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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 7.6433 [[cent]]s each.
{{Infobox ET}}
{{ED intro}}


== Theory ==
== Theory ==
157et tempers out 78732/78125 ([[sensipent comma]]) and 137438953472/134521003125 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.
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]].


[[Category:Equal divisions of the octave]]
== Regular temperament properties ==
[[Category:Prime EDO]]
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 249 -157 }}
| {{mapping| 157 249 }}
| −0.388
| 0.388
| 5.08
|-
| 2.3.5
| 78732/78125, {{val| 37 -16 -5 }}
| {{mapping| 157 249 365 }}
| −0.760
| 0.614
| 8.04
|-
| 2.3.5.7
| 2401/2400, 5120/5103, 78732/78125
| {{mapping| 157 249 365 441 }}
| −0.737
| 0.533
| 6.98
|-
| 2.3.5.7.11
| 176/175, 1331/1323, 2401/2400, 5120/5103
| {{mapping| 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
| {{mapping| 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
| {{mapping| 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
| {{mapping| 157 249 365 441 543 581 642 667 }}
| −0.420
| 0.531
| 6.95
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
| 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
| [[Warrior]]
|-
| 1
| 64\157
| 489.17
| 250/189
| [[Catafourth]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
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