User:Squib: Difference between revisions

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todo: ~~catalog & describe all 37 tunings up to n=6~~

todo: some stuff

==list of things that i think could be improved about the wiki==

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- it's hard to find a page you're looking for even if you know what it's about

-

- probably something else i forgot

==unnamed music theory==

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Just like when the layout was changed to fourths and fifths, bringing more complex intervals closer pushes larger intervals further away, but this time the range is reduced to only a couple octaves and the intervals far to the side are very complex and not very useful. We can make convenient ether very large or very complex intervals, but not both. The layout shown is in my opinion the best trade-off; it's possible to make 5/4 and especially 10/9 more convenient, but then it can barely fit a single 3/1! So is there a less complex way to approximate 5?

The other way to approximate 5/4 is to indeed add more notes between the existing ones. The tuning we already have is called Pythagorean tuning because it uses only ratios of 2 and 3. We can take two sets of Pythagorean and put them together. There are an infinite number of ways to do this, but only 3 preserve isomorphism. (In fact, the number of equal tunings with N sets of pythagorean is exactly equal to the sum of the unique factors of N. Don't ask me why, although I suspect figuring it out will help me name them.)

The other way to approximate 5/4 is to indeed add more notes between the existing ones. The tuning we already have is called Pythagorean tuning because it uses only ratios of 2 and 3. We can take two sets of Pythagorean and put them together. There are an infinite number of ways to do this, but only 3 preserve isomorphism. In fact, the number of equal tunings with N sets of pythagorean is exactly equal to the sum of the unique factors of N. Don't ask me why, although I suspect figuring it out will help me name them.

Explanation (that I still don't entirely understand) from a mathy discord server: "These arrangements of n integer lattices must be lattices themselves, so they have a basis. As a basis, we can take {(a,0), (b,c)}, where a is the least positive value such that (a,0) is in the lattice, c is the least positive value such that (x,c) is in the lattice for some x, and b is the least nonnegative value such that (b,c) is in the lattice. a has to be 1/k for integer k because (0,1) is in the lattice, and since the lattice has n copies of the integer lattice, k must be a divisor of n. This means there are n/k integer lattices with points on the x-axis, so c must be k/n. In order for the lattice to close, b must be i/n for some integer i, and since b is minimal, 0 ≤ i/n < 1/k, so 0 ≤ i < n/k and there are n/k possible values of i. Since all divisors of n are of the form n/k for some k | n, the number of lattices is the sum of the divisors of n."

[[File:First 33 tunings.jpg|thumb|Sketch of the 33 tunings from n=1 to n=6 in no particular order, and example intervals in the top right]]

===Overview===

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** what this is & why i made it

** how much i'm taking credit for

** no, i don't expect this to become widely-used. although that would be cool.

* philosophy

** creating the theory based on experience and what actually works/is helpful

@@ Line 1: / Line 1: @@
-todo: catalog & describe all 37 tunings up to n=6
+todo: some stuff
 ==list of things that i think could be improved about the wiki==
@@ Line 8: / Line 8: @@
 - it's hard to find a page you're looking for even if you know what it's about
--
+- probably something else i forgot
 ==unnamed music theory==
@@ Line 49: / Line 49: @@
 Just like when the layout was changed to fourths and fifths, bringing more complex intervals closer pushes larger intervals further away, but this time the range is reduced to only a couple octaves and the intervals far to the side are very complex and not very useful. We can make convenient ether very large or very complex intervals, but not both. The layout shown is in my opinion the best trade-off; it's possible to make 5/4 and especially 10/9 more convenient, but then it can barely fit a single 3/1! So is there a less complex way to approximate 5?
-The other way to approximate 5/4 is to indeed add more notes between the existing ones. The tuning we already have is called Pythagorean tuning because it uses only ratios of 2 and 3. We can take two sets of Pythagorean and put them together. There are an infinite number of ways to do this, but only 3 preserve isomorphism. (In fact, the number of equal tunings with N sets of pythagorean is exactly equal to the sum of the unique factors of N. Don't ask me why, although I suspect figuring it out will help me name them.)
+The other way to approximate 5/4 is to indeed add more notes between the existing ones. The tuning we already have is called Pythagorean tuning because it uses only ratios of 2 and 3. We can take two sets of Pythagorean and put them together. There are an infinite number of ways to do this, but only 3 preserve isomorphism. In fact, the number of equal tunings with N sets of pythagorean is exactly equal to the sum of the unique factors of N. Don't ask me why, although I suspect figuring it out will help me name them.
+Explanation (that I still don't entirely understand) from a mathy discord server: "These arrangements of n integer lattices must be lattices themselves, so they have a basis. As a basis, we can take {(a,0), (b,c)}, where a is the least positive value such that (a,0) is in the lattice, c is the least positive value such that (x,c) is in the lattice for some x, and b is the least nonnegative value such that (b,c) is in the lattice. a has to be 1/k for integer k because (0,1) is in the lattice, and since the lattice has n copies of the integer lattice, k must be a divisor of n. This means there are n/k integer lattices with points on the x-axis, so c must be k/n. In order for the lattice to close, b must be i/n for some integer i, and since b is minimal, 0 ≤ i/n < 1/k, so 0 ≤ i < n/k and there are n/k possible values of i. Since all divisors of n are of the form n/k for some k | n, the number of lattices is the sum of the divisors of n."
+[[File:First 33 tunings.jpg|thumb|Sketch of the 33 tunings from n=1 to n=6 in no particular order, and example intervals in the top right]]
 ===Overview===
@@ Line 57: / Line 61: @@
 ** what this is & why i made it
 ** how much i'm taking credit for
+** no, i don't expect this to become widely-used. although that would be cool.
 * philosophy
 ** creating the theory based on experience and what actually works/is helpful