52edo

← 51edo 52edo 53edo →
Prime factorization	22 × 13
Step size	23.0769 ¢
Fifth	30\52 (692.308 ¢) (→ 15\26)
Semitones (A1:m2)	2:6 (46.15 ¢ : 138.5 ¢)
Dual sharp fifth	31\52 (715.385 ¢)
Dual flat fifth	30\52 (692.308 ¢) (→ 15\26)
Dual major 2nd	9\52 (207.692 ¢)
Consistency limit	3
Distinct consistency limit	3

52 equal divisions of the octave (abbreviated 52edo or 52ed2), also called 52-tone equal temperament (52tet) or 52 equal temperament (52et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 52 equal parts of about 23.1 ¢ each. Each step represents a frequency ratio of 2^1/52, or the 52nd root of 2.

Theory

52edo has 26edo's very flat meantone fifth and a very sharp fifth close to 1/2-septimal-comma superpyth. The patent val has the same mapping for 3, 7, 11 and 13 as 26 does, but its 5 is sharp rather than flat. From this it tempers out 648/625 rather than 81/80 in the 5-limit, and 225/224 and 1029/1024 in the 7-limit, showing it supports miracle, albeit badly, and may be defined by the tempering out of both 648/625 and miracle. In the 11-limit it tempers out 99/98 and 176/175 and in the 13-limit 78/77, 144/143 and 169/168. It supplies the optimal patent val for then 12 & 40 temperament of the diminished family in the 7- and 11-limit, and also in the 13-limit where it can be defined as tempering out 78/77, 99/98, 176/175, 567/550 rather than by two patent vals. It also gives the 13-limit patent val for the 21 & 52 variant of miracle.

Using the sharp fifth rather than the flat fifth (that is, using the 52b val), it contains a version of porcupine temperament, and combining 30\52 with 31\52 leads to a whole tone of 9\52, or 208 cents, which can be used inconsistently.

The 11\52 (253.846 ¢) semifourth is a very accurate 22/19, with an error of only +0.041 ¢ and a closing error of only 9.3%.

52edo has an accurate 11/9, but unfortunately, in the 2.3.11 subgroup, neither the patent val nor the 52b val map it consistently. The 52e val does, but that would be requiring you to give up 26edo's accurate 11/8. Perhaps the best way of getting 11/9 to map correctly would be to use the 2.9.11 subgroup.

Odd harmonics

Approximation of odd harmonics in 52edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-9.6	+6.0	+0.4	+3.8	+2.5	-9.8	-3.7	+10.4	+2.5	-9.2	-5.2
Error	Relative (%)	-41.8	+26.0	+1.8	+16.4	+11.0	-42.3	-15.8	+45.2	+10.8	-40.1	-22.5
Steps (reduced)		82 (30)	121 (17)	146 (42)	165 (9)	180 (24)	192 (36)	203 (47)	213 (5)	221 (13)	228 (20)	235 (27)

Subsets and supersets

Since 52 factors into 2² × 13, 52edo contains subset edos 2, 4, 13, and 26.

Intervals

Degrees	Centss	Ups and downs notation
0	0.000	Perfect 1sn	P1	D
1	23.077	Up 1sn	^1	^D
2	46.154	Aug 1sn	A1	D#
3	69.231	Downdim 2nd, Upaug 1sn	vd2, ^A1	vEbb, ^D#
4	92.308	Dim 2nd	d2	Ebb
5	115.358	Downminor 2nd	vm2	vEb
6	138.462	Minor 2nd	m2	Eb
7	161.538	Mid 2nd	~2	vE, ^Eb
8	184.615	Major 2nd	M2	E
9	207.692	Upmajor 2nd	^M2	^E
10	230.769	Aug 2nd	A2	E#
11	253.846	Downdim 3rd, Upaug 2nd	vd3, ^A2	vFb, ^E#
12	276.923	Dim 3rd	d3	Fb
13	300.000	Downminor 3rd	vm3	vF
14	323.077	Minor 3rd	m3	F
15	346.154	Mid 3rd	~3	^F, vF#
16	369.231	Major 3rd	M3	F#
17	392.308	Upmajor 3rd	^M3	^F#
18	415.385	Aug 3rd	A3	Fx
19	438.462	Downdim 4th, Upaug 3rd	vd4, ^A4	vGb, ^Fx
20	461.538	Dim 4th	d4	Gb
21	484.615	Down 4th	v4	vG
22	507.692	Perfect 4th	P4	G
23	530.769	Up 4th	^4	^G
24	553.846	Aug 4th	A4	G#
25	576.293	Upaug 4th	^A4	^G#
26	600.000	Double-aug 4th, Double-dim 5th	AA4, dd5	Gx, Abb
27	623.077	Downdim 5th	vd5	vAb
28	646.154	Dim 5th	d5	Ab
29	669.231	Down 5th	v5	vA
30	692.308	Perfect 5th	P5	A
31	715.385	Up 5th	^5	^A
32	738.462	Aug 5th	A5	A#
33	761.538	Downdim 6th, Upaug 5th	vd6, ^A5	vBbb, ^A#
34	784.615	Dim 6th	d6	Bbb
35	807.692	Downminor 6th	vm6	vBb
36	830.769	Minor 6th	m6	Bb
37	853.846	Mid 6th	~6	vB, ^Bb
38	876.923	Major 6th	M6	B
39	900.000	Upmajor 6th	^M6	^B
40	923.077	Aug 6th	A6	B#
41	946.154	Downdim 7th, Upaug 6th	vd7, ^A6	vCb, ^B#
42	969.231	Dim 7th	d7	Cb
43	992.308	Downminor 7th	vm7	vC
44	1015.385	Minor 7th	m7	C
45	1038.462	Mid 7th	~7	^C, vC#
46	1061.538	Major 7th	M7	C#
47	1084.615	Upmajor 7th	^M7	^C#
48	1107.692	Aug 7th	A7	Cx
49	1130.769	Downdim 8ve, Upaug 7th	vd8, ^A7	vDb, ^Cx
50	1153.846	Dim 8ve	d8	Db
51	1176.923	Down 8ve	v8	vD
52	1200.000	Perfect 8ve	P8	D