643edo
| ← 642edo | 643edo | 644edo → |
643 equal divisions of the octave (abbreviated 643edo or 643ed2), also called 643-tone equal temperament (643tet) or 643 equal temperament (643et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 643 equal parts of about 1.866 ¢ each. Each step represents a frequency ratio of 21/643, or the 643rd root of 2.
Theory
643edo is distinctly consistent to the 21-odd-limit, with a generally flat tendency, but the 5th harmonic is only 0.000439 cents sharp as the denominator of a convergent to log25, after 146 and before 4004. It tempers out 32805/32768 in the 5-limit and 2401/2400 in the 7-limit, so that it supports the sesquiquartififths temperament. In the 11-limit it tempers out 3025/3024 and 151263/151250; in the 13-limit 1001/1000, 1716/1715 and 4225/4224; in the 17-limit 1089/1088, 1701/1700, 2431/2430 and 2601/2600; and in the 19-limit 1331/1330, 1521/1520, 1729/1728, 2376/2375 and 2926/2925. It provides the optimal patent val for the rank-3 13-limit vili temperament.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.000 | -0.244 | +0.000 | -0.241 | -0.774 | -0.714 | -0.445 | -0.779 | +0.653 | +0.594 | +0.843 |
| relative (%) | +0 | -13 | +0 | -13 | -41 | -38 | -24 | -42 | +35 | +32 | +45 | |
| Steps (reduced) |
643 (0) |
1019 (376) |
1493 (207) |
1805 (519) |
2224 (295) |
2379 (450) |
2628 (56) |
2731 (159) |
2909 (337) |
3124 (552) |
3186 (614) | |
Subsets and supersets
643edo is the 117th prime edo.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-1019 643⟩ | [⟨643 1019]] | +0.0771 | 0.0771 | 4.13 |
| 2.3.5 | 32805/32768, [1 99 -68⟩ | [⟨643 1019 1493]] | +0.0513 | 0.7270 | 3.90 |
| 2.3.5.7 | 2401/2400, 32805/32768, [9 21 -17 -1⟩ | [⟨643 1019 1493 1805]] | +0.0600 | 0.0647 | 3.47 |
| 2.3.5.7.11 | 2401/2400, 3025/3024, 32805/32768, 391314/390625 | [⟨643 1019 1493 1805 2224]] | +0.0927 | 0.0874 | 4.68 |
| 2.3.5.7.11.13 | 1001/1000, 1716/1715, 3025/3024, 4225/4224, 32805/32768 | [⟨643 1019 1493 1805 2224 2379]] | +0.1094 | 0.0881 | 4.72 |
| 2.3.5.7.11.13.17 | 1001/1000, 1089/1088, 1701/1700, 1716/1715, 2601/2600, 4225/4224 | [⟨643 1019 1493 1805 2224 2379 2628]] | +0.1094 | 0.0816 | 4.37 |
| 2.3.5.7.11.13.17.19 | 1001/1000, 1089/1088, 1521/1520, 1701/1700, 1716/1715, 1729/1728, 2601/2600 | [⟨643 1019 1493 1805 2224 2379 2628 2731]] | +0.1186 | 0.0801 | 4.29 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 94\643 | 175.43 | 448/405 | Sesquiquartififths |
| 1 | 267\643 | 498.29 | 4/3 | Helmholtz |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct