431edo: Difference between revisions
mNo edit summary |
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 | |||