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 cents flat), but less for the third harmonic (1.5258 cents 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 (%) | -30 | -1 | -11 | +41 | -5 | -20 | -31 | -38 | +23 | -41 | +1 | |
| 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 it is distinct