Vengeance: Difference between revisions

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{{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.

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 (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]].

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| C

|-

|3

| 3

| 383.784

| '''5/4'''

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| 767.568

| 25/16

| ~~minor 6th~~

| augmented 5th

| B

| A#

|-

| 7

@@ Line 1: / Line 1: @@
+{{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.
 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 (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]].
@@ Line 35: / Line 47: @@
 | C
 |-
-|3
+| 3
 | 383.784
 | '''5/4'''
@@ Line 56: / Line 68: @@
 | 767.568
 | 25/16
-| minor 6th
+| augmented 5th
-| B
+| A#
 |-
 | 7