1944edo: Difference between revisions
Cleanup; clarify the title row of the rank-2 temp table |
Rework theory; +subsets and supersets |
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== Theory == | == Theory == | ||
1944edo is [[consistent]] in the [[5-odd-limit]] to which it provides good approximation, and a near-perfect [[15/8]], but it is only consistent that far. In the 5-limit, it | 1944edo is [[consistent]] in the [[5-odd-limit]] to which it provides good approximation, and a near-perfect [[15/8]], but it is only consistent that far since [[harmonic]] [[7/1|7]] is about halfway between its steps. In the 5-limit, it [[tempering out|tempers out]] the [[luna comma]], {{monzo| 38 -2 -15 }}. In the 1944d [[val]] in the 7-limit, it is a [[landscape]] system, tempering out [[250047/250000]], and as a consequence it tunes the 24th-octave [[chromium]] temperament. | ||
In higher limits, 1944edo is a tuning for the [[jamala]] temperament in the 2.5.11.13.19.41.47 [[subgroup]], for which 1944edo provides good approximation (except for the 13th harmonic). Overall, the best subgroup for 1944edo is 2.3.5.11.17.19.29.31.41.47. | In higher limits, 1944edo is a tuning for the [[jamala]] temperament in the 2.5.11.13.19.41.47 [[subgroup]], for which 1944edo provides good approximation (except for the 13th harmonic). Overall, the best subgroup for 1944edo is 2.3.5.11.17.19.29.31.41.47. | ||
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=== Odd harmonics === | === Odd harmonics === | ||
{{Harmonics in equal|1944}} | {{Harmonics in equal|1944}} | ||
=== Subsets and supersets === | |||
Since 1944 factors into {{factorization|1944}}, 1944edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 81, 108, 162, 216, 243, 324, 486, 648, and 972 }}. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
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! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
Revision as of 12:19, 30 October 2023
| ← 1943edo | 1944edo | 1945edo → |
Theory
1944edo is consistent in the 5-odd-limit to which it provides good approximation, and a near-perfect 15/8, but it is only consistent that far since harmonic 7 is about halfway between its steps. In the 5-limit, it tempers out the luna comma, [38 -2 -15⟩. In the 1944d val in the 7-limit, it is a landscape system, tempering out 250047/250000, and as a consequence it tunes the 24th-octave chromium temperament.
In higher limits, 1944edo is a tuning for the jamala temperament in the 2.5.11.13.19.41.47 subgroup, for which 1944edo provides good approximation (except for the 13th harmonic). Overall, the best subgroup for 1944edo is 2.3.5.11.17.19.29.31.41.47.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.103 | +0.106 | -0.307 | -0.206 | -0.083 | +0.213 | +0.003 | -0.017 | +0.018 | +0.207 | +0.121 |
| Relative (%) | -16.7 | +17.2 | -49.8 | -33.4 | -13.5 | +34.5 | +0.5 | -2.8 | +2.9 | +33.5 | +19.6 | |
| Steps (reduced) |
3081 (1137) |
4514 (626) |
5457 (1569) |
6162 (330) |
6725 (893) |
7194 (1362) |
7595 (1763) |
7946 (170) |
8258 (482) |
8539 (763) |
8794 (1018) | |
Subsets and supersets
Since 1944 factors into 23 × 35, 1944edo has subset edos 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 81, 108, 162, 216, 243, 324, 486, 648, and 972.
Regular temperament properties
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 313\1944 | 193.210 | 262144/234375 | Luna (5-limit) |
| 24 | 944\1944 (53\1944) |
582.716 (32.716) |
7/5 (?) |
Chromium (1944d) |
| 72 | 892\1944 (1\1944) |
550.617 (0.617) |
73205/53248 (?) |
Jamala |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct