Vengeance
| Vengeance |
2023/2000, 4165/4096 (2.5.7.17)
2.5.7.17 25-odd-limit: 8.88 ¢
2.5.7.17 25-odd-limit: 16 notes
Vengeance (CompactStar's name) or pentagoth (groundfault's and Userminusone's name) is a 2.5.17 subgroup temperament. It is notable for having a structure similar to mavila with an antidiatonic scale and superdiatonic scale but being comparatively very low in error and badness, because the flat fifth generator is represented by 25/17 rather than 3/2 (or equivalently, 34/25 rather than 4/3). It is defined by tempering out the comma 78608/78125. The name "vengeance' was coined by CompactStar and derives from 25/17's name as the "vengeance subfifth". Like with mavila, 3 generators reach the major third represented by 5/4, but the minor third is represented by 20/17. The minor triad is 17:20:25, which makes it far simpler than the major triad of 68:85:100, in contrast to 5-limit major and minor triads as used in mavila and meantone.
Pentagoth was defined by ground and Userminusone as having an extension to the 2.5.13/11.17 subgroup that identifies 20/17 and 13/11 by tempering out 221/220. The exact-13/11 tuning is 672.3¢, near 14\25 (672.0¢), and the exact-20/17 tuning is 670.3¢, near 19\34 (670.6¢).
The harmonic 7 can be added in a similar way to how mavila is extended to armodue, by having 7/4 reached as -5 generators of 34/25 (or the "minor seventh" in antidiatonic terms).
For technical data, see no-threes subgroup temperaments #Vengeance.
Interval chain
In the following table, prime harmonics are labeled in bold.
| # | Cents* | Approximate ratios | Melodic antidiatonic notation | |
|---|---|---|---|---|
| 0 | 0.00 | 1/1 | perfect unison | D |
| 1 | 527.928 | 34/25 | perfect 4th | G |
| 2 | 1055.856 | 119/64, 125/68 | major 7th | C |
| 3 | 383.784 | 5/4 | major 3rd | F |
| 4 | 911.712 | 17/10 | major 6th | B# |
| 5 | 239.64 | 8/7 | major 2nd | E# |
| 6 | 767.568 | 25/16 | augmented 5th | A# |
| 7 | 95.496 | 17/16 | augmented unison | D# |
* in 2.5.7.17 subgroup CTE tuning