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 ([[128/125]])- and 7-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]]. | ||
Revision as of 04:16, 27 August 2024
Augene is a regular temperament of the augmented family, which means that 128/125 is 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 (128/125)- and 7-limit (64/63, 126/125).
The first few augene mos scales are 3L 3s, 3L 6s, 3L 9s, 12L 3s, … and the first edos that reasonably support augene are 12edo, 15edo, and especially 27edo.
Interval chain
In the following table, odd harmonics 1–11 are in bold.
| # | Period 0 | Period 1 | Period 2 | |||
|---|---|---|---|---|---|---|
| Cents* | Approx. Ratios | Cents* | Approx. Ratios | Cents* | Approx. Ratios | |
| 0 | 0.0 | 1/1 | 400.0 | 5/4, 14/11 | 800.0 | 8/5, 11/7 |
| 1 | 1113.1 | 15/8, 40/21 | 313.1 | 6/5 | 713.1 | 3/2 |
| 2 | 1026.3 | 9/5, 20/11 | 226.3 | 8/7, 9/8 | 626.3 | 10/7, 16/11 |
| 3 | 939.4 | 12/7 | 139.4 | 12/11, 15/14 | 539.4 | 15/11 |
| 4 | 852.6 | 18/11 | 52.6 | 45/44 | 452.6 | 9/7 |
| 5 | 765.7 | 54/35 | 1165.7 | 27/14, 96/49 | 365.7 | 27/22 |
* in 11-odd-limit minimax tuning
Tuning spectrum
| Edo Generator |
Eigenmonzo (Unchanged-interval)* |
Fifth (¢) | Comments |
|---|---|---|---|
| 13/7 | 671.702 | ||
| 15/8 | 688.269 | ||
| 13/11 | 689.210 | ||
| 7\12 | 700.000 | ||
| 3/2 | 701.955 | ||
| 30\51 | 705.882 | ||
| 15/14 | 706.481 | ||
| 23\39 | 707.692 | ||
| 7/5 | 708.744 | ||
| 9/7 | 708.771 | ||
| 9/5 | 708.798 | 9-odd-limit minimax | |
| 39\66 | 709.091 | ||
| 7/6 | 711.043 | ||
| 16\27 | 711.111 | ||
| 15/11 | 712.317 | ||
| 57\96 | 712.500 | ||
| 11/9 | 713.148 | 11-odd-limit minimax | |
| 25\42 | 714.286 | ||
| 7/4 | 715.587 | 7-, 13- and 15-odd-limit minimax | |
| 5/3 | 715.641 | 5-odd-limit minimax | |
| 34\57 | 715.789 | ||
| 25\72 | 716.667 | ||
| 11/6 | 716.879 | ||
| 11/10 | 717.498 | ||
| 9\15 | 720.000 | ||
| 13/9 | 721.127 | ||
| 15/13 | 723.871 | ||
| 11/8 | 724.341 | ||
| 13/12 | 730.714 | ||
| 13/10 | 745.786 | ||
| 13/8 | 759.472 |
* besides the octave
Music
- Sad Like Winter Leaves in Augene[12] tuned to 27edo
- Galticeran Sonatina – in Augene[12] tuned to 27edo