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'''Vengeance''' is a 2.5.17 [[subgroup temperament]] | {{Infobox regtemp | ||
| Title = Vengeance | |||
| Subgroups = 2.5.17, 2.5.7.17 | |||
| Comma basis = [[78608/78125]] (2.5.17) <br>[[2023/2000]], [[4165/4096]] (2.5.7.17) | |||
| Edo join 1 = 16 | Edo join 2 = 25 | |||
| Mapping = 1; 3 -5 7 | |||
| Generators = 34/25 | Generators tuning = 527.718 | Optimization method = CWE | |||
| MOS scales = [[2L 5s]], [[7L 2s]] | |||
| Pergen = (P8, M10<sup>5</sup>/3) | |||
| Odd limit 1 = 2.5.7.17 17 | Mistuning 1 = 8.88 | Complexity 1 = 16 | |||
| Odd limit 2 = 2.5.7.17 25 | Mistuning 2 = 8.88 | Complexity 2 = 16 | |||
}} | |||
'''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. | |||
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). | 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 (temperament)|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 == | == Interval chain == | ||
In the following table, prime harmonics are labeled in '''bold'''. | In the following table, prime harmonics are labeled in '''bold'''. | ||
| Line 9: | Line 24: | ||
{|class="wikitable" | {|class="wikitable" | ||
|- | |- | ||
!# | ! # | ||
!Cents* | ! Cents* | ||
!Approximate ratios | ! Approximate ratios | ||
!colspan=2|Melodic antidiatonic notation | ! colspan=2| Melodic antidiatonic notation | ||
|- | |- | ||
|0 | | 0 | ||
|0.00 | | 0.00 | ||
|'''1/1''' | | '''1/1''' | ||
|perfect unison | | perfect unison | ||
|D | | D | ||
|- | |- | ||
|1 | | 1 | ||
|527.928 | | 527.928 | ||
|34/25 | | 34/25 | ||
|perfect 4th | | perfect 4th | ||
|G | | G | ||
|- | |- | ||
|2 | | 2 | ||
|1055.856 | | 1055.856 | ||
|119/64, 125/68 | | 119/64, 125/68 | ||
| | | major 7th | ||
| | | C | ||
|- | |- | ||
|3 | | 3 | ||
|383.784 | | 383.784 | ||
|'''5/4''' | | '''5/4''' | ||
|major 3rd | | major 3rd | ||
|F | | F | ||
|- | |- | ||
|4 | | 4 | ||
|911.712 | | 911.712 | ||
|17/10 | | 17/10 | ||
|major 6th | | major 6th | ||
| | | B# | ||
|- | |- | ||
|5 | | 5 | ||
|239.64 | | 239.64 | ||
|'''8/7''' | | '''8/7''' | ||
|major 2nd | | major 2nd | ||
|E# | | E# | ||
|- | |- | ||
|6 | | 6 | ||
|767.568 | | 767.568 | ||
|25/16 | | 25/16 | ||
| | | augmented 5th | ||
| | | A# | ||
|- | |- | ||
|7 | | 7 | ||
|95.496 | | 95.496 | ||
|'''17/16''' | | '''17/16''' | ||
|augmented unison | | augmented unison | ||
|D# | | D# | ||
|} | |} | ||
<nowiki>*</nowiki> in 2.5.7.17 subgroup CTE tuning | <nowiki>*</nowiki> in 2.5.7.17 subgroup CTE tuning | ||
[[Category | [[Category:Vengeance| ]] <!-- main article --> | ||
[[Category:Subgroup temperaments]] | |||
[[Category:Rank-2 temperaments]] | |||
Latest revision as of 08:31, 17 February 2026
| 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