696edo: Difference between revisions
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{{ED intro}} | {{ED intro}} | ||
696edo is a strong 7-limit tuning, but unfortunately it is consistent only up to the [[9-odd-limit]]. In the | == Theory == | ||
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 [[support]]s 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 {{nowrap| 2<sup>3</sup> × 3 × 29 }}, 696edo has subset edos {{EDOs| 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<br>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, {{monzo| 52 80 -77 }} | |||
| {{Mapping| 696 1103 1616 }} | |||
| 0.060798 | |||
| 0.064 | |||
| 3.71 | |||
|- | |||
| 2.3.5.7 | |||
| 32805/32768, 250047/250000, {{monzo| 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 | |||
|} | |||