624edo: Difference between revisions
→Rank-2 temperaments: +sulfur |
Cleanup; clarify the title row of the rank-2 temp table; -redundant categories |
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== Theory == | == Theory == | ||
624edo is [[consistent]] to the [[27-odd-limit]] | 624edo is [[consistent]] to the [[27-odd-limit]]. The equal temperament [[tempering out|tempers out]] {{monzo| 23 6 -14 }} ([[vishnuzma]]) and {{monzo| -69 45 -1 }} ([[counterschisma]]) in the 5-limit; [[250047/250000]], 2460375/2458624, and 134217728/133984375 in the 7-limit; [[9801/9800]], 46656/46585, [[131072/130977]], and 151263/151250 in the 11-limit; [[1716/1715]], [[2080/2079]], [[4096/4095]], 34398/34375, and 39366/39325 in the 13-limit; [[936/935]], [[1701/1700]], [[2025/2023]], and [[2058/2057]] in the 17-limit; [[1521/1520]], 2376/2375, [[2432/2431]], and 3328/3325 in the 19-limit; 2024/2023, 2025/2024, and 3888/3887 in the 23-limit. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 16: | Line 16: | ||
! rowspan="2" | [[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning Error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
| Line 24: | Line 24: | ||
| 2.3 | | 2.3 | ||
| {{monzo| -989 624 }} | | {{monzo| -989 624 }} | ||
| | | {{mapping| 624 989 }} | ||
| +0.0101 | | +0.0101 | ||
| 0.0101 | | 0.0101 | ||
| Line 31: | Line 31: | ||
| 2.3.5 | | 2.3.5 | ||
| {{monzo| 23 6 -14 }}, {{monzo| -69 45 -1 }} | | {{monzo| 23 6 -14 }}, {{monzo| -69 45 -1 }} | ||
| | | {{mapping| 624 989 1449 }} | ||
| -0.0256 | | -0.0256 | ||
| 0.0510 | | 0.0510 | ||
| Line 38: | Line 38: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 250047/250000, 2460375/2458624, {{monzo| 27 0 -8 -3 }} | | 250047/250000, 2460375/2458624, {{monzo| 27 0 -8 -3 }} | ||
| | | {{mapping| 624 989 1449 1752 }} | ||
| -0.0552 | | -0.0552 | ||
| 0.0678 | | 0.0678 | ||
| Line 45: | Line 45: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 9801/9800, 46656/46585, 131072/130977, 151263/151250 | | 9801/9800, 46656/46585, 131072/130977, 151263/151250 | ||
| | | {{mapping| 624 989 1449 1752 2159 }} | ||
| -0.0792 | | -0.0792 | ||
| 0.0772 | | 0.0772 | ||
| Line 52: | Line 52: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 1716/1715, 2080/2079, 4096/4095, 34398/34375, 39366/39325 | | 1716/1715, 2080/2079, 4096/4095, 34398/34375, 39366/39325 | ||
| | | {{mapping| 624 989 1449 1752 2159 2309 }} | ||
| -0.0595 | | -0.0595 | ||
| 0.0831 | | 0.0831 | ||
| Line 59: | Line 59: | ||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 936/935, 1701/1700, 1716/1715, 2025/2023, 4096/4095, 11016/11011 | | 936/935, 1701/1700, 1716/1715, 2025/2023, 4096/4095, 11016/11011 | ||
| | | {{mapping| 624 989 1449 1752 2159 2309 2551 }} | ||
| -0.0795 | | -0.0795 | ||
| 0.0911 | | 0.0911 | ||
| Line 66: | Line 66: | ||
| 2.3.5.7.11.13.17.19 | | 2.3.5.7.11.13.17.19 | ||
| 936/935, 1521/1520, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 11016/11011 | | 936/935, 1521/1520, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 11016/11011 | ||
| | | {{mapping| 624 989 1449 1752 2159 2309 2551 2651 }} | ||
| -0.0861 | | -0.0861 | ||
| 0.0870 | | 0.0870 | ||
| 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* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 135: | Line 135: | ||
| [[Iron]] | | [[Iron]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
[[ | |||