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

The 75 equal divisions of the octave (75edo), or the 75-tone equal temperament (75tet), 75 equal temperament (75et) when viewed from a regular temperament perspective, divides the octave into 75 equal parts of exactly 16 cents each.

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 (%)	+12.8	-14.5	+44.8	+25.6	-45.7	+46.7	-1.7	+44.0	+40.5	-42.4	-26.7
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	Approximate ratios	Ups and downs notation
0	0	1/1	D
1	16		^D, v4E♭
2	32		^^D, v3E♭
3	48	35/34, 36/35, 37/36, 38/37	^3D, vvE♭
4	64	27/26, 28/27	^4D, vE♭
5	80	23/22	v3D♯, E♭
6	96	18/17, 19/18, 37/35	vvD♯, ^E♭
7	112	16/15	vD♯, ^^E♭
8	128	14/13	D♯, ^3E♭
9	144	25/23, 37/34, 38/35	^D♯, v4E
10	160	34/31	^^D♯, v3E
11	176	21/19, 31/28	^3D♯, vvE
12	192	19/17	^4D♯, vE
13	208	35/31	E
14	224	25/22, 33/29	^E, v4F
15	240	23/20, 31/27	^^E, v3F
16	256	29/25, 36/31	^3E, vvF
17	272		^4E, vF
18	288		F
19	304	31/26, 37/31	^F, v4G♭
20	320		^^F, v3G♭
21	336	17/14	^3F, vvG♭
22	352	38/31	^4F, vG♭
23	368	21/17, 26/21	v3F♯, G♭
24	384	5/4	vvF♯, ^G♭
25	400	29/23, 34/27	vF♯, ^^G♭
26	416		F♯, ^3G♭
27	432	9/7	^F♯, v4G
28	448	35/27	^^F♯, v3G
29	464	17/13	^3F♯, vvG
30	480	29/22, 33/25, 37/28	^4F♯, vG
31	496	4/3	G
32	512	35/26	^G, v4A♭
33	528	19/14	^^G, v3A♭
34	544	26/19, 37/27	^3G, vvA♭
35	560	18/13	^4G, vA♭
36	576		v3G♯, A♭
37	592	38/27	vvG♯, ^A♭
38	608	27/19, 37/26	vG♯, ^^A♭
39	624	33/23	G♯, ^3A♭
40	640	13/9, 29/20	^G♯, v4A
41	656	19/13, 35/24	^^G♯, v3A
42	672	28/19, 31/21	^3G♯, vvA
43	688		^4G♯, vA
44	704	3/2	A
45	720		^A, v4B♭
46	736	26/17	^^A, v3B♭
47	752	37/24	^3A, vvB♭
48	768	14/9	^4A, vB♭
49	784		v3A♯, B♭
50	800	27/17	vvA♯, ^B♭
51	816	8/5	vA♯, ^^B♭
52	832	21/13, 34/21	A♯, ^3B♭
53	848	31/19	^A♯, v4B
54	864	28/17, 33/20	^^A♯, v3B
55	880		^3A♯, vvB
56	896		^4A♯, vB
57	912		B
58	928		^B, v4C
59	944	31/18	^^B, v3C
60	960		^3B, vvC
61	976		^4B, vC
62	992		C
63	1008	34/19	^C, v4D♭
64	1024	38/21	^^C, v3D♭
65	1040	31/17	^3C, vvD♭
66	1056	35/19	^4C, vD♭
67	1072	13/7	v3C♯, D♭
68	1088	15/8	vvC♯, ^D♭
69	1104	17/9, 36/19	vC♯, ^^D♭
70	1120		C♯, ^3D♭
71	1136	27/14	^C♯, v4D
72	1152	35/18, 37/19	^^C♯, v3D
73	1168		^3C♯, vvD
74	1184		^4C♯, vD
75	1200	2/1	D

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