Rank-3 scale theorems: Difference between revisions

Line 118:

# from g1 (odd) g1 g3 g1 (odd) g1, get a4 = n/2 g1 + (n/2-2) g2 + g3.

Choose a tuning where g1 and g2 are both very close to but not exactly 1/2*g0 (i.e. they differ from 1/2*g0 by ε, a quantity much smaller than the chroma of the n/2-note mos generated by g0, which is |g3 - g2|). We have 4 distinct sizes for n/2-steps, a contradiction to unconditional-MV3:

Choose a tuning where g1 and g2 are both very close to but not exactly 1/2*g0 (i.e. they differ from 1/2*g0 by ε, a quantity much smaller than the chroma of the n/2-note mos generated by g0, which is |g3 - g2|). Assuming n > 4, We have 4 distinct sizes for n/2-steps, a contradiction to unconditional-MV3:

# a1, a2 and a3 are clearly distinct.

# a4 - a3 = g1 - g2 != 0, since the scale is a non-trivial AG.

Line 130:

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.

This proof shows that AG and unconditionally-MV3 scales must have odd size.

This proof shows that AG and unconditionally-MV3 scales must have odd size or size 4.

==== An AG scale is unconditionally MV3 iff its cardinality is odd ====

@@ Line 118: / Line 118: @@
 # from g1 (odd) g1 g3 g1 (odd) g1, get a4 = n/2 g1 + (n/2-2) g2 + g3.
-Choose a tuning where g1 and g2 are both very close to but not exactly 1/2*g0 (i.e. they differ from 1/2*g0 by ε, a quantity much smaller than the chroma of the n/2-note mos generated by g0, which is |g3 - g2|). We have 4 distinct sizes for n/2-steps, a contradiction to unconditional-MV3:
+Choose a tuning where g1 and g2 are both very close to but not exactly 1/2*g0 (i.e. they differ from 1/2*g0 by ε, a quantity much smaller than the chroma of the n/2-note mos generated by g0, which is |g3 - g2|). Assuming n > 4, We have 4 distinct sizes for n/2-steps, a contradiction to unconditional-MV3:
 # a1, a2 and a3 are clearly distinct.
 # a4 - a3 = g1 - g2 != 0, since the scale is a non-trivial AG.
@@ Line 130: / Line 130: @@
 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.
-This proof shows that AG and unconditionally-MV3 scales must have odd size.
+This proof shows that AG and unconditionally-MV3 scales must have odd size or size 4.
 ==== An AG scale is unconditionally MV3 iff its cardinality is odd ====