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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|+ style="font-size: 105%;" | Circle of fifths in 27edo

|- style="white-space: nowrap;"

!Cents

! Cents

! colspan="2" | Extended Pythagorean notation

! colspan="2" | Extended Pythagorean notation

! colspan="2" | Quartertone notation

! colspan="2" | Quartertone notation

|-

| 0.0

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Using standard [[chain-of-fifths notation]], a sharp (an augmented unison) raises a note by 4 edosteps, just one edostep beneath the following nominal, and the flat conversely lowers. The sharp is quite wide at about 178¢, sounding like a narrow major 2nd. C to C♯ describes the approximate 10/9 and 11/10 interval. An accidental can be divided in half, and the remaining places can then be filled in with half-sharps, half-flats, sesquisharps, and sesquiflats, reducing the need for double sharps and double flats. The half-sharp is notated as a quartertone, but at about 89¢ it sounds more like a narrow semitone. The gamut from C to D is C, D♭, C{{demisharp2}}, D{{demiflat2}}, C♯, and D, with many ascending intervals appearing to be descending on the staff.

=== ~~Ups~~ and downs notation ===

=== Stein–Zimmermann–Gould notation ===

27edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.

[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:

=== Kite's ups and downs notation ===

27edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.

~~[[Alternative symbols for ups and downs notation|Alternatively,]] sharps and flats with arrows can be used, borrowed from extended [[Helmholtz–Ellis notation]]:~~

~~{{Sharpness-sharp4}}~~

=== Sagittal notation ===

Line 1,022:

Line 1,024:

| 3

| 2\27

| [[Augene]] (27e) / ~~Eugene~~ (27)

| [[Augene]] (27e) / eugene (27)

| [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]]

|-

Line 1,032:

Line 1,034:

| 9

| 1\27

| [[Niner]] (27e~~) [[Ennealimmal]] (out of tune~~)

| [[Niner]] (27e)

| [[9L 9s]]

|}

In addition, 27edo can be used as a detempering target for [[ennealimmal]].

=== Commas ===

Line 1,404:

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; [[Brendan Byrnes]]

* [https://www.youtube.com/watch?v=sWaqlAgSWcc ''Sunspots''] (2022)

* ''27 EDO Etude'' (2022)

** [https://brendanbyrnes.bandcamp.com/track/27-edo-etude on Bandcamp]

** [https://m.youtube.com/watch?v=Lml2cfJW9QI on YouTube] (with sheet music)

* [https://www.youtube.com/watch?v=lywpWPBYQi0 ''Istril Bloom''] (2025)

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

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

* ''Flies Control My Pain - 27edo'' (2026) [https://www.youtube.com/shorts/sKnjDPEOQtc <nowiki>[short 1]</nowiki>]; [https://www.youtube.com/shorts/QEebNJkcIlE <nowiki>[short 2]</nowiki>]

* ''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)

; [[Francium]]

* [https://www.youtube.com/watch?v=3Ty3FpmAdGA ''Happy Birthday in 27edo''] (2025)

* [https://www.youtube.com/watch?v=Wfg2gWW9qZg ''~~Router~~-~~Pseudoscientist~~''] (~~2025~~)

* "Router-Pseudoscientist" from ''TOTMC 2025'' (2025) – [https://open.spotify.com/track/5qrXYuhz3XOEaUyFvP4ldp Spotify] | [https://francium223.bandcamp.com/track/router-pseudoscientist Bandcamp] | [https://www.youtube.com/watch?v=Wfg2gWW9qZg YouTube]

* [https://www.youtube.com/watch?v=hY0zo6MqQtU ''Waltz No. 11 in A flat major''] (2026)

* [https://www.youtube.com/watch?v=wY43YLa17s4 ''Plane Sonatina No. 4''] (2026)

; [[groundfault]]

* From ''A New Dusk'' (2024) – [https://groundfco.bandcamp.com/album/a-new-dusk Bandcamp] | [https://www.youtube.com/watch?v=1bnEO8vGvbo YouTube]

** "Back Stalk"

** "Superior Intermedial" – in part, the rest being in 31edo

** "Revelation of Your Forever"

* "Sakura Blade Minivan", from ''Souvenirs of the Affliction'' (2025) – [https://groundfco.bandcamp.com/track/sakura-blade-minivan-27-35edo-2 Bandcamp] | [https://www.youtube.com/watch?v=rrjuGmmodn0&t=1436 YouTube (23:56–27:58)] – in part, the rest being in 35edo

; [[Igliashon Jones]]

Line 1,467:

Line 1,485:

* "Funkrotonal" from ''Microtonal Allsorts'' (2023) – [https://open.spotify.com/track/1zjNkbm8kIkuCxtodyFCL0 Spotify] | [https://xotla.bandcamp.com/track/funkrotonal-27edo Bandcamp] | [https://www.youtube.com/watch?v=7gt1BBJuJsE YouTube]

~~[[Category:Augene]]~~

[[Category:Listen]]

[[Category:Augmented]]

[[Category:Sensi]]

[[Category:Superpyth]]

[[Category:Tetracot]]

[[Category:Twentuning]]

@@ Line 11: / Line 11: @@
 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.
@@ Line 539: / Line 539: @@
 |+ style="font-size: 105%;" | Circle of fifths in 27edo
 |- style="white-space: nowrap;"
-!Cents
+! Cents
-! colspan="2" | Extended<br />Pythagorean<br />notation
+! colspan="2" | Extended<br>Pythagorean<br>notation
-! colspan="2" | Quartertone<br />notation
+! colspan="2" | Quartertone<br>notation
 |-
 | 0.0
@@ Line 671: / Line 671: @@
 Using standard [[chain-of-fifths notation]], a sharp (an augmented unison) raises a note by 4 edosteps, just one edostep beneath the following nominal, and the flat conversely lowers. The sharp is quite wide at about 178¢, sounding like a narrow major 2nd. C to C♯ describes the approximate 10/9 and 11/10 interval. An accidental can be divided in half, and the remaining places can then be filled in with half-sharps, half-flats, sesquisharps, and sesquiflats, reducing the need for double sharps and double flats. The half-sharp is notated as a quartertone, but at about 89¢ it sounds more like a narrow semitone. The gamut from C to D is C, D♭, C{{demisharp2}}, D{{demiflat2}}, C♯, and D, with many ascending intervals appearing to be descending on the staff.
-=== Ups and downs notation ===
+=== Stein–Zimmermann–Gould notation ===
-edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
+[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
+{{Sharpness-sharp4-szg}}
+=== Kite's ups and downs notation ===
+edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
 {{Ups and downs sharpness}}
-[[Alternative symbols for ups and downs notation|Alternatively,]] sharps and flats with arrows can be used, borrowed from extended [[Helmholtz–Ellis notation]]:
-{{Sharpness-sharp4}}
 === Sagittal notation ===
@@ Line 1,022: / Line 1,024: @@
 | 3
 | 2\27
-| [[Augene]] (27e) / Eugene (27)
+| [[Augene]] (27e) / eugene (27)
 | [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]]
 |-
@@ Line 1,032: / Line 1,034: @@
 | 9
 | 1\27
-| [[Niner]] (27e)<br>[[Ennealimmal]] (out of tune)
+| [[Niner]] (27e)
 | [[9L 9s]]
 |}
+In addition, 27edo can be used as a detempering target for [[ennealimmal]].
 === Commas ===
@@ Line 1,404: / Line 1,408: @@
 ; [[Brendan Byrnes]]
 * [https://www.youtube.com/watch?v=sWaqlAgSWcc ''Sunspots''] (2022)
+* ''27 EDO Etude'' (2022)
+** [https://brendanbyrnes.bandcamp.com/track/27-edo-etude on Bandcamp]
+** [https://m.youtube.com/watch?v=Lml2cfJW9QI on YouTube] (with sheet music)
 * [https://www.youtube.com/watch?v=lywpWPBYQi0 ''Istril Bloom''] (2025)
@@ Line 1,413: / Line 1,420: @@
 * [https://www.youtube.com/shorts/FSPUebavRCQ ''27edo waltz''] (2025)
 * [https://www.youtube.com/shorts/izpEen38Sps ''27edo improv''] (2025)
-* ''Flies Control My Pain - 27edo'' (2026) [https://www.youtube.com/shorts/sKnjDPEOQtc <nowiki>[short 1]</nowiki>]; [https://www.youtube.com/shorts/QEebNJkcIlE <nowiki>[short 2]</nowiki>]
+* ''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)
 ; [[Francium]]
 * [https://www.youtube.com/watch?v=3Ty3FpmAdGA ''Happy Birthday in 27edo''] (2025)
-* [https://www.youtube.com/watch?v=Wfg2gWW9qZg ''Router-Pseudoscientist''] (2025)
+* "Router-Pseudoscientist" from ''TOTMC 2025'' (2025) – [https://open.spotify.com/track/5qrXYuhz3XOEaUyFvP4ldp Spotify] | [https://francium223.bandcamp.com/track/router-pseudoscientist Bandcamp] | [https://www.youtube.com/watch?v=Wfg2gWW9qZg YouTube]
+* [https://www.youtube.com/watch?v=hY0zo6MqQtU ''Waltz No. 11 in A flat major''] (2026)
+* [https://www.youtube.com/watch?v=wY43YLa17s4 ''Plane Sonatina No. 4''] (2026)
+; [[groundfault]]
+* From ''A New Dusk'' (2024) – [https://groundfco.bandcamp.com/album/a-new-dusk Bandcamp] | [https://www.youtube.com/watch?v=1bnEO8vGvbo YouTube]
+** "Back Stalk"
+** "Superior Intermedial" – in part, the rest being in 31edo
+** "Revelation of Your Forever"
+* "Sakura Blade Minivan", from ''Souvenirs of the Affliction'' (2025) – [https://groundfco.bandcamp.com/track/sakura-blade-minivan-27-35edo-2 Bandcamp] | [https://www.youtube.com/watch?v=rrjuGmmodn0&t=1436 YouTube (23:56–27:58)] – in part, the rest being in 35edo
 ; [[Igliashon Jones]]
@@ Line 1,467: / Line 1,485: @@
 * "Funkrotonal" from ''Microtonal Allsorts'' (2023) – [https://open.spotify.com/track/1zjNkbm8kIkuCxtodyFCL0 Spotify] | [https://xotla.bandcamp.com/track/funkrotonal-27edo Bandcamp] | [https://www.youtube.com/watch?v=7gt1BBJuJsE YouTube]
-[[Category:Augene]]
 [[Category:Listen]]
+[[Category:Augmented]]
 [[Category:Sensi]]
 [[Category:Superpyth]]
 [[Category:Tetracot]]
 [[Category:Twentuning]]