75edo: Difference between revisions

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== 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.

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 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).

=== Odd harmonics ===

Line 28:

| 2.3

| {{monzo| 119 -75 }}

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

| {{mapping| 75 119 }}

| -0.645

| 0.645

Line 35:

| 2.3.5

| 20000/19683, 2109375/2097152

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

| {{mapping| 75 119 174 }}

| -0.099

| 0.936

Line 42:

| 2.3.5.7

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

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

| {{mapping| 75 119 174 211 }}

| -0.713

| 1.337

Line 49:

== Music ==

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

; [[Claudi Meneghin]]

* [https://www.youtube.com/watch?v=oL6K6O4FBxc ''Fugue on The Lick''] (2019)

[[Category:Listen]]

@@ Line 3: / Line 3: @@
 == 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.
-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 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).
 === Odd harmonics ===
@@ Line 28: / Line 28: @@
 | 2.3
 | {{monzo| 119 -75 }}
-| [{{val| 75 119 }}]
+| {{mapping| 75 119 }}
 | -0.645
 | 0.645
@@ Line 35: / Line 35: @@
 | 2.3.5
 | 20000/19683, 2109375/2097152
-| [{{val| 75 119 174 }}]
+| {{mapping| 75 119 174 }}
 | -0.099
 | 0.936
@@ Line 42: / Line 42: @@
 | 2.3.5.7
 | 225/224, 1728/1715, 15625/15309
-| [{{val| 75 119 174 211 }}]
+| {{mapping| 75 119 174 211 }}
 | -0.713
 | 1.337
@@ Line 49: / Line 49: @@
 == Music ==
-* [https://www.youtube.com/watch?v=oL6K6O4FBxc Fugue on The Lick] by [[Claudi Meneghin]]
+; [[Claudi Meneghin]]
+* [https://www.youtube.com/watch?v=oL6K6O4FBxc ''Fugue on The Lick''] (2019)
 [[Category:Listen]]