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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 minor 6th B
7 95.496 17/16 augmented unison D#

* in subgroup CTE tuning