Amity: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
m see-also format
m Text replacement - "Eigenmonzo<br>(unchanged-interval)" to "Unchanged interval<br>(eigenmonzo)"
 
(20 intermediate revisions by 7 users not shown)
Line 1: Line 1:
{{See also|Amity family #Amity}}
'''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 }}.
{{See also|Ragismic microtemperaments #Amity}}


'''Amity''' is a temperament for the 5, 7, 11, and 13 [[Harmonic limit|prime limits]]. 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 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]].


In the 7-limit, amity tempers out 4375/4374 and 5120/5103, as well as 5-limit [[amity comma]], 1600000/1594323. But extending it 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).
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>.


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.
{{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
|-
|-
! 7-limit
! rowspan="2" | 7-limit
! 13-limit Extensions<br>Amity Mapping (53 &amp; 152)
! colspan="2" | 13-limit extensions
! 13-limit Extensions<br>Hitchcock Mapping (46 &amp; 53)
|-
! Amity ({{nowrap| 53 & 152 }})
! Hitchcock ({{nowrap| 46 & 53 }})
|-
|-
| 0
| 0
| 0.00
| 0.00
| 1/1
| '''1/1'''
|
|
|
|
Line 32: Line 35:
|-
|-
| 2
| 2
| 678.86
| 678.87
| 40/27
| 40/27
|
|
Line 44: 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 56: Line 59:
|-
|-
| 6
| 6
| 836.59
| 836.61
| 81/50
| 81/50
|
|
Line 62: 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 68: Line 71:
|-
|-
| 8
| 8
| 315.46
| 315.48
| 6/5
| 6/5
|
|
Line 74: Line 77:
|-
|-
| 9
| 9
| 654.89
| 654.91
| 35/24
| 35/24
|
|
Line 80: 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 92: Line 95:
|-
|-
| 12
| 12
| 473.19
| 473.22
| 21/16
| '''21/16'''
|
|
|
|
|-
|-
| 13
| 13
| 812.62
| 812.65
| '''8/5'''
| '''8/5'''
|
|
Line 104: Line 107:
|-
|-
| 14
| 14
| 1152.05
| 1152.09
| 35/18
| 35/18
|
|
Line 110: Line 113:
|-
|-
| 15
| 15
| 291.48
| 291.52
| 32/27
| 32/27
| 13/11
| 13/11
Line 116: Line 119:
|-
|-
| 16
| 16
| 630.92
| 630.96
| 36/25
| 36/25
|
|
Line 122: Line 125:
|-
|-
| 17
| 17
| 970.35
| 970.39
| '''7/4'''
| '''7/4'''
|
|
Line 128: Line 131:
|-
|-
| 18
| 18
| 109.78
| 109.83
| 16/15
| '''16/15'''
|
|
|
|
|-
|-
| 19
| 19
| 449.21
| 449.26
| 35/27
| 35/27
|
|
Line 140: Line 143:
|-
|-
| 20
| 20
| 788.64
| 788.70
| 63/40
| 63/40
|
|
Line 146: Line 149:
|-
|-
| 21
| 21
| 1128.08
| 1128.13
| 48/25
| 48/25
| 25/13
| 25/13
Line 152: Line 155:
|-
|-
| 22
| 22
| 267.51
| 267.57
| 7/6
| 7/6
|
|
Line 158: Line 161:
|-
|-
| 23
| 23
| 606.94
| 607.00
| 64/45
| 64/45
|
|
Line 164: Line 167:
|-
|-
| 24
| 24
| 946.37
| 946.44
| 81/70
| 81/70
|
|
Line 170: Line 173:
|-
|-
| 25
| 25
| 85.81
| 85.87
| 21/20
| 21/20
|
|
Line 176: Line 179:
|-
|-
| 26
| 26
| 425.24
| 425.31
| 32/25
| 32/25
|
|
Line 182: Line 185:
|-
|-
| 27
| 27
| 764.67
| 764.74
| 14/9
| 14/9
|
|
Line 188: Line 191:
|-
|-
| 28
| 28
| 1104.10
| 1104.18
| 256/135
| 256/135
|
|
Line 194: Line 197:
|-
|-
| 29
| 29
| 243.53
| 243.61
| 147/128
| 147/128
| 15/13
| 15/13
Line 200: Line 203:
|-
|-
| 30
| 30
| 582.97
| 583.05
| 7/5
| 7/5
|
|
Line 206: Line 209:
|-
|-
| 31
| 31
| 922.40
| 922.48
| 128/75
| 128/75
|
|
Line 212: Line 215:
|-
|-
| 32
| 32
| 61.83
| 61.92
| 28/27
| 28/27
| 27/26
| 27/26
Line 218: Line 221:
|-
|-
| 33
| 33
| 401.26
| 401.35
| 63/50
| 63/50
|
|
Line 224: Line 227:
|-
|-
| 34
| 34
| 740.69
| 740.79
| 49/32
| 49/32
| 20/13
| 20/13
Line 230: Line 233:
|-
|-
| 35
| 35
| 1080.13
| 1080.22
| 28/15
| 28/15
|
|
Line 236: Line 239:
|-
|-
| 36
| 36
| 219.56
| 219.66
| 256/225
| 256/225
| 25/22
| 25/22
Line 242: Line 245:
|-
|-
| 37
| 37
| 558.99
| 559.09
| 112/81
| 112/81
| 18/13
| 18/13
Line 248: Line 251:
|-
|-
| 38
| 38
| 898.42
| 898.53
| 42/25
| 42/25
|
|
Line 254: 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 ==
=== Spectrum of amity tunings ===


13-limit commas: 352/351, 540/539, 625/624, 729/728
== Tunings ==
{| class="wikitable center-all left-3"
=== Tunings spectra ===
==== Amity ====
{| class="wikitable center-all left-4"
|-
|-
! Eigenmonzo
! Edo<br>generator
! Generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
! Comments
|-
|-
| 10/9
| 11\39
|
| 338.462
| 39ee… val, lower bound of 7- and 9-odd-limit diamond monotone
|-
| 13\46
|
| 339.130
| 46ef val
|-
|
| 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
|}
|}


=== Spectrum of hitchcock tunings ===
==== Hitchcock ====
 
{| class="wikitable center-all left-4"
13-limit commas: 121/120, 169/168, 176/175, 325/324
{| class="wikitable center-all left-3"
|-
|-
! Eigenmonzo
! Edo<br>generator
! Generator
! Unchanged interval<br>(eigenmonzo)*
! Comment
! 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}}

Latest revision as of 06:56, 21 June 2025

Amity is a temperament that divides a perfect eleventh into 5 generators of acute minor thirds. A stack of 13 generators 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 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 (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. 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[1][2].

For technical data, see Amity family #Amity.

Interval chain

In the following table, odd harmonics 1–21 and their inversions are labeled in bold.

# Cents* Approximate ratios
7-limit 13-limit extensions
Amity (53 & 152) Hitchcock (46 & 53)
0 0.00 1/1
1 339.43 128/105 11/9
2 678.87 40/27
3 1018.30 9/5
4 157.74 35/32 12/11, 11/10
5 497.17 4/3
6 836.61 81/50 13/8, 21/13
7 1176.04 63/32, 160/81 65/33, 77/39 65/33, 77/39, 128/65
8 315.48 6/5
9 654.91 35/24 16/11, 22/15
10 994.35 16/9 39/22
11 133.78 27/25 13/12, 14/13
12 473.22 21/16
13 812.65 8/5
14 1152.09 35/18 39/20, 64/33, 88/45
15 291.52 32/27 13/11 13/11
16 630.96 36/25 13/9
17 970.39 7/4
18 109.83 16/15
19 449.26 35/27 13/10
20 788.70 63/40 52/33
21 1128.13 48/25 25/13 21/11, 52/27
22 267.57 7/6
23 607.00 64/45
24 946.44 81/70 26/15
25 85.87 21/20
26 425.31 32/25 14/11
27 764.74 14/9
28 1104.18 256/135
29 243.61 147/128 15/13
30 583.05 7/5
31 922.48 128/75 56/33
32 61.92 28/27 27/26
33 401.35 63/50
34 740.79 49/32 20/13
35 1080.22 28/15
36 219.66 256/225 25/22
37 559.09 112/81 18/13
38 898.53 42/25
39 37.96 49/48 40/39, 45/44

* In 7-limit CWE tuning, octave reduced

Tunings

Tunings spectra

Amity

Edo
generator 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
11\39 338.462 39ee… val, lower bound of 7- and 9-odd-limit diamond monotone
13\46 339.130 46ef val
9/5 339.199
13/11 339.281
7/4 339.343
28\99 339.394 99ef val, lower bound of 11-, 13-, 15-, and 13-limit 21-odd-limit diamond monotone
7/6 339.403
7/5 339.417 7-odd-limit minimax
9/7 339.441 9-odd-limit minimax
15/14 339.444
5/3 339.455
11/7 339.462 11-odd-limit minimax
11/9 339.473
43\152 339.474 152f val
15/11 339.476
11/6 339.485
11/10 339.490
11/8 339.495 13- and 15-odd-limit minimax
13/7 339.505
58\205 339.512
5/4 339.514 5-odd-limit minimax
15/8 339.541
13/9 339.551
13/12 339.558
13/8 339.563
15/13 339.577
13/10 339.582
3/2 339.609
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

Edo
generator 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
11/6 337.659
11\39 338.462 Lower bound of 7-, 9, and 11-odd-limit diamond monotone
11/8 338.742
13/7 338.936
13\46 339.130 Lower bound of 13-, 15-odd-limit and 13-limit 21-odd-limit diamond monotone
11/7 339.135
9/5 339.199
13/11 339.281
7/4 339.343
28\99 339.394
7/6 339.403
7/5 339.417 7-odd-limit minimax
9/7 339.441 9-, 11-, and 13-odd-limit minimax
15/14 339.444 15-odd-limit minimax
5/3 339.455
5/4 339.514 5-odd-limit minimax
15/8 339.541
3/2 339.609
15\53 339.623 Upper bound of 11-, 13-, 15-odd-limit and 13-limit 21-odd-limit diamond monotone
15/13 339.677
13/10 339.695
13/9 339.789
13/12 339.870
17\60 340.000 60de val, upper bound of 7- and 9-odd-limit diamond monotone
13/8 340.088
15/11 340.339
11/10 341.251
11/9 347.408

* Besides the octave

Music

Francium

Notes

  1. ↑ Yahoo! Tuning Group | Kleismic & co
  2. ↑ Yahoo! Tuning Group | 32 best 5-limit linear temperaments redux
Retrieved from "https://en.xen.wiki/index.php?title=Amity&oldid=200165"