90edo: Difference between revisions

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~~The 90~~ equal temperament ~~divides the octave into 90 equal parts of 13.333 cents each. It~~ tempers out 2048/2025 in the 5-limit, 3125/3087 ~~and 245/243~~ in the 7-limit, 121/120 and 176/175 in the 11-limit, and 275/273 ~~and 169/168~~ in the 13-limit. It provides the optimal patent val for the 31&90 temperament in the 7-, 11- and 13-~~limits~~.

== Theory ==

As an equal temperament, it [[tempering out|tempers out]] 2048/2025 ([[diaschisma]]) in the 5-limit, [[245/243]] and [[3125/3087]] in the 7-limit, [[121/120]] and [[176/175]] in the 11-limit, and [[169/168]] and [[275/273]] in the 13-limit. It provides the [[optimal patent val]] for the 31 & 90 temperament in the 7-, 11- and 13-limit.

=== Odd harmonics ===

=== Subsets and supersets ===

Since 90 factors into primes as 2 x 3<sup>2</sup> × 5, 90 has subset edos {{EDOs| 2, 3, 5, 6, 9, 10, 15, 18, 30, 45 }}.

As a composite edo, the smallest subsets it lacks are subsets of [[4edo|4]], [[7edo|7]] and [[8edo|8]], but 13\90 = 173.333{{cent}} offers a good approximation to 1\7 = 171.428{{c}}, and instead of 1\8 = 150{{cent}}, it has 27\90 = 146.667{{cent}}, serving a similar function.

Like [[80edo]], this may offer a relatively unexplored strategy of "tempered [[detempering]]", a sort of middle path between complete detempering to JI (which lacks the simplifications and unique comma pumping and structural opportunities of tempering) and not detempering the small edo at all (which can lead to challenging interpretation of harmony if one's goal is approximation to JI).

Some supersets of 90edo include: {{EDOs| 180, 270, 360, 450, 540, 630, 720, 810, 900... }}. Of these, 270edo is notable for being the first to correct the tuning of harmonics [[3/1|3]], [[7/1|7]], [[11/1|11]], and [[19/1|19]] to near-just. Temperament mergers of these might produce [[90th-octave temperaments]] ''(see [[Fractional-octave temperaments]])''.

== Interval table ==

== Scales ==

* Amulet{{idiosyncratic}}, (approximated from [[magic]] in [[25edo]]): 7 4 7 7 4 7 11 7 7 4 7 11 7

* Decimetra[20]: 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2

* [[Maeve Gutierrez|Gutierrez Moonglade scale]]: 1 6 7 1 7 1 6 8 2 5 7 2 1 5 6 1 7 1 2 4 1 2 5 2

== Instruments ==

* [[Lumatone mapping for 90edo]]

* [[Skip fretting system 90 5 17]]

== Music ==

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/0j33e4F-4jY ''microtonal improvisation in 90edo''] (2025)

* ''Fantasy in 90edo'' (2026)

** [https://www.youtube.com/shorts/LdxUD1taOqI <nowiki>[short clip]</nowiki>] (Lumatone view)

** [https://www.youtube.com/watch?v=YRNtKaWNjUU <nowiki>[full version]</nowiki>]

; [[James Kukula]]

* [https://interdependentscience.blogspot.com/2026/01/90edo-diaschismic.html ''90edo diaschismic''] (2025)

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-The 90 equal temperament divides the octave into 90 equal parts of 13.333 cents each. It tempers out 2048/2025 in the 5-limit, 3125/3087 and 245/243 in the 7-limit, 121/120 and 176/175 in the 11-limit, and 275/273 and 169/168 in the 13-limit. It provides the optimal patent val for the 31&amp;90 temperament in the 7-, 11- and 13-limits.
+{{Infobox ET}}
+{{ED intro}}
+== Theory ==
+As an equal temperament, it [[tempering out|tempers out]] 2048/2025 ([[diaschisma]]) in the 5-limit, [[245/243]] and [[3125/3087]] in the 7-limit, [[121/120]] and [[176/175]] in the 11-limit, and [[169/168]] and [[275/273]] in the 13-limit. It provides the [[optimal patent val]] for the 31 &amp; 90 temperament in the 7-, 11- and 13-limit.
+=== Odd harmonics ===
+{{Harmonics in equal|90}}
+=== Subsets and supersets ===
+Since 90 factors into primes as 2 x 3<sup>2</sup> × 5, 90 has subset edos {{EDOs| 2, 3, 5, 6, 9, 10, 15, 18, 30, 45 }}.
+As a composite edo, the smallest subsets it lacks are subsets of [[4edo|4]], [[7edo|7]] and [[8edo|8]], but 13\90 = 173.333{{cent}} offers a good approximation to 1\7 = 171.428{{c}}, and instead of 1\8 = 150{{cent}}, it has 27\90 = 146.667{{cent}}, serving a similar function.
+Like [[80edo]], this may offer a relatively unexplored strategy of "tempered [[detempering]]", a sort of middle path between complete detempering to JI (which lacks the simplifications and unique comma pumping and structural opportunities of tempering) and not detempering the small edo at all (which can lead to challenging interpretation of harmony if one's goal is approximation to JI).
+Some supersets of 90edo include: {{EDOs| 180, 270, 360, 450, 540, 630, 720, 810, 900... }}. Of these, 270edo is notable for being the first to correct the tuning of harmonics [[3/1|3]], [[7/1|7]], [[11/1|11]], and [[19/1|19]] to near-just. Temperament mergers of these might produce [[90th-octave temperaments]] ''(see [[Fractional-octave temperaments]])''.
+== Interval table ==
+{{Interval table}}
+== Scales ==
+* Amulet{{idiosyncratic}}, (approximated from [[magic]] in [[25edo]]): 7 4 7 7 4 7 11 7 7 4 7 11 7
+* Decimetra[20]: 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2
+* [[Maeve Gutierrez|Gutierrez Moonglade scale]]: 1 6 7 1 7 1 6 8 2 5 7 2 1 5 6 1 7 1 2 4 1 2 5 2
+== Instruments ==
+* [[Lumatone mapping for 90edo]]
+* [[Skip fretting system 90 5 17]]
+== Music ==
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/0j33e4F-4jY ''microtonal improvisation in 90edo''] (2025)
+* ''Fantasy in 90edo'' (2026)
+** [https://www.youtube.com/shorts/LdxUD1taOqI <nowiki>[short clip]</nowiki>] (Lumatone view)
+** [https://www.youtube.com/watch?v=YRNtKaWNjUU <nowiki>[full version]</nowiki>]
+; [[James Kukula]]
+* [https://interdependentscience.blogspot.com/2026/01/90edo-diaschismic.html ''90edo diaschismic''] (2025)