Amity: Difference between revisions

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'''Amity''' is the microtemperament tempering out the [[amity comma]], 1600000/1594323, in the 5-limit. This article also assumes the canonical mapping for 7, which means tempering out [[4375/4374]] and [[5120/5103]] in the 7-limit. Equal temperaments that support amity include [[46edo|46]], [[53edo|53]], [[99edo|99]], [[152edo|152]], and [[205edo|205]] ~~EDOs~~.

'''Amity''' is the microtemperament tempering out the [[amity comma]], 1600000/1594323, in the 5-limit. This article also assumes the canonical mapping for 7, which means tempering out [[4375/4374]] and [[5120/5103]] in the 7-limit. Equal temperaments that support amity include [[46edo|46]], [[53edo|53]], [[99edo|99]], [[152edo|152]], and [[205edo|205]].

Extending amity from the 7-limit to the 11-limit is not so simple. There are ~~two~~ mappings that are comparable in complexity and error: ~~11-limit~~ amity (53&~~amp;205~~) and hitchcock (46&~~amp;~~53). ~~Tempering~~ out 540/539 ~~leads~~ 11-~~limit amity, supported by 53, 152,~~ and ~~205 EDOs~~. ~~Tempering~~ out 121/120 ~~leads~~ 11-~~limit hitchcock, supported by 46, 53, and 99 EDOs~~. They can be extended to the 13-limit ~~by the~~ 352/351 ~~comma~~, and results in 625/624 and 729/728 being tempered out in 13-limit amity; 169/168 and 325/324 being tempered out in ~~13-limit~~ hitchcock. In the 17-limit, ~~46&53~~ tempers out 154/153, 256/255, and 273/272.

Extending amity from the 7-limit to the 11-limit is not so simple. There are three mappings that are comparable in complexity and error: undecimal amity (53 & 152), catamite (46 & 145), and hitchcock (46 & 53). Undecimal amity tempers out 540/539 and has the harmonic 11 mapped to -62 generator steps. Catamite tempers out 441/440 and has the harmonic 11 mapped to +37 generators steps. Hitchcock tempers out 121/120 and has the harmonic 11 mapped to -9 steps. They can be extended to the 13-limit through [[352/351]], and results in [[625/624]] and [[729/728]] being tempered out in 13-limit amity, [[196/195]] and [[364/363]] being tempered out in catamite, and [[169/168]] and [[325/324]] being tempered out in hitchcock. In the 17-limit, hitchcock tempers out 154/153, 256/255, and 273/272.

See [[Amity family #Amity]] or [[Ragismic microtemperaments #Amity]] for more technical data.

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Gencom mapping: [{{val| 1 3 6 -2 21 17 }}, {{val| 0 -5 -13 17 -62 -47 }}]

{| class="wikitable center-all"

{| class="wikitable center-all left-3"

|-

! [[~~eigenmonzo~~|~~eigenmonzo~~ (unchanged interval]])

! [[Eigenmonzo|Eigenmonzo (unchanged interval)]]

! ~~generator~~ (¢)

! Generator (¢)

! ~~comments~~

! Comments

|-

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Gencom mapping: [{{val| 1 3 6 -2 6 2 }}, {{val| 0 -5 -13 17 -9 6 }}]

{| class="wikitable center-all"

{| class="wikitable center-all left-3"

|-

! ~~eigenmonzo~~ (unchanged interval)

! Eigenmonzo (unchanged interval)

! ~~generator~~ (¢)

! Generator (¢)

! ~~comments~~

! Comments

|-

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~~== Scales ==~~

* [[Amity7]] – proper [[4L 3s]]

* [[Amity11]] – improper [[7L 4s]]

* [[Amity18]] – improper [[7L 11s]]

* [[Amity25]] – improper [[7L 18s]]

* [[Amity32]] – improper [[7L 25s]]

* [[Amity39]] – improper [[7L 32s]]

* [[Amity46]] – proper [[7L 39s]]

[[Category:Temperaments]]

[[Category:Amity family]]

[[Category:Ragismic microtemperaments]]

[[Category:Hemifamity temperaments]]

[[Category:Amity| ]]

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-'''Amity''' is the microtemperament tempering out the [[amity comma]], 1600000/1594323, in the 5-limit. This article also assumes the canonical mapping for 7, which means tempering out [[4375/4374]] and [[5120/5103]] in the 7-limit. Equal temperaments that support amity include [[46edo|46]], [[53edo|53]], [[99edo|99]], [[152edo|152]], and [[205edo|205]] EDOs.
+'''Amity''' is the microtemperament tempering out the [[amity comma]], 1600000/1594323, in the 5-limit. This article also assumes the canonical mapping for 7, which means tempering out [[4375/4374]] and [[5120/5103]] in the 7-limit. Equal temperaments that support amity include [[46edo|46]], [[53edo|53]], [[99edo|99]], [[152edo|152]], and [[205edo|205]].
-Extending amity from the 7-limit to the 11-limit is not so simple. There are two mappings that are comparable in complexity and error: 11-limit amity (53&amp;205) and hitchcock (46&amp;53). Tempering out 540/539 leads 11-limit amity, supported by 53, 152, and 205 EDOs. Tempering out 121/120 leads 11-limit hitchcock, supported by 46, 53, and 99 EDOs. They can be extended to the 13-limit by the 352/351 comma, and results in 625/624 and 729/728 being tempered out in 13-limit amity; 169/168 and 325/324 being tempered out in 13-limit hitchcock. In the 17-limit, 46&amp;53 tempers out 154/153, 256/255, and 273/272.
+Extending amity from the 7-limit to the 11-limit is not so simple. There are three mappings that are comparable in complexity and error: undecimal amity (53 & 152), catamite (46 & 145), and hitchcock (46 & 53). Undecimal amity tempers out 540/539 and has the harmonic 11 mapped to -62 generator steps. Catamite tempers out 441/440 and has the harmonic 11 mapped to +37 generators steps. Hitchcock tempers out 121/120 and has the harmonic 11 mapped to -9 steps. They can be extended to the 13-limit through [[352/351]], and results in [[625/624]] and [[729/728]] being tempered out in 13-limit amity, [[196/195]] and [[364/363]] being tempered out in catamite, and [[169/168]] and [[325/324]] being tempered out in hitchcock. In the 17-limit, hitchcock tempers out 154/153, 256/255, and 273/272.
 See [[Amity family #Amity]] or [[Ragismic microtemperaments #Amity]] for more technical data.
@@ Line 46: / Line 46: @@
 | 12/11, 11/10
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-| | 5
+| 5
 | 497.16
 | '''4/3'''
@@ Line 264: / Line 264: @@
 Gencom mapping: [{{val| 1 3 6 -2 21 17 }}, {{val| 0 -5 -13 17 -62 -47 }}]
-{| class="wikitable center-all"
+{| class="wikitable center-all left-3"
 |-
-! [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
+! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]
-! generator<br>(¢)
+! Generator<br>(¢)
-! comments
+! Comments
 |-
 | 10/9
@@ Line 368: / Line 368: @@
 Gencom mapping: [{{val| 1 3 6 -2 6 2 }}, {{val| 0 -5 -13 17 -9 6 }}]
-{| class="wikitable center-all"
+{| class="wikitable center-all left-3"
 |-
-! eigenmonzo<br>(unchanged interval)
+! Eigenmonzo<br>(unchanged interval)
-! generator<br>(¢)
+! Generator<br>(¢)
-! comments
+! Comments
 |-
 | 12/11
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 |
 |}
-== Scales ==
-* [[Amity7]] – proper [[4L 3s]]
-* [[Amity11]] – improper [[7L 4s]]
-* [[Amity18]] – improper [[7L 11s]]
-* [[Amity25]] – improper [[7L 18s]]
-* [[Amity32]] – improper [[7L 25s]]
-* [[Amity39]] – improper [[7L 32s]]
-* [[Amity46]] – proper [[7L 39s]]
 [[Category:Temperaments]]
 [[Category:Amity family]]
 [[Category:Ragismic microtemperaments]]
+[[Category:Hemifamity temperaments]]
 [[Category:Amity| ]] <!-- main article -->