27edo: Difference between revisions

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.

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.

Line 15:

=== 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]], and [[97ed12]] are good options if octave compression is acceptable, and these narrow the octaves by 5.75, 3.53, and 2.55{{c}}, respectively.

=== Subsets and supersets ===

Line 411:

|- 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.

@@ 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.
@@ Line 15: / Line 15: @@
 === 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]], and [[97ed12]] are good options if octave compression is acceptable, and these narrow the octaves by 5.75, 3.53, and 2.55{{c}}, respectively.
 === Subsets and supersets ===
@@ Line 411: / Line 411: @@
 |- 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.