Vengeance

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 2.5.7.17 subgroup CTE tuning