Augene: Difference between revisions
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Cleanup +1; +categories |
→Tuning spectrum: some of the edo tunings outright don't support augene. Un-normalize ratios to the octave-reduced form. +note |
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| Line 47: | Line 47: | ||
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! Edo<br>Generator | ! Edo<br>Generator | ||
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval]] | ! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]* | ||
! Fifth (¢) | ! Fifth (¢) | ||
! Comments | ! Comments | ||
|- | |- | ||
| | | | ||
| | | 13/7 | ||
| 671.702 | | 671.702 | ||
| | | | ||
|- | |- | ||
| | | | ||
| | | 15/8 | ||
| 688.269 | | 688.269 | ||
| | | | ||
| Line 74: | Line 69: | ||
| | | | ||
| 700.000 | | 700.000 | ||
| | | | ||
|- | |- | ||
| | | | ||
| | | 3/2 | ||
| 701.955 | | 701.955 | ||
| | | | ||
| Line 112: | Line 102: | ||
|- | |- | ||
| | | | ||
| | | 9/5 | ||
| 708.798 | | 708.798 | ||
| 9-odd-limit minimax | | 9-odd-limit minimax | ||
| Line 152: | Line 142: | ||
|- | |- | ||
| | | | ||
| | | 7/4 | ||
| 715.587 | | 715.587 | ||
| 7-, 13- and 15-odd-limit minimax | | 7-, 13- and 15-odd-limit minimax | ||
|- | |- | ||
| | | | ||
| | | 5/3 | ||
| 715.641 | | 715.641 | ||
| 5-odd-limit minimax | | 5-odd-limit minimax | ||
| Line 172: | Line 162: | ||
|- | |- | ||
| | | | ||
| | | 11/6 | ||
| 716.879 | | 716.879 | ||
| | | | ||
| Line 187: | Line 177: | ||
|- | |- | ||
| | | | ||
| | | 13/9 | ||
| 721.127 | | 721.127 | ||
| | | | ||
| Line 212: | Line 202: | ||
|- | |- | ||
| | | | ||
| | | 13/8 | ||
| 759.472 | | 759.472 | ||
| | | | ||
|} | |} | ||
<nowiki>*</nowiki> besides the octave | |||
== Music == | == Music == | ||
Revision as of 14:14, 13 May 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, ….
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.
11-limit interval chain
| Gen. | Period 0 | Period 1 | Period 2 |
|---|---|---|---|
| 0 | 0.000 | 400.000 | 800.000 |
| 1 | 1113.148 | 313.148 | 713.148 |
| 2 | 1026.296 | 226.296 | 626.296 |
| 3 | 939.444 | 139.444 | 539.444 |
| 4 | 852.592 | 52.592 | 452.592 |
| 5 | 765.740 | 1165.740 | 365.740 |
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