121edo: Difference between revisions
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The '''121 equal divisions of the octave''' (''' | The '''121 equal divisions of the octave''' ('''121EDO'''), or the '''121(-tone) equal temperament''' ('''121TET''', '''121ET''') when viewed from a [[regular temperament]] perspective, divides the octave into 121 [[equal]] steps of 9.9174 [[cent]]s each, and being the square closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], it has a unit step which is the quadratic (fine) relative cent of [[1edo|1EDO]]. | ||
== Theory == | == Theory == | ||
121EDO has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, 6144/6125, 10976/10935 in the [[7-limit]]; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the [[13-limit]]; 256/255, 375/374 and 442/441 in the [[17-limit]]; 190/189 and 361/360 in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai_clan #Grendel|grendel temperament]]. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]]. | |||
Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit | Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121ET, it is a system only associated with 121. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 57: | Line 57: | ||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 256/255, 325/324, 352/351, 364/363, 442/441, 540/539 | | 256/255, 325/324, 352/351, 364/363, 442/441, 540/539 | ||
| [{{val| 121 192 281 340 419 448 | | [{{val| 121 192 281 340 419 448 495 }}] | ||
| -0.787 | | -0.787 | ||
| 0.480 | | 0.480 | ||
| 4.85 | | 4.85 | ||
|- | |||
| 2.3.5.7.11.13.17.19 | |||
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374 | |||
| [{{val| 121 192 281 340 419 448 495 514 }}] | |||
| -0.689 | |||
| 0.519 | |||
| 5.23 | |||
|} | |} | ||
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| 21/20 | | 21/20 | ||
| [[Slithy]] | | [[Slithy]] | ||
|- | |||
| 1 | |||
| 10\121 | |||
| 99.17 | |||
| 18/17 | |||
| [[Quintupole]] | |||
|- | |- | ||
| 1 | | 1 | ||
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|- | |- | ||
| 1 | | 1 | ||
| 46 | | 46\121 | ||
| 456.20 | | 456.20 | ||
| 125/96 | | 125/96 | ||
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|- | |- | ||
| 1 | | 1 | ||
| 47 | | 47\121 | ||
| 466.12 | | 466.12 | ||
| 55/42 | | 55/42 | ||
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|- | |- | ||
| 1 | | 1 | ||
| 48 | | 48\121 | ||
| 476.03 | | 476.03 | ||
| 21/16 | | 21/16 | ||
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|} | |} | ||
== 13-limit detempering of | == 13-limit detempering of 121ET == | ||
{{See also| Detempering }} | {{See also| Detempering }} | ||