157edo: Difference between revisions

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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
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| 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]]

Latest revision as of 18:01, 19 February 2025

← 156edo 157edo 158edo →
Prime factorization 157 (prime)
Step size 7.64331 ¢ 
Fifth 92\157 (703.185 ¢)
Semitones (A1:m2) 16:11 (122.3 ¢ : 84.08 ¢)
Consistency limit 9
Distinct consistency limit 9

157 equal divisions of the octave (abbreviated 157edo or 157ed2), also called 157-tone equal temperament (157tet) or 157 equal temperament (157et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 157 equal parts of about 7.64 ¢ each. Each step represents a frequency ratio of 21/157, or the 157th root of 2.

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 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.

Odd harmonics

Approximation of odd harmonics in 157edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +1.23 +3.50 +1.87 +2.46 -1.00 +0.24 -2.92 +2.05 +0.58 +3.10 -1.52
Relative (%) +16.1 +45.7 +24.5 +32.2 -13.1 +3.1 -38.2 +26.8 +7.5 +40.6 -19.9
Steps
(reduced) 		249
(92) 		365
(51) 		441
(127) 		498
(27) 		543
(72) 		581
(110) 		613
(142) 		642
(14) 		667
(39) 		690
(62) 		710
(82)

Subsets and supersets

157edo is the 37th prime 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

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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