90edo: Difference between revisions

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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\80 = 146.667{{cent}}, serving a similar function.

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... }}. Temperament mergers of these might produce [[90th-octave temperaments]] ''(see [[Fractional-octave temperaments]])''.

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 ==

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; [[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)

@@ Line 11: / Line 11: @@
 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\80 = 146.667{{cent}}, serving a similar function.
+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... }}. Temperament mergers of these might produce [[90th-octave temperaments]] ''(see [[Fractional-octave temperaments]])''.
+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 ==
@@ Line 32: / Line 32: @@
 ; [[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)