75ed4

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75ed4

77ed4 →

Prime factorization	3 × 52
Step size	32¢
Octave	38\75ed4 (1216¢)
Twelfth	59\75ed4 (1888¢)
Consistency limit	1
Distinct consistency limit	1

75 equal divisions of the 4th harmonic (abbreviated 75ed4) is a nonoctave tuning system that divides the interval of 4/1 into 75 equal parts of exactly 32 ¢ each. Each step represents a frequency ratio of 4^1/75, or the 75th root of 4. As an ED4 system, it is equivalent to taking every other tone of 75edo.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	32
2	64
3	96	18/17, 19/18, 37/35
4	128	14/13
5	160	34/31
6	192	19/17
7	224	33/29
8	256	29/25, 36/31
9	288	13/11
10	320
11	352	38/31
12	384
13	416	14/11
14	448
15	480	33/25
16	512
17	544
18	576
19	608
20	640
21	672
22	704
23	736
24	768
25	800
26	832
27	864
28	896
29	928	29/17
30	960
31	992
32	1024
33	1056	35/19
34	1088
35	1120
36	1152	35/18, 37/19
37	1184
38	1216
39	1248	35/17, 37/18
40	1280	23/11
41	1312
42	1344	37/17
43	1376	31/14
44	1408
45	1440	23/10
46	1472
47	1504	31/13
48	1536	17/7
49	1568
50	1600
51	1632	18/7
52	1664	34/13
53	1696
54	1728	19/7
55	1760	36/13
56	1792	31/11
57	1824
58	1856	38/13
59	1888
60	1920
61	1952	34/11
62	1984
63	2016
64	2048
65	2080
66	2112
67	2144	38/11
68	2176
69	2208
70	2240
71	2272
72	2304
73	2336
74	2368
75	2400

Harmonics

Approximation of harmonics in 135ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+8.89	+0.27	+0.00	+4.80	-8.62	-8.83	+8.89	+0.53	-4.09	+8.68	+0.27
	Relative (%)	+50.0	+1.5	+0.0	+27.0	-48.5	-49.6	+50.0	+3.0	-23.0	+48.8	+1.5
Steps (reduced)		68 (68)	107 (107)	135 (0)	157 (22)	174 (39)	189 (54)	203 (68)	214 (79)	224 (89)	234 (99)	242 (107)

Approximation of harmonics in 135ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+3.92	+0.06	+5.06	+0.00	+1.71	-8.35	+4.71	+4.80	-8.56	-0.21	-6.05
	Relative (%)	+22.0	+0.4	+28.5	+0.0	+9.6	-47.0	+26.5	+27.0	-48.1	-1.2	-34.0
Steps (reduced)		250 (115)	257 (122)	264 (129)	270 (0)	276 (6)	281 (11)	287 (17)	292 (22)	296 (26)	301 (31)	305 (35)

Music

JUMBLE

• 4/1 Equave Blues (2024)