User:Inthar/MV3: Difference between revisions

Line 35:

== MV3 Theorem 1==

''Suppose we have an MV3 scale word S with steps x, y and z. With only one exception ("xyxzxyx"), at least two of the three steps must occur the same number of times. (The one exception to this rule is "xyxzxyx", along with their repetitions "xyxzxyx", etc.) Moreover, there always exists some "generator" interval for any max-variety-3 scale (other than two exceptions, "xyzyx" and "xyxzxyx") such that the scale can be expressed as two parallel chains of this generator which are almost equal in length (the lengths are either equal, or differ by 1).''

Theorem: ''Suppose we have an MV3 scale word S with steps x, y and z. With only one exception ("xyxzxyx"), at least two of the three steps must occur the same number of times. (The one exception to this rule is "xyxzxyx", along with their repetitions "xyxzxyx", etc.) Moreover, there always exists some "generator" interval for any max-variety-3 scale (other than two exceptions, "xyzyx" and "xyxzxyx") such that the scale can be expressed as two parallel chains of this generator which are almost equal in length (the lengths are either equal, or differ by 1).''

Restatement: ''The following are equivalent for a scale word S with steps x, y, z:''

# ''S is MV3.''

# ''S is PMOS, or S is of the form x'y'z'y'x' or its repetitions.''

# ''S is AG, or S is of the form x'y'z'y'x' or its repetitions, or x'y'x'z'x'y'x' or its repetitions.''

Below, assume that the scale word S is not multiperiod.

Line 115:

Line 120:

if a step is an odd number of generators (since the scale size is odd, we can always ensure this by taking octave complements of all the generators). The first two sizes must occur the same number of times. QED.

=== 3-DE implies MV3 ===

We prove that 3-DE + not abcba implies PMOS, which is known to imply MV3.

== MV3 Theorem 2 ==

''Once you have chosen a rank-3 temperament and a specific generator interval, there is a mechanical procedure to generate all max-variety-3 scales of a certain size (of which there are, however, infinitely many).''

~~== 3-DE implies MV3 ==~~

~~We prove that 3-DE + not abcba implies PMOS, which is known to imply MV3.~~

@@ Line 35: / Line 35: @@
 == MV3 Theorem 1==
-''Suppose we have an MV3 scale word S with steps x, y and z. With only one exception ("xyxzxyx"), at least two of the three steps must occur the same number of times. (The one exception to this rule is "xyxzxyx", along with their repetitions "xyxzxyx", etc.) Moreover, there always exists some "generator" interval for any max-variety-3 scale (other than two exceptions, "xyzyx" and "xyxzxyx") such that the scale can be expressed as two parallel chains of this generator which are almost equal in length (the lengths are either equal, or differ by 1).''
+Theorem: ''Suppose we have an MV3 scale word S with steps x, y and z. With only one exception ("xyxzxyx"), at least two of the three steps must occur the same number of times. (The one exception to this rule is "xyxzxyx", along with their repetitions "xyxzxyx", etc.) Moreover, there always exists some "generator" interval for any max-variety-3 scale (other than two exceptions, "xyzyx" and "xyxzxyx") such that the scale can be expressed as two parallel chains of this generator which are almost equal in length (the lengths are either equal, or differ by 1).''
+Restatement: ''The following are equivalent for a scale word S with steps x, y, z:''
+# ''S is MV3.''
+# ''S is PMOS, or S is of the form x'y'z'y'x' or its repetitions.''
+# ''S is AG, or S is of the form x'y'z'y'x' or its repetitions, or x'y'x'z'x'y'x' or its repetitions.''
 Below, assume that the scale word S is not multiperiod.
@@ Line 115: / Line 120: @@
 if a step is an odd number of generators (since the scale size is odd, we can always ensure this by taking octave complements of all the generators). The first two sizes must occur the same number of times. QED.
+=== 3-DE implies MV3 ===
+We prove that 3-DE + not abcba implies PMOS, which is known to imply MV3.
 == MV3 Theorem 2 ==
 ''Once you have chosen a rank-3 temperament and a specific generator interval, there is a mechanical procedure to generate all max-variety-3 scales of a certain size (of which there are, however, infinitely many).''
-== 3-DE implies MV3 ==
-We prove that 3-DE + not abcba implies PMOS, which is known to imply MV3.