157edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo| 249 -157 }}
| {{monzo| 249 -157 }}
| {{mapping| 157 249 }}
| {{mapping| 157 249 }}
| -0.388
| &minus;0.388
| 0.388
| 0.388
| 5.08
| 5.08
Line 32: Line 24:
| 78732/78125, {{val| 37 -16 -5 }}
| 78732/78125, {{val| 37 -16 -5 }}
| {{mapping| 157 249 365 }}
| {{mapping| 157 249 365 }}
| -0.760
| &minus;0.760
| 0.614
| 0.614
| 8.04
| 8.04
Line 39: Line 31:
| 2401/2400, 5120/5103, 78732/78125
| 2401/2400, 5120/5103, 78732/78125
| {{mapping| 157 249 365 441 }}
| {{mapping| 157 249 365 441 }}
| -0.737
| &minus;0.737
| 0.533
| 0.533
| 6.98
| 6.98
Line 46: Line 38:
| 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
| &minus;0.532
| 0.629
| 0.629
| 8.24
| 8.24
Line 53: Line 45:
| 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
| &minus;0.454
| 0.600
| 0.600
| 7.86
| 7.86
Line 60: Line 52:
| 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
| &minus;0.461
| 0.556
| 0.556
| 7.28
| 7.28
Line 67: Line 59:
| 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
| &minus;0.420
| 0.531
| 0.531
| 6.95
| 6.95
Line 73: Line 65:


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all right-3 left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperament
|-
|-
| 1
| 1
Line 116: Line 102:
| 250/189
| 250/189
| [[Catafourth]]
| [[Catafourth]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}

Revision as of 03:56, 16 November 2024

← 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

Template:Comma basis begin |- | 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

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf

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