75edo: Difference between revisions

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== 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]].

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.

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).

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=== 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 ==

@@ Line 3: / Line 3: @@
 == 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]].
+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.
 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).
@@ Line 11: / Line 11: @@
 === 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 ==