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| Fifth = 6\10 = 720¢ (→[[5edo|3\5]])
| Fifth = 6\10 = 720¢ (→[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢ (→1\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]].
'''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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{{Odd harmonics in edo|edo=10}}
{{Odd harmonics in edo|edo=10}}


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 ==
== Intervals ==
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| 4
| 4
| 60¢
| 60¢
| -60¢
| -60¢
|-
|-
| 6
| 6
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== Linear temperaments (with images for MOS horagrams) ==
== Linear temperaments (with images for MOS horagrams) ==
{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|-
|-

Revision as of 13:50, 15 October 2021

← 9edo 10edo 11edo →
Prime factorization 2 × 5
Step size 120 ¢ 
Fifth 6\10 (720 ¢) (→ 3\5)
Semitones (A1:m2) 2:0 (240 ¢ : 0 ¢)
Consistency limit 7
Distinct consistency limit 3

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 cents.

Theory

Approximation of odd harmonics in 10 EDO
Odd harmonic 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
Error absolute (¢) +18.0 -26.3 -8.8 +36.1 +48.7 -0.5 -8.3 +15.0 -57.5 +9.2 -28.3 -52.6 +54.1 +50.4 +55.0
relative (%) +15 -22 -7 +30 +41 -0 -7 +13 -48 +8 -24 -44 +45 +42 +46
Steps (reduced) 16 (6) 23 (3) 28 (8) 32 (2) 35 (5) 37 (7) 39 (9) 41 (1) 42 (2) 44 (4) 45 (5) 46 (6) 48 (8) 49 (9) 50 (0)

10EDO 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, 14/13 10/9, 13/12, 81/80 neutral second mid 2nd ~2 ^D, vE
2 240 8/7, 15/13, 144/125 9/8, 7/6 hemifourth, major second, minor third maj 2nd, min 3rd M2, m3 E, F
3 360 16/13 5/4 neutral third mid 3rd ~3 ^F, vG
4 480 64/49, 169/128 4/3, 9/7, 13/10 perfect 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 hemioctave up 4th, down 5th ^4, v5 ^G, vA
6 720 49/32, 256/169 3/2, 14/9, 20/13 perfect 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 hemitwelfth, major sixth, minor seventh maj 6th, min 7th M6, m7 B, C
9 1080 15/8, 28/15, 13/7 9/5, 24/13, 160/81 neutral seventh 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 10EDO 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 the diatonic notation, generated by 5ths (6\10, representing 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...

Differences between distributionally-even scales and smaller EDOs

N L-Nedo s-Nedo
3 80¢ -40¢
4 60¢ -60¢
6 40¢ -80¢
7 68.571¢ -51.429¢
8 90¢ 30¢
9 106.667¢ -13.333¢

Just approximation

Selected just intervals by error

Selected 13-limit intervals

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Temperament measures

The following table shows TE temperament measures (RMS normalized) of 10ET.

3-limit 2.3.7 2.3.7.13 2.3.7.13.17 5-limit 7-limit 2.3.5.7.13 2.3.5.7.13.17
Octave stretch (¢) -5.69 -2.77 -2.05 -2.37 -0.09 +0.72 +0.60 -0.11
Error absolute (¢) 5.66 6.23 5.54 5.00 9.27 8.15 7.30 6.85
relative (%) 4.74 5.20 4.62 4.17 7.73 6.79 6.08 5.70
  • 10ET is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively.
  • 10ET is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

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

Pathological Modes

2 1 1 1 2 1 1 1 2L 6s MOS

3 1 1 1 1 1 1 1 1L 7s MOS  

2 1 1 1 1 1 1 1 1 1L 8s MOS

Commas

10EDO tempers out the following commas. This assumes the val ⟨10 16 23 28 35 37].

Prime
Limit 		Ratio[1] Monzo Cents Color name Name(s)
3 256/243 [8 -5⟩ 90.22 Sawa Limma, Pythagorean diatonic semitone
5 25/24 [-3 -1 2⟩ 70.67 Yoyo Classic chromatic semitone, dicot comma
5 16875/16384 [-14 3 4⟩ 51.12 Laquadyo Negri comma, double augmentation diesis
5 (16 digits) [-25 7 6⟩ 31.57 Lala-tribiyo Ampersand, Ampersand's comma
5 2048/2025 [11 -4 -2⟩ 19.55 Sagugu Diaschisma
7 525/512 [-9 1 2 1⟩ 43.41 Lazoyoyo Avicennma, Avicenna's enharmonic diesis
7 49/48 [-4 -1 0 2⟩ 35.70 Zozo Slendro diesis
7 50/49 [1 0 2 -2⟩ 34.98 Biruyo Tritonic diesis, jubilisma
7 686/675 [1 -3 -2 3⟩ 27.99 Trizo-agugu Senga
7 64/63 [6 -2 0 -1⟩ 27.26 Ru Septimal comma, Archytas' comma, Leipziger Komma
7 (18 digits) [-10 7 8 -7⟩ 22.41 Lasepru-aquadbiyo Blackjackisma
7 1029/1024 [-10 1 0 3⟩ 8.43 Latrizo Gamelisma
7 225/224 [-5 2 2 -1⟩ 7.71 Ruyoyo Septimal kleisma, marvel comma
7 16875/16807 [0 3 4 -5⟩ 6.99 Quinru-aquadyo Mirkwai
7 (24 digits) [11 -10 -10 10⟩ 5.57 Saquinbizogu Linus
7 2401/2400 [-5 -1 -2 4⟩ 0.72 Bizozogu Breedsma
11 243/242 [-1 5 0 0 -2⟩ 7.14 Lulu Rastma
11 385/384 [-7 -1 1 1 1⟩ 4.50 Lozoyo Keenanisma
11 441/440 [-3 2 -1 2 -1⟩ 3.93 Luzozogu Werckisma
11 540/539 [2 3 1 -2 -1⟩ 3.21 Lururuyo Swetisma
11 3025/3024 [-4 -3 2 -1 2⟩ 0.57 Loloruyoyo Lehmerisma
13 91/90 [-1 -2 -1 1 0 1⟩ 19.13 Thozogu Superleap
13 676/675 [2 -3 -2 0 0 2⟩ 2.56 Bithogu Island comma, parizeksma
  1. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints

Images

10edo wheel.png

Instruments

10EDO 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 12EDO, 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 (10EDO) Classical Guitar

decaphonic-uke.JPG

Music

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