487edo: Difference between revisions
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== Theory == | == Theory == | ||
487edo is [[consistency|distinctly consistent]] to the [[13-odd-limit]]. | 487edo is [[consistency|distinctly consistent]] to the [[13-odd-limit]]. It [[tempers out]] {{monzo| 24 -21 4 }} ([[vulture comma]]) and {{monzo| 55 -1 -23 }} (counterwürschmidt comma) in the 5-limit, 4375/4374 ([[ragisma]]), 235298/234375 ([[triwellisma]]), and 33554432/33480783 ([[garischisma]]) in the 7-limit, [[5632/5625]], [[12005/11979]], [[19712/19683]], [[41503/41472]] in the 11-limit, [[676/675]], [[1001/1000]], [[2080/2079]], [[4096/4095]], and [[4225/4224]] in the 13-limit. It supports [[semidimfourth]], [[seniority]], and [[vulture]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 26: | Line 26: | ||
| {{monzo| 772 -487 }} | | {{monzo| 772 -487 }} | ||
| {{mapping| 487 772 }} | | {{mapping| 487 772 }} | ||
| | | −0.0958 | ||
| 0.0958 | | 0.0958 | ||
| 3.89 | | 3.89 | ||
| Line 33: | Line 33: | ||
| {{monzo| 24 -21 4 }}, {{monzo| 55 -1 -23 }} | | {{monzo| 24 -21 4 }}, {{monzo| 55 -1 -23 }} | ||
| {{mapping| 487 772 1131 }} | | {{mapping| 487 772 1131 }} | ||
| | | −0.1421 | ||
| 0.1020 | | 0.1020 | ||
| 4.14 | | 4.14 | ||
| Line 40: | Line 40: | ||
| 4375/4374, 235298/234375, 33554432/33480783 | | 4375/4374, 235298/234375, 33554432/33480783 | ||
| {{mapping| 487 772 1131 1367 }} | | {{mapping| 487 772 1131 1367 }} | ||
| | | −0.0667 | ||
| 0.1577 | | 0.1577 | ||
| 6.40 | | 6.40 | ||
| Line 47: | Line 47: | ||
| 4375/4374, 5632/5625, 12005/11979, 41503/41472 | | 4375/4374, 5632/5625, 12005/11979, 41503/41472 | ||
| {{mapping| 487 772 1131 1367 1685 }} | | {{mapping| 487 772 1131 1367 1685 }} | ||
| | | −0.0899 | ||
| 0.1485 | | 0.1485 | ||
| 6.03 | | 6.03 | ||
| Line 54: | Line 54: | ||
| 676/675, 1001/1000, 4096/4095, 4375/4374, 12005/11979 | | 676/675, 1001/1000, 4096/4095, 4375/4374, 12005/11979 | ||
| {{mapping| 487 772 1131 1367 1685 1802 }} | | {{mapping| 487 772 1131 1367 1685 1802 }} | ||
| | | −0.0623 | ||
| 0.1490 | | 0.1490 | ||
| 6.05 | | 6.05 | ||
| Line 105: | Line 105: | ||
| [[Tritriple]] (5-limit) | | [[Tritriple]] (5-limit) | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Scales == | == Scales == | ||
Revision as of 18:55, 15 January 2025
| ← 486edo | 487edo | 488edo → |
Theory
487edo is distinctly consistent to the 13-odd-limit. It tempers out [24 -21 4⟩ (vulture comma) and [55 -1 -23⟩ (counterwürschmidt comma) in the 5-limit, 4375/4374 (ragisma), 235298/234375 (triwellisma), and 33554432/33480783 (garischisma) in the 7-limit, 5632/5625, 12005/11979, 19712/19683, 41503/41472 in the 11-limit, 676/675, 1001/1000, 2080/2079, 4096/4095, and 4225/4224 in the 13-limit. It supports semidimfourth, seniority, and vulture.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.30 | +0.54 | -0.45 | +0.63 | -0.28 | +1.00 | +0.64 | +0.06 | +0.40 | +0.75 |
| Relative (%) | +0.0 | +12.3 | +22.1 | -18.2 | +25.7 | -11.4 | +40.6 | +25.9 | +2.5 | +16.3 | +30.6 | |
| Steps (reduced) |
487 (0) |
772 (285) |
1131 (157) |
1367 (393) |
1685 (224) |
1802 (341) |
1991 (43) |
2069 (121) |
2203 (255) |
2366 (418) |
2413 (465) | |
Subsets and supersets
487edo is the 93rd prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [772 -487⟩ | [⟨487 772]] | −0.0958 | 0.0958 | 3.89 |
| 2.3.5 | [24 -21 4⟩, [55 -1 -23⟩ | [⟨487 772 1131]] | −0.1421 | 0.1020 | 4.14 |
| 2.3.5.7 | 4375/4374, 235298/234375, 33554432/33480783 | [⟨487 772 1131 1367]] | −0.0667 | 0.1577 | 6.40 |
| 2.3.5.7.11 | 4375/4374, 5632/5625, 12005/11979, 41503/41472 | [⟨487 772 1131 1367 1685]] | −0.0899 | 0.1485 | 6.03 |
| 2.3.5.7.11.13 | 676/675, 1001/1000, 4096/4095, 4375/4374, 12005/11979 | [⟨487 772 1131 1367 1685 1802]] | −0.0623 | 0.1490 | 6.05 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 131\487 | 322.79 | 3087/2560 | Seniority |
| 1 | 157\487 | 386.86 | 5/4 | Counterwürschmidt |
| 1 | 182\487 | 448.46 | 35/27 | Semidimfourth |
| 1 | 193\487 | 475.56 | 320/243 | Vulture |
| 1 | 202\487 | 497.74 | 4/3 | Gary |
| 1 | 227\487 | 559.34 | 864/625 | Tritriple (5-limit) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct