1171edo: Difference between revisions
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Clarify the title row of the rank-2 temp table |
Note 2342edo; misc. cleanup |
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== Theory == | == Theory == | ||
1171edo is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is | 1171edo is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is [[consistency|distinctly consistent]] through the [[27-odd-limit]], and is very strong on the 2.3.5.11 [[subgroup]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
1171edo is the 193rd [[prime edo]]. | 1171edo is the 193rd [[prime edo]]. [[2342edo]] which doubles it, corrects its [[harmonic]] [[7/1|7]] to a near-just quality. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator* | ! Generator* | ||
Revision as of 10:16, 31 October 2023
| ← 1170edo | 1171edo | 1172edo → |
Theory
1171edo is a very strong 5-limit division, being the first one past 612 with a lower 5-limit relative error. It has a 5-limit comma basis consisting of the monzisma, [54 -37 2⟩ and whoosh, [37 25 -33⟩. While not a strong higher-limit system, it is distinctly consistent through the 27-odd-limit, and is very strong on the 2.3.5.11 subgroup.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.009 | +0.023 | -0.423 | +0.006 | -0.220 | -0.429 | -0.331 | -0.093 | +0.312 | -0.373 |
| Relative (%) | +0.0 | +0.9 | +2.2 | -41.3 | +0.6 | -21.5 | -41.9 | -32.3 | -9.1 | +30.4 | -36.4 | |
| Steps (reduced) |
1171 (0) |
1856 (685) |
2719 (377) |
3287 (945) |
4051 (538) |
4333 (820) |
4786 (102) |
4974 (290) |
5297 (613) |
5689 (1005) |
5801 (1117) | |
Subsets and supersets
1171edo is the 193rd prime edo. 2342edo which doubles it, corrects its harmonic 7 to a near-just quality.
Regular temperament properties
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 129\1171 | 132.195 | [-38 5 13⟩ | Astro |
| 1 | 243\1171 | 249.018 | [-26 18 -1⟩ | Monzismic |
| 1 | 315\1171 | 322.801 | [-6 23 -13⟩ | Senior |
| 1 | 335\1171 | 343.296 | 8000/6561 | Raider |
| 1 | 547\1171 | 560.547 | 864/625 | Whoosh |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct