120edo: Difference between revisions

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== Theory ==

120edo is the 10th highly composite EDO and the 5th factorial EDO (120 = 1*2*3~~*4*~~5 ~~= 5!)~~.

120edo shares the [[perfect fifth]] with 12edo, [[tempering out]] the [[Pythagorean comma]]. 120edo is an excellent tuning in the 2.3.7.11.13.23.29 [[subgroup]]. In the no-5's 11-limit, it tempers out [[243/242]]. In the patent val 120edo is also a tuning for the 7-limit [[decoid]] temperament.

~~120edo~~ is ~~an excellent~~ tuning ~~in the 2.3.7.11.13.23.29 subgroup. In the no-5s 11-limit, it tempers out [[243/242]].~~

The 120bdd val is a tuning for [[superpyth]] where 3/2 is tuned to exactly 710{{c}}. It may be used as a ''de facto'' dual fifth in [[Substitute harmonic#Newcome|newcome]] temperament.

~~120edo shares the perfect fifth with 12edo, tempering out the [[Pythagorean comma]]. The sharp fifth of 710 cents also has regular temperament interpretations. It is used in the 120b val~~ for ~~tuning the 5-limit~~ [[superpyth]] ~~temperament~~ where ~~it represents~~ 3/2~~, and in the 120g val as a tuning for the 19-limit [[surmarvelpyth]] temperament where it represents 675/448, which~~ is ~~[[marvel comma]] sharp of 3/2~~. ~~In the patent val 120edo is also~~ a ~~tuning for the 7-limit~~ [[~~decoid~~]] temperament.

~~The step size of this EDO is near the upper boundary of the [[just noticeable difference]]~~.

=== Prime harmonics ===

=== Subsets and supersets ===

120edo is the 10th highly composite edo and the 5th factorial edo (since {{nowrap|120 {{=}} 5!}} {{nowrap|{{=}} 1 × 2 × 3 × 4 × 5}}). It has many subsets: {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 30, 40, and 60 }}.

=== Miscellaneous properties ===

~~Being the simplest division of the octave by the Germanic~~ [[~~wikipedia:Long_hundred|long hundred~~]]~~, it has a unit step which is~~ the ~~fine relative cent~~ of ~~[[1edo]]~~.

120edo also has a [[concoctic]] generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

~~120edo also has a concoctic generator that resembles the leap day excess of earth, 29\~~120 ~~corresponding to 5 hours and 48 minutes.~~

== JI approximation ==

== ~~Interval list~~ ==

== Intervals ==

[[~~Category:Equal divisions of the octave|###~~]] ~~~~

== Notation ==

[[~~Category:Highly composite~~]]

=== Ups and downs notation ===

120edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals and Stein–Zimmerman [[24edo#Notation|quarter-tone]] accidentals:

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|120}}
+{{ED intro}}
 == Theory ==
-edo is the 10th highly composite EDO and the 5th factorial EDO (120 = 1*2*3*4*5 = 5!).
+edo shares the [[perfect fifth]] with 12edo, [[tempering out]] the [[Pythagorean comma]]. 120edo is an excellent tuning in the 2.3.7.11.13.23.29 [[subgroup]]. In the no-5's 11-limit, it tempers out [[243/242]]. In the patent val 120edo is also a tuning for the 7-limit [[decoid]] temperament.
-edo is an excellent tuning in the 2.3.7.11.13.23.29 subgroup. In the no-5s 11-limit, it tempers out [[243/242]].
+The 120bdd val is a tuning for [[superpyth]] where 3/2 is tuned to exactly 710{{c}}. It may be used as a ''de facto'' dual fifth in [[Substitute harmonic#Newcome|newcome]] temperament.
-edo shares the perfect fifth with 12edo, tempering out the [[Pythagorean comma]]. The sharp fifth of 710 cents also has regular temperament interpretations. It is used in the 120b val for tuning the 5-limit [[superpyth]] temperament where it represents 3/2, and in the 120g val as a tuning for the 19-limit [[surmarvelpyth]] temperament where it represents 675/448, which is [[marvel comma]] sharp of 3/2. In the patent val 120edo is also a tuning for the 7-limit [[decoid]] temperament.
-The step size of this EDO is near the upper boundary of the [[just noticeable difference]].
 === Prime harmonics ===
 {{Harmonics in equal|120}}
+=== Subsets and supersets ===
+edo is the 10th highly composite edo and the 5th factorial edo (since {{nowrap|120 {{=}} 5!}} {{nowrap|{{=}} 1 × 2 × 3 × 4 × 5}}). It has many subsets: {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 30, 40, and 60 }}.
 === Miscellaneous properties ===
-Being the simplest division of the octave by the Germanic [[wikipedia:Long_hundred|long hundred]], it has a unit step which is the fine relative cent of [[1edo]].
+edo also has a [[concoctic]] generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.
-edo also has a concoctic generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.
+== JI approximation ==
+{{Q-odd-limit intervals|120}}
-== Interval list ==
+== Intervals ==
 {{Interval table}}
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+== Notation ==
-[[Category:Highly composite]]
+=== Ups and downs notation ===
+edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals and Stein–Zimmerman [[24edo#Notation|quarter-tone]] accidentals:
+{{Sharpness-sharp10-qt1|120}}