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{{Infobox ET}}
{{Infobox ET}}
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 about 7.64 [[cent]]s each.
{{ED intro}}


== Theory ==
== Theory ==
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]] temperaments). 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.
157et [[tempering out|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 ([[support]]ing the [[hemififths]] and the [[catafourth]] temperaments). 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]].
=== Odd harmonics ===
{{Harmonics in equal|157}}


=== Prime harmonics ===
=== Subsets and supersets ===
{{Primes in edo|157}}
157edo is the 37th [[prime edo]].


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
Line 23: Line 25:
| 2.3
| 2.3
| {{monzo| 249 -157 }}
| {{monzo| 249 -157 }}
| [{{val| 157 249 }}]
| {{mapping| 157 249 }}
| -0.388
| −0.388
| 0.388
| 0.388
| 5.08
| 5.08
Line 30: Line 32:
| 2.3.5
| 2.3.5
| 78732/78125, {{val| 37 -16 -5 }}
| 78732/78125, {{val| 37 -16 -5 }}
| [{{val| 157 249 365 }}]
| {{mapping| 157 249 365 }}
| -0.760
| −0.760
| 0.614
| 0.614
| 8.04
| 8.04
Line 37: Line 39:
| 2.3.5.7
| 2.3.5.7
| 2401/2400, 5120/5103, 78732/78125
| 2401/2400, 5120/5103, 78732/78125
| [{{val| 157 249 365 441 }}]
| {{mapping| 157 249 365 441 }}
| -0.737
| −0.737
| 0.533
| 0.533
| 6.98
| 6.98
Line 44: Line 46:
| 2.3.5.7.11
| 2.3.5.7.11
| 176/175, 1331/1323, 2401/2400, 5120/5103
| 176/175, 1331/1323, 2401/2400, 5120/5103
| [{{val| 157 249 365 441 543 }}]
| {{mapping| 157 249 365 441 543 }}
| -0.532
| −0.532
| 0.629
| 0.629
| 8.24
| 8.24
Line 51: Line 53:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 176/175, 351/350, 847/845, 1331/1323, 2197/2187
| 176/175, 351/350, 847/845, 1331/1323, 2197/2187
| [{{val| 157 249 365 441 543 581 }}]
| {{mapping| 157 249 365 441 543 581 }}
| -0.454
| −0.454
| 0.600
| 0.600
| 7.86
| 7.86
Line 58: Line 60:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 715/714, 2197/2187
| 176/175, 256/255, 351/350, 442/441, 715/714, 2197/2187
| [{{val| 157 249 365 441 543 581 642 }}]
| {{mapping| 157 249 365 441 543 581 642 }}
| -0.461
| −0.461
| 0.556
| 0.556
| 7.28
| 7.28
Line 65: Line 67:
| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 351/350, 361/360, 442/441, 476/475
| 176/175, 256/255, 286/285, 351/350, 361/360, 442/441, 476/475
| [{{val| 157 249 365 441 543 581 642 667 }}]
| {{mapping| 157 249 365 441 543 581 642 667 }}
| -0.420
| −0.420
| 0.531
| 0.531
| 6.95
| 6.95
Line 72: Line 74:


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all right-3 left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Temperament
! Associated<br />ratio*
! Temperaments
|-
|-
| 1
| 1
Line 108: Line 111:
| 443.31
| 443.31
| 162/125
| 162/125
| [[Sensipent]]
| [[Warrior]]
|-
|-
| 1
| 1
Line 116: Line 119:
| [[Catafourth]]
| [[Catafourth]]
|}
|}
 
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Prime EDO]]
Retrieved from "https://en.xen.wiki/w/157edo"