10edo

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10edo

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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 (10tet, 10et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into ten equal steps of exactly 120 cents.

Theory

Approximation of prime harmonics in 10edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.0	+18.0	-26.3	-8.8	+48.7	-0.5	+15.0	-57.5	-28.3	+50.4	+55.0
	Relative (%)	+0.0	+15.0	-21.9	-7.4	+40.6	-0.4	+12.5	-47.9	-23.6	+42.0	+45.8
Steps (reduced)		10 (0)	16 (6)	23 (3)	28 (8)	35 (5)	37 (7)	41 (1)	42 (2)	45 (5)	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 familiar 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			Audio
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, 224/195	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, 195/112	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

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...

JI approximation

Selected just intervals by error

Selected 13-limit intervals

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3.5	25/24, 256/243	[⟨10 16 23]]	-0.09	9.27	7.73
2.3.5.7	25/24, 28/27, 49/48	[⟨10 16 23 28]]	+0.72	8.15	6.79

• 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing 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 equal temperaments doing better in those subgroups are 17, 19, 36 and 31, respectively.

Rank-2 temperaments

Periods per octave	Generator	Temperament(s)
1	1\10	Negri, miracle (out-of-tune)
1	3\10	Dicot / beatles / neutral
2	1\10	Pajara (out-of-tune)
2	2\10	Decimal, lemba (out-of-tune)
5	1\10	Blackwood / blacksmith

Scales

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 scales

• 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

10et tempers out the following commas. 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 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	196/195	[2 -1 -1 2 0 -1⟩	8.86	Thuzozogu	Mynucuma
13	676/675	[2 -3 -2 0 0 2⟩	2.56	Bithogu	Island comma, parizeksma

Diagrams

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.

A Decaphonic (10edo) Classical Guitar

Music

See also: Category:10edo tracks

Dmitry Bazhenov

• Hey, ule! (second part only) (2020)
• Струнка (Strunka) (2021)

Evan Bennet

• Microtonal Moment Musicale 2 (2018)

Cenobyte

• 10preview (2018)
• decexperiment (2018)

City of the Asleep

• "Ideas on the Waterfall of Expression", from Map of an Internal Landscape (2007)
• "Sad Mike", from CLOIIIDS (2018)

Clem Fortuna (website (archived))

• 10Tone (2015)

Andrew Heathwaite and Chris Vaisvil

• "Shimmerwing", from Newbeams (2012)

Aaron Andrew Hunt

• "Prelude in 10ET", from The Equal-Tempered Keyboard (1999-2022)
• "Fugue a3 in 10ET", from The Equal-Tempered Keyboard (1999-2022) (SoundCloud)

Aaron Krister Johnson

• 10 fantasy (≤2005)

Steve Martin

• Shall I Refuse My Dinner (2012)

Rick McGowan

• Rhino Prelude (2003) (details)

Carlo Serafini

• Blues 10 (2013) (details)
• Waltz 10 (2013) (details)
• Smooth 10 (2013) (details)
• 10 PRS (2020) (details)

Bill Sethares

• "Ten Fingers", from Xentonality (1997)
• "Circle of Thirds", from Xentonality (1997)

Userminusone

• Bit Crystals (2021)

Chris Vaisvil

• For 2 Violas and Gongs in 10 edo (2011) (details)
• The Dramatic Squirrel Overture (2011) (details)
• 10 Earwigs Invasive (2011) (details)

Stephen Weigel

• Fiat Circadia (2018)
• Three Weeping Tyrants (2020)

Randy Winchester

• "9. 10 / octave", from Comets Over Flatland (2007)

ZIA

• "Sol", from ZIA 1.5 (1992) (details)
• "Agribusiness", from ZIA 1.5 (1992) (details)
• "Future", from SHEM (1996) (details)
• "River", from SHEM (1996) (details)
• "Who Loves You.. Me?", from Drum 'n' Space (2011)
• "Champagne", from Drum 'n' Space (2011)
• "Love is the Catalyst", from Drum 'n' Space (2011)
• "Decagon Dancefloor", from Trapezoid (2019)
• "Matter Over Mind", from No Terrestrial Road" (2020)