157edo: Difference between revisions
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The '''157 equal divisions of the octave''' ('''157edo'''), or the '''157(-tone) equal temperament''' ('''157tet''', '''157et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 157 parts of 7. | {{Infobox ET | ||
| Prime factorization = 157 (prime) | |||
| Step size = 7.64331¢ | |||
| Fifth = 92\157 (703.18¢) | |||
| Major 2nd = 27\157 (206¢) | |||
| Minor 2nd = 11\157 (84¢) | |||
| Augmented 1sn = 16\157 (122¢) | |||
}} | |||
The '''157 equal divisions of the octave''' ('''157edo'''), or the '''157(-tone) equal temperament''' ('''157tet''', '''157et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 157 parts of 7.64 [[cent]]s each. | |||
== Theory == | == Theory == | ||
157et tempers out 78732/78125 ([[sensipent comma]]) and | 157et tempers out 78732/78125 ([[sensipent comma]]) and {{monzo| 37 -16 -5 }} (quinticosiennic comma) in the 5-limit; [[2401/2400]], [[5120/5103]], and 110592/109375 in the 7-limit (supporting the [[hemififths]] and the [[catafourth]]). Using the [[patent val]], it tempers out [[176/175]], 1331/1323, 3773/3750 and 8019/8000 in the 11-limit; [[351/350]], [[352/351]], [[847/845]], 1573/1568, and 2197/2187 in the 13-limit. | ||
157edo is the 37th [[prime EDO]]. | 157edo is the 37th [[prime EDO]]. | ||
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! Associated<br>ratio | ! Associated<br>ratio | ||
! Temperament | ! Temperament | ||
|- | |||
| 1 | |||
| 13\157 | |||
| 99.36 | |||
| 18/17 | |||
| [[Quinticosiennic]] | |||
|- | |||
| 1 | |||
| 23\157 | |||
| 175.80 | |||
| 72/65 | |||
| [[Quadrafifths]] | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 89: | Line 109: | ||
| 428.03 | | 428.03 | ||
| 2800/2187 | | 2800/2187 | ||
| [[ | | [[Geb]] / [[osiris]] | ||
|- | |- | ||
| 1 | | 1 | ||
Revision as of 19:51, 16 September 2021
| ← 156edo | 157edo | 158edo → |
The 157 equal divisions of the octave (157edo), or the 157(-tone) equal temperament (157tet, 157et) when viewed from a regular temperament perspective, is the equal division of the octave into 157 parts of 7.64 cents each.
Theory
157et tempers out 78732/78125 (sensipent comma) and [37 -16 -5⟩ (quinticosiennic comma) in the 5-limit; 2401/2400, 5120/5103, and 110592/109375 in the 7-limit (supporting the hemififths and the catafourth). Using the patent val, it tempers out 176/175, 1331/1323, 3773/3750 and 8019/8000 in the 11-limit; 351/350, 352/351, 847/845, 1573/1568, and 2197/2187 in the 13-limit.
157edo is the 37th prime EDO.
Prime harmonics
Script error: No such module "primes_in_edo".
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [249 -157⟩ | [⟨157 249]] | -0.388 | 0.388 | 5.08 |
| 2.3.5 | 78732/78125, ⟨37 -16 -5] | [⟨157 249 365]] | -0.760 | 0.614 | 8.04 |
| 2.3.5.7 | 2401/2400, 5120/5103, 78732/78125 | [⟨157 249 365 441]] | -0.737 | 0.533 | 6.98 |
| 2.3.5.7.11 | 176/175, 1331/1323, 2401/2400, 5120/5103 | [⟨157 249 365 441 543]] | -0.532 | 0.629 | 8.24 |
| 2.3.5.7.11.13 | 176/175, 351/350, 847/845, 1331/1323, 2197/2187 | [⟨157 249 365 441 543 581]] | -0.454 | 0.600 | 7.86 |
| 2.3.5.7.11.13.17 | 176/175, 256/255, 351/350, 442/441, 715/714, 2197/2187 | [⟨157 249 365 441 543 581 642]] | -0.461 | 0.556 | 7.28 |
| 2.3.5.7.11.13.17.19 | 176/175, 256/255, 286/285, 351/350, 361/360, 442/441, 476/475 | [⟨157 249 365 441 543 581 642 667]] | -0.420 | 0.531 | 6.95 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperament |
|---|---|---|---|---|
| 1 | 13\157 | 99.36 | 18/17 | Quinticosiennic |
| 1 | 23\157 | 175.80 | 72/65 | Quadrafifths |
| 1 | 46\157 | 351.59 | 49/40 | Hemififths |
| 1 | 56\157 | 428.03 | 2800/2187 | Geb / osiris |
| 1 | 58\157 | 443.31 | 162/125 | Sensipent |
| 1 | 64\157 | 489.17 | 250/189 | Catafourth |