1778edo: Difference between revisions
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{{ | 1778edo is [[consistent]] to the [[9-odd-limit]], but the errors of both [[harmonic]]s [[5/1|5]] and [[7/1|7]] are quite large. It is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[889edo]], [[tempering out|tempering out]] {{monzo| -29 -11 20 }} (gammic comma) and {{monzo| -69 45 -1 }} ([[counterschisma]]). In the 7-limit, the equal temperament tempers out 2401/2400 ([[breedsma]]) and 48828125/48771072 (neptunisma). It provides the [[optimal patent val]] for the 7-limit [[neptune]] temperament. | ||
== | For higher harmonics, it is suitable for a 2.3.11.19.23.43.47.61 [[subgroup]] interpretation. | ||
=== Odd harmonics === | |||
{{Harmonics in equal|1778}} | {{Harmonics in equal|1778}} | ||
=== Subsets and supersets === | |||
Since 1778 factors into {{factorization|1778}}, 1778edo has subset edos {{EDOs| 2, 7, 14, 127, 254, and 889 }}. | |||
[[Category: | [[Category:Neptune]] | ||
Latest revision as of 23:05, 20 February 2025
| ← 1777edo | 1778edo | 1779edo → |
1778 equal divisions of the octave (abbreviated 1778edo or 1778ed2), also called 1778-tone equal temperament (1778tet) or 1778 equal temperament (1778et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1778 equal parts of about 0.675 ¢ each. Each step represents a frequency ratio of 21/1778, or the 1778th root of 2.
1778edo is consistent to the 9-odd-limit, but the errors of both harmonics 5 and 7 are quite large. It is enfactored in the 5-limit, with the same tuning as 889edo, tempering out [-29 -11 20⟩ (gammic comma) and [-69 45 -1⟩ (counterschisma). In the 7-limit, the equal temperament tempers out 2401/2400 (breedsma) and 48828125/48771072 (neptunisma). It provides the optimal patent val for the 7-limit neptune temperament.
For higher harmonics, it is suitable for a 2.3.11.19.23.43.47.61 subgroup interpretation.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.043 | -0.262 | -0.322 | -0.085 | +0.088 | -0.258 | -0.305 | +0.331 | +0.125 | +0.310 | +0.072 |
| Relative (%) | -6.3 | -38.8 | -47.7 | -12.7 | +13.1 | -38.2 | -45.1 | +49.1 | +18.5 | +46.0 | +10.7 | |
| Steps (reduced) |
2818 (1040) |
4128 (572) |
4991 (1435) |
5636 (302) |
6151 (817) |
6579 (1245) |
6946 (1612) |
7268 (156) |
7553 (441) |
7810 (698) |
8043 (931) | |
Subsets and supersets
Since 1778 factors into 2 × 7 × 127, 1778edo has subset edos 2, 7, 14, 127, 254, and 889.