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Created page with "{{Infobox ET}} {{ED intro}}. 272ed23 is primarily intended to be used as 60edo but with slightly compressed octaves. == Theory == Compared to pure-[..."
 
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{{Infobox ET}}
{{Editable user page}}
{{ED intro}}.
{{Infobox ET|debug=1}}
{{ED intro}}


272ed23 is primarily intended to be used as [[60edo]] but with slightly [[Octave shrinking|compressed]] octaves.
272ed23 is primarily intended to be used as [[60edo]] but with slightly [[Octave shrinking|compressed]] octaves.


== Theory ==
== Theory ==
Compared to pure-[[octave]]s 60edo, 272ed3 features a relatively large improvement to [[7/1]] and [[11/1]], at the cost of moderate worsening of [[2/1]], [[3/1]] and [[5/1]].  
Compared to pure-[[octave]]s 60edo, 272ed23 features a relatively large improvement to [[7/1]] and [[11/1]], at the cost of moderate worsening of [[2/1]], [[3/1]] and [[5/1]].  


It also causes the [[val]]s to flip for [[5/1]], [[7/1]], [[13/1]] and [[17/1]].
It also causes the [[val]]s to flip for [[5/1]], [[7/1]], [[13/1]] and [[17/1]].
Line 16: Line 17:


=== Prime harmonics ===
=== Prime harmonics ===
In the [[47-limit]], 272ed3 has less than 40% relative error on primes 2, 3, 5, 7, 11, 17, 23, 29, 31, 37, 41, 43 and 47.  
In the [[47-limit]], 272ed3 has less than 40% [[relative error]] on primes 2, 3, 5, 7, 11, 17, 23, 29, 31, 37, 41, 43 and 47.  


This makes it a solid tuning for the entire no-13, no-19 47-limit.
This makes it a solid tuning for the entire no-13, no-19 47-limit.
Line 25: Line 26:


==== 60edo for comparison ====
==== 60edo for comparison ====
In the [[47-limit]], 60edo has less than 40% relative error on primes 2, 3, 5, 13, 17, 19, 31, 47.
In the 47-limit, 60edo has less than 40% relative error on primes 2, 3, 5, 13, 17, 19, 31, 47.


This makes it a solid tuning for the no-7, no-11 [[19-limit]] (or [[dual-n|dual]]-7, dual-11).
This makes it a solid tuning for the no-7, no-11 [[19-limit]] (or [[dual-n|dual]]-7, dual-11).
Line 35: Line 36:
272ed23 is quite composite, with subset ed23s 1, 2, 4, 8, 16, 17, 34, 68, 136.  
272ed23 is quite composite, with subset ed23s 1, 2, 4, 8, 16, 17, 34, 68, 136.  


Unlike pure-octaves 60edo, 272ed23 does not have high [[consistency]] at all.
Unlike pure-octaves 60edo, 272ed23 does not have high [[consistency]] at all. 60edo is both consistent and [[distinctly consistent]] up to the 9-[[integer limit]], while 272ed23 is consistent and distinctly consistently only up to the 4-integer limit.


== Notation ==
== Notation ==
Line 44: Line 45:


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3 left-4 mw-collapsible"
{| class="wikitable center-all right-2 left-3 left-4 left-5 mw-collapsible"
|+ Intervals of 272ed23, up to the octave
|-
|-
! Degrees
! Degrees
Line 52: Line 54:
! Additional ratios<br />in the full 47-limit
! Additional ratios<br />in the full 47-limit
|-
|-
| 0
! 0
| 0
! 0
| 1/1
| 1/1
|  
|  
|  
|  
|-
|-
| 1
! 1
| 19.96
! 19.96
| 55/54, 56/55, 64/63, 81/80
| 55/54, 56/55, 64/63, 81/80
| 51/50, 63/62, 69/68, 70/69, 75/74, 82/81, 85/84, 88/87
| 51/50, 63/62, 69/68, 70/69, 75/74, 82/81, 85/84, 88/87
| 52/51, 53/52, 57/56, 58/57, 65/64, 66/65, 76/75, 77/76, 78/77
| 52/51, 53/52, 57/56, 58/57, 65/64, 66/65, 76/75, 77/76, 78/77
|-
|-
| 2
! 2
| 39.91
! 39.91
| 33/32, 36/35
| 33/32, 36/35
| 34/33, 35/34, 37/36, 41/40, 42/41, 43/42, 44/43
| 34/33, 35/34, 37/36, 41/40, 42/41, 43/42, 44/43
| 38/37, 39/38, 40/39
| 38/37, 39/38, 40/39
|-
|-
| 3
! 3
| 59.87
! 59.87
| 25/24, 28/27
| 25/24, 28/27
| 24/23, 29/28, 30/29, 88/85
| 24/23, 29/28, 30/29, 88/85
| 26/25, 27/26
| 26/25, 27/26
|-
|-
| 4
! 4
| 79.83
! 79.83
| 21/20, 22/21
| 21/20, 22/21
| 23/22, 45/43
| 23/22, 45/43
| 19/18, 20/19
| 19/18, 20/19
|-
|-
| 5
! 5
| 99.78
! 99.78
| 16/5, 35/33
| 16/5, 35/33
| 17/16, 18/17
| 17/16, 18/17
|  
|  
|-
|-
| 6
! 6
| 119.74
! 119.74
| 15/14
| 15/14
| 29/27, 44/41, 74/69
| 29/27, 44/41, 74/69
| 14/13
| 14/13
|-
|-
| 7
! 7
| 139.70
! 139.70
| 12/11
|  
| 25/23, 51/47
| 25/23, 51/47
| 13/12, 38/35
| 13/12, 38/35
|-
|-
| 8
! 8
| 159.66
! 159.66
| 11/10, 12/11
| 11/10, 12/11
| 23/21, 34/31
| 23/21, 34/31
| 57/52
| 57/52
|-
|-
| 9
! 9
| 179.61
! 179.61
| 10/9
| 10/9
| 31/28, 41/37, 51/46
| 31/28, 41/37, 51/46
| 21/19
| 21/19
|-
|-
| 10
! 10
| 199.57
! 199.57
| 9/8, 28/25, 55/49
| 9/8, 28/25, 55/49
| 37/33, 46/41
| 37/33, 46/41
| 19/17
| 19/17
|-
|-
| 11
! 11
| 219.53
! 219.53
| 8/7
| 8/7 (patent)
| 17/15, 42/37
| 17/15, 42/37
|  
|  
|-
|-
| 12
! 12
| 239.48
! 239.48
| 8/7
| 8/7 (catnip)
| 23/20, 31/27, 54/47, 85/74
| 23/20, 31/27, 54/47, 85/74
| 15/13
| 15/13
|-
|-
| 13
! 13
| 259.44
! 259.44
| 7/6
| 7/6
| 29/25, 36/31, 43/37
| 29/25, 36/31, 43/37
| 22/19
| 22/19
|-
|-
| 14
! 14
| 279.4
! 279.4
|  
|  
| 20/17, 27/23, 47/40
| 20/17, 27/23, 47/40
| 13/11
| 13/11
|-
|-
| 15
! 15
| 299.35
! 299.35
| 25/21
| 25/21
| 44/37
| 44/37
| 19/16
| 19/16
|-
|-
| 16
! 16
| 319.31
! 319.31
| 6/5, 77/64
| 6/5, 77/64
| 29/24, 35/28, 41/34
| 29/24, 35/28, 41/34
| 23/19, 47/39, 65/54
| 23/19, 47/39, 65/54
|-
|-
| 17
! 17
| 339.27
! 339.27
| 11/9
| 11/9
| 17/14, 28/23, 45/37
| 17/14, 28/23, 45/37
|  
|  
|-
|-
| 18
! 18
| 359.22
! 359.22
| 27/22
| 27/22
| 43/35
| 43/35
| 16/13, 91/74
| 16/13, 91/74
|-
|-
| 19
! 19
| 379.18
! 379.18
| 5/4, 56/45
| 5/4, 56/45
| 21/17, 31/25, 36/29, 41/33, 46/37, 51/41
| 21/17, 31/25, 36/29, 41/33, 46/37, 51/41
| 26/21
| 26/21
|-
|-
| 20
! 20
| 399.14
! 399.14
| 63/50
| 63/50
| 29/23, 34/27
| 29/23, 34/27
| 19/15, 24/19
| 19/15, 24/19
|-
|-
| 21
! 21
| 419.09
! 419.09
| 14/11
| 14/11
| 23/18, 37/29, 51/40
| 23/18, 37/29, 51/40
| 65/51
| 65/51
|-
|-
| 22
! 22
| 439.05
! 439.05
| 9/7
| 9/7
| 22/17, 31/24, 40/31, 58/45
| 22/17, 31/24, 40/31, 58/45
| 49/38
| 49/38
|-
|-
| 23
! 23
| 459.01
! 459.01
| 176/135
| 176/135
| 30/23, 43/33
| 30/23, 43/33
| 13/10, 17/13
| 13/10, 17/13
|-
|-
| 24
! 24
| 478.97
! 478.97
| 21/16, 33/25
| 21/16, 33/25
| 29/22, 62/47
| 29/22, 62/47
| 25/19, 91/69
| 25/19, 91/69
|-
|-
| 25
! 25
| 498.92
! 498.92
| 4/3, 75/56
| 4/3, 75/56
| 43/32, 47/35, 63/47, 99/74
| 43/32, 47/35, 63/47, 99/74
| 39/29, 51/38, 87/65, 91/68
| 39/29, 51/38, 87/65, 91/68
|-
|-
| 26
! 26
| 518.88
! 518.88
| 27/20
| 27/20
| 23/17, 31/23, 58/43, 85/63
| 23/17, 31/23, 58/43, 85/63
| 19/14
| 19/14
|-
|-
| 27
! 27
| 538.84
! 538.84
| 15/11
| 15/11
| 41/30, 56/41, 86/63
| 41/30, 56/41, 86/63
| 26/19
| 26/19
|-
|-
| 28
! 28
| 558.79
! 558.79
| 11/8
| 11/8
| 29/21, 40/29, 69/50
| 29/21, 40/29, 69/50
| 18/13
| 18/13
|-
|-
| 29
! 29
| 578.75
! 578.75
| 7/5, 25/18, 88/63
| 7/5, 25/18, 88/63
| 32/23, 46/33, 60/43, 81/58
| 32/23, 46/33, 60/43, 81/58
| 39/28, 95/68
| 39/28, 95/68
|-
|-
| 30
! 30
| 598.71
! 598.71
|  
|  
| 17/12, 24/17, 41/29
| 17/12, 24/17, 41/29
| 65/46
| 65/46
|-
|-
| 31
! 31
| 618.66
! 618.66
| 10/7, 63/44
| 10/7, 63/44
| 23/16, 33/23, 43/30
| 23/16, 33/23, 43/30
| 93/65
| 93/65
|-
|-
| 32
! 32
| 638.62
! 638.62
| 81/56
| 81/56
| 29/20, 42/29, 68/47
| 29/20, 42/29, 68/47
| 13/9, 55/38, 94/65
| 13/9, 55/38, 94/65
|-
|-
| 33
! 33
| 658.58
! 658.58
| 16/11, 22/15
| 16/11, 22/15
| 41/28, 60/41
| 41/28, 60/41
| 19/13
| 19/13
|-
|-
| 34
! 34
| 678.53
! 678.53
|  
|  
| 25/17, 31/21, 34/32, 37/25
| 25/17, 31/21, 34/32, 37/25
| 28/19
| 28/19
|-
|-
| 35
! 35
| 698.49
! 698.49
| 3/2, 49/33
| 3/2, 49/33
| 55/37, 64/43, 70/47, 82/55, 94/63
| 55/37, 64/43, 70/47, 82/55, 94/63
| 52/35, 58/39, 76/51, 85/57
| 52/35, 58/39, 76/51, 85/57
|-
|-
| 36
! 36
| 718.45
! 718.45
| 32/21, 50/33
| 32/21, 50/33
| 29/19, 35/23, 41/27, 44/29, 47/31
| 29/19, 35/23, 41/27, 44/29, 47/31
| 38/25
| 38/25
|-
|-
| 37
! 37
| 738.40
! 738.40
| 49/32
| 49/32
| 23/15, 72/47
| 23/15, 72/47
| 20/13, 26/17
| 20/13, 26/17
|-
|-
| 38
! 38
| 748.36
! 748.36
| 14/9
| 14/9
| 17/11, 31/20, 48/31
| 17/11, 31/20, 48/31
|  
|  
|-
|-
| 39
! 39
| 778.32
! 778.32
| 11/7, 25/16
| 11/7, 25/16
| 36/23, 47/30, 69/44
| 36/23, 47/30, 69/44
|  
|  
|-
|-
| 40
! 40
| 798.28
! 798.28
|  
|  
| 27/17, 46/29
| 27/17, 46/29
| 19/12, 65/41
| 19/12, 65/41
|-
|-
| 41
! 41
| 818.23
! 818.23
| 8/5, 45/28, 77/48
| 8/5, 45/28, 77/48
| 29/18, 37/23, 69/43
| 29/18, 37/23, 69/43
| 21/13
| 21/13
|-
|-
| 42
! 42
| 838.19
! 838.19
|  
|  
| 34/21, 47/29, 60/37
| 34/21, 47/29, 60/37
| 13/8, 21/13
| 13/8, 21/13
|-
|-
| 43
! 43
| 858.15
! 858.15
| 18/11
| 18/11
| 23/14, 41/25
| 23/14, 41/25
| 64/39
| 64/39
|-
|-
| 44
! 44
| 878.10
! 878.10
| 5/3, 33/20
| 5/3, 33/20
| 28/17, 48/29, 58/35, 68/41, 93/56
| 28/17, 48/29, 58/35, 68/41, 93/56
| 38/23, 43/26, 63/38, 78/47
| 38/23, 43/26, 63/38, 78/47
|-
|-
| 45
! 45
| 898.06
! 898.06
| 27/16, 42/25
| 27/16, 42/25
| 37/22, 47/28
| 37/22, 47/28
| 32/19
| 32/19
|-
|-
| 46
! 46
| 918.02
! 918.02
| 56/33
| 56/33
| 17/10
| 17/10
| 22/13, 39/23
| 22/13, 39/23
|-
|-
| 47
! 47
| 937.97
! 937.97
| 12/7, 55/32
| 12/7, 55/32
| 31/18, 43/25
| 31/18, 43/25
| 19/11, 98/57
| 19/11, 98/57
|-
|-
| 48
! 48
| 958.93
! 958.93
| 7/4 (catnip)
| 7/4 (catnip)
| 19/11, 40/23
| 19/11, 40/23
| 26/15, 33/19
| 26/15, 33/19
|-
|-
| 49
! 49
| 977.89
! 977.89
| 7/4 (patent), 44/25
| 7/4 (patent), 44/25
| 30/17, 37/21, 51/29
| 30/17, 37/21, 51/29
| 23/13, 95/54
| 23/13, 95/54
|-
|-
| 50
! 50
| 997.84
! 997.84
| 16/9, 25/14
| 16/9, 25/14
| 41/23
| 41/23
| 57/32
| 57/32
|-
|-
| 51
! 51
| 1017.8
! 1017.8
| 9/5
| 9/5
| 29/16, 56/31, 74/41, 92/51
| 29/16, 56/31, 74/41, 92/51
| 38/21, 47/26, 65/36
| 38/21, 47/26, 65/36
|-
|-
| 52
! 52
| 1037.76
! 1037.76
| 20/11
| 20/11
| 31/17, 51/28
| 31/17, 51/28
|  
|  
|-
|-
| 53
! 53
| 1057.72
! 1057.72
| 11/6
| 11/6
| 94/51
| 94/51
| 24/13, 35/18
| 24/13, 35/18
|-
|-
| 54
! 54
| 1077.67
! 1077.67
| 28/15
| 28/15
| 41/22, 54/29
| 41/22, 54/29
| 13/7, 95/51
| 13/7, 95/51
|-
|-
| 55
! 55
| 1097.63
! 1097.63
| 15/8, 66/35
| 15/8, 66/35
| 17/9, 32/17
| 17/9, 32/17
| 49/26
| 49/26
|-
|-
| 56
! 56
| 1117.59
! 1117.59
| 21/11, 40/21
| 21/11, 40/21
| 82/43
| 82/43
| 19/10
| 19/10
|-
|-
| 57
! 57
| 1137.54
! 1137.54
| 27/14
| 27/14
| 23/12, 29/15, 56/29
| 23/12, 29/15, 56/29
| 25/3
| 25/3
|-
|-
| 58
! 58
| 1157.50
! 1157.50
| 35/18
| 35/18
| 31/16, 33/17, 41/21, 80/41
| 31/16, 33/17, 41/21, 80/41
| 37/19, 39/20
| 37/19, 39/20
|-
|-
| 59
! 59
| 1177.46
! 1177.46
| 49/25, 55/28, 63/32
| 49/25, 55/28, 63/32
| 47/24, 57/29, 69/35
| 47/24, 57/29, 69/35
| 51/26, 65/33, 75/38, 77/39
| 51/26, 65/33, 75/38, 77/39
|-
|-
| 60
! 60
| 1200
! 1200
| 2/1, 99/50
| 2/1, 99/50
|  
|  
|  
|  
|}
=== Plain text ===
{| class="wikitable mw-collapsible mw-collapsed"
|-
|19.96
39.91
59.87
79.83
99.78
119.74
139.7
159.66
179.61
199.57
219.53
239.48
259.44
279.4
299.35
319.31
339.27
359.22
379.18
399.14
419.09
439.05
459.01
478.97
498.92
518.88
538.84
558.79
578.75
598.71
618.66
638.62
658.58
678.53
698.49
718.45
738.4
758.36
778.32
798.28
818.23
838.19
858.15
878.1
898.06
918.02
937.97
957.93
977.89
997.84
1017.8
1037.76
1057.72
1077.67
1097.63
1117.59
1137.54
1157.5
1177.46
1197.41
|}
|}


== Regular temperament properties ==
== Regular temperament properties ==
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
(This table is incomplete.){{todo|complete table}}
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
Line 498: Line 433:
|-
|-
| 12
| 12
| 12\60<br>(2\60)
| 12\272ed23<br>(2\272ed23)
| 240.0<br>(40.0)
| 239.48<br>(39.91)
| 8/7<br>(40/39)
| 8/7<br>(36/35)
| [[Catnip]] (60cf)
| [[Catnip]] (272dg)
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
(This table is incomplete.){{todo|complete table}}


== Scales ==
== Scales ==
; 60edo scales
These are useable in 272ed23, simply apply an octave compression of 0.99784 in [[Scale Workshop]].
* [[5- to 10-tone scales in 60edo]]
* [[5- to 10-tone scales in 60edo]]


== Nearby equal-step tunings ==
== Nearby equal-step tunings ==
There are a few other useful [[equal-step tuning]]s which occur close to 60edo in step size:
There are a few other useful [[equal-step tuning]]s which occur close to 272ed23 in step size:




Line 560: Line 501:


; 272ed23
; 272ed23
{{Harmonics in equal|272|23|1|intervals=prime|columns=11|collapsed=1}}


== Instruments ==
== Instruments ==
Retrieved from "https://en.xen.wiki/w/User:BudjarnLambeth/272ed23"