Vengeance: Difference between revisions
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'''Vengeance''' is a 2.5.17 [[subgroup temperament]], which is notable for having an [[2L 5s|antidiatonic scale]] similar to [[mavila]], 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]]. Its name 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]]. | '''Vengeance''' is a 2.5.17 [[subgroup temperament]], which is notable for having an [[2L 5s|antidiatonic scale]] similar to [[mavila]], 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]]. Its name 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]]. Thhe 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). | 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). | ||
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For technical data, see [[no-threes subgroup temperaments#Vengeance]]. | For technical data, see [[no-threes subgroup temperaments#Vengeance]]. | ||
== Interval chain == | == Interval chain == | ||
In the following table, prime harmonics are labeled in '''bold'''. | |||
{|class="wikitable" | {|class="wikitable" | ||
|- | |- | ||
| Line 10: | Line 12: | ||
!Cents* | !Cents* | ||
!Approximate ratios | !Approximate ratios | ||
!Melodic antidiatonic notation | !colspan=2|Melodic antidiatonic notation | ||
|- | |- | ||
|0 | |0 | ||
|0.00 | |0.00 | ||
| | |'''1/1''' | ||
|perfect unison | |||
|D | |D | ||
|- | |- | ||
|1 | |1 | ||
|527.928 | |527.928 | ||
| | |34/25 | ||
|perfect 4th | |||
|G | |G | ||
|- | |||
|2 | |||
|1055.856 | |||
|119/64, 125/68 | |||
|minor 2nd | |||
|Eb | |||
|- | |||
|3 | |||
|383.784 | |||
|'''5/4''' | |||
|major 3rd | |||
|F | |||
|- | |||
|4 | |||
|911.712 | |||
|17/10 | |||
|major 6th | |||
|F | |||
|- | |||
|5 | |||
|239.64 | |||
|'''8/7''' | |||
|major 2nd | |||
|E# | |||
|- | |||
|6 | |||
|767.568 | |||
|25/16 | |||
|minor 6th | |||
|B | |||
|- | |||
|7 | |||
|95.496 | |||
|'''17/16''' | |||
|augmented unison | |||
|D# | |||
|} | |} | ||
<nowiki>*</nowiki> in 2.5.7.17 subgroup CTE tuning | <nowiki>*</nowiki> in 2.5.7.17 subgroup CTE tuning | ||
Revision as of 08:47, 17 November 2023
Vengeance is a 2.5.17 subgroup temperament, which is notable for having an antidiatonic scale similar to mavila, 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. Its name 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. Thhe 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).
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 | minor 2nd | Eb |
| 3 | 383.784 | 5/4 | major 3rd | F |
| 4 | 911.712 | 17/10 | major 6th | F |
| 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