108edo: Difference between revisions

(16 intermediate revisions by 8 users not shown)

Line 1:

The ''108 equal division'' divides the octave into 108 equal parts of 11.111 cents each. It tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.

~~Aside from~~ tuning, ~~108 is~~ the ~~smallest number with a prime factorization of~~ the ~~form <math>p^p \cdot q^q<~~/~~math>. Being close to 100~~, it is a good ~~substitute for~~ a ~~relative cent if you desire the measure~~ to ~~have such a property~~. ~~Also~~, ~~multiplying it by~~ 12 ~~gives the fourth power of an integer~~.

== Theory ==

[[Category:~~valentine~~]]

108et [[tempering out|tempers out]] the [[Pythagorean comma]] in the 3-limit and 1990656/1953125 ([[valentine comma]]) in the 5-limit. In the 7-limit it tempers out [[126/125]] and [[1029/1024]], [[support]]ing [[valentine]], and making for a good tuning for it and for [[starling]] temperament, the planar temperament tempering out 126/125. In the 11-limit the [[patent val]] tempers out [[540/539]] and 1344/1331, and the 108e val tempers out [[121/120]], [[176/175]], [[385/384]], and [[441/440]], supporting 11-limit valentine for which it is again a good tuning.

108edo is the smallest 12''n''-edo which offers a decent alternative fifth to 12edo's fifth, that is [[27edo]]'s superpyth fifth.

=== Prime harmonics ===

=== Subsets and supersets ===

Since 108 factors into {{factorization|108}}, 108edo has subset edos {{EDOs| 2, 3, 4, 6, 9, 12, 18, 27, 36, and 54 }}.

== Intervals ==

== Music ==

; [[Billy Stiltner]]

* [https://billystiltner.bandcamp.com/track/108-tone-equal 108 tone equal]

[[Category:Valentine]]

@@ Line 1: / Line 1: @@
-The ''108 equal division'' divides the octave into 108 equal parts of 11.111 cents each. It tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.
+{{Infobox ET}}
+{{ED intro}}
-Aside from tuning, 108 is the smallest number with a prime factorization of the form <math>p^p \cdot q^q</math>. Being close to 100, it is a good substitute for a relative cent if you desire the measure to have such a property. Also, multiplying it by 12 gives the fourth power of an integer.
+== Theory ==
-[[Category:valentine]]
+et [[tempering out|tempers out]] the [[Pythagorean comma]] in the 3-limit and 1990656/1953125 ([[valentine comma]]) in the 5-limit. In the 7-limit it tempers out [[126/125]] and [[1029/1024]], [[support]]ing [[valentine]], and making for a good tuning for it and for [[starling]] temperament, the planar temperament tempering out 126/125. In the 11-limit the [[patent val]] tempers out [[540/539]] and 1344/1331, and the 108e val tempers out [[121/120]], [[176/175]], [[385/384]], and [[441/440]], supporting 11-limit valentine for which it is again a good tuning.
+edo is the smallest 12''n''-edo which offers a decent alternative fifth to 12edo's fifth, that is [[27edo]]'s superpyth fifth.
+=== Prime harmonics ===
+{{Harmonics in equal|108}}
+=== Subsets and supersets ===
+Since 108 factors into {{factorization|108}}, 108edo has subset edos {{EDOs| 2, 3, 4, 6, 9, 12, 18, 27, 36, and 54 }}.
+== Intervals ==
+{{Interval table}}
+== Music ==
+; [[Billy Stiltner]]
+* [https://billystiltner.bandcamp.com/track/108-tone-equal 108 tone equal]
+[[Category:Valentine]]