696edo: Difference between revisions
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696edo is a strong 7-limit tuning, but unfortunately it is consistent only up to the [[9-odd-limit]]. In the | 696edo is a strong 7-limit tuning, but unfortunately it is consistent only up to the [[9-odd-limit]]. In the 5-limit, it tempers out the schisma, and in the 7-limit, the landscape comma. It supports the [[magnesium]] temperament which divides the octave in 12, as well as [[chromium]] temperament that divides it in 24. | ||
Nonetheless despite inconsistency, it is a valuable xenharmonic system. It provides the [[optimal patent val]] for the [[octant]] temperament in the 13-limit, even if its approximation of 13 is almost half a step off. Likewise, 696edo tunes [[altierran]] and [[house]] temperaments in the 11-limit. | Nonetheless despite inconsistency, it is a valuable xenharmonic system in higher limits. It provides the [[optimal patent val]] for the [[octant]] temperament in the 13-limit, even if its approximation of 13 is almost half a step off. Likewise, 696edo tunes [[altierran]] and [[house]] temperaments in the 11-limit. In the higher limits, it may be used as a 2.3.5.7.17.31 subgroup tuning. | ||
The 696cc val is also very close to the [[POTE]] tuning for the [[witcher]] temperament, while 696f tunes [[semiterm]] and the inaccurate 696d tunes [[pontic]]. | The 696cc val is also very close to the [[POTE]] tuning for the [[witcher]] temperament, while 696f tunes [[semiterm]] and the inaccurate 696d tunes [[pontic]]. | ||
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=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|696}} | {{Harmonics in equal|696}} | ||
=== Subsets and supersets === | |||
Since 696 factors as {{Factorization|696}}, 696edo has subset edos {{EDOs|1, 2, 3, 4, 6, 8, 12, 24, 29, 58, 87, 116, 174, 232, 348}}. | |||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" |[[Subgroup]] | |||
! rowspan="2" |[[Comma list]] | |||
! rowspan="2" |[[Mapping]] | |||
! rowspan="2" |Optimal | |||
8ve stretch (¢) | |||
! colspan="2" |Tuning error | |||
|- | |||
![[TE error|Absolute]] (¢) | |||
![[TE simple badness|Relative]] (%) | |||
|- | |||
|2.3 | |||
|{{monzo|-1103 696}} | |||
|{{mapping|696 1103}} | |||
|0.072829 | |||
|0.073 | |||
|4.22 | |||
|- | |||
|2.3.5 | |||
|32805/32768, 52 80 -77 | |||
|{{mapping|696 1103 1616}} | |||
|0.060798 | |||
|0.064 | |||
|3.71 | |||
|- | |||
|2.3.5.7 | |||
|32805/32768, 250047/250000, 22 10 -3 -11 | |||
|{{mapping|696 1103 1616 1954}} | |||
|0.072061 | |||
|0.035 | |||
|2.06 | |||
|- | |||
|2.3.5.7.11 | |||
|9801/9800, 32805/32768, 46656/46585, 250047/250000 | |||
|{{mapping|696 1103 1616 1954 2408}} | |||
|0.004896 | |||
|0.089 | |||
|5.15 | |||
|- | |||
|2.3.5.7.11.13 | |||
|729/728, 1575/1573, 4096/4095, 67392/67375, 250047/250000 | |||
|{{mapping|696 1103 1616 1954 2408 2576}} | |||
| -0.034283 | |||
|0.119 | |||
|6.92 | |||
|} | |||
Revision as of 23:55, 11 October 2025
| ← 695edo | 696edo | 697edo → |
696 equal divisions of the octave (abbreviated 696edo or 696ed2), also called 696-tone equal temperament (696tet) or 696 equal temperament (696et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 696 equal parts of about 1.72 ¢ each. Each step represents a frequency ratio of 21/696, or the 696th root of 2.
696edo is a strong 7-limit tuning, but unfortunately it is consistent only up to the 9-odd-limit. In the 5-limit, it tempers out the schisma, and in the 7-limit, the landscape comma. It supports the magnesium temperament which divides the octave in 12, as well as chromium temperament that divides it in 24.
Nonetheless despite inconsistency, it is a valuable xenharmonic system in higher limits. It provides the optimal patent val for the octant temperament in the 13-limit, even if its approximation of 13 is almost half a step off. Likewise, 696edo tunes altierran and house temperaments in the 11-limit. In the higher limits, it may be used as a 2.3.5.7.17.31 subgroup tuning.
The 696cc val is also very close to the POTE tuning for the witcher temperament, while 696f tunes semiterm and the inaccurate 696d tunes pontic.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.231 | -0.107 | +0.140 | +0.406 | +0.852 | +0.217 | +0.763 | -0.688 | -0.267 | -0.208 |
| Relative (%) | +0.0 | -13.4 | -6.2 | +8.1 | +23.6 | +49.4 | +12.6 | +44.2 | -39.9 | -15.5 | -12.1 | |
| Steps (reduced) |
696 (0) |
1103 (407) |
1616 (224) |
1954 (562) |
2408 (320) |
2576 (488) |
2845 (61) |
2957 (173) |
3148 (364) |
3381 (597) |
3448 (664) | |
Subsets and supersets
Since 696 factors as 23 × 3 × 29, 696edo has subset edos 1, 2, 3, 4, 6, 8, 12, 24, 29, 58, 87, 116, 174, 232, 348.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal
8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-1103 696⟩ | [⟨696 1103]] | 0.072829 | 0.073 | 4.22 |
| 2.3.5 | 32805/32768, 52 80 -77 | [⟨696 1103 1616]] | 0.060798 | 0.064 | 3.71 |
| 2.3.5.7 | 32805/32768, 250047/250000, 22 10 -3 -11 | [⟨696 1103 1616 1954]] | 0.072061 | 0.035 | 2.06 |
| 2.3.5.7.11 | 9801/9800, 32805/32768, 46656/46585, 250047/250000 | [⟨696 1103 1616 1954 2408]] | 0.004896 | 0.089 | 5.15 |
| 2.3.5.7.11.13 | 729/728, 1575/1573, 4096/4095, 67392/67375, 250047/250000 | [⟨696 1103 1616 1954 2408 2576]] | -0.034283 | 0.119 | 6.92 |