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This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.
This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.
Line 183: Line 183:
10 EDO tempers out the following commas. (Note: This assumes the val &lt; 10 16 23 28 35 37 |.)
10 EDO tempers out the following commas. (Note: This assumes the val &lt; 10 16 23 28 35 37 |.)


{| class="wikitable center-1 right-3 center-4"
{| class="wikitable center-1 center-2 right-4 center-5"
|-
|-
!Prime<br>Limit
! [[Ratios|Ratio]]
! [[Ratios|Ratio]]
! [[Monzo]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name(s)
! Name 2
! Name 3
|-
|-
|3
| 256/243
| 256/243
| <nowiki> | 8 -5 </nowiki>&gt;
| <nowiki> | 8 -5 </nowiki>&gt;
| 90.22
| 90.22
| Sawa
| Sawa
| Limma
| Limma, Pythagorean minor 2nd
| Pythagorean Minor 2nd
|
|-
|-
|5
| 25/24
| 25/24
| <nowiki> | -3 -1 2 </nowiki>&gt;
| <nowiki> | -3 -1 2 </nowiki>&gt;
| 70.67
| 70.67
| Yoyo
| Yoyo
| 5-limit large semitone
| 5-limit large semitone, 5-limit chromatic semitone
| 5-limit chromatic semitone
|
|-
|-
|"
| 16875/16384
| 16875/16384
| <nowiki> | -14 3 4 </nowiki>&gt;
| <nowiki> | -14 3 4 </nowiki>&gt;
| 51.12
| 51.12
| Laquadyo
| Laquadyo
| Negri Comma
| Negri comma, double augmentation diesis
| Double Augmentation Diesis
|
|-
|-
|"
| 9931568/9752117
| 9931568/9752117
| <nowiki> | -25 7 6 </nowiki>&gt;
| <nowiki> | -25 7 6 </nowiki>&gt;
| 31.57
| 31.57
| Lala-tribiyo
| Lala-tribiyo
| Ampersand's Comma
| Ampersand's comma
|
|
|-
|-
|"
| 2048/2025
| 2048/2025
| <nowiki> | 11 -4 -2 </nowiki>&gt;
| <nowiki> | 11 -4 -2 </nowiki>&gt;
Line 230: Line 226:
| Sagugu
| Sagugu
| Diaschisma
| Diaschisma
|
|
|-
|-
|7
| 525/512
| 525/512
| <nowiki> | -9 1 2 1 </nowiki>&gt;
| <nowiki> | -9 1 2 1 </nowiki>&gt;
| 43.41
| 43.41
| Lazoyoyo
| Lazoyoyo
| Avicennma
| Avicennma, Avicenna's enharmonic diesis
| Avicennma's Enharmonic Diesis
|
|-
|-
|"
| 49/48
| 49/48
| <nowiki> | -4 -1 0 2 </nowiki>&gt;
| <nowiki> | -4 -1 0 2 </nowiki>&gt;
| 35.70
| 35.70
| Zozo
| Zozo
| Slendro Diesis
| Slendro diesis
|
|
|-
|-
|"
| 50/49
| 50/49
| <nowiki> | 1 0 2 -2 </nowiki>&gt;
| <nowiki> | 1 0 2 -2 </nowiki>&gt;
| 34.98
| 34.98
| Biruyo
| Biruyo
| Tritonic Diesis
| Tritonic diesis, jubilisma
| Jubilisma
|
|-
|-
|"
| 686/675
| 686/675
| <nowiki> | 1 -3 -2 3 </nowiki>&gt;
| <nowiki> | 1 -3 -2 3 </nowiki>&gt;
Line 262: Line 254:
| Trizo-agugu
| Trizo-agugu
| Senga
| Senga
|
|
|-
|-
|"
| 64/63
| 64/63
| <nowiki> | 6 -2 0 -1 </nowiki>&gt;
| <nowiki> | 6 -2 0 -1 </nowiki>&gt;
| 27.26
| 27.26
| Ru
| Ru
| Septimal Comma
| Septimal comma, Archytas' comma, Leipziger Komma
| Archytas' Comma
| Leipziger Komma
|-
|-
|"
| 9859966/9733137
| 9859966/9733137
| <nowiki> | -10 7 8 -7 </nowiki>&gt;
| <nowiki> | -10 7 8 -7 </nowiki>&gt;
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| Lasepru-aquadbiyo
| Lasepru-aquadbiyo
| Blackjackisma
| Blackjackisma
|
|
|-
|-
|"
| 1029/1024
| 1029/1024
| <nowiki> | -10 1 0 3 </nowiki>&gt;
| <nowiki> | -10 1 0 3 </nowiki>&gt;
Line 286: Line 275:
| Latrizo
| Latrizo
| Gamelisma
| Gamelisma
|
|
|-
|-
|"
| 225/224
| 225/224
| <nowiki> | -5 2 2 -1 </nowiki>&gt;
| <nowiki> | -5 2 2 -1 </nowiki>&gt;
| 7.71
| 7.71
| Ruyoyo
| Ruyoyo
| Septimal Kleisma
| Septimal kleisma, marvel comma
| Marvel Comma
|
|-
|-
|"
| 16875/16807
| 16875/16807
| <nowiki> | 0 3 4 -5 </nowiki>&gt;
| <nowiki> | 0 3 4 -5 </nowiki>&gt;
Line 302: Line 289:
| Quinru-aquadyo
| Quinru-aquadyo
| Mirkwai
| Mirkwai
|
|
|-
|-
|"
| 6772805/6751042
| 6772805/6751042
| <nowiki> | 11 -10 -10 10 </nowiki>&gt;
| <nowiki> | 11 -10 -10 10 </nowiki>&gt;
Line 310: Line 296:
| Saquinbizogu
| Saquinbizogu
| Linus
| Linus
|
|
|-
|-
|"
| 2401/2400
| 2401/2400
| <nowiki> | -5 -1 -2 4 </nowiki>&gt;
| <nowiki> | -5 -1 -2 4 </nowiki>&gt;
Line 318: Line 303:
| Bizozogu
| Bizozogu
| Breedsma
| Breedsma
|
|
|-
|-
|11
| 243/242
| 243/242
| <nowiki> | -1 5 0 0 -2 </nowiki>&gt;
| <nowiki> | -1 5 0 0 -2 </nowiki>&gt;
Line 326: Line 310:
| Lulu
| Lulu
| Rastma
| Rastma
|
|
|-
|-
|"
| 385/384
| 385/384
| <nowiki> | -7 -1 1 1 1 </nowiki>&gt;
| <nowiki> | -7 -1 1 1 1 </nowiki>&gt;
Line 334: Line 317:
| Lozoyo
| Lozoyo
| Keenanisma
| Keenanisma
|
|
|-
|-
|"
| 441/440
| 441/440
| <nowiki> | -3 2 -1 2 -1 </nowiki>&gt;
| <nowiki> | -3 2 -1 2 -1 </nowiki>&gt;
Line 342: Line 324:
| Luzozogu
| Luzozogu
| Werckisma
| Werckisma
|
|
|-
|-
|"
| 540/539
| 540/539
| <nowiki> | 2 3 1 -2 -1 </nowiki>&gt;
| <nowiki> | 2 3 1 -2 -1 </nowiki>&gt;
Line 350: Line 331:
| Lururuyo
| Lururuyo
| Swetisma
| Swetisma
|
|
|-
|-
|"
| 3025/3024
| 3025/3024
| <nowiki> | -4 -3 2 -1 2 </nowiki>&gt;
| <nowiki> | -4 -3 2 -1 2 </nowiki>&gt;
Line 358: Line 338:
| Loloruyoyo
| Loloruyoyo
| Lehmerisma
| Lehmerisma
|
|
|-
|-
|13
| 91/90
| 91/90
| <nowiki> | -1 -2 -1 1 0 1 </nowiki>&gt;
| <nowiki> | -1 -2 -1 1 0 1 </nowiki>&gt;
Line 366: Line 345:
| Thozogu
| Thozogu
| Superleap
| Superleap
|
|
|-
|-
|"
| 676/675
| 676/675
| <nowiki> | 2 -3 -2 0 0 2 </nowiki>&gt;
| <nowiki> | 2 -3 -2 0 0 2 </nowiki>&gt;
Line 374: Line 352:
| Bithogu
| Bithogu
| Parizeksma
| Parizeksma
|
|
|}
|}


Line 406: Line 382:
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

Revision as of 02:38, 9 June 2020


Theory

10edo, or 10-tone equal temperament, is a tuning system which divides the octave into 10 equal parts of exactly 120 cents. It can be thought of as two circles of 5edo separated by 120 cents (or 5 circles of 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 moment of symmetry scale of the form 1 2 1 2 1 2 1 (3L 4s - mosh). While not an integral or gap edo, it is a 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

Degree Cents Approximate Ratios[1] Additional Ratios
of 3, 5 and 9[2] 		Interval Names ups and downs notation
0 0 1/1 256/243, 50/49, 25/24 unison unison, min 2nd P1, m2 D, Eb
1 120 16/15, 15/14, 13/14 10/9, 13/12, 81/80 small neutral second, large minor second mid 2nd ~2 ^D, vE
2 240 8/7, 15/13, 144/125 9/8, 7/6 second/third maj 2nd, min 3rd M2, m3 E, F
3 360 16/13 5/4 large neutral third mid 3rd ~3 ^F, vG
4 480 64/49, 169/128 4/3, 9/7, 13/10 smaller fourth maj 3rd, perf 4th M3, P4 F#, G
5 600 91/64, 128/91, 169/120, 240/169 7/5, 10/7, 13/9, 18/13 tritone up 4th, down 5th ^4,v5 ^G, vA
6 720 49/32, 256/169 3/2, 14/9, 20/13 bigger fifth perf 5th, min 6th P5, m6 A, Bb
7 840 13/8 8/5 neutral sixth mid 6th ~6 ^A, vB
8 960 7/4, 26/15, 125/72 16/9, 12/7 sixth/seventh maj 6th, min 7th M6, m7 B, C
9 1080 15/8, 28/15, 13/7 9/5, 24/13, 160/81 small major 7th mid 7th ~7 ^C, vD
10 1200 2/1 243/128, 49/25, 48/25 octave maj 7th, octave M7, P8 C#, D
  1. ↑ based on treating 10-EDO as a 2.7.13.15 subgroup temperament
  2. ↑ adding the ratios of 3, 5 and 9 introduces greater error while giving several more harmonic identities to the 10-EDO intervals

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

Pentatonic 5th-generated: D * E * G * A * C * D (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

Heptatonic 3rd-generated: D E * F G * A B * C D (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

alt : Your browser has no SVG support.

10ed2-001.svg

Linear temperaments (with images for MOS horagrams)

Periods
per octave 		Generator Temperament(s)
1 1\10 Messed-up negri (or miracle)
1 3\10 Dicot/beatles/neutral thirds scale
2 1\10 Messed-up pajara
2 2\10 Decimal / messed-up lemba
5 1\10 Blackwood/blacksmith
1\10 MOS
1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s
3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s

Commas

10 EDO tempers out the following commas. (Note: This assumes the val < 10 16 23 28 35 37 |.)

Prime
Limit 		Ratio Monzo Cents Color Name Name(s)
3 256/243 | 8 -5 > 90.22 Sawa Limma, Pythagorean minor 2nd
5 25/24 | -3 -1 2 > 70.67 Yoyo 5-limit large semitone, 5-limit chromatic semitone
" 16875/16384 | -14 3 4 > 51.12 Laquadyo Negri comma, double augmentation diesis
" 9931568/9752117 | -25 7 6 > 31.57 Lala-tribiyo Ampersand's comma
" 2048/2025 | 11 -4 -2 > 19.55 Sagugu Diaschisma
7 525/512 | -9 1 2 1 > 43.41 Lazoyoyo Avicennma, Avicenna's enharmonic diesis
" 49/48 | -4 -1 0 2 > 35.70 Zozo Slendro diesis
" 50/49 | 1 0 2 -2 > 34.98 Biruyo Tritonic diesis, jubilisma
" 686/675 | 1 -3 -2 3 > 27.99 Trizo-agugu Senga
" 64/63 | 6 -2 0 -1 > 27.26 Ru Septimal comma, Archytas' comma, Leipziger Komma
" 9859966/9733137 | -10 7 8 -7 > 22.41 Lasepru-aquadbiyo Blackjackisma
" 1029/1024 | -10 1 0 3 > 8.43 Latrizo Gamelisma
" 225/224 | -5 2 2 -1 > 7.71 Ruyoyo Septimal kleisma, marvel comma
" 16875/16807 | 0 3 4 -5 > 6.99 Quinru-aquadyo Mirkwai
" 6772805/6751042 | 11 -10 -10 10 > 5.57 Saquinbizogu Linus
" 2401/2400 | -5 -1 -2 4 > 0.72 Bizozogu Breedsma
11 243/242 | -1 5 0 0 -2 > 7.14 Lulu Rastma
" 385/384 | -7 -1 1 1 1 > 4.50 Lozoyo Keenanisma
" 441/440 | -3 2 -1 2 -1 > 3.93 Luzozogu Werckisma
" 540/539 | 2 3 1 -2 -1 > 3.21 Lururuyo Swetisma
" 3025/3024 | -4 -3 2 -1 2 > 0.57 Loloruyoyo Lehmerisma
13 91/90 | -1 -2 -1 1 0 1 > 19.13 Thozogu Superleap
" 676/675 | 2 -3 -2 0 0 2 > 2.56 Bithogu Parizeksma

Images

10edo wheel.png

Instruments

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from 20edo which approximates the 11th harmonic with relative accuracy, among other features.

Decaphonic_Classic_Guitar.png
A Decaphonic (10-EDO) Classical Guitar

decaphonic-uke.JPG

Music

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