23edo: Difference between revisions

Line 17:

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"

|-

|+ 23-EDO Approximation of Primary Intervals

|-

! colspan="2" | Prime number

! 3

! 5

! 7

! 11

! 13

! 17

! 19

! 23

|-

! rowspan="2" | Error

! absolute ([[cent|¢]])

| -23.69

| -21.10

| +22.48

| +22.60

| -5.75

| -0.61

| +15.53

| -2.19

|-

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

| -45.4

| -40.4

| +43.1

| +43.3

| -11.0

| -1.2

| +29.8

| -4.2

|-

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

| 36 (13)

| 53 (7)

| 65 (19)

| 80 (11)

| 85 (16)

| 94 (2)

| 98 (6)

| 104 (12)

|}

== Notation ==

@@ Line 17: / Line 17: @@
 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"
+|-
+|+ 23-EDO Approximation of Primary Intervals
+|-
+! colspan="2" | Prime number
+! 3
+! 5
+! 7
+! 11
+! 13
+! 17
+! 19
+! 23
+|-
+! rowspan="2" | Error
+! absolute ([[cent|¢]])
+| -23.69
+| -21.10
+| +22.48
+| +22.60
+| -5.75
+| -0.61
+| +15.53
+| -2.19
+|-
+! [[Relative error|relative]] (%)
+| -45.4
+| -40.4
+| +43.1
+| +43.3
+| -11.0
+| -1.2
+| +29.8
+| -4.2
+|-
+! colspan="2" | Degree ([[octave reduction|reduced]])
+| 36 (13)
+| 53 (7)
+| 65 (19)
+| 80 (11)
+| 85 (16)
+| 94 (2)
+| 98 (6)
+| 104 (12)
+|}
 == Notation ==