutral sixth
|align=center| 68 |align=center| {{0}}850 | {{Audio|Neutral sixth on C.mid|play}} |align=center| 18:11 |align=center| {{0}}852.59 | {{Audio|Undecimal neutral sixth on C.mid|play}} |align=center| −{{0}}2.59 |
| minor sixth
|align=center| 65 |align=center| {{0}}812.5 | |align=center| 8:5 |align=center| {{0}}813.69 | {{Audio|Just minor sixth on C.mid|play}} |align=center| −{{0}}1.19 |
| subminor sixth
|align=center| 61 |align=center| {{0}}762.5 | |align=center| 14:9{{0}} |align=center| {{0}}764.92 | {{Audio|Septimal minor sixth on C.mid|play}} |align=center| −{{0}}2.42 |
| perfect fifth
|align=center| 56 |align=center| {{0}}700 | {{Audio|Perfect fifth on C.mid|play}} |align=center| 3:2 |align=center| {{0}}701.96 | {{Audio|Just perfect fifth on C.mid|play}} |align=center| −{{0}}1.96 |
| minor fifth
|align=center| 52 |align=center| {{0}}650 | {{Audio|Thirteen quarter tones on C.mid|play}} |align=center| 16:11 |align=center| {{0}}648.68 | {{Audio|Eleventh harmonic inverse on C.mid|play}} |align=center| +{{0}}1.32 |
| lesser septimal tritone
|align=center| 47 |align=center| {{0}}587.5 | |align=center| 7:5 |align=center| {{0}}582.51 | {{Audio|Lesser septimal tritone on C.mid|play}} |align=center| +{{0}}4.99 |
| major fourth
|align=center| 44 |align=center| {{0}}550 | {{Audio|Eleven quarter tones on C.mid|play}} |align=center| 11:8{{0}} |align=center| {{0}}551.32 | {{Audio|Eleventh harmonic on C.mid|play}} |align=center| −{{0}}1.32 |
| perfect fourth
|align=center| 40 |align=center| {{0}}500 | {{Audio|Perfect fourth on C.mid|play}} |align=center| 4:3 |align=center| {{0}}498.04 | {{Audio|Just perfect fourth on C.mid|play}} |align=center| +{{0}}1.96 |
| tridecimal major third
|align=center| 36 |align=center| {{0}}450 | {{Audio|Nine quarter tones on C.mid|play}} |align=center| 13:10 |align=center| {{0}}454.21 | {{Audio|Tridecimal major third on C.mid|play}} |align=center| −{{0}}4.21 |
| septimal major third
|align=center| 35 |align=center| {{0}}437.5 | |align=center| 9:7 |align=center| {{0}}435.08 | {{Audio|Septimal major third on C.mid|play}} |align=center| +{{0}}2.42 |
| major third
|align=center| 31 |align=center| {{0}}387.5 | |align=center| 5:4 |align=center| {{0}}386.31 | {{Audio|Just major third on C.mid|play}} |align=center| +{{0}}1.19 |
| undecimal neutral third
|align=center| 28 |align=center| {{0}}350 | {{Audio|Neutral third on C.mid|play}} |align=center| {{0}}11:9 |align=center| {{0}}347.41 | {{Audio|Undecimal neutral third on C.mid|play}} |align=center| +{{0}}2.59 |
| superminor third
|align=center| 27 |align=center| {{0}}337.5 | |align=center| {{0}}17:14 |align=center| {{0}}336.13 | {{Audio|Superminor third on C.mid|play}} |align=center| +{{0}}1.37 |
| 77th harmonic
|align=center| 26 |align=center| {{0}}325 | {{Audio|13 steps in 48-et on C.mid|play}} |align=center| {{0}}77:64 |align=center| {{0}}320.14 | {{Audio|Seventy-seventh harmonic on C.mid|play}} |align=center| +{{0}}4.86 |
| minor third
|align=center| 25 |align=center| {{0}}312.5 | |align=center| 6:5 |align=center| {{0}}315.64 | {{Audio|Just minor third on C.mid|play}} |align=center| −{{0}}3.14 |
| second septimal minor third
|align=center|24 |align=center| {{0}}300 | {{Audio|Minor third on C.mid|play}} |align=center| 25:21 |align=center| {{0}}301.85 | {{Audio|Second septimal minor third on C.mid|play}} |align=center| −{{0}}1.85 |
| tridecimal minor third
|align=center| 23 |align=center| {{0}}287.5 | |align=center| 13:11 |align=center| {{0}}289.21 | {{Audio|Tridecimal minor third on C.mid|play}} |align=center| −{{0}}1.71 |
| augmented second, just
|align=center| 22 |align=center| {{0}}275 | {{Audio|11 steps in 48-et on C.mid|play}} |align=center| 75:64 |align=center| {{0}}274.58 | {{Audio|Just augmented second on C.mid|play}} |align=center| +{{0}}0.42 |
| septimal minor third
|align=center| 21 |align=center| {{0}}262.5 | |align=center| 7:6 |align=center| {{0}}266.87 | {{Audio|Septimal minor third on C.mid|play}} |align=center| −{{0}}4.37 |
| tridecimal five-quarter tone
|align=center| 20 |align=center| {{0}}250 | {{Audio|Five quarter tones on C.mid|play}} |align=center| 15:13 |align=center| {{0}}247.74 | {{Audio|Tridecimal five-quarter tone on C.mid|play}} |align=center| +{{0}}2.26 |
| septimal whole tone
|align=center| 18 |align=center| {{0}}225 | |align=center| 8:7 |align=center| {{0}}231.17 | {{Audio|Septimal major second on C.mid|play}} |align=center| −{{0}}6.17 |
| major second, major tone
|align=center| 16 |align=center| {{0}}200 | {{Audio|Major second on C.mid|play}} |align=center| 9:8 |align=center| {{0}}203.91 | {{Audio|Major tone on C.mid|play}} |align=center| −{{0}}3.91 |
| major second, minor tone
|align=center| 15 |align=center|{{0}}187.5 | |align=center| 10:9{{0}} |align=center| {{0}}182.40 | {{Audio|Minor tone on C.mid|play}} |align=center| +{{0}}5.10 |
| neutral second, greater undecimal
|align=center| 13 |align=center| {{0}}162.5 | |align=center| 11:10 |align=center| {{0}}165.00 | {{Audio|Greater undecimal neutral second on C.mid|play}} |align=center| −{{0}}2.50 |
| neutral second, lesser undecimal
|align=center| 12 |align=center| {{0}}150 | {{Audio|Neutral second on C.mid|play}} |align=center| 12:11 |align=center| {{0}}150.64 | {{Audio|Lesser undecimal neutral second on C.mid|play}} |align=center| −{{0}}0.64 |
| greater tridecimal {{frac|2|3}}-tone
|align=center| 11 |align=center| {{0}}137.5 | |align=center| 13:12 |align=center| {{0}}138.57 | {{Audio|Greater tridecimal two-third tone on C.mid|play}} |align=center| −{{0}}1.07 |
| septimal diatonic semitone
|align=center| 10 |align=center| {{0}}125 | {{Audio|5 steps in 48-et on C.mid|play}} |align=center| 15:14 |align=center| {{0}}119.44 | {{Audio|Septimal diatonic semitone on C.mid|play}} |align=center| +{{0}}5.56 |
| diatonic semitone, just
|align=center| {{0}}9 |align=center| {{0}}112.5 | |align=center| 16:15 |align=center| {{0}}111.73 | {{Audio|Just diatonic semitone on C.mid|play}} |align=center| +{{0}}0.77 |
| undecimal minor second
|align=center| {{0}}8 |align=center| {{0}}100 | {{Audio|Minor second on C.mid|play}} |align=center| 128:121 |align=center| {{0|00}}97.36 | {{Audio|121st subharmonic on C.mid|play}} |align=center| −{{0}}2.64 |
| septimal chromatic semitone
|align=center| {{0}}7 |align=center| {{0}}87.5 | |align=center| 21:20 |align=center| {{0|00}}84.47 | {{Audio|Septimal chromatic semitone on C.mid|play}} |align=center| +{{0}}3.03 |
| just chromatic semitone
|align=center| {{0}}6 |align=center| {{0}}75 | {{Audio|1 step in 16-et on C.mid|play}} |align=center| 25:24 |align=center| {{0|00}}70.67 | {{Audio|Just chromatic semitone on C.mid|play}} |align=center| +{{0}}4.33 |
| septimal minor second
|align=center| {{0}}5 |align=center| {{0}}62.5 | |align=center| 28:27 |align=center| {{0|00}}62.96 | {{Audio|Septimal minor second on C.mid|play}} |align=center| −{{0}}0.46 |
| undecimal quarter-tone
|align=center|{{0}}4 |align=center| {{0|00}}50 | {{Audio|Quarter tone on C.mid|play}} |align=center| 33:32 |align=center| {{0|00}}53.27 | {{Audio|Thirty-third harmonic on C.mid|play}} |align=center| −{{0}}3.27 |
| undecimal diesis
|align=center| {{0}}3 |align=center| {{0|00}}37.5 | |align=center| 45:44 |align=center| {{0|00}}38.91 | {{Audio|Undecimal diesis on C.mid|play}} |align=center| −{{0}}1.41 |
| septimal comma
|align=center| {{0}}2 |align=center| {{0|00}}25 | {{Audio|Eighth-tone on C.mid|play}} |align=center| 64:63 |align=center| {{0|00}}27.26 | {{Audio|Septimal comma on C.mid|play}} |align=center| −{{0}}2.26 |
| septimal semicomma
|align=center| {{0}}1 |align=center| {{0|00}}12.5 | {{Audio|Sixteenth-tone on C.mid|play}} |align=center| 126:125 |align=center| {{0|00}}13.79 | {{Audio|Septimal semicomma on C.mid|play}} |align=center| −{{0}}1.29 |
| unison
|align=center| {{0}}0 |align=center| {{0|000}}0 | {{Audio|Middle C.mid|play}} |align=center| 1:1 |align=center| {{0|000}}0.00 | {{Audio|Middle C.mid|play}} |align=center| {{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
References
{{Reflist}}