Augene: Difference between revisions
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'''Augene''' is a [[regular temperament]] of the [[augmented family]], which means that [[128/125]] is [[tempering out|tempered out]] and three [[5/4]]'s make a [[2/1]]. Augene is distinguished from its relative [[august]] by tempering out [[64/63]], which means that tunings in which the fifth is larger than 7\12 (700 cents) are optimal. Therefore the [[optimal ET sequence]] of augene goes 12, 15, 27, … in constrast to august's which goes 9, 12, 21, …. | '''Augene''' is a [[regular temperament]] of the [[augmented family]], which means that [[128/125]] is [[tempering out|tempered out]] and three [[5/4]]'s make a [[2/1]]. Augene is distinguished from its relative [[august]] by tempering out [[64/63]], which means that tunings in which the fifth is larger than 7\12 (700 cents) are optimal. Therefore the [[optimal ET sequence]] of augene goes 12, 15, 27, … in constrast to august's which goes 9, 12, 21, …. | ||
It is also the unique [[7-limit]] [[regular temperament]] that tempers out the [[essential tempering commas]] of the [[5-odd-limit|5-]] ([[128/125]]) | It is also the unique [[7-limit]] [[regular temperament]] that tempers out the [[essential tempering commas]] of the [[5-odd-limit|5-]] ([[128/125]]) and [[7-odd-limit]] ([[64/63]], [[126/125]]). | ||
The first few augene [[mos scale]]s are [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], … and the first edos that reasonably support augene are [[12edo]], [[15edo]], and especially [[27edo]]. | The first few augene [[mos scale]]s are [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], … and the first edos that reasonably support augene are [[12edo]], [[15edo]], and especially [[27edo]]. | ||
See [[Augmented family #Augene]] for technical data. See [[Augene extensions]] for a discussion on 13-limit extensions. | |||
== Interval chain == | == Interval chain == | ||
| Line 72: | Line 74: | ||
<nowiki/>* In 11-odd-limit minimax tuning, octave reduced | <nowiki/>* In 11-odd-limit minimax tuning, octave reduced | ||
== | == Tunings == | ||
{| class="wikitable mw-collapsible mw-collapsed" | |||
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit prime-optimized tunings | |||
|- | |||
! rowspan="2" | | |||
! colspan="3" | Euclidean | |||
|- | |||
! Constrained | |||
! Constrained & skewed | |||
! Destretched | |||
|- | |||
! Tenney | |||
| CTE: ~3/2 = 709.5949{{c}} | |||
| CWE: ~3/2 = 709.3249{{c}} | |||
| POTE: ~3/2 = 709.2568{{c}} | |||
|} | |||
{| class="wikitable mw-collapsible mw-collapsed" | |||
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit prime-optimized tunings | |||
|- | |||
! rowspan="2" | | |||
! colspan="3" | Euclidean | |||
|- | |||
! Constrained | |||
! Constrained & skewed | |||
! Destretched | |||
|- | |||
! Tenney | |||
| CTE: ~3/2 = 713.5701{{c}} | |||
| CWE: ~3/2 = 711.6031{{c}} | |||
| POTE: ~3/2 = 711.1766{{c}} | |||
|} | |||
=== Tuning spectrum === | |||
{| class="wikitable center-all left-4" | {| class="wikitable center-all left-4" | ||
|- | |- | ||
| Line 80: | Line 113: | ||
! Generator (¢) | ! Generator (¢) | ||
! Comments | ! Comments | ||
|- | |- | ||
| | | | ||
| 15/8 | | 15/8 | ||
| 688.269 | | 688.269 | ||
| -1/3 comma | |||
|- | |||
|- | |- | ||
| 7\12 | | 7\12 | ||
| Line 104: | Line 127: | ||
| 3/2 | | 3/2 | ||
| 701.955 | | 701.955 | ||
| | | Untempered | ||
|- | |- | ||
| 30\51 | | 30\51 | ||
| | | | ||
| 705.882 | | 705.882 | ||
| | | 51cdeee val | ||
|- | |- | ||
| | | | ||
| Line 119: | Line 142: | ||
| | | | ||
| 707.692 | | 707.692 | ||
| | | 39dee val | ||
|- | |- | ||
| | | | ||
| Line 134: | Line 157: | ||
| 9/5 | | 9/5 | ||
| 708.798 | | 708.798 | ||
| 9-odd-limit minimax | | 1/6 comma, 9-odd-limit minimax | ||
|- | |- | ||
| 39\66 | | 39\66 | ||
| | | | ||
| 709.091 | | 709.091 | ||
| | | 66cdeee val | ||
|- | |- | ||
| | | | ||
| Line 149: | Line 172: | ||
| | | | ||
| 711.111 | | 711.111 | ||
| | | 27e val | ||
|- | |- | ||
| | | | ||
| Line 156: | Line 179: | ||
| | | | ||
|- | |- | ||
| | | 41\69 | ||
| | | | ||
| 713.043 | |||
| 69bcee val | |||
|- | |- | ||
| | | | ||
| Line 169: | Line 192: | ||
| | | | ||
| 714.286 | | 714.286 | ||
| | | 42e val | ||
|- | |- | ||
| | | | ||
| 7/4 | | 7/4 | ||
| 715.587 | | 715.587 | ||
| 7 | | 7-odd-limit minimax | ||
|- | |- | ||
| | | | ||
| 5/3 | | 5/3 | ||
| 715.641 | | 715.641 | ||
| 5-odd-limit minimax | | 1/3 comma, 5-odd-limit minimax | ||
|- | |- | ||
| 34\57 | | 34\57 | ||
| | | | ||
| 715.789 | | 715.789 | ||
| | | 53bce val | ||
|- | |- | ||
| | | | ||
| Line 204: | Line 222: | ||
| | | | ||
| 720.000 | | 720.000 | ||
| | | | ||
|- | |- | ||
| Line 219: | Line 227: | ||
| 11/8 | | 11/8 | ||
| 724.341 | | 724.341 | ||
| | | | ||
|} | |} | ||