Vengeance: Difference between revisions
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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). | ||
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'''. | In the following table, prime harmonics are labeled in '''bold'''. | ||
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{|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 | | major 7th | ||
|C | | 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# | | 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 | ||
|minor 6th | | minor 6th | ||
|B | | B | ||
|- | |- | ||
|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:Subgroup temperaments]] | ||
[[Category:Rank-2 temperaments]] | [[Category:Rank-2 temperaments]] | ||
Revision as of 14:49, 28 April 2025
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