60edo: Difference between revisions
→Intervals: <code><nowiki>sed -E "s#[0-9]+/[0-9]+#\[\[&\]\]#g" 60EDOintervals.txt > 60EDOintervals.new.txt</nowiki></code> |
→Notation: SZG notation |
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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]] | 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 === | ||
| Line 94: | Line 94: | ||
| 280 | | 280 | ||
| [[20/17]] | | [[20/17]] | ||
| | | [[33/28]] | ||
|- | |- | ||
| 15 | | 15 | ||
| 300 | | 300 | ||
| [[32/27]] | | [[32/27]] | ||
| | | ''[[13/11]]'' | ||
|- | |- | ||
| 16 | | 16 | ||
| Line 124: | Line 124: | ||
| 400 | | 400 | ||
| [[81/64]] | | [[81/64]] | ||
| | | ''[[33/26]]'' | ||
|- | |- | ||
| 21 | | 21 | ||
| 420 | | 420 | ||
| | | | ||
| [[14/11]] | | [[14/11]] | ||
|- | |- | ||
| 22 | | 22 | ||
| Line 219: | Line 219: | ||
| 780 | | 780 | ||
| | | | ||
| | | [[11/7]] | ||
|- | |- | ||
| 40 | | 40 | ||
| 800 | | 800 | ||
| [[128/81]] | | [[128/81]] | ||
| | | ''[[52/33]]'' | ||
|- | |- | ||
| 41 | | 41 | ||
| Line 249: | Line 249: | ||
| 900 | | 900 | ||
| [[27/16]] | | [[27/16]] | ||
| | | ''[[22/13]]'' | ||
|- | |- | ||
| 46 | | 46 | ||
| 920 | | 920 | ||
| [[17/10]] | | [[17/10]] | ||
| | | [[56/33]] | ||
|- | |- | ||
| 47 | | 47 | ||
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== Notation == | == 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}} | ||
=== Sagittal notation === | === Sagittal notation === | ||
This notation is a superset of the notations for | This notation is a superset of the notations for edos [[12edo #Sagittal notation|12]] and [[6edo #Sagittal notation|6]]. | ||
==== Evo flavor ==== | ==== Evo flavor ==== | ||
| Line 357: | Line 360: | ||
default [[File:60-EDO_Revo_Sagittal.svg]] | default [[File:60-EDO_Revo_Sagittal.svg]] | ||
</imagemap> | </imagemap> | ||
== Approximation to JI == | |||
=== Interval mappings === | |||
{{Q-odd-limit intervals|60}} | |||
{{Q-odd-limit intervals|59.9|apx=val|header=none|tag=none|title=15-odd-limit intervals by 60e val mapping}} | |||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 448: | Line 456: | ||
| 340.0 | | 340.0 | ||
| 39/32 | | 39/32 | ||
| [[Houborizic]] (60) / houbor (60e) | | [[Houborizic]] (60) / [[houbor]] (60e) | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 454: | Line 462: | ||
| 380.0 | | 380.0 | ||
| 5/4 | | 5/4 | ||
| [[Magic]] (60) / witchcraft (60e) | | [[Magic]] (60) / [[Magic_extensions#Witchcraft|witchcraft]] (60e) | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 460: | Line 468: | ||
| 580.0 | | 580.0 | ||
| 7/5 | | 7/5 | ||
| [[Tritonic]] (60e) / tritoni (60) | | [[Tritonic]] (60e) / [[tritoni]] (60) | ||
|- | |- | ||
| 2 | | 2 | ||
| Line 514: | Line 522: | ||
| 500.0<br>(20.0) | | 500.0<br>(20.0) | ||
| 4/3<br>(91/90) | | 4/3<br>(91/90) | ||
| [[Decile]] (60e)<br>[[Decic]] (60) / splendecic (60e) / prodecic (60e) | | [[Decile]] (60e)<br>[[Decic]] (60) / [[splendecic]] (60e) / [[prodecic]] (60e) | ||
|- | |- | ||
| 12 | | 12 | ||
| Line 520: | Line 528: | ||
| 380.0<br>(20.0) | | 380.0<br>(20.0) | ||
| 5/4<br>(81/80) | | 5/4<br>(81/80) | ||
| [[Compton]] / comptone (60e) | | [[Compton]] / [[comptone]] (60e) | ||
|- | |- | ||
| 12 | | 12 | ||
| Line 532: | Line 540: | ||
| 500.0<br>(20.0) | | 500.0<br>(20.0) | ||
| 4/3<br>(126/125) | | 4/3<br>(126/125) | ||
| [[Pentadecal]] (60) / quindecal (60e) | | [[Pentadecal]] (60) / [[Cloudy_clan#Quindeca|quindecal]] (60e) | ||
|- | |- | ||
| 20 | | 20 | ||
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| [[Degrees]] (60e) | | [[Degrees]] (60e) | ||
|} | |} | ||
<nowiki/>* [[Normal | <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 | ||
== Diagrams == | == Diagrams == | ||