157edo: Difference between revisions

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== Theory ==
== Theory ==
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 (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.


=== Odd harmonics ===
=== Odd harmonics ===
Line 26: Line 26:
| {{monzo| 249 -157 }}
| {{monzo| 249 -157 }}
| {{mapping| 157 249 }}
| {{mapping| 157 249 }}
| −0.388
| −0.388
| 0.388
| 0.388
| 5.08
| 5.08
Line 33: Line 33:
| 78732/78125, {{val| 37 -16 -5 }}
| 78732/78125, {{val| 37 -16 -5 }}
| {{mapping| 157 249 365 }}
| {{mapping| 157 249 365 }}
| −0.760
| −0.760
| 0.614
| 0.614
| 8.04
| 8.04
Line 40: Line 40:
| 2401/2400, 5120/5103, 78732/78125
| 2401/2400, 5120/5103, 78732/78125
| {{mapping| 157 249 365 441 }}
| {{mapping| 157 249 365 441 }}
| −0.737
| −0.737
| 0.533
| 0.533
| 6.98
| 6.98
Line 47: Line 47:
| 176/175, 1331/1323, 2401/2400, 5120/5103
| 176/175, 1331/1323, 2401/2400, 5120/5103
| {{mapping| 157 249 365 441 543 }}
| {{mapping| 157 249 365 441 543 }}
| −0.532
| −0.532
| 0.629
| 0.629
| 8.24
| 8.24
Line 54: Line 54:
| 176/175, 351/350, 847/845, 1331/1323, 2197/2187
| 176/175, 351/350, 847/845, 1331/1323, 2197/2187
| {{mapping| 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 61: Line 61:
| 176/175, 256/255, 351/350, 442/441, 715/714, 2197/2187
| 176/175, 256/255, 351/350, 442/441, 715/714, 2197/2187
| {{mapping| 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 68: Line 68:
| 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
| {{mapping| 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 119: Line 119:
| [[Catafourth]]
| [[Catafourth]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

Revision as of 19:41, 15 January 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

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.

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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