27edo: Difference between revisions

Line 3:

== Theory ==

27edo divides the [[octave]] in 27 equal parts each exactly 44.444… [[cent]]s in size. However, 27 is a prime candidate for [[octave shrinking]], since harmonics [[3/1|3]], [[5/1|5]], and [[7/1|7]] are all tuned significantly sharp of just, and a step size of between 44.2 and 44.35 cents would be better in theory. The optimal step size for octave-shrinking in the 7-limit is 44.3071 cents, which rougly corresponds to 27.0837-edo. Furthermore, 27edo's local zeta peak is at 27.086614-edo, which corresponds to a step size of 44.3023 cents.

However, assuming pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as [[12edo]], which is 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 1/3 [[septimal comma]] superpyth in the same way that 19edo is audibly indistinguishable from [[1/3 syntonic comma meantone]], resulting in three of them reaching a near perfect minor third and major sixth in both, with 19edo reaching a near-perfect [[6/5]] and 27edo reaching a near-perfect [[7/6]].

27edo, with its 400 cent major third, tempers out the [[lesser diesis~~]] of~~ [[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 allegedly Bohlen-Pierce 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 cent major third, 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 allegedly Bohlen-Pierce 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.

Though the [[7-limit]] tuning of 27edo 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]] 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 temperament. It also approximates [[19/10]], [[19/12]], and [[19/14]], so 0-7-13-25 does quite well as a 10:12:14:19; the major seventh 25\27 is less than a cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen-Pierce triad, 3:5:7, making it the smallest edo that can simulate tritave harmony, although it rapidly becomes quite rough if extended to the 9 and above, unlike a true tritave based system.

Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[harmonic entropy]] possible and thus is, in theory, most dissonant, assuming the relatively common values of ''a'' = 2 and ''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.

The 27-note system or one similar like a well temperament can be notated by a variation on the quartertone accidentals. In this case a sharp raises a note by 4 edosteps, just one edostep beneath the following nominal (for example C to C# describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and this fill the remaining places without need for double sharps and double flats. Enharmonically then, E double flat means C half sharp. In other words, the resemblance to quarter tone notation differs in enharmonic divergence. The notes from C to D are C, D flat, C half-sharp, D half-flat, C sharp, D. Unfortunately, some ascending intervals appear to be descending on the staff. Furthermore, the 3rd of a 4:5:6 or 10:12:15 chord must be notated as either a 2nd or a 4th, since a major third from C is C–D♯ rather than C–E and a minor third from C is now C–F♭ instead of C–E♭. The composer can decide for themselves which addidional accidental pair is necessary if they will need redundancy to remedy these problems, and to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.). Otherwise it is simple enough, and the same tendency for A# to be higher than Bb is not only familiar, though here very exaggerated, to those working with the Pythagorean scale, but also to many classically trained violinists.

=== Odd harmonics ===

=== Notation ===

{| class="wikitable center-all"

! Step offset

| 9

| '''8'''

| 7

| 6

| 5

| '''4'''

| 3

| 2

| 1

| '''0'''

|-

! Sharp symbol

| [[File:Heji33.svg|18px|center]]

| [[File:Heji32.svg|18px|center]]

| [[File:Heji31.svg|18px|center]]

| [[File:HeQu3.svg|20px|center]]

| [[File:Heji26.svg|18px|center]]

| [[File:Heji25.svg|17px|center]]

| [[File:Heji24.svg|17px|center]]

| [[File:HeQu1.svg|14px|center]]

| [[File:Heji19.svg|18px|center]]

| rowspan="2" | [[File:Heji18.svg|15px|center]]

|-

! Flat symbol

| [[File:Heji3.svg|24px|center]]

| [[File:Heji4.svg|24px|center]]

| [[File:Heji5.svg|27px|center]]

| [[File:HeQd3.svg|24px|center]]

| [[File:Heji10.svg|19px|center]]

| [[File:Heji11.svg|15px|center]]

| [[File:Heji12.svg|18px|center]]

| [[File:HeQd1.svg|15px|center]]

| [[File:Heji17.svg|15px|center]]

|}

The 27-note system can be notated using [[ups and downs notation]], in which case arrows or [[Helmholtz-Ellis notation|Hrlmholtz–Ellis]] accidentals can be used, or with a variation on quarter tone accidentals. In this case a sharp raises a note by 4 steps, just one step beneath the following nominal (for example C to C♯ describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and this fill the remaining places without need for double sharps and double flats. Enharmonically then, C[[File:HeQu1.svg|28x28px]] is equal to E𝄫. In other words, the resemblance to quarter tone notation differs in enharmonic divergence. The notes from C to D are C, D♭, C[[File:HeQu1.svg|28x28px]], D[[File:HeQd1.svg|28x28px]], C♯, and D, with some ascending intervals appearing to be descending on the staff.

Furthermore, the major third of a 4:5:6 chord must be notated as an augmented second, since (for example) C–E is a 9/7 supermajor thirds and the note located one major third above C is now D♯. Conversely, the minor third of a 10:12:15 chord must be notated as a diminished fourth, since ((for example) D–F is a 7/6 subminor third and the note located one minor third above D is now G♭. The composer can decide for themselves which addidional accidental pair is necessary if they will need redundancy to remedy these problems, and to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.). Otherwise it is simple enough, and the same tendency for A♯ to be higher than B♭ is not only familiar, though here very exaggerated, to those working with the Pythagorean scale, but also to many classically trained violinists.

== Intervals ==

@@ Line 3: / Line 3: @@
 {{EDO intro|27}}
 == Theory ==
 edo divides the [[octave]] in 27 equal parts each exactly 44.444… [[cent]]s in size. However, 27 is a prime candidate for [[octave shrinking]], since harmonics [[3/1|3]], [[5/1|5]], and [[7/1|7]] are all tuned significantly sharp of just, and a step size of between 44.2 and 44.35 cents would be better in theory. The optimal step size for octave-shrinking in the 7-limit is 44.3071 cents, which rougly corresponds to 27.0837-edo. Furthermore, 27edo's local zeta peak is at 27.086614-edo, which corresponds to a step size of 44.3023 cents.
 However, assuming pure octaves, 27 has a fifth sharp by slightly more than nine cents and a 7/4 sharp by slightly less, and the same 400 cent major third as [[12edo]], which is 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 1/3 [[septimal comma]] superpyth in the same way that 19edo is audibly indistinguishable from [[1/3 syntonic comma meantone]], resulting in three of them reaching a near perfect minor third and major sixth in both, with 19edo reaching a near-perfect [[6/5]] and 27edo reaching a near-perfect [[7/6]].
-edo, with its 400 cent major third, tempers out the [[lesser diesis]] of [[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 allegedly Bohlen-Pierce 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 cent major third, 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 allegedly Bohlen-Pierce 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.
 Though the [[7-limit]] tuning of 27edo 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]] 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 temperament. It also approximates [[19/10]], [[19/12]], and [[19/14]], so 0-7-13-25 does quite well as a 10:12:14:19; the major seventh 25\27 is less than a cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen-Pierce triad, 3:5:7, making it the smallest edo that can simulate tritave harmony, although it rapidly becomes quite rough if extended to the 9 and above, unlike a true tritave based system.
 Its step, as well as the octave-inverted and octave-equivalent versions of it, holds the distinction for having around the highest [[harmonic entropy]] possible and thus is, in theory, most dissonant, assuming the relatively common values of ''a'' = 2 and ''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.
-The 27-note system or one similar like a well temperament can be notated by a variation on the quartertone accidentals. In this case a sharp raises a note by 4 edosteps, just one edostep beneath the following nominal (for example C to C# describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and this fill the remaining places without need for double sharps and double flats. Enharmonically then, E double flat means C half sharp. In other words, the resemblance to quarter tone notation differs in enharmonic divergence. The notes from C to D are C, D flat, C half-sharp, D half-flat, C sharp, D. Unfortunately, some ascending intervals appear to be descending on the staff. Furthermore, the 3rd of a 4:5:6 or 10:12:15 chord must be notated as either a 2nd or a 4th, since a major third from C is C–D♯ rather than C–E and a minor third from C is now C–F♭ instead of C–E♭. The composer can decide for themselves which addidional accidental pair is necessary if they will need redundancy to remedy these problems, and to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.). Otherwise it is simple enough, and the same tendency for A# to be higher than Bb is not only familiar, though here very exaggerated, to those working with the Pythagorean scale, but also to many classically trained violinists.
 === Odd harmonics ===
 {{Harmonics in equal|27}}
+=== Notation ===
+{| class="wikitable center-all"
+! Step offset
+| 9
+| '''8'''
+| 7
+| 6
+| 5
+| '''4'''
+| 3
+| 2
+| 1
+| '''0'''
+|-
+! Sharp symbol
+| [[File:Heji33.svg|18px|center]]
+| [[File:Heji32.svg|18px|center]]
+| [[File:Heji31.svg|18px|center]]
+| [[File:HeQu3.svg|20px|center]]
+| [[File:Heji26.svg|18px|center]]
+| [[File:Heji25.svg|17px|center]]
+| [[File:Heji24.svg|17px|center]]
+| [[File:HeQu1.svg|14px|center]]
+| [[File:Heji19.svg|18px|center]]
+| rowspan="2" | [[File:Heji18.svg|15px|center]]
+|-
+! Flat symbol
+| [[File:Heji3.svg|24px|center]]
+| [[File:Heji4.svg|24px|center]]
+| [[File:Heji5.svg|27px|center]]
+| [[File:HeQd3.svg|24px|center]]
+| [[File:Heji10.svg|19px|center]]
+| [[File:Heji11.svg|15px|center]]
+| [[File:Heji12.svg|18px|center]]
+| [[File:HeQd1.svg|15px|center]]
+| [[File:Heji17.svg|15px|center]]
+|}
+The 27-note system can be notated using [[ups and downs notation]], in which case arrows or [[Helmholtz-Ellis notation|Hrlmholtz–Ellis]] accidentals can be used, or with a variation on quarter tone accidentals. In this case a sharp raises a note by 4 steps, just one step beneath the following nominal (for example C to C♯ describes the approximate 10/9 and 11/10 interval) and the flat conversely lowers: these are augmented unisons and diminished unisons. Just so, one finds that an accidental can be divided in half, and this fill the remaining places without need for double sharps and double flats. Enharmonically then, C[[File:HeQu1.svg|28x28px]] is equal to E𝄫. In other words, the resemblance to quarter tone notation differs in enharmonic divergence. The notes from C to D are C, D♭, C[[File:HeQu1.svg|28x28px]], D[[File:HeQd1.svg|28x28px]], C♯, and D, with some ascending intervals appearing to be descending on the staff.
+Furthermore, the major third of a 4:5:6 chord must be notated as an augmented second, since (for example) C–E is a 9/7 supermajor thirds and the note located one major third above C is now D♯. Conversely, the minor third of a 10:12:15 chord must be notated as a diminished fourth, since ((for example) D–F is a 7/6 subminor third and the note located one minor third above D is now G♭. The composer can decide for themselves which addidional accidental pair is necessary if they will need redundancy to remedy these problems, and to keep the chromatic pitches within a compass on paper relative to the natural names (C, D, E etc.). Otherwise it is simple enough, and the same tendency for A♯ to be higher than B♭ is not only familiar, though here very exaggerated, to those working with the Pythagorean scale, but also to many classically trained violinists.
 == Intervals ==