User:Ganaram inukshuk/Notes: Difference between revisions

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== Mosses related to metallic mosses ==

=== Fibonacci numbers and the golden ratio ===

Let F(n) be a recursive function that returns the nth Fibonacci number.

* For the base cases of n = 1 or n = 0:

** If n = 1, then F(1) = 1.

** If n = 0, then F(0) = 0.

* For the recursive case of n > 1:

** If n > 1, then F(n) = F(n-1) + F(n-2)

Mosses whose step ratio approximates the golden ratio will have a step ratio L:s that is F(n):F(n-1), or two consecutive Fibonacci numbers. In relation to a parent mos xL ys, mosses of an arbitrarily large step ratio F(n):F(n-1) (where n is arbitrarily large) will contain a sequence of mosses of the form (F(k)x+F(k-1)y)L (F(k-1)x+F(k-2)y)s, where F(k), F(k-1), and F(k-2) are the kth, (k-1)th, and (k-2)th Fibonacci numbers. Due to mos recursion, the mos F(n)x+F(n-1)y)L (F(n-1)x+F(n-2)y)s contains xL ys, as well as every mos between xL ys and F(n)x+F(n-1)y)L (F(n-1)x+F(n-2)y)s. As an example, golden meantone describes the mos 5L 2s whose step ratio approaches the golden ratio. This also describes a series of mos descendants that contain 5L 2s as a subset, which are 7L 5s, 12L 7s, 19L 12s, 31L 19s, 50L 31s, and so on. This is to say that the aforementioned mosses are supported by golden meantone.

Any arbitrary mos is the start of a '''golden mos sequence''' (the temperament-agnostic equivalent of a golden temperament), even if it coincides with that of another mos.

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 |metaheptic
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+== Mosses related to metallic mosses ==
+=== Fibonacci numbers and the golden ratio ===
+Let F(n) be a recursive function that returns the nth Fibonacci number.
+* For the base cases of n = 1 or n = 0:
+** If n = 1, then F(1) = 1.
+** If n = 0, then F(0) = 0.
+* For the recursive case of n > 1:
+** If n > 1, then F(n) = F(n-1) + F(n-2)
+Mosses whose step ratio approximates the golden ratio will have a step ratio L:s that is F(n):F(n-1), or two consecutive Fibonacci numbers. In relation to a parent mos xL ys, mosses of an arbitrarily large step ratio F(n):F(n-1) (where n is arbitrarily large) will contain a sequence of mosses of the form (F(k)x+F(k-1)y)L (F(k-1)x+F(k-2)y)s, where F(k), F(k-1), and F(k-2) are the kth, (k-1)th, and (k-2)th Fibonacci numbers. Due to mos recursion, the mos F(n)x+F(n-1)y)L (F(n-1)x+F(n-2)y)s contains xL ys, as well as every mos between xL ys and F(n)x+F(n-1)y)L (F(n-1)x+F(n-2)y)s. As an example, golden meantone describes the mos 5L 2s whose step ratio approaches the golden ratio. This also describes a series of mos descendants that contain 5L 2s as a subset, which are 7L 5s, 12L 7s, 19L 12s, 31L 19s, 50L 31s, and so on. This is to say that the aforementioned mosses are supported by golden meantone.
+Any arbitrary mos is the start of a '''golden mos sequence''' (the temperament-agnostic equivalent of a golden temperament), even if it coincides with that of another mos.