718edo
| ← 717edo | 718edo | 719edo → |
718 equal divisions of the octave (abbreviated 718edo or 718ed2), also called 718-tone equal temperament (718tet) or 718 equal temperament (718et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 718 equal parts of about 1.67 ¢ each. Each step represents a frequency ratio of 21/718, or the 718th root of 2.
Theory
718edo is distinctly consistent in the 23-odd-limit, and does well enough in the 31-limit. It is closely related to 359edo, but the mapping differs for 5, 13, 17 and 31.
As does 359et, 718et tempers out the 359-comma in the 3-limit, rendering a very accurate harmonic 3. In the 5-limit it tempers out the gammic comma, [-29 -11 20⟩, and the monzisma, [54 -37 2⟩. In the 7-limit it tempers out 4375/4374; in the 11-limit 3025/3024, 9801/9800 and 131072/130977; in the 13-limit 1716/1715, 2080/2079, 4096/4095, 4225/4224, 6656/6655 and 10648/10647; in the 17-limit 1275/1274, 2025/2023; in the 19-limit 2432/2431, 3250/3249, 4200/4199 and 5985/5984; and in the 23-limit 2024/2023, 2025/2024, 2185/2184, 3060/3059. It supports gammic, monzismic and abigail.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.005 | -0.241 | +0.533 | +0.214 | +0.141 | +0.337 | -0.020 | +0.138 | -0.051 | -0.189 |
| Relative (%) | +0.0 | -0.3 | -14.4 | +31.9 | +12.8 | +8.4 | +20.2 | -1.2 | +8.3 | -3.0 | -11.3 | |
| Steps (reduced) |
718 (0) |
1138 (420) |
1667 (231) |
2016 (580) |
2484 (330) |
2657 (503) |
2935 (63) |
3050 (178) |
3248 (376) |
3488 (616) |
3557 (685) | |
Subsets and supersets
Since 718 factors into 2 × 359, 718edo contains 2edo and 359edo as subsets.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [-29 -11 20⟩, [54 -37 2⟩ | [⟨718 1138 1667]] | +0.0357 | 0.0482 | 2.89 |
| 2.3.5.7 | 4375/4374, 40500000/40353607, [31 -6 -2 -6⟩ | [⟨718 1138 1667 2016]] | −0.0207 | 0.1063 | 6.36 |
| 2.3.5.7.11 | 3025/3024, 4375/4374, 131072/130977, 40500000/40353607 | [⟨718 1138 1667 2016 2484]] | −0.0290 | 0.0965 | 5.77 |
| 2.3.5.7.11.13 | 1716/1715, 2080/2079, 3025/3024, 4096/4095, 7031250/7014007 | [⟨718 1138 1667 2016 2484 2657]] | −0.0305 | 0.0881 | 5.27 |
| 2.3.5.7.11.13.17 | 1275/1274, 1716/1715, 2025/2023, 2080/2079, 3025/3024, 4096/4095 | [⟨718 1138 1667 2016 2484 2657 2935]] | −0.0379 | 0.0836 | 5.00 |
| 2.3.5.7.11.13.17.19 | 1275/1274, 1716/1715, 2025/2023, 2080/2079, 2432/2431, 3025/3024, 3250/3249 | [⟨718 1138 1667 2016 2484 2657 2935 3050]] | −0.0326 | 0.0795 | 4.76 |
| 2.3.5.7.11.13.17.19.23 | 1275/1274, 1716/1715, 2024/2023, 2025/2023, 2080/2079, 2185/2184, 2432/2431, 3025/3024 | [⟨718 1138 1667 2016 2484 2657 2935 3050 3248]] | −0.0323 | 0.0749 | 4.48 |
- 718et has a lower absolute error in the 23-limit than any previous equal temperaments, past 581 and followed by 742i.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 21\718 | 35.10 | 234375/229376 | Gammic |
| 1 | 249\718 | 249.03 | [-27 11 3 1⟩ | Monzismic |
| 2 | 125\718 | 208.91 | 44/39 | Abigail |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct