157edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Plumtree (talk | contribs)
autopatrolled, emailconfirmed
1,976 edits
m Infobox ET now computes most parameters automatically
Adopt template: EDO intro; cleanup; -redundant categories
Line 1: Line 1:
{{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.
{{EDO intro|157}}


== 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 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.


157edo is the 37th [[prime EDO]].
=== Prime 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|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
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 74: Line 75:
{| class="wikitable center-all right-3 left-5"
{| class="wikitable center-all right-3 left-5"
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator<br>(Reduced)
! Cents<br>(reduced)
! Cents<br>(Reduced)
! Associated<br>ratio
! Associated<br>Ratio
! Temperament
! Temperament
|-
|-
Line 108: Line 109:
| 443.31
| 443.31
| 162/125
| 162/125
| [[Sensipent]]
| [[Warrior]]
|-
|-
| 1
| 1
Line 116: Line 117:
| [[Catafourth]]
| [[Catafourth]]
|}
|}
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Prime EDO]]

Revision as of 15:41, 1 September 2023

← 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

Template:EDO intro

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.

Prime 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
(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 Warrior
1 64\157 489.17 250/189 Catafourth
Retrieved from "https://en.xen.wiki/index.php?title=157edo&oldid=123147"