96 equal temperament
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 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 that are 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
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↑, C↑↑/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)
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↑↑4 or note 2, and written A♭/G♯2 corresponds to sounding E4 or note 32.
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)|
|neutral seventh, major tone||84||1050||11:6||1049.36||+0.64|
|neutral seventh, minor tone||83||1037.5||20:11||1035.00||+2.50|
|large just minor seventh||81||1012.5||9:5||1017.60||−5.10|
|small just minor seventh||80||1000||16:9||996.09||+3.91|
|supermajor sixth/subminor seventh||78||975||7:4||968.83||+6.17|
|lesser septimal tritone||47||587.5||7:5||582.51||+4.99|
|tridecimal major third||36||450||13:10||454.21||−4.21|
|septimal major third||35||437.5||9:7||435.08||+2.42|
|undecimal neutral third||28||350||11:9||347.41||+2.59|
|Second septimal minor third||24||300||25:21||301.85||−1.85|
|Tridecimal minor third||23||287.5||13:11||289.21||−1.71|
|augmented second, just||22||275||75:64||274.58||+0.42|
|septimal minor third||21||262.5||7:6||266.87||−4.37|
|tridecimal five-quarter tone||20||250||15:13||247.74||+2.26|
|septimal whole tone||18||225||8:7||231.17||−6.17|
|major second, major tone||16||200||9:8||203.91||−3.91|
|major second, minor tone||15||187.5||10:9||182.40||+5.10|
|neutral second, greater undecimal||13||162.5||11:10||165.00||−2.50|
|neutral second, lesser undecimal||12||150||12:11||150.64||−0.64|
|Greater tridecimal ⅔-tone||11||137.5||13:12||138.57||−1.07|
|Septimal diatonic semitone||10||125||15:14||119.44||+5.56|
|diatonic semitone, just||9||112.5||16:15||111.73||+0.77|
|Undecimal minor second (121st subharmonic)||8||100||128:121||97.36||−2.64|
|Septimal chromatic semitone||7||87.5||21:20||84.47||+3.03|
|Just chromatic semitone||6||75||25:24||70.67||+4.33|
|Septimal minor second||5||62.5||28:27||62.96||−0.46|
|Undecimal quarter-tone (33rd harmonic)||4||50||33:32||53.27||−3.27|
Moving from 12-EDO to 96-EDO allows the better approximation of a number of intervals, such as the minor third and major sixth.
96-EDO contains all of the 12-EDO modes. However, it contains better approximations to some intervals (such as the minor third).
- Monzo, Joe (2005). "Equal-Temperament". Tonalsoft Encyclopedia of Microtonal Music Theory. Joe Monzo. Retrieved 26 February 2019.
- Sonido 13, Julián Carillo's theory of 96-EDO