View source for User:BudjarnLambeth/272ed23

{{Infobox ET}}
{{ED intro}}.

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

== 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]]. 

It also causes the [[val]]s 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 [[wart]]s 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.

{{Harmonics in equal|272|23|1|intervals=prime}}
{{Harmonics in equal|272|23|1|intervals=prime|start=12|collapsed=1}}


==== 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-n|dual]]-7, dual-11).

{{Harmonics in equal|60|2|1|intervals=prime}}
{{Harmonics in equal|60|2|1|intervals=prime|start=12|collapsed=1}}

=== 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:
{{Sharpness-sharp5|60}}

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 ==
{| class="wikitable center-all right-2 left-3 left-4 left-5 mw-collapsible"
|+ Intervals of 272ed23, up to the octave
|-
! Degrees
! Cents
! Approximate ratios<br />in the 11-limit
! Additional ratios<br />in the no-13, no-19 47-limit
! Additional ratios<br />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
| 12/11
| 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
| 17/15, 42/37
| 
|-
! 12
! 239.48
| 8/7
| 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 ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
| 12
| 12\272ed23<br>(2\272ed23)
| 239.48<br>(39.91)
| 8/7<br>(36/35)
| [[Catnip]] (272dg)
|}
<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 ==

; 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]]

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


; 207ed11, 168ed7

The tunings [[207ed11]] and [[168ed7]] are almost identical. Each is [[60edo]] but with slightly ''[[Octave stretch|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 [[val]]s to flip for [[11/1]] and [[13/1]].
{{Harmonics in equal|207|11|1|intervals=prime|columns=11|collapsed=1}}
{{Harmonics in equal|168|7|1|intervals=prime|columns=11|collapsed=1}}


; 301zpi

The tuning [[301zpi]], the 301st [[zeta peak index]], is [[60edo]] but with slightly ''[[Octave stretch|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.
{{Harmonics in equal|1|38083|37645| 20.0266469020418 |intervals=prime|columns=11|title= Approximation of prime harmonics in 301zpi|collapsed=1}}


; 60edo
{{Harmonics in equal|60|2|1|intervals=prime|columns=11|collapsed=1}}


; 255ed19

The tuning [[255ed19]] is [[60edo]] but with slightly ''[[Octave shrinking|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.
{{Harmonics in equal|255|19|1|intervals=prime|columns=11|collapsed=1}}


; 208ed11

The tuning [[208ed11]] is [[60edo]] but with slightly ''[[Octave shrinking|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 [[val]]s to flip for [[5/1]], [[7/1]] and [[17/1]].
{{Harmonics in equal|208|11|1|intervals=prime|columns=11|collapsed=1}}


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

== 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]]. 

[[File:60edoguitar.jpg|alt=60edoguitar.jpg|60edoguitar.jpg]]

* [[Lumatone mapping for 60edo]]
* [[Skip fretting system 60 2 29]]
* [[Skip fretting system 60 3 19]]
* [[Skip fretting system 60 4 17]]

[[Category:Catnip]][[Category:60edo]]