10edo

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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 (abbreviated 10edo or 10ed2), also called 10-tone equal temperament (10tet) or 10 equal temperament (10et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 10 equal parts of exactly 120 ¢ each. Each step represents a frequency ratio of 2^1/10, or the 10th root of 2.

Theory

10edo contains all the intervals of 5edo, but also adds another copy of it separated by 120 cents. The new intervals have sizes of 120 ¢, 360 ¢, 600 ¢, 840 ¢, and 1080 ¢. The 120 ¢ interval can be treated a small neutral second or large minor 2nd, and its inversion a large neutral seventh or small major 7th, with the 120 ¢ and 1080 ¢ intervals being close (about 0.6 ¢ off) to 15/14 and 28/15 respectively. The 360 ¢ interval is a large neutral third, being about 0.5 ¢ sharp of 16/13, with its inversion being equally close to 13/8. Finally, the 600 ¢ interva is the tritone that appears in every even-numbered edo, including 12edo.

Taking the the 360 ¢ large neutral third as a generator produces a heptatonic moment of symmetry scale with step sizes 2 1 1 2 1 2 1 (pattern 3L 4s, or "mosh"), which is the most diatonic-like scale in 10edo excluding the 5edo collapsed diatonic scale, and can be seen as a neutralized diatonic scale.

It shares 5edo's approximation quality in the 2.3.7 subgroup, though the detuned fifth could be seen as a bigger problem with the more fine division of steps. Compared to 5edo, 10edo attains more accuracy in the full 7-limit, by including a better approximation of 5/4 at 360 ¢, resulting in the better tuning of various intervals including 5, such as 16/15 and 7/5. However, the approximation to 5/4 is still over 25 ¢ flat, and this interval is also equated with 6/5 (which is even more inaccurate, at 44 ¢ sharp), tempering out 25/24 and resulting in the dicot exotemperament. Thus, if one wishes to represent JI with 10edo, it is best to use prime 5 carefully or not at all.

Even if 10edo isn't directly used to represent JI, it could still serve as a structural archetype for the 7-limit. The fact that 25/24 is tempered out means that the 5-limit major triad 4:5:6 and minor triad 1/(6:5:4) are mapped to the same number of scale steps in the 10-form, a feature shared with 7edo and the heptatonic system used in western music. 10edo additionally sends 49/48 to the unison, meaning the 7-limit triad 4:6:7 and its inverse 1/(12:8:7) are the same number of scale steps in a decatonic system as well, and therefore also the 4:5:6:7 major and 1/(12:10:8:7) minor tetrads as well. Tempering out 25/24 and 49/48 leads to the decimal exotemperament (which is named after 10edo). A more accurate temperament based off of the 10-form that doesn't temper out 25/24 or 49/48 is pajara, which shares many desireable properties with diatonic.

Since the neutral third is very close to 16/13, 10edo is usable as a 2.3.5.7.13 temperament, which includes 5edo's representation of 2.3.7; however, it is not without high damage. For one, all of 9/7, 13/10, 21/16, and 4/3 are equated to a flat fourth (or an extremely sharp supermajor third), tempering out 28/27, 40/39, 49/48, 64/63, 91/90, and 105/104. Also, 5-limit major and minor thirds are equated as mentioned before (tempering out 25/24), and the third is also equated to 16/13, tempering out 40/39 and 65/64. Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out 225/224), and since 3/2 is tuned sharp and 5/4 is tuned flat, the syntonic comma is exaggerated to a full step, or 120 ¢. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.

By treating 360 ¢ as 11/9, we arrive at 11/8 = 600 ¢ (tempering out 144/143 and 243/242), which allows 10edo to be treated as a full 13-limit temperament. However, it is more accurate as a no-11 system.

10edo is a zeta peak edo, due to its relatively decent tunings of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains 25% or lower relative error on all of the first eight harmonics of the harmonic series.

Thanks to its sevenths, 10edo is an ideal tuning for its size for metallic harmony.

Prime harmonics

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)

Intervals

Degree	Cents	Approximate ratios	Additional ratios of 3, 5, and 9	Interval names	Ups and downs notation (EUs: vvA1 and m2)			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, tritone	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

Notation

Ups and downs notation

The interval table above shows 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 circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles 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, see 5edo notation)

Enharmonic unison: vvs3

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...

Enharmonic unison: d2

See below: 3L 4s mosh notation

3L 4s (mosh) notation

See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭.

#	Cents	Note	Name	Associated ratios
0	0	C	Perf 1sn	1/1
1	120	Db	Min 2nd	12/11
2	240	D, Eb	Maj 2nd, dim 3rd	9/8, 32/27
3	360	E	Perf 3rd	11/9, 27/22
4	480	Fb	Min 4th	4/3
5	600	F, Gb	Maj 4th, min 5th	11/8, 16/11
6	720	G	Maj 5th	3/2
7	840	A	Perf 6th	18/11, 44/27
8	960	A#, Bb	Aug 6th, min 7th	16/9, 27/16
9	1080	B	Maj 7th	11/6
10	1200	C	Perf 8ve	2/1

Sagittal notation

This notation is a subset of the notations for edos 20 and 30 and a superset of the notation for 5edo.

Evo and Revo flavors

Evo-SZ flavor

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.

Approximation to JI

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]]	-0.089	9.27	7.73
2.3.5.7	25/24, 28/27, 49/48	[⟨10 16 23 28]]	+0.718	8.15	6.79
2.3.5.7.13	25/24, 28/27, 40/39, 49/48	[⟨10 16 23 28 37]]	+0.603	7.30	6.08

• 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.

Uniform maps

13-limit uniform maps between 9.8 and 10.2

Min. size	Max. size	Wart notation	Map
9.7957	9.8637	10eff	⟨10 16 23 28 34 36]
9.8637	9.9727	10e	⟨10 16 23 28 34 37]
9.9727	10.1209	10	⟨10 16 23 28 35 37]
10.1209	10.1339	10c	⟨10 16 24 28 35 37]
10.1339	10.1519	10cf	⟨10 16 24 28 35 38]
10.1519	10.2618	10cdf	⟨10 16 24 29 35 38]

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	Blackwood comma, Pythagorean limma
5	25/24	[-3 -1 2⟩	70.67	Yoyo	Dicot comma, classic chroma
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 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	Semaphoresma, slendro diesis
7	50/49	[1 0 2 -2⟩	34.98	Biruyo	Jubilisma, tritonic diesis
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	Marvel comma, septimal kleisma
7	16875/16807	[0 3 4 -5⟩	6.99	Quinru-aquadyo	Mirkwai comma
7	(24 digits)	[11 -10 -10 10⟩	5.57	Saquinbizogu	Linus comma
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 comma, biome comma
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

Rank-2 temperaments

Periods per 8ve	Generator	Temperament(s)
1	1\10	Negri, miracle (out-of-tune)
1	3\10	Dicot / beatles (out-of-tune) / neutral (out-of-tune)
2	1\10	Pajara (out-of-tune)
2	2\10	Decimal, lemba (out-of-tune)
5	1\10	Blackwood

Octave stretch or compression

If one wishes to use 10edo as a no-5s, 19-or-lower-limit tuning, then it benefits from octave shrinking. 26zpi and 36ed12 are compressed-octave versions of 10edo.

If one wishes to use 10edo as a no-3s, 13-or-lower-limit tuning, then it benefits from octave stretching. 28ed7 is a stretched version of 10edo.

Scales

MOS scales

• Decimal/Lemba[6] 4L 2s (period = 5\10, gen = 2\10): 2 2 1 2 2 1
• Dicot[7] 3L 4s (gen = 3\10): 1 2 1 2 1 2 1
• Negri[9] 1L 8s (gen = 1\10): 1 1 1 1 2 1 1 1 1

Other scales

• Pinetone major pentatonic (subset of Dicot[7]): 2 1 3 1 3
• Pinetone minor pentatonic (subset of Dicot[7]): 3 1 2 3 1
• Marvel augmented hexatonic (subset of Dicot[7]): 2 1 3 1 2 1
• Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
• Decimal/Lemba[6] 4M: 2 1 2 2 2 1
• Dicot[7] 4M: 2 1 1 2 2 1 1

Horagrams

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

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 ¢) 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

Lumatone

See Lumatone mapping for 10edo.

Music

Main article: 10edo/Music

See also: Category:10edo tracks