10edo: Difference between revisions

Line 10:

| Fifth = 6\10 = 720¢ (→[[5edo|3\5]])

| Major 2nd = 2\10 = 240¢ (→1\5)

| Semitones = 2~~\10~~ : 0~~\10~~

| Semitones = 2 : 0

| Consistency = 7

| Monotonicity = 9

}}

'''10 equal divisions of the octave''' ('''10EDO'''), or '''10-tone equal temperament''' ('''10-TET''', '''10ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into ten equal steps of exactly 120 [[cent|cents]].

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10EDO can be thought of as two circles of [[5edo|5EDO]] separated by 120 cents (or 5 circles of [[2edo|2EDO]]). It adds to 5EDO a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta peak ~~edo~~]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

10EDO can be thought of as two circles of [[5edo|5EDO]] separated by 120 cents (or 5 circles of [[2edo|2EDO]]). It adds to 5EDO a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta peak EDO]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

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| 4

| 60¢

| -60¢

|-

| 6

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== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"

|-

@@ Line 10: / Line 10: @@
 | Fifth = 6\10 = 720¢ (&rarr;[[5edo|3\5]])
 | Major 2nd = 2\10 = 240¢ (&rarr;1\5)
-| Semitones = 2\10 : 0\10
+| Semitones = 2 : 0
+| Consistency = 7
+| Monotonicity = 9
 }}
 '''10 equal divisions of the octave''' ('''10EDO'''), or '''10-tone equal temperament''' ('''10-TET''', '''10ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into ten equal steps of exactly 120 [[cent|cents]].
@@ Line 17: / Line 19: @@
 {{Odd harmonics in edo|edo=10}}
-EDO can be thought of as two circles of [[5edo|5EDO]] separated by 120 cents (or 5 circles of [[2edo|2EDO]]). It adds to 5EDO a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.
+EDO can be thought of as two circles of [[5edo|5EDO]] separated by 120 cents (or 5 circles of [[2edo|2EDO]]). It adds to 5EDO a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta peak EDO]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.
 == Intervals ==
@@ Line 165: / Line 167: @@
 | 4
 | 60¢
-|  -60¢
+| -60¢
 |-
 | 6
@@ Line 236: / Line 238: @@
 == Linear temperaments (with images for MOS horagrams) ==
 {| class="wikitable center-1 center-2"
 |-