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 5-limit, it tempers out the schisma, and in the 7-limit, the landscape comma. It | == 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 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. | 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. | ||
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=== Subsets and supersets === | === 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 }}. | |||
Since 696 factors as {{ | |||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" |[[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
8ve stretch (¢) | ! colspan="2" | Tuning error | ||
! colspan="2" |Tuning error | |||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3 | | 2.3 | ||
|{{ | | {{Monzo| -1103 696 }} | ||
|{{ | | {{Mapping| 696 1103 }} | ||
|0.072829 | | 0.072829 | ||
|0.073 | | 0.073 | ||
|4.22 | | 4.22 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|32805/32768, 52 80 -77 | | 32805/32768, {{monzo| 52 80 -77 }} | ||
|{{ | | {{Mapping| 696 1103 1616 }} | ||
|0.060798 | | 0.060798 | ||
|0.064 | | 0.064 | ||
|3.71 | | 3.71 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|32805/32768, 250047/250000, 22 10 -3 -11 | | 32805/32768, 250047/250000, {{monzo| 22 10 -3 -11 }} | ||
|{{ | | {{Mapping| 696 1103 1616 1954 }} | ||
|0.072061 | | 0.072061 | ||
|0.035 | | 0.035 | ||
|2.06 | | 2.06 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|9801/9800, 32805/32768, 46656/46585, 250047/250000 | | 9801/9800, 32805/32768, 46656/46585, 250047/250000 | ||
|{{ | | {{Mapping| 696 1103 1616 1954 2408 }} | ||
|0.004896 | | 0.004896 | ||
|0.089 | | 0.089 | ||
|5.15 | | 5.15 | ||
|- | |- | ||
|2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
|729/728, 1575/1573, 4096/4095, 67392/67375, 250047/250000 | | 729/728, 1575/1573, 4096/4095, 67392/67375, 250047/250000 | ||
|{{ | | {{Mapping| 696 1103 1616 1954 2408 2576 }} | ||
| -0.034283 | | -0.034283 | ||
|0.119 | | 0.119 | ||
|6.92 | | 6.92 | ||
|} | |} | ||