51edo: Difference between revisions

← 50edo 51edo 52edo →
Prime factorization	3 × 17
Step size	23.5294 ¢
Fifth	30\51 (705.882 ¢) (→ 10\17)
Semitones (A1:m2)	6:3 (141.2 ¢ : 70.59 ¢)
Consistency limit	3
Distinct consistency limit	3

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VisualWikitext

Revision as of 23:14, 11 February 2026

51 equal divisions of the octave (abbreviated 51edo or 51ed2), also called 51-tone equal temperament (51tet) or 51 equal temperament (51et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 51 equal parts of about 23.5 ¢ each. Each step represents a frequency ratio of 2^1/51, or the 51st root of 2.

Theory

Since 51 = 3 × 17, 51edo shares its fifth with 17edo. Compared to other multiples of 17edo, notably 34edo and 68edo, 51edo's harmonic inventory seems lacking, getting few harmonics very well considering its step size. However, it does possess excellent approximations of 11/10 and 21/16, only about 0.3 cents off in each case.

Using the patent val, 51et tempers out 250/243 in the 5-limit, 225/224 and 2401/2400 in the 7-limit, and 55/54 and 100/99 in the 11-limit. It is the optimal patent val for sonic, the rank-3 temperament tempering out 55/54, 100/99, and 250/243, and also for the rank-4 temperament tempering out 55/54. It provides an alternative tuning to 22edo for porcupine, with a nice fifth but a rather flat major third, and the optimal patent val for the 7- and 11-limit porky temperament, which is sonic plus 225/224. It contains an archeotonic (6L 1s) scale based on repetitions of 8\51, creating a scale with a whole-tone-like drive towards the tonic through the 17edo semitone at the top.

Using the 51c val ⟨51 81 119 143], the 5/4 is mapped to 1\3 (400 cents), supporting augmented. In the 7-limit it tempers out 245/243 and supports hemiaug and rodan. Alternatively, the 51cd val ⟨51 81 119 144] takes the same 7/4 from 17edo, and supports augene. The 51ce val ⟨51 81 119 143 177 189] supports a variant of rodan called aerodino.

51edo's step is the closest direct approximation to the Pythagorean comma by edo steps, though that comma itself is mapped to a different interval.

Odd harmonics

Approximation of prime harmonics in 51edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.0	+3.9	-9.8	-4.1	-10.1	+6.5	-10.8	+8.4	+7.0	+5.7	+7.9
Error	Relative (%)	+0.0	+16.7	-41.8	-17.5	-43.1	+27.8	-46.1	+35.6	+29.8	+24.3	+33.6
Steps (reduced)		51 (0)	81 (30)	118 (16)	143 (41)	176 (23)	189 (36)	208 (4)	217 (13)	231 (27)	248 (44)	253 (49)

Subsets and supersets

51edo contains 3edo and 17edo as subsets.

One of the very powerful (but very complex) supersets of 51edo is 612edo, which divides each step of 51edo into 12 equal parts, for which the name "skisma" has been proposed.

Intervals

#	Cents	Approximate ratios*			Ups and downs notation
#	Cents	2.3.7.11/5.13 subgroup	Ratios of 5 and 11 tending flat (51 val)	Ratios of 5 and 11 tending sharp (51ce val)	Ups and downs notation
0	0.0	1/1			Perfect 1sn	P1	D
1	23.5	64/63, 49/48	40/39	81/80	Up 1sn	^1	^D
2	47.1	28/27	33/32, 25/24, 81/80	36/35, 40/39	Downminor 2nd	vm2	vEb
3	70.6	27/26	36/35	21/20, 33/32	Minor 2nd	m2	Eb
4	94.1		21/20	16/15, 25/24	Upminor 2nd	^m2	^Eb
5	117.6	14/13	15/14, 16/15		Downmid 2nd	v~2	^^Eb
6	141.2	13/12		12/11, 15/14	Mid 2nd	~2	vvvE, ^^^Eb
7	164.7	11/10	10/9, 12/11		Upmid 2nd	^~2	vvE
8	188.2			10/9	Downmajor 2nd	vM2	vE
9	211.8	9/8			Major 2nd	M2	E
10	235.3	8/7	15/13		Upmajor 2nd	^M2	^E
11	258.8	7/6		15/13	Downminor 3rd	vm3	vF
12	282.4	32/27		13/11	Minor 3rd	m3	F
13	305.9		13/11	6/5	Upminor 3rd	^m3	^F
14	329.4	40/33, 63/52	6/5, 11/9		Downmid 3rd	v~3	^^F
15	352.9	16/13, 39/32		11/9, 27/22	Mid 3rd	~3	^^^F, vvvF#
16	376.5	26/21	5/4, 27/22		Upmid 3rd	^~3	vvF#
17	400.0			5/4, 14/11	Downmajor 3rd	vM3	vF#
18	423.5	81/64	14/11		Major 3rd	M3	F#
19	447.1	9/7		13/10	Upmajor 3rd	^M3	^F#
20	470.6	21/16	13/10		Down 4th	v4	vG
21	494.1	4/3			Perfect 4th	P4	G
22	517.6			27/20	Up 4th	^4	^G
23	541.2	15/11	11/8, 27/20		Downdim 5th	vd5	vAb
24	564.7	18/13		7/5, 11/8	Dim 5th	d5	Ab
25	588.2	39/28	7/5		Updim 5th	^d5	^Ab
26	611.8	56/39	10/7		Downaug 4th	vA4	vG#
27	635.3	13/9		10/7, 16/11	Aug 4th	A4	G#
28	658.8	22/15	16/11, 40/27		Upaug 4th	^A4	^G#
29	682.4			40/27	Down 5th	v5	vA
30	705.9	3/2			Perfect 5th	P5	A
31	729.4	32/21	20/13		Up 5th	^5	^A
32	752.9	14/9		20/13	Downminor 6th	vm6	vBb
33	776.5	128/81	11/7		Minor 6th	m6	Bb
34	800.0			8/5, 11/7	Upminor 6th	^m6	^Bb
35	823.5	21/13	8/5, 44/27		Downmid 6th	v~6	^^Bb
36	847.1	13/8, 64/39		18/11, 44/27	Mid 6th	~6	vvvB, ^^^Bb
37	870.6	33/20, 104/63	5/3, 18/11		Upmid 6th	^~6	vvB
38	894.1		22/13	5/3	Downmajor 6th	vM6	vB
39	917.6	27/16		22/13	Major 6th	M6	B
40	941.2	12/7		26/15	Upmajor 6th	^M6	^B
41	964.7	7/4	26/15		Downminor 7th	vm7	vC
42	988.2	16/9			Minor 7th	m7	C
43	1011.8			9/5	Upminor 7th	^m7	^C
44	1035.3	20/11		9/5, 11/6	Downmid 7th	v~7	^^C
45	1058.8	24/13		11/6, 28/15	Mid 7th	~7	^^^C, vvvC#
46	1082.4	13/7	15/8, 28/15		Upmid 7th	^~7	vvC#
47	1105.9		40/21	15/8, 48/25	Downmajor 7th	vM7	vC#
48	1129.4	52/27	35/18	40/21, 64/33	Major 7th	M7	C#
49	1152.9	27/14	64/33, 48/25, 160/81	35/18, 39/20	Upmajor 7th	^M7	^C#
50	1176.5	63/32, 96/49	39/20	160/81	Down 8ve	v8	vD
51	1200.0	2/1			Perfect 8ve	P8	D

* inconsistent intervals in italic.

Notation

Ups and downs notation

51edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12
Sharp symbol
Flat symbol

Half-sharps and half-flats can be used to avoid triple arrows:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12
Sharp symbol
Flat symbol

In 51edo, a combination of quarter tone accidentals and arrow accidentals from Helmholtz–Ellis notation can be used.

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14
Sharp symbol
Flat symbol

If double arrows are not desirable, then arrows can be attached to quarter-tone accidentals:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13
Sharp symbol
Flat symbol

Ivan Wyschnegradsky's notation

Since a sharp raises by six steps, Wyschnegradsky accidentals borrowed from 72edo can also be used:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13
Sharp symbol
Flat symbol

Sagittal notation

In the following diagrams, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's primary comma (the comma it exactly represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it approximately represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo.

Evo flavor

Revo flavor

Evo-SZ flavor

Approximation to JI

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 51edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 51edo (direct approximation, even if inconsistent)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
11/10, 20/11	0.298	1.3
13/9, 18/13	1.324	5.6
15/14, 28/15	1.796	7.6
13/12, 24/13	2.604	11.1
3/2, 4/3	3.927	16.7
7/4, 8/7	4.120	17.5
15/11, 22/15	4.226	18.0
11/9, 18/11	5.533	23.5
7/5, 10/7	5.723	24.3
9/5, 10/9	5.832	24.8
15/8, 16/15	5.916	25.1
11/7, 14/11	6.021	25.6
13/8, 16/13	6.531	27.8
13/11, 22/13	6.857	29.1
13/10, 20/13	7.155	30.4
9/8, 16/9	7.855	33.4
7/6, 12/7	8.047	34.2
11/6, 12/11	9.461	40.2
5/3, 6/5	9.759	41.5
5/4, 8/5	9.843	41.8
11/8, 16/11	10.141	43.1
13/7, 14/13	10.651	45.3
15/13, 26/15	11.082	47.1
9/7, 14/9	11.555	49.1

15-odd-limit intervals in 51edo (patent val mapping)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
11/10, 20/11	0.298	1.3
13/9, 18/13	1.324	5.6
15/14, 28/15	1.796	7.6
13/12, 24/13	2.604	11.1
3/2, 4/3	3.927	16.7
7/4, 8/7	4.120	17.5
15/11, 22/15	4.226	18.0
7/5, 10/7	5.723	24.3
15/8, 16/15	5.916	25.1
11/7, 14/11	6.021	25.6
13/8, 16/13	6.531	27.8
9/8, 16/9	7.855	33.4
7/6, 12/7	8.047	34.2
5/4, 8/5	9.843	41.8
11/8, 16/11	10.141	43.1
13/7, 14/13	10.651	45.3
9/7, 14/9	11.975	50.9
15/13, 26/15	12.447	52.9
5/3, 6/5	13.770	58.5
11/6, 12/11	14.069	59.8
13/10, 20/13	16.374	69.6
13/11, 22/13	16.673	70.9
9/5, 10/9	17.698	75.2
11/9, 18/11	17.996	76.5

15-odd-limit intervals in 51edo (51ce val mapping)
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
11/10, 20/11	0.298	1.3
13/9, 18/13	1.324	5.6
13/12, 24/13	2.604	11.1
3/2, 4/3	3.927	16.7
7/4, 8/7	4.120	17.5
15/11, 22/15	4.226	18.0
11/9, 18/11	5.533	23.5
9/5, 10/9	5.832	24.8
13/8, 16/13	6.531	27.8
13/11, 22/13	6.857	29.1
13/10, 20/13	7.155	30.4
9/8, 16/9	7.855	33.4
7/6, 12/7	8.047	34.2
11/6, 12/11	9.461	40.2
5/3, 6/5	9.759	41.5
13/7, 14/13	10.651	45.3
15/13, 26/15	11.082	47.1
9/7, 14/9	11.975	50.9
11/8, 16/11	13.388	56.9
5/4, 8/5	13.686	58.2
11/7, 14/11	17.508	74.4
15/8, 16/15	17.614	74.9
7/5, 10/7	17.806	75.7
15/14, 28/15	21.734	92.4

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3.7	1029/1024, [17 -16 3⟩	[⟨51 81 143]]	−0.339	1.63	6.92
2.3.7.13	343/338, 512/507, 2197/2187	[⟨51 81 143]]	−0.695	1.54	6.54
2.3.5	128/125, [-13 17 -6⟩	[⟨51 81 119]] (51c)	−2.789	2.41	10.3
2.3.5.7	128/125, 245/243, 1029/1000	[⟨51 81 119 143]] (51c)	−1.730	2.79	11.9
2.3.5	250/243, 34171875/33554432	[⟨51 81 118]] (51)	+0.581	2.77	11.8
2.3.5.7	225/224, 250/243, 1029/1024	[⟨51 81 118 143]] (51)	+0.803	2.43	10.3

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperament
1	5\51	117.6	15/14	Miracle (51e, out of tune)
1	7\51	164.7	11/10	Porky (51)
1	10\51	235.3	8/7	Rodan (51cf…, out of tune) / aerodino (51ce)
1	22\51	517.6	27/20	Gravity (51ce)
1	23\51	541.2	15/11	Necromanteion (51ce) Oracle (51)
3	19\51 (2\51)	447.1 (47.1)	9/7 (36/35)	Hemiaug (51ce)
3	21\51 (4\51)	494.1 (94.1)	4/3 (16/15)	Augmented (51c)
3	21\51 (4\51)	494.1 (94.1)	4/3 (21/20)	Fog (51)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Scales

Porky[7] (Palace^{[idiosyncratic term]}): 7 7 7 9 7 7 7
UFO scale^{[idiosyncratic term]} (inflected MOS of Teefs[19]^{[idiosyncratic term]}): 2 2 4 1 2 2 2 4 2 5 2 4 4 2 2 1 4 2 2
Cosmic scale^{[idiosyncratic term]} subset of UFO scale): 21 9 4 9 8

Instruments

Lumatone: See Lumatone mapping for 51edo.

Music

Bryan Deister

51edo improv (2025-02-03)
51edo improv (2025-05-02)
Northernlight - Deltarune (microtonal cover in 51edo) (2025)
51edo prelude (2026)

Frédéric Gagné

Whalectric (2022) – YouTube | score – 7:4 semiquartal 4|4 mode

James Mulvale (FASTFAST)

STARS (Thoughts and Prayers) (2020)

Ray Perlner

Fugue (2023) – for organ in 51edo Porcupine[7] ssssssL "Pandian"

@@ Line 27: / Line 27: @@
 ! rowspan="2" colspan="3" | [[Ups and downs notation]]
 |-
-! 2.3.7.13<br>subgroup
+! 2.3.7.11/5.13<br>subgroup
 ! Ratios of 5 and 11<br>tending flat (51 val)
 ! Ratios of 5 and 11<br>tending sharp (51ce val)
@@ Line 96: / Line 96: @@
 | 7
 | 164.7
-|
+| [[11/10]]
-| [[11/10]], ''10/9'', ''12/11''
+| ''10/9'', ''12/11''
-| 11/10
+|
 | Upmid 2nd
 | ^~2
@@ Line 159: / Line 159: @@
 | 14
 | 329.4
-| [[63/52]]
+| [[40/33]], [[63/52]]
 | ''6/5'', ''11/9''
 |
@@ Line 240: / Line 240: @@
 | 23
 | 541.2
-|
+| [[15/11]]
-| [[11/8]], [[15/11]], ''27/20''
+| [[11/8]], ''27/20''
-| 15/11
+|
 | Downdim 5th
 | vd5
@@ Line 285: / Line 285: @@
 | 28
 | 658.8
-|
+| [[22/15]]
-| [[16/11]], [[22/15]], ''40/27''
+| [[16/11]], ''40/27''
-| 22/15
+|
 | Upaug 4th
 | ^A4
@@ Line 366: / Line 366: @@
 | 37
 | 870.6
-| [[104/63]]
+| [[33/20]], [[104/63]]
 | ''5/3'', ''18/11''
 |
@@ Line 429: / Line 429: @@
 | 44
 | 1035.3
-|
 | [[20/11]]
-| 20/11, ''9/5'', ''11/6''
+|
+| ''9/5'', ''11/6''
 | Downmid 7th
 | v~7

51edo: Difference between revisions

Revision as of 23:14, 11 February 2026

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