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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 tenpers [[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.
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]]
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