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Amity: Difference between revisions

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{{Infobox regtemp
| Title = Amity
| Subgroups = 2.3.5, 2.3.5.7
| Comma basis = [[1600000/1594323]] (2.3.5); <br>[[4375/4374]], [[5120/5103]] (2.3.5.7)
| Edo join 1 = 46 | Edo join 2 = 53
| Mapping = 1; -5 -13 17
| Generators = 243/200
| Generators tuning = 339.4
| Optimization method = CWE
| MOS scales = [[7L&nbsp;4s]], [[7L&nbsp;11s]], [[7L&nbsp;18s]], [[7L&nbsp;25s]]
| Pergen = (P8, cP4/5)
| Color name = Saquinyoti
| Odd limit 1 = 5 | Mistuning 1 = 0.47 | Complexity 1 = 18
| Odd limit 2 = 9 | Mistuning 2 = 1.68 | Complexity 2 = 32
}}
'''Amity''' is a [[regular temperament|temperament]] that divides a [[8/3|perfect eleventh]] into 5 [[generator]]s of acute minor thirds. A stack of 13 generators [[octave reduction|octave reduced]] represents [[8/5]], [[tempering out]] the [[amity comma]], 1600000/1594323. This article also assumes the canonical [[extension]] to the [[7-limit]],  where a stack of 17 generators octave reduced represents [[7/4]], tempering out [[4375/4374]] and [[5120/5103]]. [[Equal temperaments]] that [[support]] amity include {{EDOs| 46, 53, 99, 152, and 205 }}.
'''Amity''' is a [[regular temperament|temperament]] that divides a [[8/3|perfect eleventh]] into 5 [[generator]]s of acute minor thirds. A stack of 13 generators [[octave reduction|octave reduced]] represents [[8/5]], [[tempering out]] the [[amity comma]], 1600000/1594323. This article also assumes the canonical [[extension]] to the [[7-limit]],  where a stack of 17 generators octave reduced represents [[7/4]], tempering out [[4375/4374]] and [[5120/5103]]. [[Equal temperaments]] that [[support]] amity include {{EDOs| 46, 53, 99, 152, and 205 }}.


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 ({{nowrap| 53 & 152 }}), catamite ({{nowrap| 46 & 145 }}), and hitchcock ({{nowrap| 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. Hitchcock has an extra extension to the 17-limit where it 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 ({{nowrap| 53 & 152 }}), stalagmite ({{nowrap| 46 & 145 }}), and hitchcock ({{nowrap| 46 & 53 }}). Undecimal amity tempers out 540/539 and has the harmonic 11 mapped to −62 generator steps. Stalagmite 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 stalagmite, and [[169/168]] and [[325/324]] being tempered out in hitchcock. Hitchcock has an extra extension to the 17-limit where it tempers out [[154/153]], [[256/255]], and [[273/272]].


Amity was named by [[Gene Ward Smith]] in 2001–2002 as a restructuring of the phrase ''acute minor third''<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2064.html Yahoo! Tuning Group | ''Kleismic & co'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3481.html Yahoo! Tuning Group | ''32 best 5-limit linear temperaments redux'']</ref>.
Amity was named by [[Gene Ward Smith]] in 2001–2002 as a restructuring of the phrase ''acute minor third''<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2064.html Yahoo! Tuning Group | ''Kleismic & co'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3481.html Yahoo! Tuning Group | ''32 best 5-limit linear temperaments redux'']</ref>.
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|-
|-
! Edo<br>generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
| 11\39
|
| 338.462
| 39ee… val, lower bound of 7- and 9-odd-limit diamond monotone
|-
|-
| 13\46
| 13\46
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|  
|  
| 339.394
| 339.394
| 99ef val
| 99ef val, lower bound of 11-, 13-, 15-, and 13-limit 21-odd-limit diamond monotone
|-
|-
|  
|  
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|  
|  
| 339.623
| 339.623
| Upper bound of 11-, 13-, 15-odd-limit and 13-limit 21-odd-limit diamond monotone
|-
| 17\60
|  
|  
| 340.000
| 60deee… val, upper bound of 7- and 9-odd-limit diamond monotone
|}
|}


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|-
|-
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)*
! Unchanged interval<br>(eigenmonzo)*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
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|  
|  
| 338.462
| 338.462
|  
| Lower bound of 7-, 9, and 11-odd-limit diamond monotone
|-
|-
|  
|  
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|  
|  
| 339.130
| 339.130
|  
| Lower bound of 13-, 15-odd-limit and 13-limit 21-odd-limit diamond monotone
|-
|-
|  
|  
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|  
|  
| 339.623
| 339.623
|  
| Upper bound of 11-, 13-, 15-odd-limit and 13-limit 21-odd-limit diamond monotone
|-
|-
|  
|  
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|  
|  
| 340.000
| 340.000
| 60de val
| 60de val, upper bound of 7- and 9-odd-limit diamond monotone
|-
|-
|  
|  
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[[Category:Amity family]]
[[Category:Amity family]]
[[Category:Ragismic microtemperaments]]
[[Category:Ragismic microtemperaments]]
[[Category:Hemifamity temperaments]]
[[Category:Aberschismic temperaments]]
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