1889edo: Difference between revisions
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m →Regular temperament properties: cleanup |
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! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
| Line 26: | Line 26: | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{ | | {{Monzo| 2994 -1889 }} | ||
| {{ | | {{Mapping| 1889 2994 }} | ||
| −0.0012 | | −0.0012 | ||
| 0.0012 | | 0.0012 | ||
| Line 33: | Line 33: | ||
|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| {{ | | {{Monzo| 54 -37 2 }}, {{monzo| -66 -36 53 }} | ||
| {{ | | {{Mapping| 1889 2994 4386 }} | ||
| +0.0104 | | +0.0104 | ||
| 0.0163 | | 0.0163 | ||
| Line 40: | Line 40: | ||
|- | |- | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 4375/4374, | | 4375/4374, {{monzo| -1 4 11 -1 }}, {{monzo| -57 16 10 3 }} | ||
| {{ | | {{Mapping| 1889 2994 4386 5303 }} | ||
| +0.0131 | | +0.0131 | ||
| 0.0149 | | 0.0149 | ||
| Line 47: | Line 47: | ||
|- | |- | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 4375/4374, 151263/151250, 820125/819896, | | 4375/4374, 151263/151250, 820125/819896, {{monzo| -28 7 7 -1 1 }} | ||
| {{ | | {{Mapping| 1889 2994 4386 5303 6535 }} | ||
| +0.0055 | | +0.0055 | ||
| 0.0201 | | 0.0201 | ||
| Line 54: | Line 54: | ||
|- | |- | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 4375/4374, | | 4225/4224, 4375/4374, 6656/6655, 151263/151250, 4100625/4100096 | ||
| {{ | | {{Mapping| 1889 2994 4386 5303 6535 6990 }} | ||
| +0.0084 | | +0.0084 | ||
| 0.0194 | | 0.0194 | ||
| Line 61: | Line 61: | ||
|- | |- | ||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 4375/4374, 12376/12375 | | 4225/4224, 4375/4374, 6656/6655, 12376/12375, 14875/14872, 194481/194480 | ||
| {{ | | {{Mapping| 1889 2994 4386 5303 6535 6990 7721 }} | ||
| +0.0120 | | +0.0120 | ||
| 0.0200 | | 0.0200 | ||
| Line 68: | Line 68: | ||
|- | |- | ||
| 2.3.5.7.11.13.17.19 | | 2.3.5.7.11.13.17.19 | ||
| 4375/4374 | | 4225/4224, 4375/4374, 5985/5984, 6175/6174, 6656/6655, 12376/12375, 61965/61952 | ||
| {{ | | {{Mapping| 1889 2994 4386 5303 6535 6990 7721 8024 }} | ||
| +0.0167 | | +0.0167 | ||
| 0.0226 | | 0.0226 | ||
| Line 79: | Line 79: | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |- | ||
! Periods<br | ! Periods<br>per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 88: | Line 88: | ||
| 392\1889 | | 392\1889 | ||
| 249.021 | | 249.021 | ||
| {{ | | {{Monzo| -26 18 1 }} | ||
| [[Monzismic]] | | [[Monzismic]] | ||
|- | |- | ||
| Line 97: | Line 97: | ||
| [[Semidimi]] | | [[Semidimi]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[ | <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct | ||
== Music == | == Music == | ||
Revision as of 15:54, 2 March 2025
| ← 1888edo | 1889edo | 1890edo → |
1889 equal divisions of the octave (abbreviated 1889edo or 1889ed2), also called 1889-tone equal temperament (1889tet) or 1889 equal temperament (1889et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1889 equal parts of about 0.635 ¢ each. Each step represents a frequency ratio of 21/1889, or the 1889th root of 2.
Theory
1889edo is strong in the 23-limit, though 1578, which among other things has a lower 23-limit relative error, rather puts it in the shade. It is distinctly consistent through the 27-odd-limit, but not, unlike 1578, to the 29-odd-limit. Even so, it should be noted that it supplies the optimal patent val for the 7-limit monzismic temperament.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.004 | -0.078 | -0.059 | +0.085 | -0.083 | -0.138 | -0.213 | -0.005 | +0.174 | -0.303 |
| Relative (%) | +0.0 | +0.6 | -12.2 | -9.3 | +13.4 | -13.1 | -21.7 | -33.5 | -0.9 | +27.4 | -47.7 | |
| Steps (reduced) |
1889 (0) |
2994 (1105) |
4386 (608) |
5303 (1525) |
6535 (868) |
6990 (1323) |
7721 (165) |
8024 (468) |
8545 (989) |
9177 (1621) |
9358 (1802) | |
Subsets and supersets
1889edo is the 290th prime edo.
Intervals
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [2994 -1889⟩ | [⟨1889 2994]] | −0.0012 | 0.0012 | 0.19 |
| 2.3.5 | [54 -37 2⟩, [-66 -36 53⟩ | [⟨1889 2994 4386]] | +0.0104 | 0.0163 | 2.57 |
| 2.3.5.7 | 4375/4374, [-1 4 11 -1⟩, [-57 16 10 3⟩ | [⟨1889 2994 4386 5303]] | +0.0131 | 0.0149 | 2.35 |
| 2.3.5.7.11 | 4375/4374, 151263/151250, 820125/819896, [-28 7 7 -1 1⟩ | [⟨1889 2994 4386 5303 6535]] | +0.0055 | 0.0201 | 3.16 |
| 2.3.5.7.11.13 | 4225/4224, 4375/4374, 6656/6655, 151263/151250, 4100625/4100096 | [⟨1889 2994 4386 5303 6535 6990]] | +0.0084 | 0.0194 | 3.05 |
| 2.3.5.7.11.13.17 | 4225/4224, 4375/4374, 6656/6655, 12376/12375, 14875/14872, 194481/194480 | [⟨1889 2994 4386 5303 6535 6990 7721]] | +0.0120 | 0.0200 | 3.15 |
| 2.3.5.7.11.13.17.19 | 4225/4224, 4375/4374, 5985/5984, 6175/6174, 6656/6655, 12376/12375, 61965/61952 | [⟨1889 2994 4386 5303 6535 6990 7721 8024]] | +0.0167 | 0.0226 | 3.56 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 392\1889 | 249.021 | [-26 18 1⟩ | Monzismic |
| 1 | 707\1889 | 449.127 | 35/27 | Semidimi |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct