60edo: Difference between revisions

{{ED intro}}

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.

| [[1/1]]

| [[81/80]], ''[[49/48]]''

| [[50/49]], ''[[64/63]]''

| ''[[33/32]]''

| ''[[25/24]]'', [[28/27]], ''[[36/35]]''

| [[21/20]]

| [[17/16]], [[18/17]]

| [[16/15]], [[15/14]], [[14/13]]

| [[13/12]]

| [[12/11]], [[11/10]]

| [[10/9]]

| [[9/8]]

| [[17/15]]

| [[8/7]], [[15/13]]

| [[7/6]]

| [[20/17]]

| [[33/28]]

| [[32/27]]

| ''[[13/11]]''

| [[6/5]]

| [[39/32]], [[17/14]]

| [[11/9]]

| [[16/13]], [[21/17]]

| [[27/22]]

| [[5/4]]

| [[81/64]]

| ''[[33/26]]''

| [[14/11]]

| [[9/7]]

| [[22/17]]

| ''[[21/16]]'', [[13/10]], [[17/13]]

| [[4/3]]

| [[27/20]]

| ''[[11/8]]'', [[15/11]]

| [[18/13]]

| [[7/5]]

| [[17/12]], [[24/17]]

| [[10/7]]

| [[13/9]]

| ''[[16/11]]'', [[22/15]]

| [[40/27]]

| [[3/2]]

| ''[[32/21]]'', [[20/13]], [[26/17]]

| [[14/9]]

| [[17/11]]

| [[11/7]]

| [[128/81]]

| ''[[52/33]]''

| [[8/5]]

| [[13/8]], [[34/21]]

| [[44/27]]

| [[64/39]], [[28/17]]

| [[18/11]]

| [[5/3]]

| [[27/16]]

| ''[[22/13]]''

| [[17/10]]

| [[56/33]]

| [[12/7]]

| [[7/4]], [[26/15]]

| [[30/17]]

| [[16/9]]

| [[9/5]]

| [[11/6]], [[20/11]]

| [[24/13]]

| [[15/8]], [[28/15]], [[13/7]]

| [[17/9]], [[32/17]]

| [[40/21]]

| ''[[48/25]]'', [[27/14]], ''[[35/18]]''

| [[49/25]], ''[[63/32]]''

| ''[[64/33]]''

| [[160/81]], ''[[96/49]]''

| [[2/1]]

=== Stein–Zimmermann–Gould notation ===

[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:

{{Sharpness-sharp5-szg|60}}

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).

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

=== 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}}

| [[Houborizic]] (60) / [[houbor]] (60e)

| [[Magic]] (60) / [[Magic_extensions#Witchcraft|witchcraft]] (60e)

| [[Tritonic]] (60e) / [[tritoni]] (60)

| [[Quintile]] (60)

| [[Decile]] (60e)<br>[[Decic]] (60) / [[splendecic]] (60e) / [[prodecic]] (60e)

| [[Compton]] / [[comptone]] (60e)

| [[Pentadecal]] (60) / [[Cloudy_clan#Quindeca|quindecal]] (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

* Approximations of [[gamelan]] scales:

** 5-tone pelog: 6 8 20 5 21

** 7-tone pelog: 6 8 12 8 5 14 7

** 5-tone slendro: 12 12 12 12 12