75edo: Difference between revisions

(13 intermediate revisions by 3 users not shown)

Line 1:

== 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]].

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.

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 {{w|Fibonacci sequence}} beginning with 5 and 12, it closely approximates the [[peppermint]] temperament. The size of its fifth is exactly 704 ~~cents~~, 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).

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 ===

Line 13:

=== 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]]

Line 25:

Line 24:

== Notation ==

===Sagittal notation===

=== Sagittal notation ===

In the following diagrams, 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.

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

~~====Evo flavor====~~

==== Evo flavor ====

File:75-EDO_Evo_Sagittal.svg

Line 43:

Line 40:

</imagemap>

====Revo flavor====

==== Revo flavor ====

File:75-EDO_Revo_Sagittal.svg

Line 57:

Line 53:

</imagemap>

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

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

File:75-EDO_Evo-SZ_Sagittal.svg

Line 70:

Line 65:

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 85:

Line 88:

| {{monzo| 119 -75 }}

| {{mapping| 75 119 }}

| -0.645

| −0.645

| 0.645

| 4.03

Line 92:

Line 95:

| 20000/19683, 2109375/2097152

| {{mapping| 75 119 174 }}

| -0.099

| −0.099

| 0.936

| 5.85

Line 99:

Line 102:

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

| {{mapping| 75 119 174 211 }}

| -0.713

| −0.713

| 1.337

| 8.36

|}

== 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]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro}}
+{{ED intro}}
 == Theory ==
 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''' }} (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.
+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 {{w|Fibonacci sequence}} beginning with 5 and 12, it closely approximates the [[peppermint]] temperament. The size of its fifth is exactly 704 cents, 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).
+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 ===
@@ Line 13: / Line 13: @@
 === 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]]
@@ Line 25: / Line 24: @@
 == Notation ==
-===Sagittal notation===
+=== Sagittal notation ===
-In the following diagrams, 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.
 This notation uses the same sagittal sequence as [[68edo#Sagittal notation|68-EDO]].
-====Evo flavor====
+==== Evo flavor ====
 <imagemap>
 File:75-EDO_Evo_Sagittal.svg
@@ Line 43: / Line 40: @@
 </imagemap>
-====Revo flavor====
+==== Revo flavor ====
 <imagemap>
 File:75-EDO_Revo_Sagittal.svg
@@ Line 57: / Line 53: @@
 </imagemap>
-====Evo-SZ flavor====
+==== Evo-SZ flavor ====
 <imagemap>
 File:75-EDO_Evo-SZ_Sagittal.svg
@@ Line 70: / Line 65: @@
 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 85: / Line 88: @@
 | {{monzo| 119 -75 }}
 | {{mapping| 75 119 }}
-| -0.645
+| −0.645
 | 0.645
 | 4.03
@@ Line 92: / Line 95: @@
 | 20000/19683, 2109375/2097152
 | {{mapping| 75 119 174 }}
-| -0.099
+| −0.099
 | 0.936
 | 5.85
@@ Line 99: / Line 102: @@
 | 225/224, 1728/1715, 15625/15309
 | {{mapping| 75 119 174 211 }}
-| -0.713
+| −0.713
 | 1.337
 | 8.36
 |}
+== 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]]