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Kleismic: Difference between revisions

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moved DR optimizations to separate table
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== Tunings ==
== Tunings ==
=== Optimized tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Prime-Optimized Tunings
|-
! Weight-skew\Order !! Euclidean
|-
| Tenney || (2.3.5) CTE: ~6/5 = 317.0308¢
|-
| Tenney || (2.3.5) POTE: ~6/5 = 317.007¢
|-
| Tenney || (2.3.5.13) CTE: ~6/5 = 317.1110¢
|-
| Tenney || (2.3.5.13) POTE: ~6/5 = 317.0756¢
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+style="font-size: 105%; white-space: nowrap;" | [[Delta-rational chord|DR]] and equal-beating tunings
|-
! Optimized chord !! Generator value !! Polynomial !! Further notes
|-
| 3:4:5 (+1 +1) || ~6/5 = 317.1496 || ''g''<sup>6</sup> + 2''g''<sup>5</sup> - 8 = 0 || 1-3-5 equal-beating tuning
|-
| 4:5:6 (+1 +1) || ~6/5 = 317.9593 || ''g''<sup>6</sup> - 2''g''<sup>5</sup> + 2 = 0 || 1-3-5 equal-beating tuning, close to 2/7-kleisma
|-
| 10:12:15 (+2 +3) || ~6/5 = 317.6675 || ''g''<sup>6</sup> - 5''g'' + 3 = 0 || Close to 1/4-kleisma
|-
| 9:13:15 (+2 +1) || ~6/5 = 317.5679 || 3''g''<sup>3</sup> + 4''g'' - 10 = 0 || Close to 13/36-marveltwin comma
|-
| 13:15:18 (+2 +3) || ~6/5 = 317.0010 || 3''g''<sup>3</sup> - ''g'' - 4 = 0 || Close to 13/51-marveltwin comma
|}
=== Tuning spectrum ===
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
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| 317.1429
| 317.1429
|  
|  
|-
|
| ''f''<sup>6</sup> + 2''f''<sup>5</sup> - 8 = 0
| 317.1496
| 1–3–5 equal-beating tuning, [[Delta-rational chord|DR]] 3:4:5
|-
|-
|  
|  
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| 317.6681
| 317.6681
| 1/4-kleisma, virtually [[Delta-rational chord|DR]] 10:12:15
| 1/4-kleisma, virtually [[Delta-rational chord|DR]] 10:12:15
|-
|
| ''f''<sup>6</sup> - 2''f''<sup>5</sup> + 2 = 0
| 317.9593
| 1–3–5 equal-beating tuning, [[Delta-rational chord|DR]] 4:5:6, close to 2/7-kleisma
|-
|-
| [[83edo|22\83]]
| [[83edo|22\83]]
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=== Other tunings ===
=== Other tunings ===
* [[DKW theory|DKW]] (2.3.5): ~2 = 1\1, ~6/5 = 317.1983
* [[DKW theory|DKW]] (2.3.5): ~2 = 1\1, ~6/5 = 317.1983
* [[Delta-rational chord|DR]] 9:13:15: ~2 = 1\1, ~6/5 = 317.0010 (real root of 3''f''<sup>3</sup> - ''f'' - 4 = 0)
* [[Delta-rational chord|DR]] 13:15:18: ~2 = 1\1, ~6/5 = 317.5679 (real root of 3''f''<sup>3</sup> + 4''f'' - 10 = 0)


== Scales ==
== Scales ==
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