Generator-offset property: Difference between revisions

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These are equivalent, since the separating interval can be taken to be g1 and the generator of each chain = g1 + g2.

== Theorems ==

'''Theorem 1''': If a 3-step-size scale word ''S'' in L, M, and s is both AG and unconditionally MV3, then the scale is of the form ax by bz for (x,y,z) some permutation of (L, M, s); and the scale's cardinality is either odd, or 4 (and is of the form xyxz). Moreover, any odd-cardinality AG scale is MV3.

'''Theorem 1''': If a 3-step-size scale word ''S'' in L, M, and s is both AG and unconditionally MV3 (i.e. MV3 regardless of tuning), then the scale is of the form ax by bz for (x,y,z) some permutation of (L, M, s); and the scale's cardinality is either odd, or 4 (and is of the form xyxz). Moreover, any odd-cardinality AG scale is unconditionally MV3.

=== Proof of Thm 1 ===

'''Assuming both AG and ~~unconditional~~ MV3''', we have two chains of generator g0 (going right). The two cases are:

'''Assuming both AG and unconditionally MV3''', we have two chains of generator g0 (going right). The two cases are:

O-O-...-O (m notes)

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# a4 - a2 = g1 - 2 g2 + g3 = (g3 - g2) + (g1 - g2) = (chroma ± ε) != 0 by choice of tuning.

By applying this argument to 1-steps, we see that there must be 4 step sizes in some tuning, a contradiction. Thus g1 and g2 must themselves be step sizes. Thus we see that an even-cardinality, MV3, AG scale must be of the form xy...xyxz. But this pattern is not MV3 if n >=6, 3-steps come in 4 sizes: xyx, yxy, yxz and xzx. Thus n = 4 and the scale is xyxz.

By applying this argument to 1-steps, we see that there must be 4 step sizes in some tuning, a contradiction. Thus g1 and g2 must themselves be step sizes. Thus we see that an even-cardinality, unconditionally MV3, AG scale must be of the form xy...xyxz. But this pattern is not unconditionally MV3 if n >=6, 3-steps come in 4 sizes: xyx, yxy, yxz and xzx. Thus n = 4 and the scale is xyxz.

In case 2, let (2,1)-(1,1) = g1, (1,2)-(2,1) = g2 be the two alternating generators. Let g3 be the leftover generator after stacking alternating g1 and g2. Then the generator circle looks like g1 g2 g1 g2 ... g1 g2 g3. Then the generators corresponding to a step are:

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(The above holds for any odd n >= 3.)

Now we only need to see that AG + odd cardinality => MV3. But the argument in case 2 above works for any interval class (MV3 wasn't used), hence any interval class comes in at most 3 sizes regardless of tuning.

Now we only need to see that AG + odd cardinality => unconditionally MV3. But the argument in case 2 above works for any interval class (unconditional MV3 wasn't used), hence any interval class comes in at most 3 sizes regardless of tuning.

[[Category:Theory]]

[[Category:AG scales| ]]

@@ Line 10: / Line 10: @@
 These are equivalent, since the separating interval can be taken to be g1 and the generator of each chain = g1 + g2.
 == Theorems ==
-'''Theorem 1''': If a 3-step-size scale word ''S'' in L, M, and s is both AG and unconditionally MV3, then the scale is of the form ax by bz for (x,y,z) some permutation of (L, M, s); and the scale's cardinality is either odd, or 4 (and is of the form xyxz). Moreover, any odd-cardinality AG scale is MV3.
+'''Theorem 1''': If a 3-step-size scale word ''S'' in L, M, and s is both AG and unconditionally MV3 (i.e. MV3 regardless of tuning), then the scale is of the form ax by bz for (x,y,z) some permutation of (L, M, s); and the scale's cardinality is either odd, or 4 (and is of the form xyxz). Moreover, any odd-cardinality AG scale is unconditionally MV3.
 === Proof of Thm 1 ===
-'''Assuming both AG and unconditional MV3''', we have two chains of generator g0 (going right). The two cases are:
+'''Assuming both AG and unconditionally MV3''', we have two chains of generator g0 (going right). The two cases are:
   O-O-...-O (m notes)
   O-O-...-O (m notes)
@@ Line 35: / Line 35: @@
 # a4 - a2 = g1 - 2 g2 + g3 = (g3 - g2) + (g1 - g2) = (chroma ± ε) != 0 by choice of tuning.
-By applying this argument to 1-steps, we see that there must be 4 step sizes in some tuning, a contradiction. Thus g1 and g2 must themselves be step sizes. Thus we see that an even-cardinality, MV3, AG scale must be of the form xy...xyxz. But this pattern is not MV3 if n >=6, 3-steps come in 4 sizes: xyx, yxy, yxz and xzx. Thus n = 4 and the scale is xyxz.
+By applying this argument to 1-steps, we see that there must be 4 step sizes in some tuning, a contradiction. Thus g1 and g2 must themselves be step sizes. Thus we see that an even-cardinality, unconditionally MV3, AG scale must be of the form xy...xyxz. But this pattern is not unconditionally MV3 if n >=6, 3-steps come in 4 sizes: xyx, yxy, yxz and xzx. Thus n = 4 and the scale is xyxz.
 In case 2, let (2,1)-(1,1) = g1, (1,2)-(2,1) = g2 be the two alternating generators. Let g3 be the leftover generator after stacking alternating g1 and g2. Then the generator circle looks like g1 g2 g1 g2 ... g1 g2 g3. Then the generators corresponding to a step are:
@@ Line 45: / Line 45: @@
 (The above holds for any odd n >= 3.)
-Now we only need to see that AG + odd cardinality => MV3. But the argument in case 2 above works for any interval class (MV3 wasn't used), hence any interval class comes in at most 3 sizes regardless of tuning.
+Now we only need to see that AG + odd cardinality => unconditionally MV3. But the argument in case 2 above works for any interval class (unconditional MV3 wasn't used), hence any interval class comes in at most 3 sizes regardless of tuning.
 [[Category:Theory]]
 [[Category:AG scales| ]]<!--Main article-->