468edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[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 | ||
| Line 55: | Line 64: | ||
| 0.1533 | | 0.1533 | ||
| 5.98 | | 5.98 | ||
|} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {| class="wikitable center-all left-5" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 83: | Line 99: | ||
| 231/200<br />(99/98) | | 231/200<br />(99/98) | ||
| [[Hemiennealimmal]] | | [[Hemiennealimmal]] | ||
|} | |||
<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 13:11, 16 November 2024
| ← 467edo | 468edo | 469edo → |
Theory
468et tempers out 78125000/78121827, 4375/4374, 250047/250000, 2401/2400, 420175/419904, 200120949/200000000 and 40353607/40310784 in the 7-limit; 21437500/21434787, 151263/151250, 117649/117612, 514714375/514434888, 47265625/47258883, 9801/9800, 3025/3024, 1890625/1889568, 160083/160000, 41503/41472, 3294225/3294172, 43923/43904, 102487/102400 and 1771561/1769472 in the 11-limit. It provides the optimal patent val for unlit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.61 | +0.87 | +0.40 | +1.22 | -0.04 | +0.50 | -1.09 | +0.17 | -0.08 | +1.01 | -0.07 |
| Relative (%) | +23.8 | +33.8 | +15.8 | +47.5 | -1.4 | +19.4 | -42.5 | +6.7 | -3.0 | +39.5 | -2.7 | |
| Steps (reduced) |
742 (274) |
1087 (151) |
1314 (378) |
1484 (80) |
1619 (215) |
1732 (328) |
1828 (424) |
1913 (41) |
1988 (116) |
2056 (184) |
2117 (245) | |
Subsets and supersets
468 factors into 22 × 32 × 13, with subset edos 2, 3, 4, 6, 9, 12, 13, 18, 26, 36, 39, 52, 78, 117, 156, and 234.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [371 -234⟩ | [⟨468 742]] | −0.1922 | 0.1921 | 7.49 |
| 2.3.5 | 2199023255552/2179240250625, 7629394531250/7625597484987 | [⟨468 742 1087]] | −0.2524 | 0.1785 | 6.96 |
| 2.3.5.7 | 4375/4374, 2401/2400, 47141561040896/46708681640625 | [⟨468 742 1087 1314]] | −0.2253 | 0.1615 | 6.30 |
| 2.3.5.7.11 | 3025/3024, 4375/4374, 2401/2400, 5767168/5740875 | [⟨468 742 1087 1314 1619]] | −0.1782 | 0.1725 | 6.73 |
| 2.3.5.7.11.13 | 2080/2079, 1001/1000, 3025/3024, 1716/1715, 1982464/1974375 | [⟨468 742 1087 1314 1619 1732]] | −0.1709 | 0.1583 | 6.17 |
| 2.3.5.7.11.13.17 | 2080/2079, 1001/1000, 3025/3024, 1716/1715, 17920/17901, 30464/30375 | [⟨468 742 1087 1314 1619 1732 1913]] | −0.1526 | 0.1533 | 5.98 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 179\468 | 458.97 | 125/96 | Majvam |
| 4 | 194\468 (40\468) |
497.44 (102.56) |
4/3 (35/33) |
Undim |
| 9 | 123\468 (19\468) |
315.38 (48.72) |
6/5 (36/35) |
Ennealimmal |
| 18 | 97\468 (7\468) |
248.72 (17.95) |
231/200 (99/98) |
Hemiennealimmal |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct