75edo: Difference between revisions

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~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~

~~: This revision was by author~~ [[~~User:genewardsmith~~|~~genewardsmith~~]] and ~~made on <tt>2011~~-08-~~25 17:31:31 UTC</tt>~~.~~<br>~~

~~: The original revision id was <tt>248512535<~~/~~tt>~~.~~<br>~~

== Theory ==

~~: The revision comment was: <tt><~~/~~tt><br>~~

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]].

~~The revision contents are below~~, ~~presented both~~ in the ~~original Wikispaces Wikitext format~~, and in ~~HTML~~ exactly ~~as Wikispaces rendered~~ it.~~<br>~~

~~<h4>Original Wikitext content:</h4>~~

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.

~~<div style~~=~~"width~~:~~100%; max~~-~~height:400pt; overflow:auto; background~~-~~color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word~~-~~wrap:break~~-~~word;white~~-~~space: pre~~-~~wrap ! important" class="old~~-~~revision~~-~~html">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~~]].</~~pre>~~</~~div~~>

<h4>~~Original HTML content~~:</h4>

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.

<~~div style="width~~:~~100%; max~~-~~height:400pt; overflow:auto; background~~-~~color~~:#~~f8f9fa; border~~: ~~1px solid~~ #~~eaecf0; padding~~:~~0em"><pre style~~="~~margin:0px;border:none;background:none;word~~-~~wrap:break~~-~~word;width:200%;white~~-~~space: pre~~-~~wrap~~ ! ~~important~~" ~~class~~="~~old~~-~~revision~~-~~html"><html><head><title>75edo</title></head><body>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 ~~&lt~~;~~a class~~=~~"wiki_link" href~~=~~"/Tetracot%20family"~~&~~gt;tetracot temperament~~&~~lt;~~/~~a>~~.~~<~~/~~body>&lt~~;/~~html><~~/~~pre><~~/~~div>~~

=== 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]]

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

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

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3

| {{monzo| 119 -75 }}

| {{mapping| 75 119 }}

| −0.645

| 0.645

| 4.03

|-

| 2.3.5

| 20000/19683, 2109375/2097152

| {{mapping| 75 119 174 }}

| −0.099

| 0.936

| 5.85

|-

| 2.3.5.7

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

| {{mapping| 75 119 174 211 }}

| −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: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{Infobox ET}}
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+{{ED intro}}
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-25 17:31:31 UTC</tt>.<br>
-: The original revision id was <tt>248512535</tt>.<br>
+== Theory ==
-: The revision comment was: <tt></tt><br>
+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]].
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
-<h4>Original Wikitext content:</h4>
+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.
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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]].</pre></div>
-<h4>Original HTML content:</h4>
+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.
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;75edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;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 &lt;a class="wiki_link" href="/Tetracot%20family"&gt;tetracot temperament&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
+=== 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]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br>8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.3
+| {{monzo| 119 -75 }}
+| {{mapping| 75 119 }}
+| −0.645
+| 0.645
+| 4.03
+|-
+| 2.3.5
+| 20000/19683, 2109375/2097152
+| {{mapping| 75 119 174 }}
+| −0.099
+| 0.936
+| 5.85
+|-
+| 2.3.5.7
+| 225/224, 1728/1715, 15625/15309
+| {{mapping| 75 119 174 211 }}
+| −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]]