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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]].
'''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 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 53et, 152et, and 205et. Tempering out 121/120 leads 11-limit hitchcock, supported by 46et, 53et, and 99et. 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 ({{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]].


See [[Amity family #Amity]] or [[Ragismic microtemperaments #Amity]] for more technical data.
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>.
 
{{Tdhat|Amity family #Amity}}


== Interval chain ==
== Interval chain ==
In the following table, odd harmonics 1–21 and their inversions are labeled in '''bold'''.
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
! rowspan="2"| #
! rowspan="3" | #
! rowspan="2"| Cents*  
! rowspan="3" | Cents*  
! colspan="3" | Approximate Ratios
! colspan="3" | Approximate ratios
|-
! rowspan="2" | 7-limit
! colspan="2" | 13-limit extensions
|-
|-
! 7-limit
! Amity ({{nowrap| 53 & 152 }})
! 13-limit Extensions<br>Amity Mapping (53 &amp; 152)
! Hitchcock ({{nowrap| 46 & 53 }})
! 13-limit Extensions<br>Hitchcock Mapping (46 &amp; 53)
|-
|-
| 0
| 0
| 0.00
| 0.00
| 1/1
| '''1/1'''
|
|
|
|
Line 29: Line 35:
|-
|-
| 2
| 2
| 678.86
| 678.87
| 40/27
| 40/27
|
|
Line 41: Line 47:
|-
|-
| 4
| 4
| 157.73
| 157.74
| 35/32
| 35/32
|
|
| 12/11, 11/10
| 12/11, 11/10
|-
|-
| | 5
| 5
| 497.16
| 497.17
| '''4/3'''
| '''4/3'''
|
|
Line 53: Line 59:
|-
|-
| 6
| 6
| 836.59
| 836.61
| 81/50
| 81/50
|
|
Line 59: Line 65:
|-
|-
| 7
| 7
| 1176.03
| 1176.04
| 63/32, 160/81
| 63/32, 160/81
| 65/33, 77/39
| 65/33, 77/39
Line 65: Line 71:
|-
|-
| 8
| 8
| 315.46
| 315.48
| 6/5
| 6/5
|
|
Line 71: Line 77:
|-
|-
| 9
| 9
| 654.89
| 654.91
| 35/24
| 35/24
|
|
Line 77: Line 83:
|-
|-
| 10
| 10
| 994.32
| 994.35
| 16/9
| '''16/9'''
|
|
| 39/22
| 39/22
|-
|-
| 11
| 11
| 133.75
| 133.78
| 27/25
| 27/25
|
|
Line 89: Line 95:
|-
|-
| 12
| 12
| 473.19
| 473.22
| 21/16
| '''21/16'''
|
|
|
|
|-
|-
| 13
| 13
| 812.62
| 812.65
| '''8/5'''
| '''8/5'''
|
|
Line 101: Line 107:
|-
|-
| 14
| 14
| 1152.05
| 1152.09
| 35/18
| 35/18
|
|
Line 107: Line 113:
|-
|-
| 15
| 15
| 291.48
| 291.52
| 32/27
| 32/27
| 13/11
| 13/11
Line 113: Line 119:
|-
|-
| 16
| 16
| 630.92
| 630.96
| 36/25
| 36/25
|
|
Line 119: Line 125:
|-
|-
| 17
| 17
| 970.35
| 970.39
| '''7/4'''
| '''7/4'''
|
|
Line 125: Line 131:
|-
|-
| 18
| 18
| 109.78
| 109.83
| 16/15
| '''16/15'''
|
|
|
|
|-
|-
| 19
| 19
| 449.21
| 449.26
| 35/27
| 35/27
|
|
Line 137: Line 143:
|-
|-
| 20
| 20
| 788.64
| 788.70
| 63/40
| 63/40
|
|
Line 143: Line 149:
|-
|-
| 21
| 21
| 1128.08
| 1128.13
| 48/25
| 48/25
| 25/13
| 25/13
Line 149: Line 155:
|-
|-
| 22
| 22
| 267.51
| 267.57
| 7/6
| 7/6
|
|
Line 155: Line 161:
|-
|-
| 23
| 23
| 606.94
| 607.00
| 64/45
| 64/45
|
|
Line 161: Line 167:
|-
|-
| 24
| 24
| 946.37
| 946.44
| 81/70
| 81/70
|
|
Line 167: Line 173:
|-
|-
| 25
| 25
| 85.81
| 85.87
| 21/20
| 21/20
|
|
Line 173: Line 179:
|-
|-
| 26
| 26
| 425.24
| 425.31
| 32/25
| 32/25
|
|
Line 179: Line 185:
|-
|-
| 27
| 27
| 764.67
| 764.74
| 14/9
| 14/9
|
|
Line 185: Line 191:
|-
|-
| 28
| 28
| 1104.10
| 1104.18
| 256/135
| 256/135
|
|
Line 191: Line 197:
|-
|-
| 29
| 29
| 243.53
| 243.61
| 147/128
| 147/128
| 15/13
| 15/13
Line 197: Line 203:
|-
|-
| 30
| 30
| 582.97
| 583.05
| 7/5
| 7/5
|
|
Line 203: Line 209:
|-
|-
| 31
| 31
| 922.40
| 922.48
| 128/75
| 128/75
|
|
Line 209: Line 215:
|-
|-
| 32
| 32
| 61.83
| 61.92
| 28/27
| 28/27
| 27/26
| 27/26
Line 215: Line 221:
|-
|-
| 33
| 33
| 401.26
| 401.35
| 63/50
| 63/50
|
|
Line 221: Line 227:
|-
|-
| 34
| 34
| 740.69
| 740.79
| 49/32
| 49/32
| 20/13
| 20/13
Line 227: Line 233:
|-
|-
| 35
| 35
| 1080.13
| 1080.22
| 28/15
| 28/15
|
|
Line 233: Line 239:
|-
|-
| 36
| 36
| 219.56
| 219.66
| 256/225
| 256/225
| 25/22
| 25/22
Line 239: Line 245:
|-
|-
| 37
| 37
| 558.99
| 559.09
| 112/81
| 112/81
| 18/13
| 18/13
Line 245: Line 251:
|-
|-
| 38
| 38
| 898.42
| 898.53
| 42/25
| 42/25
|
|
Line 251: Line 257:
|-
|-
| 39
| 39
| 37.86
| 37.96
| 49/48
| 49/48
| 40/39, 45/44
| 40/39, 45/44
|
|
|}
|}
<nowiki>*</nowiki> in 7-limit POTE tuning, octave reduced
<nowiki/>* In 7-limit CWE tuning, octave reduced


== Tuning spectra ==
== Tunings ==
=== Amity ===
=== Tunings spectra ===
13-limit commas: 352/351, 540/539, 625/624, 729/728
==== Amity ====
 
{| class="wikitable center-all left-4"
{| class="wikitable center-all"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
| 11\39
|
| 338.462
| 39ee… val, lower bound of 7- and 9-odd-limit diamond monotone
|-
|-
! [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
| 13\46
! generator<br>(¢)
|
! comments
| 339.130
| 46ef val
|-
|-
| 10/9
|  
| 9/5
| 339.199
| 339.199
|  
|  
|-
|-
|
| 13/11
| 13/11
| 339.281
| 339.281
|  
|  
|-
|-
| 8/7
|  
| 7/4
| 339.343
| 339.343
|  
|  
|-
|-
| 28\99
|
| 339.394
| 99ef val, lower bound of 11-, 13-, 15-, and 13-limit 21-odd-limit diamond monotone
|-
|
| 7/6
| 7/6
| 339.403
| 339.403
|  
|  
|-
|-
|
| 7/5
| 7/5
| 339.417
| 339.417
| 7-odd-limit minimax
| 7-odd-limit minimax
|-
|-
|
| 9/7
| 9/7
| 339.441
| 339.441
| 9-odd-limit minimax
| 9-odd-limit minimax
|-
|-
|
| 15/14
| 15/14
| 339.444
| 339.444
|  
|  
|-
|-
| 6/5
|  
| 5/3
| 339.455
| 339.455
|  
|  
|-
|-
| 14/11
|  
| 11/7
| 339.462
| 339.462
| 11-odd-limit minimax
| 11-odd-limit minimax
|-
|-
|
| 11/9
| 11/9
| 339.473
| 339.473
|  
|  
|-
|-
| 43\152
|
| 339.474
| 152f val
|-
|
| 15/11
| 15/11
| 339.476
| 339.476
|  
|  
|-
|-
| 12/11
|  
| 11/6
| 339.485
| 339.485
|  
|  
|-
|-
|
| 11/10
| 11/10
| 339.490
| 339.490
|  
|  
|-
|-
|
| 11/8
| 11/8
| 339.495
| 339.495
| 13- and 15-odd-limit minimax
| 13- and 15-odd-limit minimax
|-
|-
| 14/13
|  
| 13/7
| 339.505
| 339.505
|  
|  
|-
|-
| 58\205
|
| 339.512
|
|-
|
| 5/4
| 5/4
| 339.514
| 339.514
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
| 16/15
|  
| 15/8
| 339.541
| 339.541
|  
|  
|-
|-
| 18/13
|  
| 13/9
| 339.551
| 339.551
|  
|  
|-
|-
|
| 13/12
| 13/12
| 339.558
| 339.558
|  
|  
|-
|-
| 16/13
|  
| 13/8
| 339.563
| 339.563
|  
|  
|-
|-
|
| 15/13
| 15/13
| 339.577
| 339.577
|  
|  
|-
|-
|
| 13/10
| 13/10
| 339.582
| 339.582
|  
|  
|-
|-
| 4/3
|  
| 3/2
| 339.609
| 339.609
|  
|  
|-
| 15\53
|
| 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
|}
|}


=== Hitchcock ===
==== Hitchcock ====
13-limit commas: 121/120, 169/168, 176/175, 325/324
{| class="wikitable center-all left-4"
 
{| class="wikitable center-all"
|-
|-
! eigenmonzo<br>(unchanged interval)
! Edo<br>generator
! generator<br>(¢)
! Unchanged interval<br>(eigenmonzo)*
! comments
! Generator (¢)
! Comments
|-
|-
| 12/11
|  
| 11/6
| 337.659
| 337.659
|  
|  
|-
|-
| 11\39
|
| 338.462
| Lower bound of 7-, 9, and 11-odd-limit diamond monotone
|-
|
| 11/8
| 11/8
| 338.742
| 338.742
|  
|  
|-
|-
| 14/13
|  
| 13/7
| 338.936
| 338.936
|  
|  
|-
|-
| 14/11
| 13\46
|
| 339.130
| Lower bound of 13-, 15-odd-limit and 13-limit 21-odd-limit diamond monotone
|-
|
| 11/7
| 339.135
| 339.135
|  
|  
|-
|-
| 10/9
|  
| 9/5
| 339.199
| 339.199
|  
|  
|-
|-
|
| 13/11
| 13/11
| 339.281
| 339.281
|  
|  
|-
|-
| 8/7
|  
| 7/4
| 339.343
| 339.343
|  
|  
|-
|-
| 28\99
|
| 339.394
|
|-
|
| 7/6
| 7/6
| 339.403
| 339.403
|  
|  
|-
|-
|
| 7/5
| 7/5
| 339.417
| 339.417
| 7-odd-limit minimax
| 7-odd-limit minimax
|-
|-
|
| 9/7
| 9/7
| 339.441
| 339.441
| 9-, 11-, and 13-odd-limit minimax
| 9-, 11-, and 13-odd-limit minimax
|-
|-
|
| 15/14
| 15/14
| 339.444
| 339.444
| 15-odd-limit minimax
| 15-odd-limit minimax
|-
|-
| 6/5
|  
| 5/3
| 339.455
| 339.455
|  
|  
|-
|-
|
| 5/4
| 5/4
| 339.514
| 339.514
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
| 16/15
|  
| 15/8
| 339.541
| 339.541
|  
|  
|-
|-
| 4/3
|  
| 3/2
| 339.609
| 339.609
|  
|  
|-
|-
| 15\53
|
| 339.623
| Upper bound of 11-, 13-, 15-odd-limit and 13-limit 21-odd-limit diamond monotone
|-
|
| 15/13
| 15/13
| 339.677
| 339.677
|  
|  
|-
|-
|
| 13/10
| 13/10
| 339.695
| 339.695
|  
|  
|-
|-
| 18/13
|  
| 13/9
| 339.789
| 339.789
|  
|  
|-
|-
|
| 13/12
| 13/12
| 339.870
| 339.870
|  
|  
|-
|-
| 16/13
| 17\60
|
| 340.000
| 60de val, upper bound of 7- and 9-odd-limit diamond monotone
|-
|
| 13/8
| 340.088
| 340.088
|  
|  
|-
|-
|
| 15/11
| 15/11
| 340.339
| 340.339
|  
|  
|-
|-
|
| 11/10
| 11/10
| 341.251
| 341.251
|  
|  
|-
|-
|
| 11/9
| 11/9
| 347.408
| 347.408
|  
|  
|}
|}
<nowiki/>* Besides the octave


== Scales ==
== Music ==
* [[Amity7]] – proper [[4L 3s]]
; [[User:Francium|Francium]]
* [[Amity11]] – improper [[7L 4s]]
* [https://www.youtube.com/watch?v=AsDaJXCBd_w ''For Amity''] (2023) – in 463edo tuning
* [[Amity18]] – improper [[7L 11s]]
* [[Amity25]] – improper [[7L 18s]]
* [[Amity32]] – improper [[7L 25s]]
* [[Amity39]] – improper [[7L 32s]]
* [[Amity46]] – proper [[7L 39s]]


== Notes ==
[[Category:Amity| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Amity family]]
[[Category:Amity family]]
[[Category:Ragismic microtemperaments]]
[[Category:Ragismic microtemperaments]]
[[Category:Amity| ]] <!-- main article -->
[[Category:Hemifamity temperaments]]
{{IoT}}
Retrieved from "https://en.xen.wiki/w/Amity"