User:Ganaram inukshuk/Notes: Difference between revisions

Line 1,920:

** 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.

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 with an arbitrarily large step ratio F(n):F(n-1) (where n is arbitrarily large) there is a sequence of mosses of the form (xF(k)+yF(k-1))L (xF(k-1)+yF(k-2))s (where F(k), F(k-1), and F(k-2) are the kth, (k-1)th, and (k-2)th Fibonacci numbers) that descend from xL ys. Due to mos recursion, the mos (xF(n)+yF(n-1))L (xF(n-1)+yF(n-2))s contains xL ys, as well as every mos between xL ys and (xF(n)+yF(n-1))L (xF(n-1)+yF(n-2))s. The table below illustrates these mosses.

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, or rather, approximated by golden meantone if n sufficiently large.

{| class="wikitable"

|+Golden mos sequence, with golden meantone example

! rowspan="2" |k

! colspan="2" |General form

! colspan="3" |Example for 5L 2s (diatonic, golden meantone)

|-

!Mos

!Step ratio in relation to parent of xL ys

!Mos

!Step ratio of parent (5L 2s) needed to produce mos with L:s = 2:1

!Edo

|-

|0

|xL ys

|L:s (self; L and s are two consecutive Fibonacci numbers)

|5L 2s

|2:1 (self)

|12edo

|-

|1

|(x+y)L xs

|(L+s):L

|7L 5s

|3:2

|19edo

|-

|2

|(2x+y)L (x+y)s

|(2L+s):(L+s)

|7L 12s

|5:3

|31edo

|-

|3

|(3x+2y)L (2x+y)s

|(3L+2s):(2L+s)

|19L 12s

|8:5

|50edo

|-

|n

|(xF(n)+yF(n-1))L (xF(n-1)+yF(n-2))s

|(LF(n)+sF(n-1)):(LF(n-1)+sF(n-2))

|(5F(n)+2F(n-1))L (5F(n-1)+2F(n-2))s

|F(n):F(n-1)

|(2(5F(n)+2F(n-1))+(5F(n-1)+2F(n-2)))-edo

|}

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.

@@ Line 1,920: / Line 1,920: @@
 ** 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.
+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 with an arbitrarily large step ratio F(n):F(n-1) (where n is arbitrarily large) there is a sequence of mosses of the form (xF(k)+yF(k-1))L (xF(k-1)+yF(k-2))s (where F(k), F(k-1), and F(k-2) are the kth, (k-1)th, and (k-2)th Fibonacci numbers) that descend from xL ys. Due to mos recursion, the mos (xF(n)+yF(n-1))L (xF(n-1)+yF(n-2))s contains xL ys, as well as every mos between xL ys and (xF(n)+yF(n-1))L (xF(n-1)+yF(n-2))s. The table below illustrates these mosses.
+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, or rather, approximated by golden meantone if n sufficiently large.
+{| class="wikitable"
+|+Golden mos sequence, with golden meantone example
+! rowspan="2" |k
+! colspan="2" |General form
+! colspan="3" |Example for 5L 2s (diatonic, golden meantone)
+|-
+!Mos
+!Step ratio in relation to parent of xL ys
+!Mos
+!Step ratio of parent (5L 2s) needed to produce mos with L:s = 2:1
+!Edo
+|-
+|0
+|xL ys
+|L:s (self; L and s are two consecutive Fibonacci numbers)
+|5L 2s
+|2:1 (self)
+|12edo
+|-
+|1
+|(x+y)L xs
+|(L+s):L
+|7L 5s
+|3:2
+|19edo
+|-
+|2
+|(2x+y)L (x+y)s
+|(2L+s):(L+s)
+|7L 12s
+|5:3
+|31edo
+|-
+|3
+|(3x+2y)L (2x+y)s
+|(3L+2s):(2L+s)
+|19L 12s
+|8:5
+|50edo
+|-
+|n
+|(xF(n)+yF(n-1))L (xF(n-1)+yF(n-2))s
+|(LF(n)+sF(n-1)):(LF(n-1)+sF(n-2))
+|(5F(n)+2F(n-1))L (5F(n-1)+2F(n-2))s
+|F(n):F(n-1)
+|(2(5F(n)+2F(n-1))+(5F(n-1)+2F(n-2)))-edo
+|}
 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.