Vengeance: Difference between revisions

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'''Vengeance''' 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]]. ~~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.

{{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]].

~~For technical data, see [[no-threes subgroup temperaments#Vengeance]].~~

== Interval chain ==

In the following table, prime harmonics are labeled in '''bold'''.

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{|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

|~~minor 2nd~~

| major 7th

|Eb

| 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

|F

| 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~~

| augmented 5th

|B

| 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

[[Category:~~Temperaments~~]]

[[Category:Vengeance| ]]

[[Category:Subgroup temperaments]]

[[Category:Rank-2 temperaments]]

@@ Line 1: / Line 1: @@
-'''Vengeance''' 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]]. 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.
+{{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]].
-For technical data, see [[no-threes subgroup temperaments#Vengeance]].
 == Interval chain ==
 In the following table, prime harmonics are labeled in '''bold'''.
@@ Line 9: / Line 24: @@
 {|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
-|minor 2nd
+| major 7th
-|Eb
+| 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
-|F
+| 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
+| augmented 5th
-|B
+| 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
-[[Category:Temperaments]]
+[[Category:Vengeance| ]] <!-- main article -->
+[[Category:Subgroup temperaments]]
+[[Category:Rank-2 temperaments]]