10edo
| ← 9edo | 10edo | 11edo → |
10edo, or 10-tone equal temperament, is a tuning system which divides the octave into 10 equal parts of exactly 120 cents.
Theory
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, 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 |
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...
Just approximation
Selected just intervals by error
| prime 2 | prime 3 | prime 5 | prime 7 | prime 11 | prime 13 | prime 17 | ||
|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | 0.0 | +18.0 | -26.3 | -8.8 | +48.7 | -0.5 | +12.5 |
| relative (%) | 0.0 | +15.0 | -21.9 | -7.3 | +40.6 | -0.4 | +15.0 | |
Selected 13-limit intervals
Temperament measures
The following table shows TE temperament measures (RMS normalized by the rank) 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 has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, 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 ET that does better in these subgroups is 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 |
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
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.
|
| A Decaphonic (10-EDO) Classical Guitar |
Music
- ZIA Space "Who Loves You, Me?", "Champagne", and "Avatar" by Elaine Walker
- Fiat Circadia by Stephen Weigel
- Ten Fingers play by Bill Sethares (synth guitar)
- Circle of Thirds play by Bill Sethares (synth ens.)
- 10_fantasy play by Aaron Krister Johnson (synth monody)
- Prelude in 10ET by Aaron Andrew Hunt
- Fugue in 10ET | SoundCloud by Aaron Andrew Hunt, 2015
- Future play and Sol play by ZIA (synths and voice in 10)
- Prelude by Rick McGowan (Rhino synthesizer)
- Ideas on the Waterfall of Expression by Igliashon Jones (synth)
- For two violas and gongs by Chris Vaisvil Details
- Blues 10 by Carlo Serafini (blog entry)
- Waltz 10 by Carlo Serafini (blog entry)
- Smooth 10 by Carlo Serafini (blog entry)
- 10 PRS by Carlo Serafini (blog entry)
- 10preview.ogg A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
- decexperiment.ogg 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
- 10 Earwigs Invasive by Chris Vaisvil Details
- Comets Over Flatland 9 by Randy Winchester
- The Dramatic Squirrel Overture by Chris Vaisvil Details
- Shimmerwing by Andrew Heathwaite and Chris Vaisvil
- Shall I Refuse My Dinner by Steve Martin on SoundCloud
- 10tone demo by Clem Fortuna
- Hey, ule! by Dmitriy Bazhenov (second part in 10-edo)