Augene: Difference between revisions
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== 11-limit interval chain == | == 11-limit interval chain == | ||
{| class="wikitable" | {| class="wikitable center-1 right-all" | ||
|+Generator based off of 11-limit minimax | |+Generator based off of 11-limit minimax | ||
! Gen. | ! Gen. | ||
! Period 0 | |||
! Period 1 | ! Period 1 | ||
! Period 2 | ! Period 2 | ||
|- | |- | ||
| 0 | | 0 | ||
| 0.000 | |||
| 400.000 | | 400.000 | ||
| 800.000 | | 800.000 | ||
|- | |- | ||
| 1 | | 1 | ||
| 1113.148 | |||
| 313.148 | | 313.148 | ||
| 713.148 | | 713.148 | ||
|- | |- | ||
| 2 | | 2 | ||
| 1026.296 | |||
| 226.296 | | 226.296 | ||
| 626.296 | | 626.296 | ||
|- | |- | ||
| 3 | | 3 | ||
| 939.444 | |||
| 139.444 | | 139.444 | ||
| 539.444 | | 539.444 | ||
|- | |- | ||
| 4 | | 4 | ||
| 852.592 | |||
| 52.592 | | 52.592 | ||
| 452.592 | | 452.592 | ||
|- | |- | ||
| 5 | | 5 | ||
| 765.740 | |||
| 1165.740 | | 1165.740 | ||
| 365.740 | | 365.740 | ||
|} | |} | ||
== Tuning spectrum == | == Tuning spectrum == | ||
{| class="wikitable center-all" | {| class="wikitable center-all left-4" | ||
|- | |- | ||
! Edo<br>Generator | ! Edo<br>Generator | ||
| Line 114: | Line 114: | ||
| 10/9 | | 10/9 | ||
| 708.798 | | 708.798 | ||
| 9 limit minimax | | 9-odd-limit minimax | ||
|- | |- | ||
| 39\66 | | 39\66 | ||
| Line 138: | Line 138: | ||
| 57\96 | | 57\96 | ||
| | | | ||
| 712. | | 712.500 | ||
| | | | ||
|- | |- | ||
| Line 144: | Line 144: | ||
| 11/9 | | 11/9 | ||
| 713.148 | | 713.148 | ||
| 11 limit minimax | | 11-odd-limit minimax | ||
|- | |- | ||
| 25\42 | | 25\42 | ||
| Line 154: | Line 154: | ||
| 8/7 | | 8/7 | ||
| 715.587 | | 715.587 | ||
| 7, 13 and 15 limit minimax | | 7-, 13- and 15-odd-limit minimax | ||
|- | |- | ||
| | | | ||
| 6/5 | | 6/5 | ||
| 715.641 | | 715.641 | ||
| 5 limit minimax | | 5-odd-limit minimax | ||
|- | |- | ||
| 34\57 | | 34\57 | ||
| Line 222: | Line 222: | ||
; [[Joel Grant Taylor]] | ; [[Joel Grant Taylor]] | ||
* | * [https://web.archive.org/web/20201127012922/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 ''Galticeran Sonatina''] – in Augene[12] tuned to 27edo | ||
[[Category:Temperaments]] | [[Category:Temperaments]] | ||
[[Category:Augene| ]] <!-- main article --> | [[Category:Augene| ]] <!-- main article --> | ||
[[Category:Augmented family]] | |||
[[Category:Archytas clan]] | |||
[[Category:Starling temperaments]] | |||
Revision as of 09:56, 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 |
|---|---|---|---|
| 14/13 | 671.702 | ||
| 12\28 | 685.714 | ||
| 16/15 | 688.269 | ||
| 13/11 | 689.210 | ||
| 7\12 | 700.000 | ||
| 31\53 | 701.887 | ||
| 4/3 | 701.955 | ||
| 30\51 | 705.882 | ||
| 15/14 | 706.481 | ||
| 23\39 | 707.692 | ||
| 7/5 | 708.744 | ||
| 9/7 | 708.771 | ||
| 10/9 | 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 | ||
| 8/7 | 715.587 | 7-, 13- and 15-odd-limit minimax | |
| 6/5 | 715.641 | 5-odd-limit minimax | |
| 34\57 | 715.789 | ||
| 25\72 | 716.667 | ||
| 12/11 | 716.879 | ||
| 11/10 | 717.498 | ||
| 9\15 | 720.000 | ||
| 18/13 | 721.127 | ||
| 15/13 | 723.871 | ||
| 11/8 | 724.341 | ||
| 13/12 | 730.714 | ||
| 13/10 | 745.786 | ||
| 16/13 | 759.472 |
Music
- Sad Like Winter Leaves in Augene[12] tuned to 27edo
- Galticeran Sonatina – in Augene[12] tuned to 27edo