Generator-offset property: Difference between revisions
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== Theorems == | == Theorems == | ||
=== Theorem 1 === | <!--=== Theorem 1 === | ||
Let ''S'' be a 3-step-size scale word in L, M, and s, and suppose ''S'' is alt-gen. Then: | Let ''S'' be a 3-step-size scale word in L, M, and s, and suppose ''S'' is alt-gen. Then: | ||
# ''S'' is unconditionally MV3 (i.e. MV3 regardless of tuning). | # ''S'' is unconditionally MV3 (i.e. MV3 regardless of tuning). | ||
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(The above holds for any odd ''n'' ≥ 3.) | (The above holds for any odd ''n'' ≥ 3.) | ||
Now we only need to see that alt-gen + 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. | Now we only need to see that alt-gen + 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.--> | ||
== Conjectures == | == Conjectures == | ||
Revision as of 07:09, 16 August 2021
A scale satisfies the alternating generator property (also alt-gen or AG) if it satisfies the following equivalent properties:
- the scale can be built by stacking alternating generators
- the scale is generated by two chains of generators separated by a fixed interval, and the lengths of the chains differ by at most one.
The Zarlino (3L 2M 2S) JI scale is an example of an alt-gen scale, because it is built by stacking alternating 5/4 and 6/5 generators. Diasem (5L 2M 2S) is another example, with generators 7/6 and 8/7.
More formally, a cyclic word S (representing a periodic scale) of size n is alt-gen if it satisfies the following equivalent properties:
- S can be built by stacking a single chain of alternating generators g1 and g2, resulting in a circle of the form either g1 g2 ... g1 g2 g1 g3 or g1 g2 ... g1 g2 g3.
- S is generated by two chains of generators separated by a fixed interval; either both chains are of size n/2, or one chain has size (n + 1)/2 and the second has size (n − 1)/2.
These are equivalent, since the separating interval can be taken to be g1 and the generator of each chain = g1 + g2. This doesn't imply that g1 and g2 are the same number of scale steps.
Theorems
Conjectures
Conjecture 2
If a non-multiperiod 3-step size scale word is
- unconditionally MV3,
- has odd cardinality,
- is not of the form mx my mz,
- and is not of the form xyxzxyx,
then it is alt-gen. (a converse to Theorem 1)