90edo: Difference between revisions

(5 intermediate revisions by 2 users not shown)

Line 3:

== 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~~. Notably, it is the second lowest in a series of four consecutive edos to temper out the [[quartisma]]~~.

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 ===

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:

; [[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 3: / Line 3: @@
 == 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. Notably, it is the second lowest in a series of four consecutive edos to temper out the [[quartisma]].
+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 ===
@@ 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)