27edo: Difference between revisions

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== Theory ==

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 ~~cents~~, and the major third is the same 400-cent major third as [[12edo]], sharp by 13.7 ~~cents~~. 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]], where three fifths in 19edo reach a near-perfect [[6/5]] and [[5/3]] and three fifths in 27edo reaching a near-perfect [[7/6]] and [[12/7]].

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]], where three fifths in 19edo reach a near-perfect [[6/5]] and [[5/3]] and three fifths in 27edo reaching a near-perfect [[7/6]] and [[12/7]].

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. 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 and 5:7:9, making 27 the smallest edo that can simulate tritave harmony, although it rapidly becomes rough if extended to the 11 and above, unlike a true tritave based system.

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. 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 and 5:7:9, making 27 the smallest edo that can simulate tritave harmony, although it rapidly becomes rough if extended to the 11 and above, unlike a true tritave based system.

27edo, with its 400~~-cent~~ major third, [[tempering out|tempers out]] the lesser diesis, [[128/125]], and the septimal comma, [[64/63]], and hence [[126/125]] as well. These it shares with 12edo, making some relationships familiar, and they both support the [[augene]] temperament. It shares with [[22edo]] tempering out the sensamagic comma [[245/243]] as well as 64/63, so that they both support the [[superpyth]] temperament, with four quite sharp "superpythagorean" fifths giving a sharp [[9/7]] in place of meantone's 5/4.

27edo, with its 400{{c}} major third, [[tempering out|tempers out]] the lesser diesis, [[128/125]], and the septimal comma, [[64/63]], and hence [[126/125]] as well. These it shares with 12edo, making some relationships familiar, and they both support the [[augene]] temperament. It shares with [[22edo]] tempering out the sensamagic comma [[245/243]] as well as 64/63, so that they both support the [[superpyth]] temperament, with four quite sharp "superpythagorean" fifths giving a sharp [[9/7]] in place of meantone's 5/4.

Its step, as well as the octave-inverted and octave-equivalent versions of it, ~~has some of~~ the ~~highest~~ [[harmonic entropy]] ~~possible and thus~~ is~~, in theory, one~~ of the most dissonant intervals ~~possible, assuming the relatively common values of {{nowrap|''a'' {{=}} 2}} and {{nowrap|''s'' {{=}} 1%}}~~. This property is shared with all edos between around 24 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.

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.

The [[chromatic semitone]] of 27edo, at 178{{c}}, is equal to a submajor second in size, meaning 27edo is a candidate for [[extraclassical tonality]] due to its sharp major third of 444 cents.

=== Odd harmonics ===

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=== Octave stretch ===

Since the harmonics whose intervals it approximates well (3, 5, 7, 13, and 19) are all tuned sharp of just, 27edo is a prime candidate for [[stretched and compressed tuning|octave compression]]. The local zeta peak around 27 is at 27.086614, which corresponds to a step size of 44.3023 ~~cents~~. More generally, narrowing the steps to between 44.2 and 44.35 ~~cents~~ would be better in theory; [[43edt]], [[70ed6]], and [[97ed12]] are good options if octave compression is acceptable, and these narrow the octaves by 5.75, 3.53, and 2.55 ~~cents~~, respectively.

Since the harmonics whose intervals it approximates well (3, 5, 7, 13, and 19) are all tuned sharp of just, 27edo is a prime candidate for [[stretched and compressed tuning|octave compression]]. The local zeta peak around 27 is at 27.086614, which corresponds to a step size of 44.3023{{c}}. More generally, narrowing the steps to between 44.2 and 44.35{{c}} would be better in theory; [[43edt]], [[70ed6]], [[90ed10]], and [[97ed12]] are good options if octave compression is acceptable, and these narrow the octaves by 5.75, 3.53, 4.11, and 2.55{{c}}, respectively.

=== Subsets and supersets ===

Since 27 factors into primes as 33, 27edo contains [[3edo]] and [[9edo]] as subsets.

Since 27 factors into primes as 33, 27edo contains [[3edo]] and [[9edo]] as subsets. Multiplying it by 3 gives [[81edo]], which is a good [[meantone]] tuning.

== Intervals ==

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! Cents

! Approximate ratios<ref group="note">{{sg|27et|limit=2.3.5.7.13.19-[[subgroup]]}}</ref>

! colspan="3" | [[Ups and downs notation]]

! colspan="3" | [[Ups and downs notation]] ([[Enharmonic unisons in ups and downs notation|EUs]]: v4A1 and vm2)

! [[Interval region]]s

! colspan="2" | [[Solfege]]s

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| do

|}

=== Interval quality and chord names in color notation ===

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| rowspan="2" | minor

| zo

| {a, b, 0, 1}

| {{monzo| a, b, 0, 1 }}

| 7/6, 7/4

|-

| fourthward wa

| {a, b}, b < −1

| {{monzo| a, b }}, {{nowrap|b < −1}}

| 32/27, 16/9

|-

| upminor

| gu

| {a, b, -1}

| {{monzo| a, b, −1 }}

| 6/5, 9/5

|-

| rowspan="2" | mid

| tho

| {a, b, 0, 0, 0, 1}

| {{monzo| a, b, 0, 0, 0, 1 }}

| 13/12, 13/8

|-

| thu

| {a, b, 0, 0, 0, -1}

| {{monzo| a, b, 0, 0, 0, −1 }}

| 16/13, 24/13

|-

| downmajor

| yo

| {a, b, 1}

| {{monzo| a, b, 1 }}

| 5/4, 5/3

|-

| rowspan="2" | major

| fifthward wa

| {a, b}, b > 1

| {{monzo| a, b }}, {{nowrap|b > 1}}

| 9/8, 27/16

|-

| ru

| {a, b, 0, -1}

| {{monzo| a, b, 0, −1 }}

| 9/7, 12/7

|}

All 27edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, ilo, yo and ru triads:

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

!Cents

! colspan="2" |Extended

! colspan="2" | Extended Pythagorean notation

~~pythagorean~~ notation

! colspan="2" | Quartertone notation

! colspan="2" |Quartertone notation

|-

|0.0

| 0.0

| colspan="2" |C

| colspan="2" | C

| colspan="2" |A{{sesquisharp2}}

| colspan="2" | A{{sesquisharp2}}

|-

|711.1

| 711.1

| colspan="2" |G

| colspan="2" | G

| colspan="2" |E{{sesquisharp2}}

| colspan="2" | E{{sesquisharp2}}

|-

|222.2

| 222.2

| colspan="2" |D

| colspan="2" | D

|B{{sesquisharp2}}

| B{{sesquisharp2}}

|F{{sesquiflat2}}

| F{{sesquiflat2}}

|-

|933.3

| 933.3

| colspan="2" |A

| colspan="2" | A

| colspan="2" |C{{sesquiflat2}}

| colspan="2" | C{{sesquiflat2}}

|-

|444.4

| 444.4

| colspan="2" |E

| colspan="2" | E

| colspan="2" |G{{sesquiflat2}}

| colspan="2" | G{{sesquiflat2}}

|-

|1155.6

| 1155.6

| colspan="2" |B

| colspan="2" | B

| colspan="2" |D{{sesquiflat2}}

| colspan="2" | D{{sesquiflat2}}

|-

|666.7

| 666.7

| colspan="2" |F♯

| colspan="2" | F♯

| colspan="2" |A{{sesquiflat2}}

| colspan="2" | A{{sesquiflat2}}

|-

|177.8

| 177.8

| colspan="2" |C♯

| colspan="2" | C♯

| colspan="2" |E{{sesquiflat2}}

| colspan="2" | E{{sesquiflat2}}

|-

|888.9

| 888.9

| colspan="2" |G♯

| colspan="2" | G♯

| colspan="2" |B{{sesquiflat2}}

| colspan="2" | B{{sesquiflat2}}

|-

|400.0

| 400.0

| colspan="2" |D♯

| colspan="2" | D♯

| colspan="2" |F{{demiflat2}}

| colspan="2" | F{{demiflat2}}

|-

|1111.1

| 1111.1

| colspan="2" |A♯

| colspan="2" | A♯

| colspan="2" |C{{demiflat2}}

| colspan="2" | C{{demiflat2}}

|-

|622.2

| 622.2

| colspan="2" |E♯

| colspan="2" | E♯

| colspan="2" |G{{demiflat2}}

| colspan="2" | G{{demiflat2}}

|-

|133.3

| 133.3

|B♯

| B♯

|F𝄫

| F𝄫

| colspan="2" |D{{demiflat2}}

| colspan="2" | D{{demiflat2}}

|-

|844.4

| 844.4

|F𝄪

| F𝄪

|C𝄫

| C𝄫

| colspan="2" |A{{demiflat2}}

| colspan="2" | A{{demiflat2}}

|-

|355.6

| 355.6

|C𝄪

| C𝄪

|G𝄫

| G𝄫

| colspan="2" |E{{demiflat2}}

| colspan="2" | E{{demiflat2}}

|-

|1066.7

| 1066.7

|G𝄪

| G𝄪

|D𝄫

| D𝄫

| colspan="2" |B{{demiflat2}}

| colspan="2" | B{{demiflat2}}

|-

|577.8

| 577.8

|D𝄪

| D𝄪

|A𝄫

| A𝄫

| colspan="2" |F{{demisharp2}}

| colspan="2" | F{{demisharp2}}

|-

|88.9

| 88.9

|A𝄪

| A𝄪

|E𝄫

| E𝄫

| colspan="2" |C{{demisharp2}}

| colspan="2" | C{{demisharp2}}

|-

|800.0

| 800.0

|E𝄪

| E𝄪

|B𝄫

| B𝄫

| colspan="2" |G{{demisharp2}}

| colspan="2" | G{{demisharp2}}

|-

|311.1

| 311.1

|B𝄪

| B𝄪

|F♭

| F♭

| colspan="2" |D{{demisharp2}}

| colspan="2" | D{{demisharp2}}

|-

|1022.2

| 1022.2

| colspan="2" |C♭

| colspan="2" | C♭

| colspan="2" |A{{demisharp2}}

| colspan="2" | A{{demisharp2}}

|-

|533.3

| 533.3

| colspan="2" |G♭

| colspan="2" | G♭

| colspan="2" |E{{demisharp2}}

| colspan="2" | E{{demisharp2}}

|-

|44.4

| 44.4

| colspan="2" |D♭

| colspan="2" | D♭

| colspan="2" |B{{demisharp2}}

| colspan="2" | B{{demisharp2}}

|-

|755.6

| 755.6

| colspan="2" |A♭

| colspan="2" | A♭

| colspan="2" |F{{sesquisharp2}}

| colspan="2" | F{{sesquisharp2}}

|-

|266.7

| 266.7

| colspan="2" |E♭

| colspan="2" | E♭

| colspan="2" |C{{sesquisharp2}}

| colspan="2" | C{{sesquisharp2}}

|-

|977.8

| 977.8

| colspan="2" |B♭

| colspan="2" | B♭

| colspan="2" |G{{sesquisharp2}}

| colspan="2" | G{{sesquisharp2}}

|-

|488.9

| 488.9

| colspan="2" |F

| colspan="2" | F

| colspan="2" |D~~{{sesquisharp2}}~~

| colspan="2" | D{{sesquisharp2}}

|-

~~|0.0~~

~~| colspan="2" |C~~

~~| colspan="2" |A~~{{sesquisharp2}}

|-

| 0.0

| colspan="2" | C

| colspan="2" | A{{sesquisharp2}}

|}

=== Extended ~~pythagorean~~ notation ===

=== Extended Pythagorean notation ===

27edo being a superpythagorean system, the 5/4 major third present in the 4:5:6 chord is technically an augmented second, since (for example) C–E is a 9/7 supermajor third and so the note located 5/4 above C must be notated as D♯. Conversely, the 6/5 minor third of a 10:12:15 chord is actually reached by a diminished fourth, since (for example) D–F is a 7/6 subminor third and so the note located 6/5 above D must be notated as G♭. The diminished 2nd is a descending interval, thus A♯ is higher than B♭. Though here very exaggerated, this should be familiar to those working with the Pythagorean scale (see [[53edo]]), and also to many classically trained violinists.

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===Ups and downs notation===

~~34edo~~ 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.

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.

[[Alternative symbols for ups and downs notation]] ~~uses~~ sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:

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

=== Sagittal notation ===

This notation is a subset of the notation for [[54edo #Sagittal notation|54edo]].

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=== 6L 1s (archeotonic) notation ===

The notation of Tetracot[7]. The generator is the perfect 2nd. Notes are denoted as {{nowrap|LLLLLLs {{=}} CDEFGABC}}, and raising and lowering by a chroma {{nowrap|(L − s)}}, 1 edostep in this instance, is denoted by ♯ and ♭.

The notation of Tetracot[7]. The generator is the perfect 2nd. Notes are denoted as {{nowrap|LLLLLLs {{=}} CDEFGABC}}, and raising and lowering by a chroma ({{nowrap|L − s}}), 1 edostep in this instance, is denoted by ♯ and ♭.

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"

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{{Q-odd-limit intervals|27.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 27e val mapping}}

~~=== Zeta peak index ===~~

~~{| class="wikitable center-all"~~

|-

~~! colspan="3" | Tuning~~

~~! colspan="3" | Strength~~

~~! colspan="2" | Closest edo~~

~~! colspan="2" | Integer limit~~

|-

~~! ZPI~~

~~! Steps per octave~~

~~! Step size (cents)~~

~~! Height~~

~~! Integral~~

~~! Gap~~

~~! Edo~~

~~! Octave (cents)~~

~~! Consistent~~

~~! Distinct~~

|-

~~| [[106zpi]]~~

~~| 27.0866140827635~~

~~| 44.3023257293579~~

~~| 6.069233~~

~~| 1.185939~~

~~| 16.215619~~

~~| 27edo~~

~~| 1196.16279469266~~

~~| 10~~

~~| 8~~

|}

== Regular temperament properties ==

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| 64/63, 126/125, 245/243

| {{mapping| 27 43 63 76 }}

| −3.70

| −3.71

| 2.39

| 5.40

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* 27et (27eg val) is lower in relative error than any previous equal temperaments in the 13-, 17-, and 19-limit. The next equal temperaments doing better in those subgroups are [[31edo|31]], 31, and [[46edo|46]], respectively.

* 27et is particularly strong in the 2.3.5.7.13.19 subgroup. The next equal temperament that does better in this subgroup is [[53edo|53]].

=== Uniform maps ===

=== Rank-2 temperaments ===

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| 19th-partial chroma

|}

== Scales ==

=== MOS scales ===

* Superpyth ~~pentatonic~~ – Superpyth[5] [[2L 3s]] (gen = 11\27): 5 5 6 5 6

* Superpyth pentic – Superpyth[5] [[2L 3s]] (gen = 11\27): 5 5 6 5 6

* Superpyth diatonic – Superpyth[7] [[5L 2s]] (gen = 11\27): 5 5 1 5 5 5 1

* Superpyth chromatic – Superpyth[12] [[5L 7s]] (gen = 11\27): 4 1 1 4 1 4 1 4 1 1 4 1

* Superpyth ~~hyperchromatic~~ – Superpyth[17] [[5L 12s]] (gen = 11\27): 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1

* Superpyth enharmonic – Superpyth[17] [[5L 12s]] (gen = 11\27): 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1

* Augene[6] [[3L 3s]] (period = 9\27, gen = 2\27): 7 2 7 2 7 2

* Augene[9] [[3L 6s]] (period = 9\27, gen = 2\27): 5 2 2 5 2 2 5 2 2

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* [[SNS (2/1, 3/2, 5/4)-7|5-limit / pental double harmonic major]]: 2 7 2 5 2 7 2

* enharmonic tetrachord octave species: 9 1 1 5 9 1 1, 1 9 1 5 1 9 1 (also Superpyth double harmonic major), 1 1 9 5 1 1 9

* [[The Pinetone System#The Pinetone diatonic|Pinetone diatonic]]: 4 3 4 5 4 3 4

* [[The Pinetone System #The Pinetone diatonic|Pinetone diatonic]]: 4 3 4 5 4 3 4

* [[The Pinetone System#Pinetone octatonic scales|Pinetone major-harmonic octatonic]]: 4 3 4 2 3 4 3 4

* [[The Pinetone System #Pinetone octatonic scales|Pinetone major-harmonic octatonic]]: 4 3 4 2 3 4 3 4

* [[The Pinetone System#Pinetone octatonic scales|Pinetone minor-harmonic octatonic]]: 4 3 2 4 3 4 4 3

* [[The Pinetone System #Pinetone octatonic scales|Pinetone minor-harmonic octatonic]]: 4 3 2 4 3 4 4 3

* [[The Pinetone System#Pinetone diminished octatonic|Pinetone diminished octatonic]] / [[Porcusmine]]: 3 4 2 4 3 4 3 4

* [[The Pinetone System #Pinetone diminished octatonic|Pinetone diminished octatonic]] / [[Porcusmine]]: 3 4 2 4 3 4 3 4

* [[The Pinetone System#Pinetone harmonic diminished octatonic|Pinetone harmonic diminished]]: 3 4 2 5 2 4 3 4

* [[The Pinetone System #Pinetone harmonic diminished octatonic|Pinetone harmonic diminished]]: 3 4 2 5 2 4 3 4

* [[The Pinetone System#Pinetone chromatic|Pinetone chromatic]] / pinechrome: 1 3 3 1 3 2 3 1 3 3 1 3

* [[The Pinetone System #Pinetone chromatic|Pinetone chromatic]] / pinechrome: 1 3 3 1 3 2 3 1 3 3 1 3

* 5-limit / pental double harmonic nonatonic (subset of Augene[12]): 2 5 2 2 5 2 5 2 2, 2 2 5 2 5 2 2 5 2 (Augene[9] [[4M]])

* 5-limit / pental double harmonic decatonic (subset of Augene[12]): 2 5 2 2 3 2 2 5 2 2

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* [[Blackdye]] / [[syntonic dipentatonic]] (superset of [[Zarlino]]): 1 4 2 4 1 4 2 4 1 4

* [[Blackville]] / [[SNS ((2/1, 3/2)-5, 16/15)-10|5-limit dipentatonic]] (superset of [[Zarlino]]): 3 2 4 2 3 2 4 2 3 2

* Direct sunlight (original/default tuning; subset of [[Sensi]][19]): 1 2 8 5 1 9 1 ((1, 3, 11, 16, 17, 26, 27)\27)

Direct sunlight (~~this is its~~ original/default tuning; subset of [[Sensi]][19])

* Hypersakura (original/default tuning; subset of Sensi[19]): 1 10 5 1 10 ((1 11 16 17 27)\27)

* ~~44.444~~

* 133.333

* 488.889

* 711.111

* 755.555

* 1155.555

* 1200.000

Hypersakura (~~this is its~~ original/default tuning; subset of Sensi[19])

* 44.444

* 488.889

* 711.111

* 755.555

* 1200.000

== Instruments ==

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; [[Bryan Deister]]

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

; [[Francium]]

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

; [[Igliashon Jones]]

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; [[Xotla]]

* "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]

~~== Notes ==~~

~~<references group="note" />~~

[[Category:Augene]]

@@ Line 3: / Line 3: @@
 == Theory ==
-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 cents, and the major third is the same 400-cent major third as [[12edo]], sharp by 13.7 cents. 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]], where three fifths in 19edo reach a near-perfect [[6/5]] and [[5/3]] and three fifths in 27edo reaching a near-perfect [[7/6]] and [[12/7]].
+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]], where three fifths in 19edo reach a near-perfect [[6/5]] and [[5/3]] and three fifths in 27edo reaching a near-perfect [[7/6]] and [[12/7]].
-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. 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 and 5:7:9, making 27 the smallest edo that can simulate tritave harmony, although it rapidly becomes rough if extended to the 11 and above, unlike a true tritave based system.
+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. 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 and 5:7:9, making 27 the smallest edo that can simulate tritave harmony, although it rapidly becomes rough if extended to the 11 and above, unlike a true tritave based system.
-edo, with its 400-cent major third, [[tempering out|tempers out]] the lesser diesis, [[128/125]], and the septimal comma, [[64/63]], and hence [[126/125]] as well. These it shares with 12edo, making some relationships familiar, and they both support the [[augene]] temperament. It shares with [[22edo]] tempering out the sensamagic comma [[245/243]] as well as 64/63, so that they both support the [[superpyth]] temperament, with four quite sharp "superpythagorean" fifths giving a sharp [[9/7]] in place of meantone's 5/4.
+edo, with its 400{{c}} major third, [[tempering out|tempers out]] the lesser diesis, [[128/125]], and the septimal comma, [[64/63]], and hence [[126/125]] as well. These it shares with 12edo, making some relationships familiar, and they both support the [[augene]] temperament. It shares with [[22edo]] tempering out the sensamagic comma [[245/243]] as well as 64/63, so that they both support the [[superpyth]] temperament, with four quite sharp "superpythagorean" fifths giving a sharp [[9/7]] in place of meantone's 5/4.
-Its step, as well as the octave-inverted and octave-equivalent versions of it, has some of the highest [[harmonic entropy]] possible and thus is, in theory, one of the most dissonant intervals possible, assuming the relatively common values of {{nowrap|''a'' {{=}} 2}} and {{nowrap|''s'' {{=}} 1%}}. This property is shared with all edos between around 24 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.
+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.
+The [[chromatic semitone]] of 27edo, at 178{{c}}, is equal to a submajor second in size, meaning 27edo is a candidate for [[extraclassical tonality]] due to its sharp major third of 444 cents.
 === Odd harmonics ===
@@ Line 15: / Line 17: @@
 === Octave stretch ===
-Since the harmonics whose intervals it approximates well (3, 5, 7, 13, and 19) are all tuned sharp of just, 27edo is a prime candidate for [[stretched and compressed tuning|octave compression]]. The local zeta peak around 27 is at 27.086614, which corresponds to a step size of 44.3023 cents. More generally, narrowing the steps to between 44.2 and 44.35 cents would be better in theory; [[43edt]], [[70ed6]], and [[97ed12]] are good options if octave compression is acceptable, and these narrow the octaves by 5.75, 3.53, and 2.55 cents, respectively.
+Since the harmonics whose intervals it approximates well (3, 5, 7, 13, and 19) are all tuned sharp of just, 27edo is a prime candidate for [[stretched and compressed tuning|octave compression]]. The local zeta peak around 27 is at 27.086614, which corresponds to a step size of 44.3023{{c}}. More generally, narrowing the steps to between 44.2 and 44.35{{c}} would be better in theory; [[43edt]], [[70ed6]], [[90ed10]], and [[97ed12]] are good options if octave compression is acceptable, and these narrow the octaves by 5.75, 3.53, 4.11, and 2.55{{c}}, respectively.
 === Subsets and supersets ===
-Since 27 factors into primes as 3<sup>3</sup>, 27edo contains [[3edo]] and [[9edo]] as subsets.
+Since 27 factors into primes as 3<sup>3</sup>, 27edo contains [[3edo]] and [[9edo]] as subsets.  Multiplying it by 3 gives [[81edo]], which is a good [[meantone]] tuning.
 == Intervals ==
@@ Line 26: / Line 28: @@
 ! Cents
 ! Approximate ratios<ref group="note">{{sg|27et|limit=2.3.5.7.13.19-[[subgroup]]}}</ref>
-! colspan="3" | [[Ups and downs notation]]
+! colspan="3" | [[Ups and downs notation]] ([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>4</sup>A1 and vm2)
 ! [[Interval region]]s
 ! colspan="2" | [[Solfege]]s
@@ Line 310: / Line 312: @@
 | do
 |}
+<references group="note" />
 === Interval quality and chord names in color notation ===
@@ Line 323: / Line 326: @@
 | rowspan="2" | minor
 | zo
-| {a, b, 0, 1}
+| {{monzo| a, b, 0, 1 }}
 | 7/6, 7/4
 |-
 | fourthward wa
-| {a, b}, b < −1
+| {{monzo| a, b }}, {{nowrap|b &lt; −1}}
 | 32/27, 16/9
 |-
 | upminor
 | gu
-| {a, b, -1}
+| {{monzo| a, b, −1 }}
 | 6/5, 9/5
 |-
 | rowspan="2" | mid
 | tho
-| {a, b, 0, 0, 0, 1}
+| {{monzo| a, b, 0, 0, 0, 1 }}
 | 13/12, 13/8
 |-
 | thu
-| {a, b, 0, 0, 0, -1}
+| {{monzo| a, b, 0, 0, 0, −1 }}
 | 16/13, 24/13
 |-
 | downmajor
 | yo
-| {a, b, 1}
+| {{monzo| a, b, 1 }}
 | 5/4, 5/3
 |-
 | rowspan="2" | major
 | fifthward wa
-| {a, b}, b > 1
+| {{monzo| a, b }}, {{nowrap|b &gt; 1}}
 | 9/8, 27/16
 |-
 | ru
-| {a, b, 0, -1}
+| {{monzo| a, b, 0, −1 }}
 | 9/7, 12/7
 |}
 All 27edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. An up or down after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, ilo, yo and ru triads:
@@ Line 411: / Line 415: @@
 |- style="white-space: nowrap;"
 !Cents
-! colspan="2" |Extended
+! colspan="2" | Extended<br />Pythagorean<br />notation
-pythagorean<br>notation
+! colspan="2" | Quartertone<br />notation
-! colspan="2" |Quartertone<br>notation
 |-
-|0.0
+| 0.0
-| colspan="2" |C
+| colspan="2" | C
-| colspan="2" |A{{sesquisharp2}}
+| colspan="2" | A{{sesquisharp2}}
 |-
-|711.1
+| 711.1
-| colspan="2" |G
+| colspan="2" | G
-| colspan="2" |E{{sesquisharp2}}
+| colspan="2" | E{{sesquisharp2}}
 |-
-|222.2
+| 222.2
-| colspan="2" |D
+| colspan="2" | D
-|B{{sesquisharp2}}
+| B{{sesquisharp2}}
-|F{{sesquiflat2}}
+| F{{sesquiflat2}}
 |-
-|933.3
+| 933.3
-| colspan="2" |A
+| colspan="2" | A
-| colspan="2" |C{{sesquiflat2}}
+| colspan="2" | C{{sesquiflat2}}
 |-
-|444.4
+| 444.4
-| colspan="2" |E
+| colspan="2" | E
-| colspan="2" |G{{sesquiflat2}}
+| colspan="2" | G{{sesquiflat2}}
 |-
-|1155.6
+| 1155.6
-| colspan="2" |B
+| colspan="2" | B
-| colspan="2" |D{{sesquiflat2}}
+| colspan="2" | D{{sesquiflat2}}
 |-
-|666.7
+| 666.7
-| colspan="2" |F♯
+| colspan="2" | F♯
-| colspan="2" |A{{sesquiflat2}}
+| colspan="2" | A{{sesquiflat2}}
 |-
-|177.8
+| 177.8
-| colspan="2" |C♯
+| colspan="2" | C♯
-| colspan="2" |E{{sesquiflat2}}
+| colspan="2" | E{{sesquiflat2}}
 |-
-|888.9
+| 888.9
-| colspan="2" |G♯
+| colspan="2" | G♯
-| colspan="2" |B{{sesquiflat2}}
+| colspan="2" | B{{sesquiflat2}}
 |-
-|400.0
+| 400.0
-| colspan="2" |D♯
+| colspan="2" | D♯
-| colspan="2" |F{{demiflat2}}
+| colspan="2" | F{{demiflat2}}
 |-
-|1111.1
+| 1111.1
-| colspan="2" |A♯
+| colspan="2" | A♯
-| colspan="2" |C{{demiflat2}}
+| colspan="2" | C{{demiflat2}}
 |-
-|622.2
+| 622.2
-| colspan="2" |E♯
+| colspan="2" | E♯
-| colspan="2" |G{{demiflat2}}
+| colspan="2" | G{{demiflat2}}
 |-
-|133.3
+| 133.3
-|B♯
+| B♯
-|F𝄫
+| F𝄫
-| colspan="2" |D{{demiflat2}}
+| colspan="2" | D{{demiflat2}}
 |-
-|844.4
+| 844.4
-|F𝄪
+| F𝄪
-|C𝄫
+| C𝄫
-| colspan="2" |A{{demiflat2}}
+| colspan="2" | A{{demiflat2}}
 |-
-|355.6
+| 355.6
-|C𝄪
+| C𝄪
-|G𝄫
+| G𝄫
-| colspan="2" |E{{demiflat2}}
+| colspan="2" | E{{demiflat2}}
 |-
-|1066.7
+| 1066.7
-|G𝄪
+| G𝄪
-|D𝄫
+| D𝄫
-| colspan="2" |B{{demiflat2}}
+| colspan="2" | B{{demiflat2}}
 |-
-|577.8
+| 577.8
-|D𝄪
+| D𝄪
-|A𝄫
+| A𝄫
-| colspan="2" |F{{demisharp2}}
+| colspan="2" | F{{demisharp2}}
 |-
-|88.9
+| 88.9
-|A𝄪
+| A𝄪
-|E𝄫
+| E𝄫
-| colspan="2" |C{{demisharp2}}
+| colspan="2" | C{{demisharp2}}
 |-
-|800.0
+| 800.0
-|E𝄪
+| E𝄪
-|B𝄫
+| B𝄫
-| colspan="2" |G{{demisharp2}}
+| colspan="2" | G{{demisharp2}}
 |-
-|311.1
+| 311.1
-|B𝄪
+| B𝄪
-|F♭
+| F♭
-| colspan="2" |D{{demisharp2}}
+| colspan="2" | D{{demisharp2}}
 |-
-|1022.2
+| 1022.2
-| colspan="2" |C♭
+| colspan="2" | C♭
-| colspan="2" |A{{demisharp2}}
+| colspan="2" | A{{demisharp2}}
 |-
-|533.3
+| 533.3
-| colspan="2" |G♭
+| colspan="2" | G♭
-| colspan="2" |E{{demisharp2}}
+| colspan="2" | E{{demisharp2}}
 |-
-|44.4
+| 44.4
-| colspan="2" |D♭
+| colspan="2" | D♭
-| colspan="2" |B{{demisharp2}}
+| colspan="2" | B{{demisharp2}}
 |-
-|755.6
+| 755.6
-| colspan="2" |A♭
+| colspan="2" | A♭
-| colspan="2" |F{{sesquisharp2}}
+| colspan="2" | F{{sesquisharp2}}
 |-
-|266.7
+| 266.7
-| colspan="2" |E♭
+| colspan="2" | E♭
-| colspan="2" |C{{sesquisharp2}}
+| colspan="2" | C{{sesquisharp2}}
 |-
-|977.8
+| 977.8
-| colspan="2" |B♭
+| colspan="2" | B♭
-| colspan="2" |G{{sesquisharp2}}
+| colspan="2" | G{{sesquisharp2}}
 |-
-|488.9
+| 488.9
-| colspan="2" |F
+| colspan="2" | F
-| colspan="2" |D{{sesquisharp2}}
+| colspan="2" | D{{sesquisharp2}}
-|-
-|0.0
-| colspan="2" |C
-| colspan="2" |A{{sesquisharp2}}
 |-
+| 0.0
+| colspan="2" | C
+| colspan="2" | A{{sesquisharp2}}
 |}
-=== Extended pythagorean notation ===
+=== Extended Pythagorean notation ===
 edo being a superpythagorean system, the 5/4 major third present in the 4:5:6 chord is technically an augmented second, since (for example) C–E is a 9/7 supermajor third and so the note located 5/4 above C must be notated as D♯. Conversely, the 6/5 minor third of a 10:12:15 chord is actually reached by a diminished fourth, since (for example) D–F is a 7/6 subminor third and so the note located 6/5 above D must be notated as G♭. The diminished 2nd is a descending interval, thus A♯ is higher than B♭. Though here very exaggerated, this should be familiar to those working with the Pythagorean scale (see [[53edo]]), and also to many classically trained violinists.
@@ Line 545: / Line 547: @@
 ===Ups and downs 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.
 {{Sharpness-sharp4a}}
-[[Alternative symbols for ups and downs notation]] uses sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
+[[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 ===
 This notation is a subset of the notation for [[54edo #Sagittal notation|54edo]].
@@ Line 591: / Line 594: @@
 === 6L 1s (archeotonic) notation ===
-The notation of Tetracot[7]. The generator is the perfect 2nd. Notes are denoted as {{nowrap|LLLLLLs {{=}} CDEFGABC}}, and raising and lowering by a chroma {{nowrap|(L − s)}}, 1 edostep in this instance, is denoted by ♯ and ♭.
+The notation of Tetracot[7]. The generator is the perfect 2nd. Notes are denoted as {{nowrap|LLLLLLs {{=}} CDEFGABC}}, and raising and lowering by a chroma ({{nowrap|L − s}}), 1 edostep in this instance, is denoted by ♯ and ♭.
 {| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
@@ Line 777: / Line 780: @@
 {{Q-odd-limit intervals|27}}
 {{Q-odd-limit intervals|27.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 27e val mapping}}
-=== Zeta peak index ===
-{| class="wikitable center-all"
-|-
-! colspan="3" | Tuning
-! colspan="3" | Strength
-! colspan="2" | Closest edo
-! colspan="2" | Integer limit
-|-
-! ZPI
-! Steps per octave
-! Step size (cents)
-! Height
-! Integral
-! Gap
-! Edo
-! Octave (cents)
-! Consistent
-! Distinct
-|-
-| [[106zpi]]
-| 27.0866140827635
-| 44.3023257293579
-| 6.069233
-| 1.185939
-| 16.215619
-| 27edo
-| 1196.16279469266
-| 10
-| 8
-|}
 == Regular temperament properties ==
@@ Line 838: / Line 810: @@
 | 64/63, 126/125, 245/243
 | {{mapping| 27 43 63 76 }}
-| −3.70
+| −3.71
 | 2.39
 | 5.40
@@ Line 858: / Line 830: @@
 * 27et (27eg val) is lower in relative error than any previous equal temperaments in the 13-, 17-, and 19-limit. The next equal temperaments doing better in those subgroups are [[31edo|31]], 31, and [[46edo|46]], respectively.
 * 27et is particularly strong in the 2.3.5.7.13.19 subgroup. The next equal temperament that does better in this subgroup is [[53edo|53]].
+=== Uniform maps ===
+{{Uniform map|edo=27}}
 === Rank-2 temperaments ===
@@ Line 1,158: / Line 1,133: @@
 | 19th-partial chroma
 |}
+<references group="note" />
 == Scales ==
 === MOS scales ===
 {{Main|List of MOS scales in 27edo}}
-* Superpyth pentatonic – Superpyth[5] [[2L 3s]] (gen = 11\27): 5 5 6 5 6
+* Superpyth pentic – Superpyth[5] [[2L 3s]] (gen = 11\27): 5 5 6 5 6
 * Superpyth diatonic – Superpyth[7] [[5L 2s]] (gen = 11\27): 5 5 1 5 5 5 1
 * Superpyth chromatic – Superpyth[12] [[5L 7s]] (gen = 11\27): 4 1 1 4 1 4 1 4 1 1 4 1
-* Superpyth hyperchromatic – Superpyth[17] [[5L 12s]] (gen = 11\27): 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1
+* Superpyth enharmonic – Superpyth[17] [[5L 12s]] (gen = 11\27): 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1
 * Augene[6] [[3L 3s]] (period = 9\27, gen = 2\27): 7 2 7 2 7 2
 * Augene[9] [[3L 6s]] (period = 9\27, gen = 2\27): 5 2 2 5 2 2 5 2 2
@@ Line 1,212: / Line 1,188: @@
 * [[SNS (2/1, 3/2, 5/4)-7|5-limit / pental double harmonic major]]: 2 7 2 5 2 7 2
 * enharmonic tetrachord octave species: 9 1 1 5 9 1 1, 1 9 1 5 1 9 1 (also Superpyth double harmonic major), 1 1 9 5 1 1 9
-* [[The Pinetone System#The Pinetone diatonic|Pinetone diatonic]]: 4 3 4 5 4 3 4
+* [[The Pinetone System #The Pinetone diatonic|Pinetone diatonic]]: 4 3 4 5 4 3 4
-* [[The Pinetone System#Pinetone octatonic scales|Pinetone major-harmonic octatonic]]: 4 3 4 2 3 4 3 4
+* [[The Pinetone System #Pinetone octatonic scales|Pinetone major-harmonic octatonic]]: 4 3 4 2 3 4 3 4
-* [[The Pinetone System#Pinetone octatonic scales|Pinetone minor-harmonic octatonic]]: 4 3 2 4 3 4 4 3
+* [[The Pinetone System #Pinetone octatonic scales|Pinetone minor-harmonic octatonic]]: 4 3 2 4 3 4 4 3
-* [[The Pinetone System#Pinetone diminished octatonic|Pinetone diminished octatonic]] / [[Porcusmine]]: 3 4 2 4 3 4 3 4
+* [[The Pinetone System #Pinetone diminished octatonic|Pinetone diminished octatonic]] / [[Porcusmine]]: 3 4 2 4 3 4 3 4
-* [[The Pinetone System#Pinetone harmonic diminished octatonic|Pinetone harmonic diminished]]: 3 4 2 5 2 4 3 4
+* [[The Pinetone System #Pinetone harmonic diminished octatonic|Pinetone harmonic diminished]]: 3 4 2 5 2 4 3 4
-* [[The Pinetone System#Pinetone chromatic|Pinetone chromatic]] / pinechrome: 1 3 3 1 3 2 3 1 3 3 1 3
+* [[The Pinetone System #Pinetone chromatic|Pinetone chromatic]] / pinechrome: 1 3 3 1 3 2 3 1 3 3 1 3
 * 5-limit / pental double harmonic nonatonic (subset of Augene[12]): 2 5 2 2 5 2 5 2 2, 2 2 5 2 5 2 2 5 2 (Augene[9] [[4M]])
 * 5-limit / pental double harmonic decatonic (subset of Augene[12]): 2 5 2 2 3 2 2 5 2 2
@@ Line 1,223: / Line 1,199: @@
 * [[Blackdye]] / [[syntonic dipentatonic]] (superset of [[Zarlino]]): 1 4 2 4 1 4 2 4 1 4
 * [[Blackville]] / [[SNS ((2/1, 3/2)-5, 16/15)-10|5-limit dipentatonic]] (superset of [[Zarlino]]): 3 2 4 2 3 2 4 2 3 2
+* Direct sunlight (original/default tuning; subset of [[Sensi]][19]): 1 2 8 5 1 9 1 ((1, 3, 11, 16, 17, 26, 27)\27)
-Direct sunlight (this is its original/default tuning; subset of [[Sensi]][19])
+* Hypersakura (original/default tuning; subset of Sensi[19]): 1 10 5 1 10 ((1 11 16 17 27)\27)
-* 44.444
-* 133.333
-* 488.889
-* 711.111
-* 755.555
-* 1155.555
-* 1200.000
-Hypersakura (this is its original/default tuning; subset of Sensi[19])
-* 44.444
-* 488.889
-* 711.111
-* 755.555
-* 1200.000
 == Instruments ==
@@ Line 1,272: / Line 1,234: @@
 ; [[Bryan Deister]]
 * [https://www.youtube.com/watch?v=hDP8cfJqWOI ''microtonal improvisation in 27edo''] (2023)
+; [[Francium]]
+* [https://www.youtube.com/watch?v=3Ty3FpmAdGA ''Happy Birthday in 27edo''] (2025)
 ; [[Igliashon Jones]]
@@ Line 1,313: / Line 1,278: @@
 ; [[Xotla]]
 * "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]
-== Notes ==
-<references group="note" />
 [[Category:Augene]]