Vengeance: Difference between revisions
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Added information about the 2.5.13/11.17 extension of Pentagoth. |
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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 [[2L 5s|antidiatonic scale]] and [[7L 2s|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 [[User:CompactStar|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. | '''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 [[2L 5s|antidiatonic scale]] and [[7L 2s|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 [[User:CompactStar|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 [[eigenmonzo|exact]]-13/11 tuning is 672.3¢, near [[25edo|14\25]] (672.0¢), and the exact-20/17 tuning is 670.3¢, near [[34edo|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). | 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). | ||
Revision as of 01:12, 28 February 2024
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