# 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 21/52, 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-limits, 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 5\52 interval approximates 31/29 well, and when used as a generator produces tricesimoprimal miracloid temperament. The relationship is also preserved exactly in the period-52 french deck temperament.

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%.

### 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
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)

## Intervals

Degrees Cents 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

Lumatone

Claudi Meneghin
Jon Lyle Smith