75edo: Difference between revisions

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== Theory ==

~~In the 5-limit,~~ 75et tempers out 20000/19683 ([[tetracot comma]]) and 2109375/2097152 ([[semicomma]]), and provides a good tuning for the [[tetracot]] temperament. ~~In the 7-limit, it~~ tempers [[225/224]] and [[1728/1715]]~~. 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~~.

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]].

~~It provides~~ the [[~~optimal~~ patent val]] ~~for~~ the [[~~fog~~]] ~~temperament in the 7~~-limit and ~~the~~ [[~~Temperament merging|31 & 44 temperament~~]] ~~in the~~ 19-limit.

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.

Since 75 is part of the Fibonacci sequence beginning with 5 and 12, it closely approximates peppermint temperament. The size of its fifth is exactly ~~704c~~, which is very close to the peppermint fifth of 704.~~096c~~. 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 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>).

=== Odd harmonics ===

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 == Theory ==
-In the 5-limit, 75et tempers out 20000/19683 ([[tetracot comma]]) and 2109375/2097152 ([[semicomma]]), and provides a good tuning for the [[tetracot]] temperament. In the 7-limit, it tempers [[225/224]] and [[1728/1715]]. 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.
+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]].
-It provides the [[optimal patent val]] for the [[fog]] temperament in the 7-limit and the [[Temperament merging|31 & 44 temperament]] in the 19-limit.
+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.
-Since 75 is part of the Fibonacci sequence beginning with 5 and 12, it closely approximates peppermint temperament. The size of its fifth is exactly 704c, which is very close to the peppermint fifth of 704.096c. 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 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>).
 === Odd harmonics ===