718edo
|
This page presents a novelty topic.
It may contain ideas which are less likely to find practical applications in music, or numbers or structures that are arbitrary or exceedingly small, large, or complex. Novelty topics are often developed by a single person or a small group. As such, this page may also contain idiosyncratic terms, notation, or conceptual frameworks. |
|
This page is a stub. You can help the Xenharmonic Wiki by expanding it. |
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 perfect fifth. 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
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 relative error in the 23-limit than any previous equal temperaments, past 581 and followed by 742i.
Rank-2 temperaments
| Periods per Octave |
Generator (Reduced) |
Cents (Reduced) |
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 |