75edo: Difference between revisions

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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 ==

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, [[support]]ing [[bunya]] and [[orwell]], and providing the [[optimal patent val]] for [[fog]].

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]].

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.

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 the [[peppermint]] temperament. The size of its fifth is exactly 704 {{~~cent~~}}, which is very close to the peppermint fifth of 704.096 ~~{{cent}}~~. 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 ===

=== 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 ==

== Notation ==

=== Sagittal notation ===

This notation uses the same sagittal sequence as [[68edo#Sagittal notation|68-EDO]].

==== Evo flavor ====

File:75-EDO_Evo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 735 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 120 106 [[64/63]]

rect 120 80 220 106 [[81/80]]

rect 220 80 340 106 [[33/32]]

rect 340 80 460 106 [[27/26]]

default [[File:75-EDO_Evo_Sagittal.svg]]

</imagemap>

==== Revo flavor ====

File:75-EDO_Revo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 703 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 120 106 [[64/63]]

rect 120 80 220 106 [[81/80]]

rect 220 80 340 106 [[33/32]]

rect 340 80 460 106 [[27/26]]

default [[File:75-EDO_Revo_Sagittal.svg]]

</imagemap>

==== Evo-SZ flavor ====

File:75-EDO_Evo-SZ_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 727 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 120 106 [[64/63]]

rect 120 80 220 106 [[81/80]]

rect 220 80 340 106 [[33/32]]

rect 340 80 460 106 [[27/26]]

default [[File:75-EDO_Evo-SZ_Sagittal.svg]]

</imagemap>

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.

=== Ups and downs notation ===

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

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal<br>8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal<br>8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

Line 28:

Line 87:

| 2.3

| {{monzo| 119 -75 }}

| [{{~~val~~| 75 119 }}]

| {{mapping| 75 119 }}

| -0.645

| −0.645

| 0.645

| 4.03

Line 35:

Line 94:

| 2.3.5

| 20000/19683, 2109375/2097152

| [{{~~val~~| 75 119 174 }}]

| {{mapping| 75 119 174 }}

| -0.099

| −0.099

| 0.936

| 5.85

Line 42:

Line 101:

| 2.3.5.7

| 225/224, 1728/1715, 15625/15309

| [{{~~val~~| 75 119 174 211 }}]

| {{mapping| 75 119 174 211 }}

| -0.713

| −0.713

| 1.337

| 8.36

|}

== Instruments ==

A [[Lumatone mapping for 75edo]] is available.

== Music ==

* [https://www.youtube.com/watch?v=oL6K6O4FBxc Fugue on The Lick] by [[~~Claudi Meneghin~~]]

; [[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]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-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.
+{{ED intro}}
 == Theory ==
-et 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]].
+et [[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]].
-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.
+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 the [[peppermint]] temperament. The size of its fifth is exactly 704 {{cent}}, which is very close to the peppermint fifth of 704.096 {{cent}}. 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 ===
 {{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 ==
 {{Interval table}}
+== Notation ==
+=== Sagittal notation ===
+This notation uses the same sagittal sequence as [[68edo#Sagittal notation|68-EDO]].
+==== Evo flavor ====
+<imagemap>
+File:75-EDO_Evo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 735 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 120 106 [[64/63]]
+rect 120 80 220 106 [[81/80]]
+rect 220 80 340 106 [[33/32]]
+rect 340 80 460 106 [[27/26]]
+default [[File:75-EDO_Evo_Sagittal.svg]]
+</imagemap>
+==== Revo flavor ====
+<imagemap>
+File:75-EDO_Revo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 703 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 120 106 [[64/63]]
+rect 120 80 220 106 [[81/80]]
+rect 220 80 340 106 [[33/32]]
+rect 340 80 460 106 [[27/26]]
+default [[File:75-EDO_Revo_Sagittal.svg]]
+</imagemap>
+==== Evo-SZ flavor ====
+<imagemap>
+File:75-EDO_Evo-SZ_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 727 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 120 106 [[64/63]]
+rect 120 80 220 106 [[81/80]]
+rect 220 80 340 106 [[33/32]]
+rect 340 80 460 106 [[27/26]]
+default [[File:75-EDO_Evo-SZ_Sagittal.svg]]
+</imagemap>
+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.
+=== Ups and downs notation ===
+edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals:
+{{Sharpness-sharp8}}
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 28: / Line 87: @@
 | 2.3
 | {{monzo| 119 -75 }}
-| [{{val| 75 119 }}]
+| {{mapping| 75 119 }}
-| -0.645
+| −0.645
 | 0.645
 | 4.03
@@ Line 35: / Line 94: @@
 | 2.3.5
 | 20000/19683, 2109375/2097152
-| [{{val| 75 119 174 }}]
+| {{mapping| 75 119 174 }}
-| -0.099
+| −0.099
 | 0.936
 | 5.85
@@ Line 42: / Line 101: @@
 | 2.3.5.7
 | 225/224, 1728/1715, 15625/15309
-| [{{val| 75 119 174 211 }}]
+| {{mapping| 75 119 174 211 }}
-| -0.713
+| −0.713
 | 1.337
 | 8.36
 |}
+== Instruments ==
+A [[Lumatone mapping for 75edo]] is available.
 == Music ==
-* [https://www.youtube.com/watch?v=oL6K6O4FBxc Fugue on The Lick] by [[Claudi Meneghin]]
+; [[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]]