719edo
| ← 718edo | 719edo | 720edo → |
719 equal divisions of the octave (abbreviated 719edo or 719ed2), also called 719-tone equal temperament (719tet) or 719 equal temperament (719et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 719 equal parts of about 1.67 ¢ each. Each step represents a frequency ratio of 21/719, or the 719th root of 2.
Theory
719edo is only consistent to the 3-odd-limit and the error of its harmonic 3 is quite large. Its harmonics 5 and 7 are also about halfway between its steps. Its distinction, however, is that it is the largest EDO to temper out the kleisma (15625/15552) and therefore support hanson in its patent val; it also supports the extension marfifths that tempers out the hemimage comma, 10976/10935 in addition, and in fact is the optimal patent val for marfifths. For other purposes than tempering out the kleisma, it can be used in the 2.9.15.21.11.17.19.23.29 subgroup, tempering out 1701/1700, 3025/3024, 2376/2375, 8625/8624, 21888/21875, 72171/72128, 2001/2000 and 116127/116000.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.688 | -0.778 | -0.815 | -0.294 | -0.553 | +0.641 | -0.091 | +0.191 | -0.434 | -0.127 | -0.736 |
| Relative (%) | +41.2 | -46.6 | -48.8 | -17.6 | -33.1 | +38.4 | -5.4 | +11.4 | -26.0 | -7.6 | -44.1 | |
| Steps (reduced) |
1140 (421) |
1669 (231) |
2018 (580) |
2279 (122) |
2487 (330) |
2661 (504) |
2809 (652) |
2939 (63) |
3054 (178) |
3158 (282) |
3252 (376) | |
Subsets and supersets
719edo is the 128th prime EDO. 1438edo, which doubles it, gives a good correction to the harmonics 3, 5 and 7.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [-2279 719⟩ | [⟨719 2279]] | 0.0464 | 0.0464 | 2.78 |
Music
- "Mar-A-Lago & FBI-Raid" from The Scallop Disco Accident (2025) – Spotify | Bandcamp | YouTube – in Marfifths, 719edo tuning
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Todo: explain its xenharmonic value |