1861edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
| Line 13: | Line 13: | ||
== 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]] (¢) | ||
Latest revision as of 13:29, 21 February 2025
| ← 1860edo | 1861edo | 1862edo → |
1861 equal divisions of the octave (abbreviated 1861edo or 1861ed2), also called 1861-tone equal temperament (1861tet) or 1861 equal temperament (1861et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1861 equal parts of about 0.645 ¢ each. Each step represents a frequency ratio of 21/1861, or the 1861st root of 2.
Theory
1861edo is only consistent to the 5-odd-limit, and the harmonic 3 is about halfway between its steps. It has a reasonable approximation of the 2.9.5.21.11 subgroup.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.248 | -0.070 | -0.314 | -0.149 | -0.001 | +0.311 | +0.178 | +0.149 | -0.253 | -0.066 | -0.225 |
| Relative (%) | +38.5 | -10.8 | -48.8 | -23.0 | -0.2 | +48.2 | +27.7 | +23.2 | -39.3 | -10.3 | -34.9 | |
| Steps (reduced) |
2950 (1089) |
4321 (599) |
5224 (1502) |
5899 (316) |
6438 (855) |
6887 (1304) |
7271 (1688) |
7607 (163) |
7905 (461) |
8174 (730) |
8418 (974) | |
Subsets and supersets
1861edo is the 284th prime edo. 3722edo, which doubles it, provides a good correction to harmonics 3, 7, and 13.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [-5899 1861⟩ | [⟨1861 5899]] | +0.0234 | 0.0234 | 3.63 |