23edo: Difference between revisions

Line 8:

| Prime factorization = 23

| Subgroup = 2.9.15.21.33.13.17

| Step size = 52.~~174~~

| Step size = 52.174¢

| Fifth type = [[mavila]] 13\23 678.26¢

| Fifth type = [[mavila]] 13\23 = 678.26¢

| Major 2nd = 3\23 = 157¢

| Minor 2nd = 4\23 = 209¢

| Augmented 1sn = -1\23 = -52¢

| Common uses = mavila two sets of distorted 12edo extreme xenharmony in e.g. metal

| Important MOS = mavila 2L5s 4334333 (13\23, 1\1) mavila 7L2s 133313333 (13\23, 1\1) sephiroth 3L4s 2525252 (7\23, 1\1) "semaphore" 5L4s 332323232 (5\23, 1\1)

}}

== Theory ==

23-TET, or 23-EDO, is a tempered musical system which divides the [[octave]] into 23 equal parts of approximately 52.173913 cents, which is also called with the neologism Icositriphony ''(Icositrifonía)''. It has good approximations for [[5/3]], [[11/7]], 13 and 17, allowing it to represent the 2.5/3.11/7.13.17 [[just intonation subgroup]]. If to this subgroup is added the commas of [[17-limit]] [[46edo]], the larger 17-limit [[k*N_subgroups|2*23 subgroup]] 2.9.15.21.33.13.17 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does 17-limit·46edo, and may be regarded as a basis for analyzing the harmony of 23-EDO so far, as approximations to just intervals goes. 23edo is the 9th [[prime numbers|prime]] edo, following [[19edo]] and coming before [[29edo]].

23-EDO was proposed by ethnomusicologist [http://en.wikipedia.org/wiki/Erich_von_Hornbostel Erich von Hornbostel] as the result of continuing a circle of "blown" fifths of ~678-cent fifths that (he argued) resulted from "overblowing" a bamboo pipe.

23-EDO is also significant in that it is the largest EDO that fails to approximate the 3rd, 5th, 7th, and 11th harmonics within 20 cents, which makes it well-suited for musicians seeking to explore harmonic territory that is unusual even for the average microtonalist. Oddly, despite the fact that it fails to approximate these harmonics, it approximates the intervals between them (5/3, 7/3, 11/3, 7/5, 11/7, and 11/5) very well. The lowest harmonics well-approximated by 23-EDO are 9, 13, 15, 17, 21, and 23. See [[Harmony of 23edo|here]] for more details. Also note that some approximations can be improved by [[23edo and octave stretching|octave stretching]].

As with[[9edo| 9-EDO]], [[16edo|16-EDO]], and [[25edo|25-EDO]], one way to treat 23-EDO is as a Pelogic temperament, tempering out the "comma" of 135/128 and equating three 'acute [[4/3]]'s with 5/1 (related to the Armodue system). This means mapping '[[3/2]]' to 13 degrees of 23, and results in a 7 notes [[2L 5s|Anti-diatonic scale]] of 3 3 4 3 3 3 4 (in steps of 23-EDO), which extends to 9 notes [[7L 2s|Superdiatonic scale]] (3 3 3 1 3 3 3 3 1). One can notate 23-EDO using the Armodue system, but just like notating 17-EDO with familiar diatonic notation, flats will be lower in pitch than enharmonic sharps, because in 23-EDO, the "Armodue 6th" is sharper than it is in 16-EDO, just like the Diatonic 5th in 17-EDO is sharper than in 12-EDO. In other words, 2b is lower in pitch than 1#, just like how in 17-EDO, Eb is lower than D#.

However, one can also map 3/2 to 14 degrees of 23-EDO without significantly increasing the error, taking us to a [[7-limit]] temperament where two 'broad 3/2's equals 7/3, meaning 28/27 is tempered out, and six 4/3's octave-reduced equals 5/4, meaning 4096/3645 is tempered out. Both of these are very large commas, so this is not at all an accurate temperament, but it is related to [[13edo|13-EDO]] and [[18edo|18-EDO]] and produces [[MOSScales|MOS scales]] of 5 and 8 notes: 5 5 4 5 4 (the [[3L 2s|"anti-pentatonic"]]) and 4 1 4 1 4 4 1 4 (the "quarter-tone" version of the Blackwood/[http://en.wikipedia.org/wiki/Paul_Rapoport_%28music_critic%29 Rapoport]/Wilson 13-EDO "subminor" scale). Alternatively we can treat this temperament as a 2.9.21 subgroup, and instead of calling 9 degrees of 23-EDO a Sub-"4/3", we can call it 21/16. Here three 21/16's gets us to 9/4, meaning 1029/1024 is tempered out. This allows us to treat a triad of 0-4-9 degrees of 23-EDO as an approximation to 16:18:21, and 0-5-9 as 1/(16:18:21); both of these triads are abundant in the 8-note MOS scale.

~~== Selected just intervals ==~~

{| class="wikitable center-all"

|-

|+ ~~Approximation of [[primary interval]]s in 23 EDO~~

|+

|-

! colspan="2" | Prime number

! colspan="2" | Prime number --->

!2

! 3

! 5

Line 40:

Line 32:

! 23

|-

! rowspan="2" | Error

! absolute ([[cent|¢]])

| -23.69

|0

| -21.10

| -23.69

| +22.48

| -21.1

| +22.60

| +22.5

| -5.75

| +22.6

| -0.61

| -5.7

| +15.53

| -0.6

| -2.19

| +15.5

| -2.2

|-

! [[Relative error|relative]] (%)

![[Relative error|relative]] (%)

| -45.4

|0

| -40.4

| -45

| +43.1

| -40

| +43.3

| +43

| -11.0

| +43

| -1.2

| -11

| +~~29.8~~

| -1.2

| -4.2

| +30

| -4.2

|-

! colspan="2" | ~~Degree (~~[[~~octave reduction|reduced~~]])

! colspan="2" | [[nearest edomapping]]

| ~~36 (~~13)

|23

| ~~53 (~~7)

| 13

| ~~65 (~~19)

| 7

| ~~80 (~~11)

| 19

| ~~85 (~~16)

| 11

| ~~94 (~~2)

| 16

| ~~98 (~~6)

| 2

| ~~104 (~~12)

| 6

| 12

|-

!

![[fifthspan]]

|0

| +1

| -3

| +5

| -8

| +3

| +9

| +4

| +8

|}

23-TET, or 23-EDO, is a tempered musical system which divides the [[octave]] into 23 equal parts of approximately 52.173913 cents, which is also called with the neologism Icositriphony ''(Icositrifonía)''. It has good approximations for [[5/3]], [[11/7]], 13 and 17, allowing it to represent the 2.5/3.11/7.13.17 [[just intonation subgroup]]. If to this subgroup is added the commas of [[17-limit]] [[46edo]], the larger 17-limit [[k*N_subgroups|2*23 subgroup]] 2.9.15.21.33.13.17 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does 17-limit·46edo, and may be regarded as a basis for analyzing the harmony of 23-EDO so far, as approximations to just intervals goes. 23edo is the 9th [[prime numbers|prime]] edo, following [[19edo]] and coming before [[29edo]].

23-EDO was proposed by ethnomusicologist [http://en.wikipedia.org/wiki/Erich_von_Hornbostel Erich von Hornbostel] as the result of continuing a circle of "blown" fifths of ~678-cent fifths that (he argued) resulted from "overblowing" a bamboo pipe.

23-EDO is also significant in that it is the largest EDO that fails to approximate the 3rd, 5th, 7th, and 11th harmonics within 20 cents, which makes it well-suited for musicians seeking to explore harmonic territory that is unusual even for the average microtonalist. Oddly, despite the fact that it fails to approximate these harmonics, it approximates the intervals between them (5/3, 7/3, 11/3, 7/5, 11/7, and 11/5) very well. The lowest harmonics well-approximated by 23-EDO are 9, 13, 15, 17, 21, and 23. See [[Harmony of 23edo|here]] for more details. Also note that some approximations can be improved by [[23edo and octave stretching|octave stretching]].

As with[[9edo| 9-EDO]], [[16edo|16-EDO]], and [[25edo|25-EDO]], one way to treat 23-EDO is as a Pelogic temperament, tempering out the "comma" of 135/128 and equating three 'acute [[4/3]]'s with 5/1 (related to the Armodue system). This means mapping '[[3/2]]' to 13 degrees of 23, and results in a 7 notes [[2L 5s|Anti-diatonic scale]] of 3 3 4 3 3 3 4 (in steps of 23-EDO), which extends to 9 notes [[7L 2s|Superdiatonic scale]] (3 3 3 1 3 3 3 3 1). One can notate 23-EDO using the Armodue system, but just like notating 17-EDO with familiar diatonic notation, flats will be lower in pitch than enharmonic sharps, because in 23-EDO, the "Armodue 6th" is sharper than it is in 16-EDO, just like the Diatonic 5th in 17-EDO is sharper than in 12-EDO. In other words, 2b is lower in pitch than 1#, just like how in 17-EDO, Eb is lower than D#.

However, one can also map 3/2 to 14 degrees of 23-EDO without significantly increasing the error, taking us to a [[7-limit]] temperament where two 'broad 3/2's equals 7/3, meaning 28/27 is tempered out, and six 4/3's octave-reduced equals 5/4, meaning 4096/3645 is tempered out. Both of these are very large commas, so this is not at all an accurate temperament, but it is related to [[13edo|13-EDO]] and [[18edo|18-EDO]] and produces [[MOSScales|MOS scales]] of 5 and 8 notes: 5 5 4 5 4 (the [[3L 2s|"anti-pentatonic"]]) and 4 1 4 1 4 4 1 4 (the "quarter-tone" version of the Blackwood/[http://en.wikipedia.org/wiki/Paul_Rapoport_%28music_critic%29 Rapoport]/Wilson 13-EDO "subminor" scale). Alternatively we can treat this temperament as a 2.9.21 subgroup, and instead of calling 9 degrees of 23-EDO a Sub-"4/3", we can call it 21/16. Here three 21/16's gets us to 9/4, meaning 1029/1024 is tempered out. This allows us to treat a triad of 0-4-9 degrees of 23-EDO as an approximation to 16:18:21, and 0-5-9 as 1/(16:18:21); both of these triads are abundant in the 8-note MOS scale.

== Selected just intervals ==

{| class="wikitable center-all"

|+Direct mapping (even if inconsistent)

@@ Line 8: / Line 8: @@
 | Prime factorization = 23
 | Subgroup = 2.9.15.21.33.13.17
-| Step size = 52.174
+| Step size = 52.174¢
-| Fifth type = [[mavila]] 13\23 678.26¢
+| Fifth type = [[mavila]] 13\23 = 678.26¢
+| Major 2nd = 3\23 = 157¢
+| Minor 2nd = 4\23 = 209¢
+| Augmented 1sn = -1\23 = -52¢
 | Common uses = mavila<br/>two sets of distorted 12edo<br/>extreme xenharmony in e.g. metal
 | Important MOS = mavila 2L5s 4334333 (13\23, 1\1)<br/>mavila 7L2s 133313333 (13\23, 1\1)<br/>sephiroth 3L4s 2525252 (7\23, 1\1)<br/>"semaphore" 5L4s 332323232 (5\23, 1\1)
 }}
 == Theory ==
-<b>23-TET</b>, or <b>23-EDO</b>, is a tempered musical system which divides the [[octave]] into 23 equal parts of approximately 52.173913 cents, which is also called with the neologism Icositriphony ''(Icositrifonía)''. It has good approximations for [[5/3]], [[11/7]], 13 and 17, allowing it to represent the 2.5/3.11/7.13.17 [[just intonation subgroup]]. If to this subgroup is added the commas of [[17-limit]] [[46edo]], the larger 17-limit [[k*N_subgroups|2*23 subgroup]] 2.9.15.21.33.13.17 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does 17-limit·46edo, and may be regarded as a basis for analyzing the harmony of 23-EDO so far, as approximations to just intervals goes. 23edo is the 9th [[prime numbers|prime]] edo, following [[19edo]] and coming before [[29edo]].
--EDO was proposed by ethnomusicologist [http://en.wikipedia.org/wiki/Erich_von_Hornbostel Erich von Hornbostel] as the result of continuing a circle of "blown" fifths of ~678-cent fifths that (he argued) resulted from "overblowing" a bamboo pipe.
--EDO is also significant in that it is the largest EDO that fails to approximate the 3rd, 5th, 7th, and 11th harmonics within 20 cents, which makes it well-suited for musicians seeking to explore harmonic territory that is unusual even for the average microtonalist. Oddly, despite the fact that it fails to approximate these harmonics, it approximates the intervals between them (5/3, 7/3, 11/3, 7/5, 11/7, and 11/5) very well. The lowest harmonics well-approximated by 23-EDO are 9, 13, 15, 17, 21, and 23. See [[Harmony of 23edo|here]] for more details. Also note that some approximations can be improved by [[23edo and octave stretching|octave stretching]].
-As with[[9edo| 9-EDO]], [[16edo|16-EDO]], and [[25edo|25-EDO]], one way to treat 23-EDO is as a Pelogic temperament, tempering out the "comma" of 135/128 and equating three 'acute [[4/3]]'s with 5/1 (related to the Armodue system). This means mapping '[[3/2]]' to 13 degrees of 23, and results in a 7 notes [[2L 5s|Anti-diatonic scale]] of 3 3 4 3 3 3 4 (in steps of 23-EDO), which extends to 9 notes [[7L 2s|Superdiatonic scale]] (3 3 3 1 3 3 3 3 1). One can notate 23-EDO using the Armodue system, but just like notating 17-EDO with familiar diatonic notation, flats will be lower in pitch than enharmonic sharps, because in 23-EDO, the "Armodue 6th" is sharper than it is in 16-EDO, just like the Diatonic 5th in 17-EDO is sharper than in 12-EDO. In other words, 2b is lower in pitch than 1#, just like how in 17-EDO, Eb is lower than D#.
-However, one can also map 3/2 to 14 degrees of 23-EDO without significantly increasing the error, taking us to a [[7-limit]] temperament where two 'broad 3/2's equals 7/3, meaning 28/27 is tempered out, and six 4/3's octave-reduced equals 5/4, meaning 4096/3645 is tempered out. Both of these are very large commas, so this is not at all an accurate temperament, but it is related to [[13edo|13-EDO]] and [[18edo|18-EDO]] and produces [[MOSScales|MOS scales]] of 5 and 8 notes: 5 5 4 5 4 (the [[3L 2s|"anti-pentatonic"]]) and 4 1 4 1 4 4 1 4 (the "quarter-tone" version of the Blackwood/[http://en.wikipedia.org/wiki/Paul_Rapoport_%28music_critic%29 Rapoport]/Wilson 13-EDO "subminor" scale). Alternatively we can treat this temperament as a 2.9.21 subgroup, and instead of calling 9 degrees of 23-EDO a Sub-"4/3", we can call it 21/16. Here three 21/16's gets us to 9/4, meaning 1029/1024 is tempered out. This allows us to treat a triad of 0-4-9 degrees of 23-EDO as an approximation to 16:18:21, and 0-5-9 as 1/(16:18:21); both of these triads are abundant in the 8-note MOS scale.
-== Selected just intervals ==
 {| class="wikitable center-all"
 |-
-|+ Approximation of [[primary interval]]s in 23 EDO
+|+
 |-
-! colspan="2" | Prime number
+! colspan="2" | Prime number --->
+!2
 ! 3
 ! 5
@@ Line 40: / Line 32: @@
 ! 23
 |-
 ! rowspan="2" | Error
 ! absolute ([[cent|¢]])
-| -23.69
+|0
-| -21.10
+|  -23.69
-| +22.48
+|  -21.1
-| +22.60
+|  +22.5
-| -5.75
+|  +22.6
-| -0.61
+|  -5.7
-| +15.53
+|  -0.6
-| -2.19
+|  +15.5
+|  -2.2
 |-
-! [[Relative error|relative]] (%)
+![[Relative error|relative]] (%)
-| -45.4
+|0
-| -40.4
+|  -45
-| +43.1
+|  -40
-| +43.3
+|  +43
-| -11.0
+|  +43
-| -1.2
+|  -11
-| +29.8
+|  -1.2
-| -4.2
+|  +30
+|  -4.2
 |-
-! colspan="2" | Degree ([[octave reduction|reduced]])
+! colspan="2" | [[nearest edomapping]]
-| 36 (13)
+|23
-| 53 (7)
+| 13
-| 65 (19)
+| 7
-| 80 (11)
+| 19
-| 85 (16)
+| 11
-| 94 (2)
+| 16
-| 98 (6)
+| 2
-| 104 (12)
+| 6
+| 12
+|-
+!
+![[fifthspan]]
+|0
+| +1
+| -3
+| +5
+| -8
+| +3
+| +9
+| +4
+| +8
 |}
+<b>23-TET</b>, or <b>23-EDO</b>, is a tempered musical system which divides the [[octave]] into 23 equal parts of approximately 52.173913 cents, which is also called with the neologism Icositriphony ''(Icositrifonía)''. It has good approximations for [[5/3]], [[11/7]], 13 and 17, allowing it to represent the 2.5/3.11/7.13.17 [[just intonation subgroup]]. If to this subgroup is added the commas of [[17-limit]] [[46edo]], the larger 17-limit [[k*N_subgroups|2*23 subgroup]] 2.9.15.21.33.13.17 is obtained. This is the largest subgroup on which 23 has the same tuning and commas as does 17-limit·46edo, and may be regarded as a basis for analyzing the harmony of 23-EDO so far, as approximations to just intervals goes. 23edo is the 9th [[prime numbers|prime]] edo, following [[19edo]] and coming before [[29edo]].
+-EDO was proposed by ethnomusicologist [http://en.wikipedia.org/wiki/Erich_von_Hornbostel Erich von Hornbostel] as the result of continuing a circle of "blown" fifths of ~678-cent fifths that (he argued) resulted from "overblowing" a bamboo pipe.
+-EDO is also significant in that it is the largest EDO that fails to approximate the 3rd, 5th, 7th, and 11th harmonics within 20 cents, which makes it well-suited for musicians seeking to explore harmonic territory that is unusual even for the average microtonalist. Oddly, despite the fact that it fails to approximate these harmonics, it approximates the intervals between them (5/3, 7/3, 11/3, 7/5, 11/7, and 11/5) very well. The lowest harmonics well-approximated by 23-EDO are 9, 13, 15, 17, 21, and 23. See [[Harmony of 23edo|here]] for more details. Also note that some approximations can be improved by [[23edo and octave stretching|octave stretching]].
+As with[[9edo| 9-EDO]], [[16edo|16-EDO]], and [[25edo|25-EDO]], one way to treat 23-EDO is as a Pelogic temperament, tempering out the "comma" of 135/128 and equating three 'acute [[4/3]]'s with 5/1 (related to the Armodue system). This means mapping '[[3/2]]' to 13 degrees of 23, and results in a 7 notes [[2L 5s|Anti-diatonic scale]] of 3 3 4 3 3 3 4 (in steps of 23-EDO), which extends to 9 notes [[7L 2s|Superdiatonic scale]] (3 3 3 1 3 3 3 3 1). One can notate 23-EDO using the Armodue system, but just like notating 17-EDO with familiar diatonic notation, flats will be lower in pitch than enharmonic sharps, because in 23-EDO, the "Armodue 6th" is sharper than it is in 16-EDO, just like the Diatonic 5th in 17-EDO is sharper than in 12-EDO. In other words, 2b is lower in pitch than 1#, just like how in 17-EDO, Eb is lower than D#.
+However, one can also map 3/2 to 14 degrees of 23-EDO without significantly increasing the error, taking us to a [[7-limit]] temperament where two 'broad 3/2's equals 7/3, meaning 28/27 is tempered out, and six 4/3's octave-reduced equals 5/4, meaning 4096/3645 is tempered out. Both of these are very large commas, so this is not at all an accurate temperament, but it is related to [[13edo|13-EDO]] and [[18edo|18-EDO]] and produces [[MOSScales|MOS scales]] of 5 and 8 notes: 5 5 4 5 4 (the [[3L 2s|"anti-pentatonic"]]) and 4 1 4 1 4 4 1 4 (the "quarter-tone" version of the Blackwood/[http://en.wikipedia.org/wiki/Paul_Rapoport_%28music_critic%29 Rapoport]/Wilson 13-EDO "subminor" scale). Alternatively we can treat this temperament as a 2.9.21 subgroup, and instead of calling 9 degrees of 23-EDO a Sub-"4/3", we can call it 21/16. Here three 21/16's gets us to 9/4, meaning 1029/1024 is tempered out. This allows us to treat a triad of 0-4-9 degrees of 23-EDO as an approximation to 16:18:21, and 0-5-9 as 1/(16:18:21); both of these triads are abundant in the 8-note MOS scale.
+== Selected just intervals ==
 {| class="wikitable center-all"
 |+Direct mapping (even if inconsistent)