Xenharmonic Wiki

75edo: Difference between revisions

← Older edit
VisualWikitext
Jkarimak (talk | contribs)
191 edits
→Intervals: Removed 7mus column
Overthink (talk | contribs)
autopatrolled
2,783 edits
→Theory: a few notes
 
(44 intermediate revisions by 16 users not shown)
Line 1: Line 1:
'''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.
|-
 
| '''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.
|'''Cents'''
 
|-
=== Odd harmonics ===
| 0
{{Harmonics in equal|75}}
|0
 
|-
=== Riemann zeta function ===
| | 1
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.
| | 16
[[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:
| | 2
{{Harmonics in cet|16.0211986487005|title=Approximation of harmonics in 400zpi|intervals=prime|columns=11}}
| | 32
{{Harmonics in cet|15.9805820697015|title=Approximation of harmonics in 401zpi|intervals=prime|columns=11}}
|-
 
| | 3
== Intervals ==
| | 48
{{Interval table}}
|-
 
| | 4
== Notation ==
| | 64
 
|-
=== Sagittal notation ===
| | 5
This notation uses the same sagittal sequence as [[68edo#Sagittal notation|68-EDO]].
| | 80
 
|-
==== Evo flavor ====
| | 6
<imagemap>
| | 96
File:75-EDO_Evo_Sagittal.svg
|-
desc none
| | 7
rect 80 0 300 50 [[Sagittal_notation]]
| | 112
rect 300 0 735 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
|-
rect 20 80 120 106 [[64/63]]
| | 8
rect 120 80 220 106 [[81/80]]
| | 128
rect 220 80 340 106 [[33/32]]
|-
rect 340 80 460 106 [[27/26]]
| | 9
default [[File:75-EDO_Evo_Sagittal.svg]]
| | 144
</imagemap>
|-
 
| | 10
==== Revo flavor ====
| | 160
<imagemap>
|-
File:75-EDO_Revo_Sagittal.svg
| | 11
desc none
| | 176
rect 80 0 300 50 [[Sagittal_notation]]
|-
rect 300 0 703 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]]
|-
rect 220 80 340 106 [[33/32]]
| | 13
rect 340 80 460 106 [[27/26]]
| | 208
default [[File:75-EDO_Revo_Sagittal.svg]]
|-
</imagemap>
| | 14
 
| | 224
==== Evo-SZ flavor ====
|-
<imagemap>
| | 15
File:75-EDO_Evo-SZ_Sagittal.svg
| | 240
desc none
|-
rect 80 0 300 50 [[Sagittal_notation]]
| | 16
rect 300 0 727 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| | 256
rect 20 80 120 106 [[64/63]]
|-
rect 120 80 220 106 [[81/80]]
| | 17
rect 220 80 340 106 [[33/32]]
| | 272
rect 340 80 460 106 [[27/26]]
|-
default [[File:75-EDO_Evo-SZ_Sagittal.svg]]
| | 18
</imagemap>
| | 288
 
|-
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.
| | 19
 
| | 304
=== Ups and downs notation ===
|-
75edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals:
| | 20
 
| | 320
{{Sharpness-sharp8}}
|-
 
| | 21
== Regular temperament properties ==
| | 336
{| class="wikitable center-4 center-5 center-6"
|-
| | 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
|-
| | 38
| | 608
|-
| | 39
| | 624
|-
| | 40
| | 640
|-
|-
| | 41
! rowspan="2" | [[Subgroup]]
| | 656
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| | 42
! [[TE error|Absolute]] (¢)
| | 672
! [[TE simple badness|Relative]] (%)
|-
|-
| | 43
| 2.3
| | 688
| {{monzo| 119 -75 }}
| {{mapping| 75 119 }}
| −0.645
| 0.645
| 4.03
|-
|-
| | 44
| 2.3.5
| | 704
| 20000/19683, 2109375/2097152
| {{mapping| 75 119 174 }}
| −0.099
| 0.936
| 5.85
|-
|-
| | 45
| 2.3.5.7
| | 720
| 225/224, 1728/1715, 15625/15309
|-
| {{mapping| 75 119 174 211 }}
| | 46
| −0.713
| | 736
| 1.337
|-
| 8.36
| | 47
| | 752
|-
| | 48
| | 768
|-
| | 49
| | 784
|-
| | 50
| | 800
|-
| | 51
| | 816
|-
| | 52
| | 832
|-
| | 53
| | 848
|-
| | 54
| | 864
|-
| | 55
| | 880
|-
| | 56
| | 896
|-
| | 57
| | 912
|-
| | 58
| | 928
|-
| | 59
| | 944
|-
| | 60
| | 960
|-
| | 61
| | 976
|-
| | 62
| | 992
|-
| | 63
| | 1008
|-
| | 64
| | 1024
|-
| | 65
| | 1040
|-
| | 66
| | 1056
|-
| | 67
| | 1072
|-
| | 68
| | 1088
|-
| | 69
| | 1104
|-
| | 70
| | 1120
|-
| | 71
| | 1136
|-
| | 72
| | 1152
|-
| | 73
| | 1168
|-
| | 74
| | 1184
|-
|75
|1200
|}
|}
== 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"