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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 3 × 5<sup>2</sup>
{{ED intro}}
| Step size = 16.0000¢
| Fifth = 44\75 (704.0¢)
| Semitones = 8:5 (120.0¢ : 80.0¢)
| Consistency = 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 [[cent]]s each.


== Theory ==
== Theory ==
In the 5-limit, 75et tempers out 20000/19683 ([[tetracot comma]]) and 2109375/2097152 ([[semicomma]]), and provides a good tuning for the [[tetracot]] temperament. In the 7-limit, it tempers [[225/224]] and [[1728/1715]]. In the 11-limit, 75e [[val]] {{val| 75 119 174 211 '''260''' }} scores lower in [[TE error|error]], and tempers [[100/99]] and [[243/242]], whereas the [[patent val]] {{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.
75et [[tempering out|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]], [[support]]ing [[bunya]] and [[orwell]], and providing the [[optimal patent val]] for [[fog]].  


It provides the [[optimal patent val]] for the [[fog]] temperament in the 7-limit and the [[Temperament merging|31 & 44 temperament]] in the 19-limit.  
In the [[11-limit]], 75e [[val]] {{val| 75 119 174 211 '''260''' }} (corresponding to [[#Riemann zeta function|401zpi]]) scores lower in [[TE error|error]], and tempers [[100/99]] and [[243/242]], whereas the [[patent val]] {{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. It is an excellent tuning for 2.3.5.11.13 [[tetracot]], and its extension [[bunya]] up to the full 19-limit.


Since 75 is part of the Fibonacci sequence beginning with 5 and 12, it closely approximates peppermint temperament. The size of its fifth is exactly 704c, which is very close to the peppermint fifth of 704.096c. This makes it suitable for neo-Gothic tunings. It also approximates the [[Carlos Beta]] scale well (<code>4\75 ≈ 1\[Carlos Beta]</code>).
Since 75 is part of the {{w|Fibonacci sequence}} beginning with [[5edo|5]] and [[12edo|12]], after [[46edo|46]] and before [[121edo|121]], it closely approximates the [[peppermint]] temperament. The size of its fifth is exactly 704{{c}}, 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 ({{nowrap|4\75 ≈ 1\Carlos Beta}}), though [[94edo]] does even better.


=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|75}}
{{Harmonics in equal|75}}
=== Riemann zeta function ===
The [[The_Riemann_zeta_function_and_tuning|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.
[[File:401zpi.png|200px|thumb|right|The Riemann zeta function around 75edo, showing 400zpi and 401zpi]]
Compare how prime harmonics are mapped in each zeta peak:
{{Harmonics in cet|16.0211986487005|title=Approximation of harmonics in 400zpi|intervals=prime|columns=11}}
{{Harmonics in cet|15.9805820697015|title=Approximation of harmonics in 401zpi|intervals=prime|columns=11}}


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2"
{{Interval table}}
|-
 
! #
== Notation ==
! Cents
 
|-
=== Sagittal notation ===
| 0
This notation uses the same sagittal sequence as [[68edo#Sagittal notation|68-EDO]].
|0
 
|-
==== Evo flavor ====
| 1
<imagemap>
| 16
File:75-EDO_Evo_Sagittal.svg
|-
desc none
| 2
rect 80 0 300 50 [[Sagittal_notation]]
| 32
rect 300 0 735 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
|-
rect 20 80 120 106 [[64/63]]
| 3
rect 120 80 220 106 [[81/80]]
| 48
rect 220 80 340 106 [[33/32]]
|-
rect 340 80 460 106 [[27/26]]
| 4
default [[File:75-EDO_Evo_Sagittal.svg]]
| 64
</imagemap>
|-
 
| 5
==== Revo flavor ====
| 80
<imagemap>
|-
File:75-EDO_Revo_Sagittal.svg
| 6
desc none
| 96
rect 80 0 300 50 [[Sagittal_notation]]
|-
rect 300 0 703 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| 7
rect 20 80 120 106 [[64/63]]
| 112
rect 120 80 220 106 [[81/80]]
|-
rect 220 80 340 106 [[33/32]]
| 8
rect 340 80 460 106 [[27/26]]
| 128
default [[File:75-EDO_Revo_Sagittal.svg]]
|-
</imagemap>
| 9
 
| 144
==== Evo-SZ flavor ====
|-
<imagemap>
| 10
File:75-EDO_Evo-SZ_Sagittal.svg
| 160
desc none
|-
rect 80 0 300 50 [[Sagittal_notation]]
| 11
rect 300 0 727 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| 176
rect 20 80 120 106 [[64/63]]
|-
rect 120 80 220 106 [[81/80]]
| 12
rect 220 80 340 106 [[33/32]]
| 192
rect 340 80 460 106 [[27/26]]
|-
default [[File:75-EDO_Evo-SZ_Sagittal.svg]]
| 13
</imagemap>
| 208
 
|-
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
| 14
 
| 224
=== Ups and downs notation ===
|-
75edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals:
| 15
 
| 240
{{Sharpness-sharp8}}
|-
| 16
| 256
|-
| 17
| 272
|-
| 18
| 288
|-
| 19
| 304
|-
| 20
| 320
|-
| 21
| 336
|-
| 22
| 352
|-
| 23
| 368
|-
| 24
| 384
|-
| 25
| 400
|-
| 26
| 416
|-
| 27
| 432
|-
| 28
| 448
|-
| 29
| 464
|-
| 30
| 480
|-
| 31
| 496
|-
| 32
| 512
|-
| 33
| 528
|-
| 34
| 544
|-
| 35
| 560
|-
| 36
| 576
|-
| 37
| 592
|-
| …
| …
|}


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
Line 155: Line 87:
| 2.3
| 2.3
| {{monzo| 119 -75 }}
| {{monzo| 119 -75 }}
| [{{val| 75 119 }}]
| {{mapping| 75 119 }}
| -0.645
| −0.645
| 0.645
| 0.645
| 4.03
| 4.03
Line 162: Line 94:
| 2.3.5
| 2.3.5
| 20000/19683, 2109375/2097152
| 20000/19683, 2109375/2097152
| [{{val| 75 119 174 }}]
| {{mapping| 75 119 174 }}
| -0.099
| −0.099
| 0.936
| 0.936
| 5.85
| 5.85
Line 169: Line 101:
| 2.3.5.7
| 2.3.5.7
| 225/224, 1728/1715, 15625/15309
| 225/224, 1728/1715, 15625/15309
| [{{val| 75 119 174 211 }}]
| {{mapping| 75 119 174 211 }}
| -0.713
| −0.713
| 1.337
| 1.337
| 8.36
| 8.36
|}
|}


[[Category:Equal divisions of the octave]]
== Instruments ==
 
A [[Lumatone mapping for 75edo]] is available.
 
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=5-G2KkYfKLs&list=WL&index=343&pp=gAQBiAQB8AUB ''microtonal improvisation in 75edo''] (2025-06-22)
* [https://www.youtube.com/shorts/QflMtKRmlSI ''microtonal improvisation in 75edo''] (2025-06-24)
* [https://www.youtube.com/watch?v=LsqNqHOfrBU ''Waltz in 75edo''] (2025) [https://www.youtube.com/shorts/sdN-5y3jhDY short clip demonstrating diatonic Lumatone mapping]
* [https://www.youtube.com/shorts/nlurS-3VYkA ''75edo improv''] (2025)
* [https://www.youtube.com/watch?v=GW-afWikisI ''Caprice in 75edo''] (2025)
 
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=oL6K6O4FBxc ''Fugue on The Lick''] (2019)
 
[[Category:Listen]]

Latest revision as of 23:36, 16 January 2026

← 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 21/75, 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. It is an excellent tuning for 2.3.5.11.13 tetracot, and its extension bunya up to the full 19-limit.

Since 75 is part of the Fibonacci sequence beginning with 5 and 12, after 46 and before 121, 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 cents. This makes it suitable for neo-Gothic tunings. It also approximates the Carlos Beta scale well (4\75 ≈ 1\Carlos Beta), though 94edo does even better.

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
Step 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
Step 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 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

Notation

Sagittal notation

This notation uses the same sagittal sequence as 68-EDO.

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation64/6381/8033/3227/26

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation64/6381/8033/3227/26

Evo-SZ flavor

Sagittal notationPeriodic table of EDOs with sagittal notation64/6381/8033/3227/26

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's primary comma (the comma it exactly represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it approximately represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

Ups and downs notation

75edo can be notated using ups and downs notation using Helmholtz–Ellis accidentals:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Sharp symbol  
Flat symbol

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

Instruments

A Lumatone mapping for 75edo is available.

Music

Bryan Deister
Claudi Meneghin
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