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60edo: Difference between revisions

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== Theory ==
== Theory ==
Since {{nowrap| 60 {{=}} 5 × 12 }}, 60edo belongs to the family of edos which contain [[12edo]], and like the other small edos of this kind, it [[tempering out|tempers out]] the [[Pythagorean comma]], 531441/524288 ({{monzo| -19 12 }}). In the [[5-limit]], it tempers out both the [[magic comma]], 3125/3072, and the [[amity comma]], 1600000/1594323, and supplies the [[optimal patent val]] for 5-limit [[magic]], tempering out 3125/3072. In the [[7-limit]] it tempers out [[225/224]], [[245/243]], [[875/864]], and [[10976/10935]], and [[support]]s [[magic]], [[compton]] and [[tritonic]] temperaments. In the [[11-limit]], the 60e [[val]] {{val| 60 95 139 168 '''207''' }} scores lower in [[badness]] than the [[patent val]] {{val| 60 95 139 168 '''208''' }} and makes for an excellent tritonic tuning. It tempers out [[121/120]] and [[441/440]], whereas the patent val tempers out [[100/99]], [[385/384]] and [[540/539]]. The tuning of 13 is superb at half a cent flat, and the 60e val also works excellently for [[13-limit]] tritonic. As a no-fives [[subgroup temperament]], it is also excellent for the 2.3.7.11.13-subgroup [[bleu]] temperament, using the 60d val.
Since {{nowrap| 60 {{=}} 5 × 12 }}, 60edo belongs to the family of edos which contain [[12edo]], and like the other small edos of this kind, it [[tempering out|tempers out]] the [[Pythagorean comma]], 531441/524288 ({{monzo| -19 12 }}). In the [[5-limit]], it tempers out both the [[magic comma]], 3125/3072, and the [[amity comma]], 1600000/1594323, and supplies the [[optimal patent val]] for 5-limit [[magic]]. In the [[7-limit]] it tempers out [[225/224]], [[245/243]], [[875/864]], and [[10976/10935]], and [[support]]s [[magic]], [[compton]] and [[tritonic]] temperaments. In the [[11-limit]], the 60e [[val]] {{val| 60 95 139 168 '''207''' }} scores lower in [[badness]] than the [[patent val]] {{val| 60 95 139 168 '''208''' }} and makes for an excellent tritonic tuning. It tempers out [[121/120]] and [[441/440]], whereas the patent val tempers out [[100/99]], [[385/384]] and [[540/539]]. The tuning of 13 is superb at half a cent flat, and the 60e val also works excellently for [[13-limit]] tritonic. As a no-fives [[subgroup temperament]], it is also excellent for the 2.3.7.11.13-subgroup [[bleu]] temperament, using the 60d val.


=== Odd harmonics ===
=== Odd harmonics ===
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== Notation ==
== Notation ==
===Ups and downs notation===
=== Stein–Zimmermann–Gould notation ===
60edo can be notated with [[ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
{{Sharpness-sharp5-szg|60}}
 
=== Kite's ups and downs notation ===
60edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
{{Sharpness-sharp5a}}
{{Sharpness-sharp5a}}
Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:{{Sharpness-sharp5|60}}


=== Sagittal notation ===
=== Sagittal notation ===
This notation is a superset of the notations for EDOs [[12edo#Sagittal notation|12]] and [[6edo#Sagittal notation|6]].
This notation is a superset of the notations for edos [[12edo #Sagittal notation|12]] and [[6edo #Sagittal notation|6]].


==== Evo flavor ====
==== Evo flavor ====
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default [[File:60-EDO_Revo_Sagittal.svg]]
default [[File:60-EDO_Revo_Sagittal.svg]]
</imagemap>
</imagemap>
== Approximation to JI ==
== Approximation to JI ==
=== Interval mappings ===
=== Interval mappings ===
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| 9.44
| 9.44
|}
|}
=== Uniform maps ===
{{Uniform map|edo=60}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
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| 380.0
| 380.0
| 5/4
| 5/4
| [[Magic]] (60) / [[Magic_extensions#Witchcraft|witchcraft]] (60e)
| [[Magic]] (60) / [[witchcraft]] (60e)
|-
|-
| 1
| 1
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|-
|-
| 2
| 2
| 19\60<br>(11\60)
| 11\60
| 380.0<br>(220.0)
| 220.0
| 5/4<br>(25/22)
| 25/22
| [[Astrology]] (60de) / [[divination]] (60e)
| [[Astrology]] (60de) / [[divination]] (60e)
|-
|-
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|-
|-
| 5
| 5
| 19\60<br>(5\60)
| 1\60
| 380.0<br>(100.0)
| 20.0
| 5/4<br>(256/245)
| 81/80
| [[Warlock]]
| [[Quintile]] (60)
|-
|-
| 5
| 5
| 25\60<br>(1\60)
| 5\60
| 500.0<br>(20.0)
| 100.0
| 4/3<br>(81/80)
| 256/245
| [[Quintile]] (60)
| [[Warlock]]
|-
|-
| 6
| 6
| 17\60<br>(3\60)
| 3\60
| 340.0<br>(60.0)
| 60.0
| 375/308<br>(1760/1701)
| 1760/1701
| [[Semiseptichrome]] (11-limit, 60e)
| [[Semiseptichrome]] (11-limit, 60e)
|-
|-
| 10
| 10
| 25\60<br>(1\60)
| 1\60
| 500.0<br>(20.0)
| 20.0
| 4/3<br>(91/90)
| 91/90
| [[Decile]] (60e)<br>[[Decic]] (60) / [[splendecic]] (60e) / [[prodecic]] (60e)
| [[Decile]] (60e)<br>[[Decic]] (60) / [[splendecic]] (60e) / [[prodecic]] (60e)
|-
|-
| 12
| 12
| 19\60<br>(1\60)
| 1\60
| 380.0<br>(20.0)
| 20.0
| 5/4<br>(81/80)
| 81/80
| [[Compton]] / [[comptone]] (60e)
| [[Compton]] / [[comptone]] (60e)
|-
|-
| 12
| 12
| 12\60<br>(2\60)
| 2\60
| 240.0<br>(40.0)
| 40.0
| 8/7<br>(40/39)
| 40/39
| [[Catnip]] (60cf)
| [[Catnip]] (60cf)
|-
|-
| 15
| 15
| 25\60<br>(3\60)
| 3\60
| 500.0<br>(20.0)
| 20.0
| 4/3<br>(126/125)
| 91/90
| [[Pentadecal]] (60) / [[Cloudy_clan#Quindeca|quindecal]] (60e)
| [[Pentadecal]] / [[quindecal]] (60e)
|-
|-
| 20
| 20
| 25\60<br>(2\60)
| 2\60
| 500.0<br>(20.0)
| 20.0
| 4/3<br>(99/98)
| 99/98
| [[Degrees]] (60e)
| [[Degrees]] (60e)
|}
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


== Diagrams ==
== Diagrams ==
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