390edo: Difference between revisions
m changed EDO intro to ED intro |
ArrowHead294 (talk | contribs) mNo edit summary |
||
| Line 15: | Line 15: | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list | ! rowspan="2" | [[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]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
| Line 27: | Line 28: | ||
| 118098/117649, 34451725707/34359738368 | | 118098/117649, 34451725707/34359738368 | ||
| {{mapping| 390 618 1095 }} | | {{mapping| 390 618 1095 }} | ||
| 0.0395 | | +0.0395 | ||
| 0.1685 | | 0.1685 | ||
| 5.48 | | 5.48 | ||
| Line 34: | Line 35: | ||
| 118098/117649, 1362944/1361367, 235782657/234881024 | | 118098/117649, 1362944/1361367, 235782657/234881024 | ||
| {{mapping| 390 618 1095 1349 }} | | {{mapping| 390 618 1095 1349 }} | ||
| 0.0693 | | +0.0693 | ||
| 0.1548 | | 0.1548 | ||
| 5.03 | | 5.03 | ||
| Line 41: | Line 42: | ||
| 729/728, 10648/10647, 16848/16807, 1574573/1572864 | | 729/728, 10648/10647, 16848/16807, 1574573/1572864 | ||
| {{mapping| 390 618 1095 1349 1443 }} | | {{mapping| 390 618 1095 1349 1443 }} | ||
| 0.0839 | | +0.0839 | ||
| 0.1415 | | 0.1415 | ||
| 4.60 | | 4.60 | ||
| Line 48: | Line 49: | ||
| 729/728, 1089/1088, 16848/16807, 65637/65536, 95823/95744 | | 729/728, 1089/1088, 16848/16807, 65637/65536, 95823/95744 | ||
| {{mapping| 390 618 1095 1349 1443 1594 }} | | {{mapping| 390 618 1095 1349 1443 1594 }} | ||
| 0.0838 | | +0.0838 | ||
| 0.1292 | | 0.1292 | ||
| 4.20 | | 4.20 | ||
|} | |} | ||
Latest revision as of 12:28, 21 February 2025
| ← 389edo | 390edo | 391edo → |
390 equal divisions of the octave (abbreviated 390edo or 390ed2), also called 390-tone equal temperament (390tet) or 390 equal temperament (390et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 390 equal parts of about 3.08 ¢ each. Each step represents a frequency ratio of 21/390, or the 390th root of 2.
Theory
390et is enfactored in the 5-limit, with the same tuning as 65edo. But its approximation to higher harmonics are improved, so that it is suitable for use in the 2.3.7.11.13.17.23.31.41 subgroup.
Using the patent val nonetheless, it tempers out 2401/2400 and 3136/3125 in the 7-limit, supporting hemiwürschmidt.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.42 | +1.38 | +0.40 | -0.55 | -0.53 | -0.34 | +0.95 | -0.58 | +1.19 | -0.42 |
| Relative (%) | +0.0 | -13.5 | +44.8 | +13.2 | -17.8 | -17.1 | -11.1 | +30.8 | -18.9 | +38.7 | -13.7 | |
| Steps (reduced) |
390 (0) |
618 (228) |
906 (126) |
1095 (315) |
1349 (179) |
1443 (273) |
1594 (34) |
1657 (97) |
1764 (204) |
1895 (335) |
1932 (372) | |
Subsets and supersets
Since 390 factors into 2 × 3 × 5 × 13, 390edo has subset edos 2, 3, 5, 6, 10, 13, 15, 26, 30, 39, 65, 78, 130, and 195. 780edo, which doubles it, gives a good correction to the harmonic 5.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.7 | 118098/117649, 34451725707/34359738368 | [⟨390 618 1095]] | +0.0395 | 0.1685 | 5.48 |
| 2.3.7.11 | 118098/117649, 1362944/1361367, 235782657/234881024 | [⟨390 618 1095 1349]] | +0.0693 | 0.1548 | 5.03 |
| 2.3.7.11.13 | 729/728, 10648/10647, 16848/16807, 1574573/1572864 | [⟨390 618 1095 1349 1443]] | +0.0839 | 0.1415 | 4.60 |
| 2.3.7.11.13.17 | 729/728, 1089/1088, 16848/16807, 65637/65536, 95823/95744 | [⟨390 618 1095 1349 1443 1594]] | +0.0838 | 0.1292 | 4.20 |