User:BudjarnLambeth/272ed23: Difference between revisions

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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 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 ==

Latest revision as of 14:43, 1 August 2025

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← 271ed23 272ed23 273ed23 →
Prime factorization 24 × 17
Step size 19.9569 ¢ 
Octave 60\272ed23 (1197.41 ¢) (→ 15\68ed23)
Twelfth 95\272ed23 (1895.9 ¢)
Consistency limit 4
Distinct consistency limit 4

272 equal divisions of the 23rd harmonic (abbreviated 272ed23) is a nonoctave tuning system that divides the interval of 23/1 into 272 equal parts of about 20 ¢ each. Each step represents a frequency ratio of 231/272, or the 272nd root of 23.

272ed23 is primarily intended to be used as 60edo but with slightly compressed octaves.

Theory

Compared to pure-octaves 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 vals to flip for 5/1, 7/1, 13/1 and 17/1.

These characteristics, particularly the flipped 5/1 and 13/1, are preferred over pure-octaves 60edo for catnip temperament specifically. They change catnip’s warts from 60cf to 272dg (later letters in the alphabet are better).

Given this, 272ed23 can be thought of as a 60edo clone tailor-made for catnip.

60edo is already on the optimal ET sequence for catnip, so 272ed23 is one of the most optimal catnip tunings concievable.

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.

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


Approximation of prime harmonics in 272ed23
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -2.59 -6.05 +7.65 +3.89 -0.28 +9.86 +4.44 -8.51 +0.00 -2.17 +2.12
Relative (%) -13.0 -30.3 +38.3 +19.5 -1.4 +49.4 +22.2 -42.6 +0.0 -10.8 +10.6
Steps
(reduced) 		60
(60) 		95
(95) 		140
(140) 		169
(169) 		208
(208) 		223
(223) 		246
(246) 		255
(255) 		272
(0) 		292
(20) 		298
(26)
Approximation of prime harmonics in 272ed23
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) -4.84 -2.94 -5.57 +0.09 -8.33 +5.57 +7.73 +4.96 +4.35 -3.83 -0.88
Relative (%) -24.2 -14.7 -27.9 +0.5 -41.8 +27.9 +38.7 +24.8 +21.8 -19.2 -4.4
Steps
(reduced) 		313
(41) 		322
(50) 		326
(54) 		334
(62) 		344
(72) 		354
(82) 		357
(85) 		365
(93) 		370
(98) 		372
(100) 		379
(107)


60edo for comparison

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-7, dual-11).


Approximation of prime harmonics in 60edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -1.96 -6.31 -8.83 +8.68 -0.53 -4.96 +2.49 -8.27 -9.58 -5.04
Relative (%) +0.0 -9.8 -31.6 -44.1 +43.4 -2.6 -24.8 +12.4 -41.4 -47.9 -25.2
Steps
(reduced) 		60
(0) 		95
(35) 		139
(19) 		168
(48) 		208
(28) 		222
(42) 		245
(5) 		255
(15) 		271
(31) 		291
(51) 		297
(57)
Approximation of prime harmonics in 60edo
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) +8.66 -9.06 +8.48 -5.51 +6.50 +0.83 +3.12 +0.69 +0.30 -7.79 -4.54
Relative (%) +43.3 -45.3 +42.4 -27.5 +32.5 +4.1 +15.6 +3.5 +1.5 -38.9 -22.7
Steps
(reduced) 		313
(13) 		321
(21) 		326
(26) 		333
(33) 		344
(44) 		353
(53) 		356
(56) 		364
(4) 		369
(9) 		371
(11) 		378
(18)

Subsets and supersets

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. 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

272ed23 can be notated using ups and downs notation using Helmholtz-Ellis accidentals as if it were 60edo:

Semitones 0 1⁄5 2⁄5 3⁄5 4⁄5 1 1+1⁄5 1+2⁄5 1+3⁄5 1+4⁄5 2 2+1⁄5 2+2⁄5
Sharp symbol
Flat symbol

If arrows are taken to have their own layer of enharmonic spellings, then in some cases certain notes may be best spelled with three arrows.

Intervals

Intervals of 272ed23, up to the octave
Degrees Cents Approximate ratios
in the 11-limit 		Additional ratios
in the no-13, no-19 47-limit 		Additional ratios
in the full 47-limit
0 0 1/1
1 19.96 55/54, 56/55, 64/63, 81/80 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
2 39.91 33/32, 36/35 34/33, 35/34, 37/36, 41/40, 42/41, 43/42, 44/43 38/37, 39/38, 40/39
3 59.87 25/24, 28/27 24/23, 29/28, 30/29, 88/85 26/25, 27/26
4 79.83 21/20, 22/21 23/22, 45/43 19/18, 20/19
5 99.78 16/5, 35/33 17/16, 18/17
6 119.74 15/14 29/27, 44/41, 74/69 14/13
7 139.70 25/23, 51/47 13/12, 38/35
8 159.66 11/10, 12/11 23/21, 34/31 57/52
9 179.61 10/9 31/28, 41/37, 51/46 21/19
10 199.57 9/8, 28/25, 55/49 37/33, 46/41 19/17
11 219.53 8/7 (patent) 17/15, 42/37
12 239.48 8/7 (catnip) 23/20, 31/27, 54/47, 85/74 15/13
13 259.44 7/6 29/25, 36/31, 43/37 22/19
14 279.4 20/17, 27/23, 47/40 13/11
15 299.35 25/21 44/37 19/16
16 319.31 6/5, 77/64 29/24, 35/28, 41/34 23/19, 47/39, 65/54
17 339.27 11/9 17/14, 28/23, 45/37
18 359.22 27/22 43/35 16/13, 91/74
19 379.18 5/4, 56/45 21/17, 31/25, 36/29, 41/33, 46/37, 51/41 26/21
20 399.14 63/50 29/23, 34/27 19/15, 24/19
21 419.09 14/11 23/18, 37/29, 51/40 65/51
22 439.05 9/7 22/17, 31/24, 40/31, 58/45 49/38
23 459.01 176/135 30/23, 43/33 13/10, 17/13
24 478.97 21/16, 33/25 29/22, 62/47 25/19, 91/69
25 498.92 4/3, 75/56 43/32, 47/35, 63/47, 99/74 39/29, 51/38, 87/65, 91/68
26 518.88 27/20 23/17, 31/23, 58/43, 85/63 19/14
27 538.84 15/11 41/30, 56/41, 86/63 26/19
28 558.79 11/8 29/21, 40/29, 69/50 18/13
29 578.75 7/5, 25/18, 88/63 32/23, 46/33, 60/43, 81/58 39/28, 95/68
30 598.71 17/12, 24/17, 41/29 65/46
31 618.66 10/7, 63/44 23/16, 33/23, 43/30 93/65
32 638.62 81/56 29/20, 42/29, 68/47 13/9, 55/38, 94/65
33 658.58 16/11, 22/15 41/28, 60/41 19/13
34 678.53 25/17, 31/21, 34/32, 37/25 28/19
35 698.49 3/2, 49/33 55/37, 64/43, 70/47, 82/55, 94/63 52/35, 58/39, 76/51, 85/57
36 718.45 32/21, 50/33 29/19, 35/23, 41/27, 44/29, 47/31 38/25
37 738.40 49/32 23/15, 72/47 20/13, 26/17
38 748.36 14/9 17/11, 31/20, 48/31
39 778.32 11/7, 25/16 36/23, 47/30, 69/44
40 798.28 27/17, 46/29 19/12, 65/41
41 818.23 8/5, 45/28, 77/48 29/18, 37/23, 69/43 21/13
42 838.19 34/21, 47/29, 60/37 13/8, 21/13
43 858.15 18/11 23/14, 41/25 64/39
44 878.10 5/3, 33/20 28/17, 48/29, 58/35, 68/41, 93/56 38/23, 43/26, 63/38, 78/47
45 898.06 27/16, 42/25 37/22, 47/28 32/19
46 918.02 56/33 17/10 22/13, 39/23
47 937.97 12/7, 55/32 31/18, 43/25 19/11, 98/57
48 958.93 7/4 (catnip) 19/11, 40/23 26/15, 33/19
49 977.89 7/4 (patent), 44/25 30/17, 37/21, 51/29 23/13, 95/54
50 997.84 16/9, 25/14 41/23 57/32
51 1017.8 9/5 29/16, 56/31, 74/41, 92/51 38/21, 47/26, 65/36
52 1037.76 20/11 31/17, 51/28
53 1057.72 11/6 94/51 24/13, 35/18
54 1077.67 28/15 41/22, 54/29 13/7, 95/51
55 1097.63 15/8, 66/35 17/9, 32/17 49/26
56 1117.59 21/11, 40/21 82/43 19/10
57 1137.54 27/14 23/12, 29/15, 56/29 25/3
58 1157.50 35/18 31/16, 33/17, 41/21, 80/41 37/19, 39/20
59 1177.46 49/25, 55/28, 63/32 47/24, 57/29, 69/35 51/26, 65/33, 75/38, 77/39
60 1200 2/1, 99/50

Regular temperament properties

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
12 12\272ed23
(2\272ed23) 		239.48
(39.91) 		8/7
(36/35) 		Catnip (272dg)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

(This table is incomplete.)

Scales

60edo scales

These are useable in 272ed23, simply apply an octave compression of 0.99784 in Scale Workshop.

Nearby equal-step tunings

There are a few other useful equal-step tunings which occur close to 272ed23 in step size:


207ed11, 168ed7

The tunings 207ed11 and 168ed7 are almost identical. Each is 60edo but with slightly stretched octaves.

Each causes relatively large improvement to 5/1, 7/1 and 11/1 at the cost of moderate worsening of 2/1 and 3/1.

Each also causes the vals to flip for 11/1 and 13/1.

Approximation of prime harmonics in 207ed11
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +3.28 +3.24 +1.29 +0.36 +0.00 -8.44 +8.44 -3.63 +6.54 +6.33 -8.85
Relative (%) +16.4 +16.2 +6.4 +1.8 +0.0 -42.1 +42.1 -18.1 +32.6 +31.6 -44.1
Steps
(reduced) 		60
(60) 		95
(95) 		139
(139) 		168
(168) 		207
(0) 		221
(14) 		245
(38) 		254
(47) 		271
(64) 		291
(84) 		296
(89)
Approximation of prime harmonics in 168ed7
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +3.15 +3.04 +0.99 +0.00 -0.44 -8.92 +7.92 -4.17 +5.96 +5.71 -9.49
Relative (%) +15.7 +15.1 +4.9 +0.0 -2.2 -44.5 +39.5 -20.8 +29.7 +28.5 -47.3
Steps
(reduced) 		60
(60) 		95
(95) 		139
(139) 		168
(0) 		207
(39) 		221
(53) 		245
(77) 		254
(86) 		271
(103) 		291
(123) 		296
(128)


301zpi

The tuning 301zpi, the 301st zeta peak index, is 60edo but with slightly stretched octaves.

It causes relatively large improvement to 3/1, 5/1, 7/1, 11/1 and 17/1 at the cost of relatively small worsening of 2/1 and relatively large worsening of 13/1.

It also causes the val for 11/1 to flip from 208 steps to 207 steps.

301zpi is both consistent and distinctly consistent up to the 10-integer-limit, which is unusually high for a two digit edo or three digit zpi.

Approximation of prime harmonics in 301zpi
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +1.60 +0.58 -2.61 -4.35 -5.80 +5.39 +1.57 +9.28 -1.05 -1.82 +2.88
Relative (%) +8.0 +2.9 -13.0 -21.7 -29.0 +26.9 +7.9 +46.3 -5.3 -9.1 +14.4
Step 60 95 139 168 207 222 245 255 271 291 297


60edo
Approximation of prime harmonics in 60edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -1.96 -6.31 -8.83 +8.68 -0.53 -4.96 +2.49 -8.27 -9.58 -5.04
Relative (%) +0.0 -9.8 -31.6 -44.1 +43.4 -2.6 -24.8 +12.4 -41.4 -47.9 -25.2
Steps
(reduced) 		60
(0) 		95
(35) 		139
(19) 		168
(48) 		208
(28) 		222
(42) 		245
(5) 		255
(15) 		271
(31) 		291
(51) 		297
(57)


255ed19

The tuning 255ed19 is 60edo but with slightly compressed octaves.

It causes a relatively large improvement to 11/1, at the cost of relatively small worsening of every smaller prime.

It also causes the val for 7/1 to flip from 168 steps to 169.

Approximation of prime harmonics in 255ed19
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -0.59 -2.88 -7.67 +9.53 +6.65 -2.69 -7.34 +0.00 +9.07 +7.57 -7.93
Relative (%) -2.9 -14.4 -38.4 +47.7 +33.3 -13.5 -36.7 +0.0 +45.4 +37.9 -39.7
Steps
(reduced) 		60
(60) 		95
(95) 		139
(139) 		169
(169) 		208
(208) 		222
(222) 		245
(245) 		255
(0) 		272
(17) 		292
(37) 		297
(42)


208ed11

The tuning 208ed11 is 60edo but with slightly compressed octaves.

It causes 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 vals to flip for 5/1, 7/1 and 17/1.

Approximation of prime harmonics in 208ed11
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -2.50 -5.92 +7.84 +4.12 +0.00 -9.79 +4.78 -8.16 +0.37 -1.77 +2.53
Relative (%) -12.5 -29.7 +39.3 +20.6 +0.0 -49.1 +23.9 -40.9 +1.9 -8.8 +12.7
Steps
(reduced) 		60
(60) 		95
(95) 		140
(140) 		169
(169) 		208
(0) 		222
(14) 		246
(38) 		255
(47) 		272
(64) 		292
(84) 		298
(90)


272ed23
Approximation of prime harmonics in 272ed23
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -2.59 -6.05 +7.65 +3.89 -0.28 +9.86 +4.44 -8.51 +0.00 -2.17 +2.12
Relative (%) -13.0 -30.3 +38.3 +19.5 -1.4 +49.4 +22.2 -42.6 +0.0 -10.8 +10.6
Steps
(reduced) 		60
(60) 		95
(95) 		140
(140) 		169
(169) 		208
(208) 		223
(223) 		246
(246) 		255
(255) 		272
(0) 		292
(20) 		298
(26)

Instruments

Due to 60edo's highly composite nature, 272ed23 has an unusually high number of ways it can be subdivided. This means it has multiple good skip-fretting systems which can be used to create stringed instruments with playable fret spacings that still span the full gamut. Probably the best of these is tuning a compressed-octaves 20edo guitar to major thirds, emulating the pure-octaves one by Robin Perry in the image below. This is very closely related to the Kite Guitar.

60edoguitar.jpg

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