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" | ||
|- | |- | ||
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!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 | ||