27edo: Difference between revisions

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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 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]] 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-11s, no-17s 19-limit) temperament. It also approximates [[19/10]], [[19/12]], and [[19/14]], so {{dash|0, 7, 13, 25|~~s=hair|d=~~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 triad, 3:5:7, and a passable approximation of 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]] 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-11s, no-17s 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 triad, 3:5:7, and a passable approximation of 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.

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 ''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.

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.

=== Odd harmonics ===

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== Notation ==

{| class="wikitable center-all floatright"

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

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

|-

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

! rowspan="2" | [[Cent]]s !! colspan="6" | Note from C

|-

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

! colspan="2" | Standard notation !! colspan="2" | Quarter tone notation

|-

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{| class="wikitable center-all right-2 left-3"

|-

! #

! Cents

! Approximate Ratios*

! Approximate Ratios<ref group="note">{{sg|2.3.5.7.13.19 [[subgroup]]}}</ref>

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

! [[Walker Brightness Notation]]

! colspan="2" | [[6L 1s]] Notation

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

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

|-

| 0

Line 499:

| do

|}

* based on treating 27edo as a 2.3.5.7.13.19 subgroup temperament; other approaches are possible.

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

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! rowspan="2" | [[Comma list|Comma List]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve Stretch (¢)

! rowspan="2" | Optimal 8ve Stretch (¢)

! colspan="2" | Tuning Error

|-

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| {{monzo| 43 -27 }}

| [{{val| 27 43 }}]

| −2.89

| 2.88

| 6.50

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| 128/125, 20000/19683

| [{{val| 27 43 63 }}]

| −3.88

| 2.74

| 6.19

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

| [{{val| 27 43 63 76 }}]

| −3.70

| 2.39

| 5.40

Line 637:

Line 636:

| 64/63, 91/90, 126/125, 169/168

| [{{val| 27 43 63 76 100 }}]

| −3.18

| 2.39

| 5.39

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Line 643:

| 64/63, 76/75, 91/90, 126/125, 169/168

| [{{val| 27 43 63 76 100 115 }}]

| −3.18

| 2.18

| 4.92

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{| class="wikitable center-all left-3 left-4"

|-

! Periods per 8ve

! Periods per 8ve

! Generator

! Temperaments

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| 1

| 5\27

| [[Machine]] (27) [[Kumonga]] (27e)

| [[Machine]] (27) [[Kumonga]] (27e)

| [[1L 4s]], [[5L 1s]], [[5L 6s]], [[11L 5s]]

|-

| 1

| 7\27

| [[Myna]] (27e) / coleto (27e) / myno (27) [[Oolong]] (27e)

| [[Myna]] (27e) / coleto (27e) / myno (27) [[Oolong]] (27e)

| [[4L 3s]], [[4L 7s]], [[4L 11s]], [[4L 15s]], [[4L 19s]]

|-

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| 3

| 4\27

| [[Oodako]] (27e) [[Terrain]]

| [[Oodako]] (27e) [[Terrain]]

| [[3L 3s]], [[6L 3s]], [[6L 9s]], [[6L 15s]]

|-

| 9

| 1\27

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

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

| [[9L 9s]]

|}

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{| class="commatable wikitable center-all left-3 right-4 left-6"

|-

! [[Harmonic Limit|Prime Limit]]

! [[Harmonic Limit|Prime Limit]]

! [[Ratio]]<ref group="note">{{rd}}</ref>

! [[Monzo]]

@@ Line 8: / Line 8: @@
 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 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]] 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-11s, no-17s 19-limit) temperament. It also approximates [[19/10]], [[19/12]], and [[19/14]], so {{dash|0, 7, 13, 25|s=hair|d=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 triad, 3:5:7, and a passable approximation of 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]] 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-11s, no-17s 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 triad, 3:5:7, and a passable approximation of 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.
-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 ''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.
+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.
 === Odd harmonics ===
@@ Line 17: / Line 17: @@
 == Notation ==
 {| class="wikitable center-all floatright"
-|+ style="font-size: 105%; white-space: nowrap;" | Circle of fifths in 27edo
+|+ style="font-size: 105%;" | Circle of fifths in 27edo
-|-
+|- style="white-space: nowrap;"
 ! rowspan="2" | [[Cent]]s !! colspan="6" | Note from C
-|-
+|- style="white-space: nowrap;"
 ! colspan="2" | Standard<br />notation !! colspan="2" | Quarter tone<br />notation
 |-
@@ Line 155: / Line 155: @@
 {| class="wikitable center-all right-2 left-3"
 |-
-! #
+! &#35;
 ! Cents
-! Approximate Ratios*
+! Approximate Ratios<ref group="note">{{sg|2.3.5.7.13.19&nbsp;[[subgroup]]}}</ref>
 ! colspan="3" | [[Ups and downs notation|Ups and Downs Notation]]
 ! [[Walker Brightness Notation]]
 ! colspan="2" | [[6L 1s]] Notation
-! colspan="2" |[[Solfege|Solfeges]]
+! colspan="2" | [[Solfege|Solfeges]]
 |-
 | 0
@@ Line 499: / Line 499: @@
 | do
 |}
-* based on treating 27edo as a 2.3.5.7.13.19 subgroup temperament; other approaches are possible.
 === Interval quality and chord names in color notation ===
@@ Line 607: / Line 606: @@
 ! rowspan="2" | [[Comma list|Comma List]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br />8ve Stretch (¢)
 ! colspan="2" | Tuning Error
 |-
@@ Line 616: / Line 615: @@
 | {{monzo| 43 -27 }}
 | [{{val| 27 43 }}]
-| −2.89
+| &minus;2.89
 | 2.88
 | 6.50
@@ Line 623: / Line 622: @@
 | 128/125, 20000/19683
 | [{{val| 27 43 63 }}]
-| −3.88
+| &minus;3.88
 | 2.74
 | 6.19
@@ Line 630: / Line 629: @@
 | 64/63, 126/125, 245/243
 | [{{val| 27 43 63 76 }}]
-| −3.70
+| &minus;3.70
 | 2.39
 | 5.40
@@ Line 637: / Line 636: @@
 | 64/63, 91/90, 126/125, 169/168
 | [{{val| 27 43 63 76 100 }}]
-| −3.18
+| &minus;3.18
 | 2.39
 | 5.39
@@ Line 644: / Line 643: @@
 | 64/63, 76/75, 91/90, 126/125, 169/168
 | [{{val| 27 43 63 76 100 115 }}]
-| −3.18
+| &minus;3.18
 | 2.18
 | 4.92
@@ Line 657: / Line 656: @@
 {| class="wikitable center-all left-3 left-4"
 |-
-! Periods<br>per 8ve
+! Periods<br />per 8ve
 ! Generator
 ! Temperaments
@@ Line 679: / Line 678: @@
 | 1
 | 5\27
-| [[Machine]] (27)<br>[[Kumonga]] (27e)
+| [[Machine]] (27)<br />[[Kumonga]] (27e)
 | [[1L 4s]], [[5L 1s]], [[5L 6s]], [[11L 5s]]
 |-
 | 1
 | 7\27
-| [[Myna]] (27e) / coleto (27e) / myno (27)<br>[[Oolong]] (27e)
+| [[Myna]] (27e) / coleto (27e) / myno (27)<br />[[Oolong]] (27e)
 | [[4L 3s]], [[4L 7s]], [[4L 11s]], [[4L 15s]], [[4L 19s]]
 |-
@@ Line 719: / Line 718: @@
 | 3
 | 4\27
-| [[Oodako]] (27e)<br>[[Terrain]]
+| [[Oodako]] (27e)<br />[[Terrain]]
 | [[3L 3s]], [[6L 3s]], [[6L 9s]], [[6L 15s]]
 |-
 | 9
 | 1\27
-| [[Niner]] (27e)<br>[[Ennealimmal]] (out of tune)
+| [[Niner]] (27e)<br />[[Ennealimmal]] (out of tune)
 | [[9L 9s]]
 |}
@@ Line 733: / Line 732: @@
 {| class="commatable wikitable center-all left-3 right-4 left-6"
 |-
-! [[Harmonic Limit|Prime<br>Limit]]
+! [[Harmonic Limit|Prime<br />Limit]]
 ! [[Ratio]]<ref group="note">{{rd}}</ref>
 ! [[Monzo]]