2684edo: Difference between revisions
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== Theory == | |||
2684edo is an extremely strong 13-limit system, with a lower 13-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any division until we reach [[5585edo]]. It is distinctly [[consistent]] through the [[17-odd-limit]], and is both a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak and zeta integral edo]]. It is [[enfactoring|enfactored]] in the 2.3.5.13 subgroup, with the same tuning as [[1342edo]], tempering out kwazy, {{monzo| -53 10 16 }}, senior, {{monzo| -17 62 -35 }} and egads, {{monzo| -36 52 51 }}. A 13-limit [[comma basis]] is {9801/9800, 10648/10647, 140625/140608, 196625/196608, 823680/823543}; it also tempers out 123201/123200. It is less accurate, but still quite accurate in the 17-limit; a comma basis is {4914/4913, 5832/5831, 9801/9800, 10648/10647, 28561/28560, 140625/140608}. | 2684edo is an extremely strong 13-limit system, with a lower 13-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any division until we reach [[5585edo]]. It is distinctly [[consistent]] through the [[17-odd-limit]], and is both a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak and zeta integral edo]]. It is [[enfactoring|enfactored]] in the 2.3.5.13 subgroup, with the same tuning as [[1342edo]], tempering out kwazy, {{monzo| -53 10 16 }}, senior, {{monzo| -17 62 -35 }} and egads, {{monzo| -36 52 51 }}. A 13-limit [[comma basis]] is {9801/9800, 10648/10647, 140625/140608, 196625/196608, 823680/823543}; it also tempers out 123201/123200. It is less accurate, but still quite accurate in the 17-limit; a comma basis is {4914/4913, 5832/5831, 9801/9800, 10648/10647, 28561/28560, 140625/140608}. | ||
Revision as of 21:09, 13 January 2023
| ← 2683edo | 2684edo | 2685edo → |
Theory
2684edo is an extremely strong 13-limit system, with a lower 13-limit relative error than any division until we reach 5585edo. It is distinctly consistent through the 17-odd-limit, and is both a zeta peak and zeta integral edo. It is enfactored in the 2.3.5.13 subgroup, with the same tuning as 1342edo, tempering out kwazy, [-53 10 16⟩, senior, [-17 62 -35⟩ and egads, [-36 52 51⟩. A 13-limit comma basis is {9801/9800, 10648/10647, 140625/140608, 196625/196608, 823680/823543}; it also tempers out 123201/123200. It is less accurate, but still quite accurate in the 17-limit; a comma basis is {4914/4913, 5832/5831, 9801/9800, 10648/10647, 28561/28560, 140625/140608}.
2684edo sets the septimal comma, 64/63, to an exact 1/44th of the octave (61 steps). As a corollary, it supports the period-44 ruthenium temperament.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.018 | -0.025 | +0.027 | -0.051 | +0.009 | +0.112 | -0.196 | -0.107 | +0.080 | -0.028 |
| Relative (%) | +0.0 | -3.9 | -5.5 | +5.9 | -11.4 | +2.0 | +25.0 | -43.7 | -24.0 | +17.9 | -6.3 | |
| Steps (reduced) |
2684 (0) |
4254 (1570) |
6232 (864) |
7535 (2167) |
9285 (1233) |
9932 (1880) |
10971 (235) |
11401 (665) |
12141 (1405) |
13039 (2303) |
13297 (2561) | |
Divisors
Since 2684 factors as 22 × 11 × 61, 2684edo has subset edos 2, 4, 11, 22, 44, 61, 122, 244, 671, and 1342.
Regular temperament properties
2684edo holds a record for the lowest relative error in the 13-limit. It is only bettered by 5585edo.
Rank-2 temperaments
Note: 5-limit temperaments represented by 1342edo are not included.
| Periods per 8ve |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 44 | 1114\2684 (16\2684) |
498.063 (7.154) |
4/3 (18375/18304) |
Ruthenium |