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*	Additional Ratios* of 3, 5 and 9	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

based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

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...

10ed2-001.svg

Linear temperaments

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 |.)

Ratio	Monzo	Cents	Color Name	Name 1	Name 2	Name 3
256/243	\| 8 -5 >	90.22	Sawa	Limma	Pythagorean Minor 2nd
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
525/512	\| -9 1 2 1 >	43.41	Lazoyoyo	Avicennma	Avicennma'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
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
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.