52edo

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← 51edo52edo53edo →
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 23.077 ¢ 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 (%) -42 +26 +2 +16 +11 -42 -16 +45 +11 -40 -23
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

Instruments

Lumatone

See Lumatone mapping for 52edo

Music

Claudi Meneghin
Jon Lyle Smith