431edo: Difference between revisions
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Rework; cleanup; clarify the title row of the rank-2 temp table |
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== Theory == | == Theory == | ||
431edo is [[consistent]] to the [[15-odd-limit]] | 431edo is [[consistent]] to the [[15-odd-limit]]. The equal temperament [[tempering out|tempers out]] the [[schisma]] in the 5-limit; [[2401/2400]] in the 7-limit; [[5632/5625]] and [[8019/8000]] in the 11-limit; [[729/728]], [[1001/1000]], [[1716/1715]], [[4096/4095]], [[6656/6655]] and [[10648/10647]] in the 13-limit. It [[support]]s the [[sesquiquartififths]] temperament. | ||
It allows [[essentially tempered chord]]s of [[squbemic chords]] and [[sinbadmic chords]] in the [[13-odd-limit]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 24: | Line 26: | ||
| 2.3 | | 2.3 | ||
| {{monzo| -683 431 }} | | {{monzo| -683 431 }} | ||
| | | {{mapping| 431 683 }} | ||
| +0.1044 | | +0.1044 | ||
| 0.1044 | | 0.1044 | ||
| Line 31: | Line 33: | ||
| 2.3.5 | | 2.3.5 | ||
| 32805/32768, {{monzo| 7 63 -46 }} | | 32805/32768, {{monzo| 7 63 -46 }} | ||
| | | {{mapping| 431 683 1001 }} | ||
| -0.0230 | | -0.0230 | ||
| 0.2082 | | 0.2082 | ||
| Line 38: | Line 40: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 2401/2400, 32805/32768, {{monzo| 3 16 -11 -1 }} | | 2401/2400, 32805/32768, {{monzo| 3 16 -11 -1 }} | ||
| | | {{mapping| 431 683 1001 1210 }} | ||
| -0.0299 | | -0.0299 | ||
| 0.1803 | | 0.1803 | ||
| Line 45: | Line 47: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 2401/2400, 5632/5625, 8019/8000, 43923/43904 | | 2401/2400, 5632/5625, 8019/8000, 43923/43904 | ||
| | | {{mapping| 431 683 1001 1210 1491 }} | ||
| -0.0215 | | -0.0215 | ||
| 0.1621 | | 0.1621 | ||
| Line 52: | Line 54: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 729/728, 1001/1000, 1716/1715, 4096/4095, 6656/6655 | | 729/728, 1001/1000, 1716/1715, 4096/4095, 6656/6655 | ||
| | | {{mapping| 431 683 1001 1210 1491 1595 }} | ||
| -0.0318 | | -0.0318 | ||
| 0.1498 | | 0.1498 | ||
| Line 62: | Line 64: | ||
|+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* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 91: | Line 93: | ||
| [[Ostara]] | | [[Ostara]] | ||
|} | |} | ||
<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 08:22, 3 November 2023
| ← 430edo | 431edo | 432edo → |
Theory
431edo is consistent to the 15-odd-limit. The equal temperament tempers out the schisma in the 5-limit; 2401/2400 in the 7-limit; 5632/5625 and 8019/8000 in the 11-limit; 729/728, 1001/1000, 1716/1715, 4096/4095, 6656/6655 and 10648/10647 in the 13-limit. It supports the sesquiquartififths temperament.
It allows essentially tempered chords of squbemic chords and sinbadmic chords in the 13-odd-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.33 | +0.69 | +0.08 | -0.04 | +0.31 | +0.85 | +0.40 | +0.96 | +0.59 | -0.72 |
| Relative (%) | +0.0 | -11.9 | +24.9 | +3.0 | -1.5 | +11.0 | +30.4 | +14.3 | +34.5 | +21.0 | -25.9 | |
| Steps (reduced) |
431 (0) |
683 (252) |
1001 (139) |
1210 (348) |
1491 (198) |
1595 (302) |
1762 (38) |
1831 (107) |
1950 (226) |
2094 (370) |
2135 (411) | |
Subsets and supersets
431edo is the 83rd prime edo.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-683 431⟩ | [⟨431 683]] | +0.1044 | 0.1044 | 3.75 |
| 2.3.5 | 32805/32768, [7 63 -46⟩ | [⟨431 683 1001]] | -0.0230 | 0.2082 | 7.48 |
| 2.3.5.7 | 2401/2400, 32805/32768, [3 16 -11 -1⟩ | [⟨431 683 1001 1210]] | -0.0299 | 0.1803 | 6.48 |
| 2.3.5.7.11 | 2401/2400, 5632/5625, 8019/8000, 43923/43904 | [⟨431 683 1001 1210 1491]] | -0.0215 | 0.1621 | 5.82 |
| 2.3.5.7.11.13 | 729/728, 1001/1000, 1716/1715, 4096/4095, 6656/6655 | [⟨431 683 1001 1210 1491 1595]] | -0.0318 | 0.1498 | 5.38 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 63\431 | 175.41 | 448/405 | Sesquiquartififths |
| 1 | 176\431 | 490.02 | 65/49 | Surmarvelpyth |
| 1 | 179\431 | 498.55 | 4/3 | Helmholtz |
| 1 | 190\431 | 529.00 | 19/14 | Ostara |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct