348edo: Difference between revisions
m changed EDO intro to ED intro |
Contribution (talk | contribs) No edit summary |
||
| (2 intermediate revisions by 2 users not shown) | |||
| Line 6: | Line 6: | ||
Using the [[patent val]], it tempers out [[2401/2400]], [[15625/15552]], [[390625/388962]] and 156250000/155649627 and in the 7-limit. It [[support]]s [[quadritikleismic]] and [[subneutral]]. | Using the [[patent val]], it tempers out [[2401/2400]], [[15625/15552]], [[390625/388962]] and 156250000/155649627 and in the 7-limit. It [[support]]s [[quadritikleismic]] and [[subneutral]]. | ||
In 348edo, the prime harmonics up to 13 map the same way as in [[87edo]], except the 7th harmonic, which is corrected. | |||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 15: | Line 17: | ||
== 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]] (¢) | ||
| Line 55: | Line 58: | ||
| 729/728, 1575/1573, 2200/2197, 32805/32768, 31250/31213 | | 729/728, 1575/1573, 2200/2197, 32805/32768, 31250/31213 | ||
| {{mapping| 348 1103 808 977 1204 1288 }} | | {{mapping| 348 1103 808 977 1204 1288 }} | ||
| | | −0.0343 | ||
| 0.1194 | | 0.1194 | ||
| 3.46 | | 3.46 | ||
|} | |} | ||
Latest revision as of 14:41, 15 December 2025
| ← 347edo | 348edo | 349edo → |
348 equal divisions of the octave (abbreviated 348edo or 348ed2), also called 348-tone equal temperament (348tet) or 348 equal temperament (348et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 348 equal parts of about 3.45 ¢ each. Each step represents a frequency ratio of 21/348, or the 348th root of 2.
Theory
348et is consistent to the 7-odd-limit, but the error of the harmonic 3 is quite large, commending itself as a 2.9.5.7.11.13 subgroup temperament.
Using the patent val, it tempers out 2401/2400, 15625/15552, 390625/388962 and 156250000/155649627 and in the 7-limit. It supports quadritikleismic and subneutral.
In 348edo, the prime harmonics up to 13 map the same way as in 87edo, except the 7th harmonic, which is corrected.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.49 | -0.11 | +0.14 | -0.46 | +0.41 | +0.85 | +1.39 | -1.51 | -0.96 | +1.63 | -0.69 |
| Relative (%) | +43.3 | -3.1 | +4.0 | -13.4 | +11.8 | +24.7 | +40.2 | -43.7 | -27.9 | +47.4 | -20.0 | |
| Steps (reduced) |
552 (204) |
808 (112) |
977 (281) |
1103 (59) |
1204 (160) |
1288 (244) |
1360 (316) |
1422 (30) |
1478 (86) |
1529 (137) |
1574 (182) | |
Subsets and supersets
Since 348 factors into 22 × 3 × 29, 348edo has subset edos 2, 3, 4, 6, 12, 29, 58, 87, 116, and 174. 696edo, which doubles it, gives a good correction to the harmonic 3.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [-1103 348⟩ | [⟨348 1103]] | +0.0728 | 0.0728 | 2.11 |
| 2.9.5 | 32805/32768, [7 52 -74⟩ | [⟨348 1103 808]] | +0.0639 | 0.0608 | 1.76 |
| 2.9.5.7 | 32805/32768, 250047/250000, [7 9 -2 -11⟩ | [⟨348 1103 808 977]] | +0.0355 | 0.0721 | 2.09 |
| 2.9.5.7.11 | 9801/9800, 32805/32768, 46656/46585, 151263/151250 | [⟨348 1103 808 977 1204]] | +0.0049 | 0.0889 | 2.58 |
| 2.9.5.7.11.13 | 729/728, 1575/1573, 2200/2197, 32805/32768, 31250/31213 | [⟨348 1103 808 977 1204 1288]] | −0.0343 | 0.1194 | 3.46 |