Generator-offset property: Difference between revisions

Line 221:

=== Conjecture ("SV3 Structure Theorem") ===

==== Conjecture 1 ====

If a 3-step size scale word ''~~ax by cz~~'' is

If a 3-step size scale word that is ''a''x ''b''y ''c''z is

# abstractly SV3 (strict variety 3, i.e. every k-step except multiples of the equave comes in ''exactly'' 3 sizes, for almost all tunings),

# has ''odd'' length,

# has gcd(''a'', ''b'', ''c'') = 1,

# is not of the form ''xyzyx'' or ''xyxzxyx'',

# is not of the form xyzyx or xyxzxyx,

then it is GO (and therefore SGA). (a converse to Proposition 1)

Line 231:

==== Conjecture 2 ====

If a 3-step size scale word that is ~~ax by cz~~ is abstractly SV3, then two of a, b, c are equal.

If a 3-step size scale word that is ''a''x ''b''y ''c''z is abstractly SV3, then two of ''a'', ''b'', ''c'' are equal.

=== Conjecture ("MV3 Sequences") ===

Given any two generators, we can iterate them to any number of notes and see what the maximum-variety of the resulting scale is. In particular, we can look at those scale sizes which are MV3, and thus compute the '''MV3 sequence''' for the pair of generators (similar to the "MOS sequence" one can compute for one generator). Thus, for any pair of generators, we can form the associated sequence of increasingly large MV3 scales.

Surprisingly, for almost all pairs of generators, this sequence seems to terminate after some (usually relatively small) scale. That is, if we simply take all possible pairs of generators between 0 and 1200 cents, and for each pair we compute the MV3 sequence for all generator pairs up to some maximum N, such as 1000, we can easily see that most points will have only a few entries in it, after which no MV3 scales are apparently generated. It would seem to be true that as the two generators get closer and closer in size, the MV3 sequence gets longer and longer, until when the two generators are equal you have an infinite-length sequence (corresponding to MOS).

Surprisingly, for almost all pairs of generators, this sequence seems to terminate after some (usually relatively small) scale. That is, if we simply take all possible pairs of generators between 0 and 1200 cents, and for each pair we compute the MV3 sequence for all generator pairs up to some maximum ''N'', such as 1000, we can easily see that most points will have only a few entries in it, after which no MV3 scales are apparently generated. It would seem to be true that as the two generators get closer and closer in size, the MV3 sequence gets longer and longer, until when the two generators are equal you have an infinite-length sequence (corresponding to MOS).

It is pretty easy to see this behavior is true if we simply compute the MV3 sequences up to any very large N, far beyond the scale sizes we typically use in music theory, but it would be good to have a proof.

It is pretty easy to see this behavior is true if we simply compute the MV3 sequences up to any very large ''N'', far beyond the scale sizes we typically use in music theory, but it would be good to have a proof.

== Open questions ==

@@ Line 221: / Line 221: @@
 === Conjecture ("SV3 Structure Theorem") ===
 ==== Conjecture 1 ====
-If a 3-step size scale word ''ax by cz'' is
+If a 3-step size scale word that is ''a''x ''b''y ''c''z is
 # abstractly SV3 (strict variety 3, i.e. every k-step except multiples of the equave comes in ''exactly'' 3 sizes, for almost all tunings),
 # has ''odd'' length,
 # has gcd(''a'', ''b'', ''c'') = 1,
-# is not of the form ''xyzyx'' or ''xyxzxyx'',
+# is not of the form xyzyx or xyxzxyx,
 then it is GO (and therefore SGA). (a converse to Proposition 1)
@@ Line 231: / Line 231: @@
 ==== Conjecture 2 ====
-If a 3-step size scale word that is ax by cz is abstractly SV3, then two of a, b, c are equal.
+If a 3-step size scale word that is ''a''x ''b''y ''c''z is abstractly SV3, then two of ''a'', ''b'', ''c'' are equal.
 === Conjecture ("MV3 Sequences") ===
 Given any two generators, we can iterate them to any number of notes and see what the maximum-variety of the resulting scale is. In particular, we can look at those scale sizes which are MV3, and thus compute the '''MV3 sequence''' for the pair of generators (similar to the "MOS sequence" one can compute for one generator). Thus, for any pair of generators, we can form the associated sequence of increasingly large MV3 scales.
-Surprisingly, for almost all pairs of generators, this sequence seems to terminate after some (usually relatively small) scale. That is, if we simply take all possible pairs of generators between 0 and 1200 cents, and for each pair we compute the MV3 sequence for all generator pairs up to some maximum N, such as 1000, we can easily see that most points will have only a few entries in it, after which no MV3 scales are apparently generated. It would seem to be true that as the two generators get closer and closer in size, the MV3 sequence gets longer and longer, until when the two generators are equal you have an infinite-length sequence (corresponding to MOS).
+Surprisingly, for almost all pairs of generators, this sequence seems to terminate after some (usually relatively small) scale. That is, if we simply take all possible pairs of generators between 0 and 1200 cents, and for each pair we compute the MV3 sequence for all generator pairs up to some maximum ''N'', such as 1000, we can easily see that most points will have only a few entries in it, after which no MV3 scales are apparently generated. It would seem to be true that as the two generators get closer and closer in size, the MV3 sequence gets longer and longer, until when the two generators are equal you have an infinite-length sequence (corresponding to MOS).
-It is pretty easy to see this behavior is true if we simply compute the MV3 sequences up to any very large N, far beyond the scale sizes we typically use in music theory, but it would be good to have a proof.
+It is pretty easy to see this behavior is true if we simply compute the MV3 sequences up to any very large ''N'', far beyond the scale sizes we typically use in music theory, but it would be good to have a proof.
 == Open questions ==