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'''88-cent equal temperament''' ('''88cET''', also known as '''1ed88¢''' or '''APS88¢''') uses equal steps of 88 [[cent]]s each. It is equivalent to 13.6364edo, and is a subset of [[150edo]] (every eleventh step). | |||
'''88-cent equal temperament''' (also known as '''1ed88¢''' or '''APS88¢''') uses equal steps of 88 [[cent]]s each. It is equivalent to 13.6364edo, and is a subset of [[150edo]] (every eleventh step). | |||
== Theory == | == Theory == | ||
88-cent [[Equal-step tuning|equal temperament]] uses 88 cents, or 11\150 of an octave, to generate a [[nonoctave]] rank-1 scale. Since the 88-cent step is an excellent generator for the [[octacot]] temperament, it can be viewed as the generator chain of octacot, stripped of octaves. However viewed, octacot and 88-cent equal temperament are very closely related, and the chords of 88-cent equal temperament are listed on the page [[Chords of octacot]]. From this it may be seen that octacot, and hence 88 cent equal temperament , share an abundance of [[essentially tempered | 88-cent [[Equal-step tuning|equal temperament]] uses 88 cents, or 11\150 of an octave, to generate a [[nonoctave]] rank-1 scale. Since the 88-cent step is an excellent generator for the [[octacot]] temperament, it can be viewed as the generator chain of octacot, stripped of octaves. However viewed, octacot and 88-cent equal temperament are very closely related, and the chords of 88-cent equal temperament are listed on the page [[Chords of octacot]]. From this it may be seen that octacot, and hence 88 cent equal temperament , share an abundance of [[essentially tempered chord]]s. | ||
Eight steps of 88 cents gives 704 cents, two cents sharp of 3/2, and eighteen gives 1584 cents, two cents flat of 5/2. Taken together this tells us that (5/2)<sup>4</sup>/(3/2)<sup>9</sup> = [[20000/19683]], the minimal diesis or tetracot comma, must be being tempered out. Eleven steps of 88 cents gives 968 cents, less than a cent flat of 7/4, and this tells us that (7/4)<sup>8</sup>/(3/2)<sup>11</sup> = 5764801/5668704 must be tempered out also. Taking this, multiplying it by the tetracot comma and taking the fourth root yields [[245/243]], which therefore must be tempered out also. The tetracot comma and 245/243 taken together define 7-limit octacot. | Eight steps of 88 cents gives 704 cents, two cents sharp of 3/2, and eighteen gives 1584 cents, two cents flat of 5/2. Taken together this tells us that (5/2)<sup>4</sup>/(3/2)<sup>9</sup> = [[20000/19683]], the minimal diesis or tetracot comma, must be being tempered out. Eleven steps of 88 cents gives 968 cents, less than a cent flat of 7/4, and this tells us that (7/4)<sup>8</sup>/(3/2)<sup>11</sup> = 5764801/5668704 must be tempered out also. Taking this, multiplying it by the tetracot comma and taking the fourth root yields [[245/243]], which therefore must be tempered out also. The tetracot comma and 245/243 taken together define 7-limit octacot. | ||
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== Intervals == | == Intervals == | ||
{{todo|cleanup|inline=true}} | |||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
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! Some Nearby <br>JI Intervals | ! Some Nearby <br>JI Intervals | ||
|- | |- | ||
! colspan="6" | | ! colspan="6" | first octave | ||
! | ! | ||
|- | |- | ||
| Line 141: | Line 141: | ||
| 27/14=1137.039, 31/16=1145.036 | | 27/14=1137.039, 31/16=1145.036 | ||
|- | |- | ||
! colspan="6" | | ! colspan="6" | second octave | ||
! | ! | ||
|- | |- | ||
| Line 167: | Line 167: | ||
| re | | re | ||
| 9/8=203.910 | | 9/8=203.910 | ||
|- | |- | ||
| 17 | | 17 | ||
| Line 259: | Line 256: | ||
| 63/32=1172.736, 160/81=1178.494 | | 63/32=1172.736, 160/81=1178.494 | ||
|- | |- | ||
! colspan="6" | | ! colspan="6" | third octave | ||
! | ! | ||
|- | |- | ||
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| maa | | maa | ||
| 81/64=407.820, 33/26=412.745, 14/11=417.508 | | 81/64=407.820, 33/26=412.745, 14/11=417.508 | ||
|- | |- | ||
| 33 | | 33 | ||
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| 36/19=1106.397, 243/128=1109.775, 19/10=1111.199, 21/11=1119.463 | | 36/19=1106.397, 243/128=1109.775, 19/10=1111.199, 21/11=1119.463 | ||
|- | |- | ||
! colspan="6" | | ! colspan="6" | fourth octave (near match) | ||
! | ! | ||
|- | |- | ||
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== Scales == | == Scales == | ||
* [[ | * [[Symmetrical scales of 88cET]] | ||
== Music == | == Music == | ||
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[[Category:Equal-step tuning]] | [[Category:Equal-step tuning]] | ||