51edo

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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

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

* As a 2.3.7.13-subgroup temperament, 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

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3.7	1029/1024, [17 -16 3⟩	[⟨51 81 143]]	−0.339	1.63	6.92
2.3.5	128/125, [-13 17 -6⟩	[⟨51 81 119]] (51c)	−2.789	2.41	10.3
2.3.5	250/243, 34171875/33554432	[⟨51 81 118]] (51)	+0.581	2.77	11.8

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) / oracle (51)
1	7\51	164.7	11/10	Porky (51)
1	10\51	235.3	8/7	Rodan (51cf…, out of tune) / aerodino (51ce)
1	5\51	541.2	15/11	Necromanteion (51ce)
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)

* 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)

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"