27edo: Difference between revisions

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Assuming pure octaves, 27edo divides the [[octave]] in 27 equal parts each exactly 44{{frac|4|9}} [[cent]]s in size. Its fifth and harmonic seventh are both sharp by 9{{c}}, and the major third is the same 400-cent major third as [[12edo]], sharp by 13.7{{c}}. The result is that [[6/5]], [[7/5]], and especially [[7/6]] are all tuned more accurately than this. It can be considered the superpythagorean counterpart of [[19edo]], as its 5th is audibly indistinguishable from [[superpyth|1/3-septimal-comma superpyth]] in the same way that 19edo is audibly indistinguishable from [[1/3-comma meantone|1/3-syntonic-comma meantone]]: Three fourths (C-Eb) in 19edo reach a near-perfect [[6/5]] and the same distance in 27edo reaches a near-perfect [[7/6]].

Though 27edo's [[7-limit]] tuning is not highly accurate, it nonetheless is the smallest equal division to represent the [[7-odd-limit]] both [[consistent]]ly and distinctly—that is, everything in the [[7-odd-limit]] [[tonality diamond]] is uniquely represented by a certain number of steps of 27edo. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13.19 (no-11's, no-17's 19-limit) temperament, if a highly sharp-tending one. It also approximates [[19/10]], [[19/12]], and [[19/14]], so {{dash|0, 7, 13, 25|med}} does quite well as a 10:12:14:19 chord, with the major seventh 25\27 being less than one cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen–Pierce triads, 3:5:7 ([0 20 33]) and 5:7:9 ([0 13 23]), via the [[BPS]] scale in [[43edt]], although approximations of the odd harmonic series rapidly become rough if extended to prime 11 and above.

Though 27edo's [[7-limit]] tuning is not highly accurate, it nonetheless is the smallest equal division to represent the [[7-odd-limit]] both [[consistent]]ly and distinctly—that is, everything in the [[7-odd-limit]] [[tonality diamond]] is uniquely represented by a certain number of steps of 27edo. It is also the highest edo for which the mapping of [[8/7]] and [[9/8]] to the same interval is [[consistent]]. It represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13.19 (no-11's, no-17's 19-limit) temperament, if a highly sharp-tending one. It also approximates [[19/10]], [[19/12]], and [[19/14]], so {{dash|0, 7, 13, 25|med}} does quite well as a 10:12:14:19 chord, with the major seventh 25\27 being less than one cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen–Pierce triads, 3:5:7 ([0 20 33]) and 5:7:9 ([0 13 23]), via the [[BPS]] scale in [[43edt]], although approximations of the odd harmonic series rapidly become rough if extended to prime 11 and above.

Its step of 44.4{{c}}, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having very high [[harmonic entropy]]. In other words, there is a general perception of quartertones as being the most dissonant intervals. This property is shared with all edos between around 20 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.

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* [[Zarlino]] / Ptolemy diatonic, "just" major: 5 4 2 5 4 5 2

* "Just" minor (inverse of "just" major): 5 2 4 5 2 5 49

* Baobab{{idio}} (approximated from [[30afdo]])

* Direct sunlight{{idio}} (original/default tuning; subset of [[Sensi]][19]): 1 2 8 5 1 9 1

* Hypersakura{{idio}} (original/default tuning; subset of Sensi[19]): 1 10 5 1 10

* [[Maeve Gutierrez#Gutierrez wisp scale|Gutierrez wisp scale]]{{idio}} ''(scale's [[period]] is [[nonoctave]])''

* [[Maeve Gutierrez#Will-o-wisps' scale|Lambeth will-o-wisps' scale]]{{idio}} ''(scale's [[period]] is [[nonoctave]])''

* [[User:BudjarnLambeth/Augene18 subsets in 97ed12]]

* ''[[User:BudjarnLambeth/Augene18 subsets in 97ed12]]''

== Instruments ==

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* [https://www.youtube.com/watch?v=hDP8cfJqWOI ''microtonal improvisation in 27edo''] (2023)

* [https://www.youtube.com/shorts/FSPUebavRCQ ''27edo waltz''] (2025)

* [https://www.youtube.com/shorts/izpEen38Sps ''27edo improv''] (2025)

* [https://www.youtube.com/shorts/izpEen38Sps ''27edo improv''] (2025-09-11)

* ''Flies Control My Pain - 27edo'' (2026)

** [https://www.youtube.com/shorts/sKnjDPEOQtc <nowiki>[short 1]</nowiki>] (using [[tetracot]] Lumatone mapping)

** [https://www.youtube.com/shorts/QEebNJkcIlE <nowiki>[short 2]</nowiki>] (using [[Starling_temperaments#Kumonga|kumonga]] Lumatone mapping)

* [https://www.youtube.com/shorts/U1uBwrq76pE ''27edo improv''] (2026-10-10)

; [[Francium]]

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 Assuming pure octaves, 27edo divides the [[octave]] in 27 equal parts each exactly 44{{frac|4|9}} [[cent]]s in size. Its fifth and harmonic seventh are both sharp by 9{{c}}, and the major third is the same 400-cent major third as [[12edo]], sharp by 13.7{{c}}. The result is that [[6/5]], [[7/5]], and especially [[7/6]] are all tuned more accurately than this. It can be considered the superpythagorean counterpart of [[19edo]], as its 5th is audibly indistinguishable from [[superpyth|1/3-septimal-comma superpyth]] in the same way that 19edo is audibly indistinguishable from [[1/3-comma meantone|1/3-syntonic-comma meantone]]: Three fourths (C-Eb) in 19edo reach a near-perfect [[6/5]] and the same distance in 27edo reaches a near-perfect [[7/6]].
-Though 27edo's [[7-limit]] tuning is not highly accurate, it nonetheless is the smallest equal division to represent the [[7-odd-limit]] both [[consistent]]ly and distinctly—that is, everything in the [[7-odd-limit]] [[tonality diamond]] is uniquely represented by a certain number of steps of 27edo. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13.19 (no-11's, no-17's 19-limit) temperament, if a highly sharp-tending one. It also approximates [[19/10]], [[19/12]], and [[19/14]], so {{dash|0, 7, 13, 25|med}} does quite well as a 10:12:14:19 chord, with the major seventh 25\27 being less than one cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen–Pierce triads, 3:5:7 ([0 20 33]) and 5:7:9 ([0 13 23]), via the [[BPS]] scale in [[43edt]], although approximations of the odd harmonic series rapidly become rough if extended to prime 11 and above.
+Though 27edo's [[7-limit]] tuning is not highly accurate, it nonetheless is the smallest equal division to represent the [[7-odd-limit]] both [[consistent]]ly and distinctly—that is, everything in the [[7-odd-limit]] [[tonality diamond]] is uniquely represented by a certain number of steps of 27edo. It is also the highest edo for which the mapping of [[8/7]] and [[9/8]] to the same interval is [[consistent]]. It represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13.19 (no-11's, no-17's 19-limit) temperament, if a highly sharp-tending one. It also approximates [[19/10]], [[19/12]], and [[19/14]], so {{dash|0, 7, 13, 25|med}} does quite well as a 10:12:14:19 chord, with the major seventh 25\27 being less than one cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen–Pierce triads, 3:5:7 ([0 20 33]) and 5:7:9 ([0 13 23]), via the [[BPS]] scale in [[43edt]], although approximations of the odd harmonic series rapidly become rough if extended to prime 11 and above.
 Its step of 44.4{{c}}, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having very high [[harmonic entropy]]. In other words, there is a general perception of quartertones as being the most dissonant intervals. This property is shared with all edos between around 20 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant.
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 * [[Zarlino]] / Ptolemy diatonic, "just" major: 5 4 2 5 4 5 2
 * "Just" minor (inverse of "just" major): 5 2 4 5 2 5 49
+* Baobab{{idio}} (approximated from [[30afdo]])
 * Direct sunlight{{idio}} (original/default tuning; subset of [[Sensi]][19]): 1 2 8 5 1 9 1
 * Hypersakura{{idio}} (original/default tuning; subset of Sensi[19]): 1 10 5 1 10
 * [[Maeve Gutierrez#Gutierrez wisp scale|Gutierrez wisp scale]]{{idio}} ''(scale's [[period]] is [[nonoctave]])''
 * [[Maeve Gutierrez#Will-o-wisps' scale|Lambeth will-o-wisps' scale]]{{idio}} ''(scale's [[period]] is [[nonoctave]])''
-* [[User:BudjarnLambeth/Augene18 subsets in 97ed12]]
+* ''[[User:BudjarnLambeth/Augene18 subsets in 97ed12]]''
 == Instruments ==
@@ Line 1,419: / Line 1,420: @@
 * [https://www.youtube.com/watch?v=hDP8cfJqWOI ''microtonal improvisation in 27edo''] (2023)
 * [https://www.youtube.com/shorts/FSPUebavRCQ ''27edo waltz''] (2025)
-* [https://www.youtube.com/shorts/izpEen38Sps ''27edo improv''] (2025)
+* [https://www.youtube.com/shorts/izpEen38Sps ''27edo improv''] (2025-09-11)
 * ''Flies Control My Pain - 27edo'' (2026)
 ** [https://www.youtube.com/shorts/sKnjDPEOQtc <nowiki>[short 1]</nowiki>] (using [[tetracot]] Lumatone mapping)
 ** [https://www.youtube.com/shorts/QEebNJkcIlE <nowiki>[short 2]</nowiki>] (using [[Starling_temperaments#Kumonga|kumonga]] Lumatone mapping)
+* [https://www.youtube.com/shorts/U1uBwrq76pE ''27edo improv''] (2026-10-10)
 ; [[Francium]]