10edo: Difference between revisions

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10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It 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|13/8]] and its inversion [[16/13|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 [[MOSScales|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.

__FORCETOC__

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! | Degree

! | Cents

! | Approximate Ratios*

! | Additional Ratios*

|-

| | 0

| style="text-align:center;" | 1/1

| | 256/243, 50/49, 25/24

| | 1

| style="text-align:right;" | 120

| style="text-align:center;" | 16/15, 15/14, 13/14

| | 10/9, 13/12, 81/80

| | 2

| style="text-align:right;" | 240

| style="text-align:center;" | 8/7, 15/13, 144/125

| | 9/8, 7/6

| | second/third

| | maj 2nd, min 3rd

| style="text-align:center;" | E, F

|-

| | 3

| style="text-align:right;" | 360

| style="text-align:center;" | 16/13

| | 5/4

| | 4

| style="text-align:right;" | 480

| style="text-align:center;" | 64/49, 169/128

| | 4/3, 9/7, 13/10

| | smaller fourth

| | maj 3rd, perf 4th

| style="text-align:center;" | F#, G

|-

| | 5

| style="text-align:right;" | 600

| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169

| | 7/5, 10/7, 13/9, 18/13

| | 6

| style="text-align:right;" | 720

| style="text-align:center;" | 49/32, 256/169

| | 3/2, 14/9, 20/13

| | bigger fifth

| | perf 5th, min 6th

| style="text-align:center;" | A, Bb

|-

| | 7

| style="text-align:right;" | 840

| style="text-align:center;" | 13/8

| | 8/5

| | 8

| style="text-align:right;" | 960

| style="text-align:center;" | 7/4, 26/15, 125/72

| | 16/9, 12/7

| | sixth/seventh

| | maj 6th, min 7th

| style="text-align:center;" | B, C

|-

| | 9

| style="text-align:right;" | 1080

| style="text-align:center;" | 15/8, 28/15, 13/7

| | 9/5, 24/13, 160/81

| | 10

| style="text-align:right;" | 1200

| style="text-align:center;" | 2/1

| | 243/128, 49/25, 48/25

|-

| style="text-align:center;" | 256/243

| style="text-align:right;" | 90.22

| style="text-align:center;" | Limma

|-

| style="text-align:center;" | 25/24

| style="text-align:right;" | 70.67

| style="text-align:center;" | 5-limit large semitone

|-

| style="text-align:center;" | 16875/16384

| style="text-align:right;" | 51.12

| style="text-align:center;" | Negri Comma

|-

| style="text-align:center;" | 9931568/9752117

| style="text-align:right;" | 31.57

| style="text-align:center;" | Ampersand's Comma

|-

| style="text-align:center;" | 2048/2025

| style="text-align:right;" | 19.55

| style="text-align:center;" | Diaschisma

|-

| style="text-align:center;" | 525/512

| style="text-align:right;" | 43.41

| style="text-align:center;" | Avicennma

|-

| style="text-align:center;" | 49/48

| style="text-align:right;" | 35.70

| style="text-align:center;" | Slendro Diesis

|-

| style="text-align:center;" | 50/49

| style="text-align:right;" | 34.98

| style="text-align:center;" | Tritonic Diesis

|-

| style="text-align:center;" | 686/675

| style="text-align:right;" | 27.99

| style="text-align:center;" | Senga

|-

| style="text-align:center;" | 64/63

| style="text-align:right;" | 27.26

| style="text-align:center;" | Septimal Comma

|-

| style="text-align:center;" | 9859966/9733137

| style="text-align:right;" | 22.41

| style="text-align:center;" | Blackjackisma

|-

| style="text-align:center;" | 1029/1024

| style="text-align:right;" | 8.43

| style="text-align:center;" | Gamelisma

|-

| style="text-align:center;" | 225/224

| style="text-align:right;" | 7.71

| style="text-align:center;" | Septimal Kleisma

|-

| style="text-align:center;" | 16875/16807

| style="text-align:right;" | 6.99

| style="text-align:center;" | Mirkwai

|-

| style="text-align:center;" | 6772805/6751042

| style="text-align:right;" | 5.57

| style="text-align:center;" | Linus

|-

| style="text-align:center;" | 2401/2400

| style="text-align:right;" | 0.72

| style="text-align:center;" | Breedsma

|-

| style="text-align:center;" | 243/242

| style="text-align:right;" | 7.14

| style="text-align:center;" | Rastma

|-

| style="text-align:center;" | 385/384

| style="text-align:right;" | 4.50

| style="text-align:center;" | Keenanisma

|-

| style="text-align:center;" | 441/440

| style="text-align:right;" | 3.93

| style="text-align:center;" | Werckisma

|-

| style="text-align:center;" | 540/539

| style="text-align:right;" | 3.21

| style="text-align:center;" | Swetisma

|-

| style="text-align:center;" | 3025/3024

| style="text-align:right;" | 0.57

| style="text-align:center;" | Lehmerisma

|-

| style="text-align:center;" | 91/90

| style="text-align:right;" | 19.13

| style="text-align:center;" | Superleap

|-

| style="text-align:center;" | 676/675

| style="text-align:right;" | 2.56

| style="text-align:center;" | Parizeksma

| | A Decaphonic (10-EDO) Classical Guitar

|}

[[Category:10edo]]

[[Category:edo]]