60edo: Difference between revisions

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.

== Octave stretch or compression ==

What follows is a comparison of compressed- and stretched-octave 60edo tunings.

60edo can benefit from slightly [[stretched and compressed tuning|stretching the octave]], especially when using it as a no-11 17-limit equal temperament. With the right amount of stretch we can find better harmonics 3, 5, and 7 at the expense of somewhat less accurate approximations of 2 and 13. Tunings such as [[95edt]] or [[155ed6]] are great demonstrations of this.

; [[zpi|303zpi]]

* Step size: 19.913{{c}}, octave size: 1194.78{{c}}

; [[ed7|169ed7]]  

* Step size: 19.958{{c}}, octave size: 1197.50{{c}}

Compressing the octave of 60edo by around 2.5{{c}} results in improved primes 7 and 11, but worse primes 2, 3, 5 and 13. This approximates all harmonics up to 16 within 9.94{{c}}. The tuning 169ed7 does this.

{{Harmonics in equal|169|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 169ed7 (continued)}}

; [[zpi|302zpi]]  

* Step size: 19.962{{c}}, octave size: 1197.72{{c}}

Compressing the octave of 60edo by around 2{{c}} results in improved primes 7 and 11, but worse primes 2, 3, 5 and 13. This approximates all harmonics up to 16 within 9.84{{c}}. The tuning 202zpi does this. So does the tuning [[equal tuning|208ed11]] whose octave is identical within 0.3{{c}}.

{{Harmonics in cet|19.962|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 302zpi (continued)}}

302zpi is particularly well suited to [[catnip]] temperament specifically: in 60edo, catnip's mappings of 5 and 13 both differ from the [[patent val]]s, but in 19.95cet, only it's mapping of 7 differs. The tuning 169ed7 also does this, but 302zpi approximates most simple harmonics better than 169ed7.

{{Harmonics in equal|60|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60edo}}

{{Harmonics in equal|60|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 60edo (continued)}}

; [[ed12|215ed12]]

; [[WE|60et, 13-limit WE tuning]] / [[155ed6]]

Stretching the octave of 60edo by just under a cent results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 8.63{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this. So does 155ed6 whose octaves differ by only 0.02{{c}}.

{{Harmonics in cet|20.013|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 60et, 13-limit WE tuning}}

; [[95edt]]  

* Step size: 20.021{{c}}, octave size: 1201.23{{c}}

Stretching the octave of 60edo by just over a cent results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 7.06{{c}}. The tuning 95edt does this.

{{Harmonics in equal|95|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 95edt}}

; [[zpi|301zpi]]  

Stretching the octave of 60edo by around 1.5{{c}} results in improved primes 3, 5, 7, 11 and 13, but worse primes 2. This approximates all harmonics up to 16 within 6.48{{c}}. The tuning 301zpi does this.

{{Harmonics in cet|20.027|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 301zpi}}

{{Harmonics in cet| 20.027 |intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 301zpi (continued)}}

; [[139ed5]]

; [[35edf]]  

Stretching the octave of 60edo by a little over 3{{c}} results in improved primes 5, 7 and 11 but worse primes 2, 3 and 13. This approximates all harmonics up to 16 within 10.00{{c}}. The tuning 35edf does this.

{{Harmonics in equal|35|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 35edf (continued)}}

* [[5- to 10-tone scales in 60edo]]

* Amulet{{idiosyncratic}} (approximated from [[25edo]], subset of [[magic]]): 5 2 5 5 2 5 7 5 5 2 5 7 5

* Approximations of [[gamelan]] scales:

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

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