51edf

← 50edf

51edf

52edf →

Prime factorization	3 × 17
Step size	13.7638 ¢
Octave	87\51edf (1197.45 ¢) (→ 29\17edf)
Twelfth	138\51edf (1899.41 ¢) (→ 46\17edf)
Consistency limit	6
Distinct consistency limit	6

Division of the just perfect fifth into 51 equal parts (51EDF) is related to 87 edo, but with the 3/2 rather than the 2/1 being just. The octave is compressed by about 2.5474 cents and the step size is about 13.7638 cents (corresponding to 87.1851 edo).

Unlike 87edo, it is only consistent up to the 6-integer-limit, with discrepancy for the 7th harmonic.

Lookalikes: 87edo, 138edt

Harmonics

Approximation of prime harmonics in 51edf

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	-2.55	-2.55	-6.02	+3.31	+5.36	+5.19	-5.03	-4.90	-5.33	+6.28	+0.94
	Relative (%)	-18.5	-18.5	-43.7	+24.1	+38.9	+37.7	-36.6	-35.6	-38.7	+45.7	+6.8
Steps (reduced)		87 (36)	138 (36)	202 (49)	245 (41)	302 (47)	323 (17)	356 (50)	370 (13)	394 (37)	424 (16)	432 (24)

Approximation of prime harmonics in 51edf

Harmonic		37	41	43	47	53	59	61	67	71	73	79
Error	Absolute (¢)	-2.57	-1.36	-1.23	-3.82	-5.36	+1.67	-0.99	+1.76	-2.29	+4.68	+5.57
	Relative (%)	-18.7	-9.9	-8.9	-27.7	-38.9	+12.1	-7.2	+12.8	-16.6	+34.0	+40.4
Steps (reduced)		454 (46)	467 (8)	473 (14)	484 (25)	499 (40)	513 (3)	517 (7)	529 (19)	536 (26)	540 (30)	550 (40)

Intervals

Degree	Cents
0
1	13.7638
2	27.52765
3	41.2915
4	55.0555
5	68.8191
6	82.5829
7·	96.3468
8	110.1106
9	123.8744
10	137.6382
11	151.0206
12·	165.1659
13	178.9297
14	192.6935
15	206.45735
16	220.2212
17·	233.985
18	248.7488
19	261.51265
20	275.2765
21	289.0403
22·	302.8041
23	316.5679
24	330.3318
25	344.0955
26	357.8594
27	371.6232
28	385.3871
29	399.1509
30	412.9147
31	426.6785
32	440.44253
33	455.2062
34	467.97
35	481.7338
36	495.49765
37	509.2615
38	523.0253
39	536.7891
40	550.5529
41	564.3168
42	578.0806
43	591.8444
44	605.6082
45	619.3721
46	633.1359
47	646.8997
48	660.6635
49	674.42735
50	688.1912
51	701.955
52	715.7188
53	729.48365
54	743.2465
55	757.0103
56	770.7741
57	784.5379
58	798.3018
59	812.0656
60	825.8294
61	839.5932
62	853.3571
63	867.1209
64	880.8847
65	894.6485
66	908.41235
67	922.1762
68	935.94
69	949.7038
70	963.46765
71	977.2315
72	990.9952
73	1004.7591
74	1018.5229
75	1032.32868
76	1046.0506
77	1059.8144
78	1073.5782
79	1087.3421
80	1101.1059
81	1114.8697
82	1128.6335
83	1142.39735
84	1156.1612
85	1169.925
86	1183.6888
87	1197.45265
88	1211.2165
89	1224.9803
90	1238.7441
91	1252.5079
92	1266.2718
93	1280.0356
94	1293.7994
95	1307.5632
96	1321.3271
97	1335.0909
98	1348.8547
99	1362.6185
100	1376.3824
101	1390.1462
102	1403.91

Todo: complete table Add a third column that comments on the intervals, either what JI they approximate, what they are named, or how they can be used musically.