75edo

← 74edo

75edo

76edo →

Prime factorization	3 × 52
Step size	16¢
Fifth	44\75 (704¢)
Semitones (A1:m2)	8:5 (128¢ : 80¢)
Consistency limit	5
Distinct consistency limit	5

75 equal divisions of the octave (abbreviated 75edo or 75ed2), also called 75-tone equal temperament (75tet) or 75 equal temperament (75et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 75 equal parts of exactly 16 ¢ each. Each step represents a frequency ratio of 2^1/75, or the 75th root of 2.

Theory

75et tempers out 20000/19683 (tetracot comma) and 2109375/2097152 (semicomma) in the 5-limit, and provides a good tuning for the tetracot temperament. It tempers out 225/224 and 1728/1715 in the 7-limit, supporting bunya and orwell, and providing the optimal patent val for fog.

In the 11-limit, 75e val ⟨75 119 174 211 260] scores lower in error, and tempers 100/99 and 243/242, whereas the patent val ⟨75 119 174 211 259] tempers 99/98 and 121/120. In the 13-limit, it tempers 325/324 and 512/507, 17-limit 120/119 and 256/255 and 19-limit 190/189 and 250/247.

Since 75 is part of the Fibonacci sequence beginning with 5 and 12, it closely approximates the peppermint temperament. The size of its fifth is exactly 704  ¢, which is very close to the peppermint fifth of 704.096  ¢. This makes it suitable for neo-Gothic tunings. It also approximates the Carlos Beta scale well (4\75 ≈ 1\[Carlos Beta]).

Odd harmonics

Approximation of odd harmonics in 75edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	+2.04	-2.31	+7.17	+4.09	-7.32	+7.47	-0.27	+7.04	+6.49	-6.78	-4.27
	relative (%)	+13	-14	+45	+26	-46	+47	-2	+44	+41	-42	-27
Steps (reduced)		119 (44)	174 (24)	211 (61)	238 (13)	259 (34)	278 (53)	293 (68)	307 (7)	319 (19)	329 (29)	339 (39)

Intervals

Steps	Cents	Ups and downs notation	Approximate ratios
0	0	D	1/1
1	16	↑D, ↓4E♭
2	32	↑↑D, ↓3E♭	65/64
3	48	↑3D, ↓↓E♭	33/32, 36/35, 65/63, 77/75
4	64	↑4D, ↓E♭	27/26, 28/27, 80/77
5	80	↑5D, E♭
6	96	↑6D, ↓7E
7	112	↑7D, ↓6E	16/15, 77/72
8	128	D♯, ↓5E	14/13
9	144	↑D♯, ↓4E	13/12
10	160	↑↑D♯, ↓3E	11/10, 35/32
11	176	↑3D♯, ↓↓E	72/65
12	192	↑4D♯, ↓E	39/35
13	208	E	9/8
14	224	↑E, ↓4F	25/22
15	240	↑↑E, ↓3F
16	256	↑3E, ↓↓F	52/45, 65/56, 81/70
17	272	↑4E, ↓F	7/6, 75/64
18	288	F	32/27, 77/65
19	304	↑F, ↓4G♭
20	320	↑↑F, ↓3G♭	6/5, 65/54, 77/64
21	336	↑3F, ↓↓G♭	40/33, 63/52
22	352	↑4F, ↓G♭
23	368	↑5F, G♭	26/21
24	384	↑6F, ↓7G	5/4, 56/45, 81/65
25	400	↑7F, ↓6G	49/39
26	416	F♯, ↓5G
27	432	↑F♯, ↓4G	9/7, 32/25, 77/60
28	448	↑↑F♯, ↓3G	35/27
29	464	↑3F♯, ↓↓G
30	480	↑4F♯, ↓G	33/25
31	496	G	4/3
32	512	↑G, ↓4A♭	35/26
33	528	↑↑G, ↓3A♭	65/48
34	544	↑3G, ↓↓A♭	48/35
35	560	↑4G, ↓A♭	18/13
36	576	↑5G, A♭	39/28
37	592	↑6G, ↓7A	45/32
38	608	↑7G, ↓6A	64/45, 77/54
39	624	G♯, ↓5A	56/39
40	640	↑G♯, ↓4A	13/9, 81/56
41	656	↑↑G♯, ↓3A	35/24
42	672	↑3G♯, ↓↓A
43	688	↑4G♯, ↓A	52/35
44	704	A	3/2
45	720	↑A, ↓4B♭	50/33
46	736	↑↑A, ↓3B♭
47	752	↑3A, ↓↓B♭	54/35, 65/42, 77/50
48	768	↑4A, ↓B♭	14/9, 25/16, 81/52
49	784	↑5A, B♭
50	800	↑6A, ↓7B	78/49
51	816	↑7A, ↓6B	8/5, 45/28, 77/48
52	832	A♯, ↓5B	21/13
53	848	↑A♯, ↓4B
54	864	↑↑A♯, ↓3B	33/20, 81/49
55	880	↑3A♯, ↓↓B	5/3
56	896	↑4A♯, ↓B
57	912	B	27/16
58	928	↑B, ↓4C	12/7, 75/44, 77/45
59	944	↑↑B, ↓3C	45/26
60	960	↑3B, ↓↓C
61	976	↑4B, ↓C	44/25
62	992	C	16/9
63	1008	↑C, ↓4D♭	70/39
64	1024	↑↑C, ↓3D♭	65/36
65	1040	↑3C, ↓↓D♭	20/11, 64/35
66	1056	↑4C, ↓D♭	24/13
67	1072	↑5C, D♭	13/7
68	1088	↑6C, ↓7D	15/8
69	1104	↑7C, ↓6D
70	1120	C♯, ↓5D
71	1136	↑C♯, ↓4D	27/14, 52/27, 77/40
72	1152	↑↑C♯, ↓3D	35/18, 64/33
73	1168	↑3C♯, ↓↓D
74	1184	↑4C♯, ↓D
75	1200	D	2/1

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[119 -75⟩	[⟨75 119]]	-0.645	0.645	4.03
2.3.5	20000/19683, 2109375/2097152	[⟨75 119 174]]	-0.099	0.936	5.85
2.3.5.7	225/224, 1728/1715, 15625/15309	[⟨75 119 174 211]]	-0.713	1.337	8.36

Music

• Fugue on The Lick by Claudi Meneghin