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'''75edo''' divides the octave into 75 equal parts of exactly 16 cents each. In the 5-limit, it tempers out the tetracot comma, 20000/19683 and the semicomma 2109375/2097152, and provides a good tuning for [[Tetracot_family|tetracot temperament]]. It provides the optimal patent val for the [[M&N_temperaments|12&51 temperament]] in the 7-limit and the [[M&N_temperaments|31&75 temperament]] in the 13-limit.
{{Infobox ET}}
{{ED intro}}


===Intervals===
== Theory ==
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]].


{| class="wikitable"
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.
|-
 
| rowspan="2" | '''Step'''
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.
| colspan="2" | '''Size in'''
 
|-
=== Odd harmonics ===
|'''Cents'''
{{Harmonics in equal|75}}
|'''7mus'''
 
|-
=== Riemann zeta function ===
| colspan="3" | 0
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]]
| | 1
Compare how prime harmonics are mapped in each zeta peak:
| | 16
{{Harmonics in cet|16.0211986487005|title=Approximation of harmonics in 400zpi|intervals=prime|columns=11}}
|20.48 (14.7B<sub>16</sub>)
{{Harmonics in cet|15.9805820697015|title=Approximation of harmonics in 401zpi|intervals=prime|columns=11}}
|-
 
| | 2
== Intervals ==
| | 32
{{Interval table}}
|40.96 (28.F6<sub>16</sub>)
 
|-
== Notation ==
| | 3
 
| | 48
=== Sagittal notation ===
|61.44 (3D.71<sub>16</sub>)
This notation uses the same sagittal sequence as [[68edo#Sagittal notation|68-EDO]].
|-
 
| | 4
==== Evo flavor ====
| | 64
<imagemap>
|81.92 (51.EB8<sub>16</sub>)
File:75-EDO_Evo_Sagittal.svg
|-
desc none
| | 5
rect 80 0 300 50 [[Sagittal_notation]]
| | 80
rect 300 0 735 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
|102.4 (66.66<sub>16</sub>)
rect 20 80 120 106 [[64/63]]
|-
rect 120 80 220 106 [[81/80]]
| | 6
rect 220 80 340 106 [[33/32]]
| | 96
rect 340 80 460 106 [[27/26]]
|122.88 (7A.E1<sub>16</sub>)
default [[File:75-EDO_Evo_Sagittal.svg]]
|-
</imagemap>
| | 7
 
| | 112
==== Revo flavor ====
|143.36 (8F.5C<sub>16</sub>)
<imagemap>
|-
File:75-EDO_Revo_Sagittal.svg
| | 8
desc none
| | 128
rect 80 0 300 50 [[Sagittal_notation]]
|163.84 (A3.D7<sub>16</sub>)
rect 300 0 703 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
|-
rect 20 80 120 106 [[64/63]]
| | 9
rect 120 80 220 106 [[81/80]]
| | 144
rect 220 80 340 106 [[33/32]]
|184.32 (B8.52<sub>16</sub>)
rect 340 80 460 106 [[27/26]]
|-
default [[File:75-EDO_Revo_Sagittal.svg]]
| | 10
</imagemap>
| | 160
 
|204.8 (CC.CD<sub>16</sub>)
==== Evo-SZ flavor ====
|-
<imagemap>
| | 11
File:75-EDO_Evo-SZ_Sagittal.svg
| | 176
desc none
|225.28 (E1.48<sub>16</sub>)
rect 80 0 300 50 [[Sagittal_notation]]
|-
rect 300 0 727 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| | 12
rect 20 80 120 106 [[64/63]]
| | 192
rect 120 80 220 106 [[81/80]]
|245.76 (F5.C3<sub>16</sub>)
rect 220 80 340 106 [[33/32]]
|-
rect 340 80 460 106 [[27/26]]
| | 13
default [[File:75-EDO_Evo-SZ_Sagittal.svg]]
| | 208
</imagemap>
|266.24 (10A.3D<sub>16</sub>)
 
|-
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 ===
|286.72 (11E.B8<sub>16</sub>)
75edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals:
|-
 
| | 15
{{Sharpness-sharp8}}
| | 240
 
|307.2 (133.33<sub>16</sub>)
== Regular temperament properties ==
|-
{| class="wikitable center-4 center-5 center-6"
| | 16
| | 256
|327.68 (147.AE<sub>16</sub>)
|-
| | 17
| | 272
|348.16 (15C.29<sub>16</sub>)
|-
|-
| | 18
! rowspan="2" | [[Subgroup]]
| | 288
! rowspan="2" | [[Comma list]]
|368.64 (170.A4<sub>16</sub>)
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| | 19
! [[TE error|Absolute]] (¢)
| | 304
! [[TE simple badness|Relative]] (%)
|389.12 (185.1F<sub>16</sub>)
|-
|-
| | 20
| 2.3
| | 320
| {{monzo| 119 -75 }}
|409.6 (199.9A<sub>16</sub>)
| {{mapping| 75 119 }}
| −0.645
| 0.645
| 4.03
|-
|-
| | 21
| 2.3.5
| | 336
| 20000/19683, 2109375/2097152
|430.08 (1AE.148<sub>16</sub>)
| {{mapping| 75 119 174 }}
| −0.099
| 0.936
| 5.85
|-
|-
| | 22
| 2.3.5.7
| | 352
| 225/224, 1728/1715, 15625/15309
|450.56 (1C2.8F<sub>16</sub>)
| {{mapping| 75 119 174 211 }}
|-
| −0.713
| | 23
| 1.337
| | 368
| 8.36
|471.04 (1D7.0A<sub>16</sub>)
|-
| | 24
| | 384
|491.52 (1E9.85<sub>16</sub>)
|-
| | 25
| | 400
|512 (200<sub>16</sub>)
|-
| | 26
| | 416
|532.48 (214.7B<sub>16</sub>)
|-
| | 27
| | 432
|552.96 (228.F6<sub>16</sub>)
|-
| | 28
| | 448
|573.44 (23D.71<sub>16</sub>)
|-
| | 29
| | 464
|593.92 (251.EB8<sub>16</sub>)
|-
| | 30
| | 480
|614.4 (266.66<sub>16</sub>)
|-
| | 31
| | 496
|634.88 (27A.E1<sub>16</sub>)
|-
| | 32
| | 512
|655.36 (28F.5C<sub>16</sub>)
|-
| | 33
| | 528
|675.84 (2A3.D7<sub>16</sub>)
|-
| | 34
| | 544
|696.32 (2B8.52<sub>16</sub>)
|-
| | 35
| | 560
|716.8 (2CC.CD<sub>16</sub>)
|-
| | 36
| | 576
|737.28 (2E1.48<sub>16</sub>)
|-
| | 37
| | 592
|757.76 (2F5.C3<sub>16</sub>)
|-
| | 38
| | 608
|778.24 (30A.3D<sub>16</sub>)
|-
| | 39
| | 624
|798.72 (31E.B8<sub>16</sub>)
|-
| | 40
| | 640
|819.2 (333.33<sub>16</sub>)
|-
| | 41
| | 656
|839.68 (347.AE<sub>16</sub>)
|-
| | 42
| | 672
|860.16 (35C.29<sub>16</sub>)
|-
| | 43
| | 688
|880.64 (370.A4<sub>16</sub>)
|-
| | 44
| | 704
|901.12 (385.1F<sub>16</sub>)
|-
| | 45
| | 720
|921.6 (399.9A<sub>16</sub>)
|-
| | 46
| | 736
|942.08 (3AE.148<sub>16</sub>)
|-
| | 47
| | 752
|962.56 (3C2.8F<sub>16</sub>)
|-
| | 48
| | 768
|983.04 (3D7.0A<sub>16</sub>)
|-
| | 49
| | 784
|1003.52 (3E9.85<sub>16</sub>)
|-
| | 50
| | 800
|1024 (400<sub>16</sub>)
|-
| | 51
| | 816
|1044.48 (414.7B<sub>16</sub>)
|-
| | 52
| | 832
|1064.96 (428.F6<sub>16</sub>)
|-
| | 53
| | 848
|1085.44 (43D.71<sub>16</sub>)
|-
| | 54
| | 864
|1105.92 (551.EB8<sub>16</sub>)
|-
| | 55
| | 880
|1126.4 (466.66<sub>16</sub>)
|-
| | 56
| | 896
|1146.88 (47A.E1<sub>16</sub>)
|-
| | 57
| | 912
|1167.36 (48F.5C<sub>16</sub>)
|-
| | 58
| | 928
|1187.84 (4A3.D7<sub>16</sub>)
|-
| | 59
| | 944
|1208.32 (4B8.52<sub>16</sub>)
|-
| | 60
| | 960
|1228.8 (4CC.CD<sub>16</sub>)
|-
| | 61
| | 976
|1249.28 (4E1.48<sub>16</sub>)
|-
| | 62
| | 992
|1269.76 (4F5.C3<sub>16</sub>)
|-
| | 63
| | 1008
|1290.24 (50A.3D<sub>16</sub>)
|-
| | 64
| | 1024
|1310.72 (51E.B8<sub>16</sub>)
|-
| | 65
| | 1040
|1331.2 (533.33<sub>16</sub>)
|-
| | 66
| | 1056
|1351.68 (547.AE<sub>16</sub>)
|-
| | 67
| | 1072
|1372.16 (55C.29<sub>16</sub>)
|-
| | 68
| | 1088
|1392.64 (570.A4<sub>16</sub>)
|-
| | 69
| | 1104
|1413.12 (585.1F<sub>16</sub>)
|-
| | 70
| | 1120
|1433.6 (599.9A<sub>16</sub>)
|-
| | 71
| | 1136
|1454.08 (5AE.148<sub>16</sub>)
|-
| | 72
| | 1152
|1474.56 (5C2.8F<sub>16</sub>)
|-
| | 73
| | 1168
|1495.04 (5D7.0A<sub>16</sub>)
|-
| | 74
| | 1184
|1515.52 (5E9.85<sub>16</sub>)
|-
|75
|1200
|1536 (600<sub>16</sub>)
|}
|}
== 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]]
Retrieved from "https://en.xen.wiki/w/75edo"