233edo
| ← 232edo | 233edo | 234edo → |
233 equal divisions of the octave (abbreviated 233edo or 233ed2), also called 233-tone equal temperament (233tet) or 233 equal temperament (233et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 233 equal parts of about 5.15 ¢ each. Each step represents a frequency ratio of 21/233, or the 233rd root of 2.
Theory
233et has a generally flat tendency, in the sense that if the octave is pure, prime harmonics 3 through 17 are all flat. 233edo is accurate for the 5th harmonic (only 0.0476 ¢ flat), but less for the third harmonic (1.5258 ¢ flat).
The equal temperament tempers out 78732/78125 and [-53 32 1⟩ in the 5-limit; 2401/2400, 65625/65536, and 177147/175616 in the 7-limit (supporting tertiaseptal and catafourth). Using the patent val, it tempers out 243/242, 441/440, 35937/35840, and 78408/78125 in the 11-limit; 351/350, 1001/1000, 1575/1573, 4225/4224, and 6656/6655 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.53 | -0.05 | -0.59 | +2.10 | -0.24 | -1.04 | -1.57 | -1.95 | +1.20 | -2.11 | +0.05 |
| Relative (%) | -29.6 | -0.9 | -11.4 | +40.7 | -4.8 | -20.2 | -30.6 | -37.9 | +23.3 | -41.0 | +1.0 | |
| Steps (reduced) |
369 (136) |
541 (75) |
654 (188) |
739 (40) |
806 (107) |
862 (163) |
910 (211) |
952 (20) |
990 (58) |
1023 (91) |
1054 (122) | |
Subsets and supersets
233edo is the 51st prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-369 233⟩ | [⟨233 369]] | +0.4813 | 0.4815 | 9.35 |
| 2.3.5 | 78732/78125, [-53 32 1⟩ | [⟨233 369 541]] | +0.3277 | 0.4492 | 8.72 |
| 2.3.5.7 | 2401/2400, 65625/65536, 78732/78125 | [⟨233 369 541 654]] | +0.2979 | 0.3924 | 7.62 |
| 2.3.5.7.11 | 243/242, 441/440, 540/539, 2401/2400 | [⟨233 369 541 654 806]] | +0.2525 | 0.3625 | 7.04 |
| 2.3.5.7.11.13 | 243/242, 351/350, 441/440, 540/539, 1001/1000 | [⟨233 369 541 654 806 862]] | +0.2574 | 0.3311 | 6.43 |
| 2.3.5.7.11.13.17 | 351/350, 441/440, 540/539, 561/560, 936/935, 1156/1155 | [⟨233 369 541 654 806 862 952]] | +0.2888 | 0.3161 | 6.14 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 15\233 | 77.25 | 256/245 | Tertiaseptal |
| 1 | 22\233 | 113.30 | 16/15 | Misneb |
| 1 | 55\233 | 283.26 | 189/160 | Neominor |
| 1 | 77\233 | 396.57 | 98304/78125 | Squarschmidt |
| 1 | 86\233 | 442.92 | 9/7 | Sensi |
| 1 | 95\233 | 489.27 | 250/189 | Catafourth |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct