1553edo: Difference between revisions
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Created page with "{{Infobox ET}} {{EDO intro|1553}} == Theory == 1553et tempers out 200120949/200000000 in the 7-limit; 759375/758912, 2359296/2358125, 369140625/369098752 and 102487/102400 in..." |
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{{EDO intro|1553}} | {{EDO intro|1553}} | ||
== Theory == | == Theory == | ||
1553et tempers out 200120949/200000000 in the 7-limit; 759375/758912, 2359296/2358125, 369140625/369098752 | 1553edo is only [[consistent]] to the 5-odd-limit and harmonic 3 is about halfway between its steps. Assume the [[patent val]], 1553et tempers out 200120949/200000000 in the 7-limit; 102487/102400, 759375/758912, 2359296/2358125, and 369140625/369098752 in the 11-limit; 1716/1715, 40656/40625, 59535/59488, 105644/105625, 196625/196608, 200000/199927, 823875/823543, 8859375/8859136, 34034175/34027136, and 75000000/74942413, in the 13-limit. | ||
=== Odd harmonics === | |||
===Odd harmonics=== | |||
{{Harmonics in equal|1553}} | {{Harmonics in equal|1553}} | ||
=== Subsets and supersets === | |||
1553edo is the 245th [[prime edo]]. 3106edo, which doubles it, provides a good correction to the harmonic 3. | |||
==Regular temperament properties== | ==Regular temperament properties== | ||
Revision as of 04:32, 18 April 2023
| ← 1552edo | 1553edo | 1554edo → |
Theory
1553edo is only consistent to the 5-odd-limit and harmonic 3 is about halfway between its steps. Assume the patent val, 1553et tempers out 200120949/200000000 in the 7-limit; 102487/102400, 759375/758912, 2359296/2358125, and 369140625/369098752 in the 11-limit; 1716/1715, 40656/40625, 59535/59488, 105644/105625, 196625/196608, 200000/199927, 823875/823543, 8859375/8859136, 34034175/34027136, and 75000000/74942413, in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.345 | +0.035 | +0.137 | +0.082 | -0.384 | +0.168 | -0.310 | +0.132 | -0.024 | -0.208 | -0.071 |
| Relative (%) | -44.7 | +4.6 | +17.8 | +10.6 | -49.7 | +21.7 | -40.1 | +17.0 | -3.1 | -26.9 | -9.2 | |
| Steps (reduced) |
2461 (908) |
3606 (500) |
4360 (1254) |
4923 (264) |
5372 (713) |
5747 (1088) |
6067 (1408) |
6348 (136) |
6597 (385) |
6821 (609) |
7025 (813) | |
Subsets and supersets
1553edo is the 245th prime edo. 3106edo, which doubles it, provides a good correction to the harmonic 3.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-2461 1553⟩ | ⟨1553 2461] | 0.1089 | 0.1089 | 14.09 |
| 2.3.5 | 1224440064/1220703125, [-201 99 19⟩ | ⟨1553 2461 3606] | 0.0675 | 0.1064 | 13.77 |
| 2.3.5.7 | 200120949/200000000, 2579890176/2573571875, 23066015625/23018340352 | ⟨1553 2461 3606 4360] | 0.0384 | 0.1051 | 13.60 |
| 2.3.5.7.11 | 24057/24010, 759375/758912, 2359296/2358125, 992436543/991232000 | ⟨1553 2461 3606 4360 5372] | 0.0529 | 0.0984 | 12.73 |
| 2.3.5.7.11.13 | 729/728, 1716/1715, 40656/40625, 196625/196608, 14085981/14080000 | ⟨1553 2461 3606 4360 5372 5747] | 0.0366 | 0.0970 | 12.55 |
| 2.3.5.7.11.13.17 | 729/728, 1716/1715, 1089/1088, 14400/14399, 27648/27625, 14085981/14080000 | ⟨1553 2461 3606 4360 5372 5747 6348] | 0.0267 | 0.0930 | 12.04 |
| 2.3.5.7.11.13.17.19 | 729/728, 1716/1715, 1089/1088, 5985/5984, 4200/4199, 21888/21875, 287469/287375 | ⟨1553 2461 3606 4360 5372 5747 6348 6597] | 0.0241 | 0.0872 | 11.29 |