User:Squib: Difference between revisions

Line 51:

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]]

[[File:First 33 tunings.jpg|left|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 51: / Line 51: @@
 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]]
+[[File:First 33 tunings.jpg|left|thumb|Sketch of the 33 tunings from n=1 to n=6 in no particular order, and example intervals in the top right]]
 ===Overview===