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] (corresponding to 401zpi) 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. It tempers out 325/324 and 512/507 in the 13-limit, 120/119 and 256/255 in the 17-limit, and 190/189 and 250/247 in the 19-limit.

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 cents, which is very close to the peppermint fifth of 704.096 cents. 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)

Riemann zeta function

The Riemann zeta function includes two peaks of similar magnitude around 75edo: 400zpi and 401zpi, corresponding to the 75dfghk and 75eij vals, with differing mappings for all primes above 5. 400zpi tempers out 686/675, 875/864, and 5120/5103 in the 7-limit, 121/120 and 441/440 in the 11-limit, 91/90, 352/351, and 2080/2079 in the 13-limit, 136/135 in the 17-limit, 190/189 in the 19-limit, and 161/160 in the 23-limit. 401zpi tempers out 20000/19683, 1728/1715, and 225/224 in the 7-limit, 100/99 and 2200/2187 in the 11-limit, 144/143 and 275/273 in the 13-limit, 120/119 and 1225/1224 in the 17-limit, 190/189 in the 19-limit, and 162/161 in the 23-limit. Its step is mapped to 49/48 (the slendro diesis) in 400zpi, but 64/63 (Archytas' comma) in 401zpi and 75p.

The Riemann zeta function around 75edo, showing 400zpi and 401zpi

Compare how prime harmonics are mapped in each zeta peak:

Approximation of harmonics in 400zpi

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+1.59	+4.57	+1.37	-4.37	-1.83	-2.66	-2.47	-2.77	+2.91	+2.14	-1.17
	Relative (%)	+9.9	+28.5	+8.6	-27.3	-11.4	-16.6	-15.4	-17.3	+18.2	+13.4	-7.3
Steps		75	119	174	210	259	277	306	318	339	364	371

Approximation of harmonics in 401zpi

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	-1.46	-0.27	-5.69	+3.08	+3.63	+2.07	+1.08	+0.29	+5.12	+3.34	-0.26
	Relative (%)	-9.1	-1.7	-35.6	+19.3	+22.7	+13.0	+6.8	+1.8	+32.1	+20.9	-1.6
Steps		75	119	174	211	260	278	307	319	340	365	372

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	^5D, E♭
6	96	18/17, 19/18, 37/35	^6D, v7E
7	112	16/15	^7D, v6E
8	128	14/13	D♯, v5E
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	^5F, G♭
24	384	5/4	^6F, v7G
25	400	29/23, 34/27	^7F, v6G
26	416		F♯, v5G
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		^5G, A♭
37	592	38/27	^6G, v7A
38	608	27/19, 37/26	^7G, v6A
39	624	33/23	G♯, v5A
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		^5A, B♭
50	800	27/17	^6A, v7B
51	816	8/5	^7A, v6B
52	832	21/13, 34/21	A♯, v5B
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	^5C, D♭
68	1088	15/8	^6C, v7D
69	1104	17/9, 36/19	^7C, v6D
70	1120		C♯, v5D
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

Music

Claudi Meneghin

• Fugue on The Lick (2019)