157edo: Difference between revisions
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Adopt template: EDO intro; cleanup; -redundant categories |
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== 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 (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 === | ||
{{Harmonics in equal|157}} | {{Harmonics in equal|157}} | ||
| Line 24: | Line 24: | ||
| 2.3 | | 2.3 | ||
| {{monzo| 249 -157 }} | | {{monzo| 249 -157 }} | ||
| | | {{mapping| 157 249 }} | ||
| -0.388 | | -0.388 | ||
| 0.388 | | 0.388 | ||
| Line 31: | Line 31: | ||
| 2.3.5 | | 2.3.5 | ||
| 78732/78125, {{val| 37 -16 -5 }} | | 78732/78125, {{val| 37 -16 -5 }} | ||
| | | {{mapping| 157 249 365 }} | ||
| -0.760 | | -0.760 | ||
| 0.614 | | 0.614 | ||
| Line 38: | Line 38: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 2401/2400, 5120/5103, 78732/78125 | | 2401/2400, 5120/5103, 78732/78125 | ||
| | | {{mapping| 157 249 365 441 }} | ||
| -0.737 | | -0.737 | ||
| 0.533 | | 0.533 | ||
| Line 45: | Line 45: | ||
| 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 | ||
| | | {{mapping| 157 249 365 441 543 }} | ||
| -0.532 | | -0.532 | ||
| 0.629 | | 0.629 | ||
| Line 52: | Line 52: | ||
| 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 | ||
| | | {{mapping| 157 249 365 441 543 581 }} | ||
| -0.454 | | -0.454 | ||
| 0.600 | | 0.600 | ||
| Line 59: | Line 59: | ||
| 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 | ||
| | | {{mapping| 157 249 365 441 543 581 642 }} | ||
| -0.461 | | -0.461 | ||
| 0.556 | | 0.556 | ||
| Line 66: | Line 66: | ||
| 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 | ||
| | | {{mapping| 157 249 365 441 543 581 642 667 }} | ||
| -0.420 | | -0.420 | ||
| 0.531 | | 0.531 | ||
| Line 76: | Line 76: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio* | ||
! Temperament | ! Temperament | ||
|- | |- | ||
| Line 117: | Line 117: | ||
| [[Catafourth]] | | [[Catafourth]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
Revision as of 08:56, 11 May 2024
| ← 156edo | 157edo | 158edo → |
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
| 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
| Periods per 8ve |
Generator* | Cents* | 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 |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct