Generator-offset property: Difference between revisions
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== Related properties == | == Related properties == | ||
* A strengthening of the AG property, tentatively named ''alternating split-well-formed-generator'' (ASWFG), states that ''g''<sub>1</sub> and ''g''<sub>2</sub> can be taken to be the same number of scale steps, thus both representing "detemperings" of a generator of a single-period [[mos]] scale (otherwise known as a well-formed scale). Only odd AG scales can satisfy this property. The Zarlino and diasem scales above are both ASWFG. | * A strengthening of the AG property, tentatively named ''alternating split-well-formed-generator'' (ASWFG), states that ''g''<sub>1</sub> and ''g''<sub>2</sub> can be taken to be the same number of scale steps, thus both representing "detemperings" of a generator of a single-period [[mos]] scale (otherwise known as a well-formed scale). Only odd AG scales and xyxz can satisfy this property. The Zarlino and diasem scales above are both ASWFG. | ||
== Theorems == | == Theorems == | ||
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== Conjectures == | == Conjectures == | ||
=== Conjecture 2 === | === Conjecture 2 === | ||
An odd AG scale is ASWFG. | |||
=== Conjecture 3 === | |||
If a non-multiperiod 3-step size scale word is | If a non-multiperiod 3-step size scale word is | ||
# unconditionally MV3, | # unconditionally MV3, | ||
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# is not of the form ''mx my mz'', | # is not of the form ''mx my mz'', | ||
# and is not of the form ''xyxzxyx'', | # and is not of the form ''xyxzxyx'', | ||
then it is alt-gen. (a converse to Theorem 1) | then it is alt-gen. (a converse to Theorem 1 + Conjecture 2) | ||
[[Category:Theory]] | [[Category:Theory]] | ||
[[Category:AG scales| ]]<!--Main article--> | [[Category:AG scales| ]]<!--Main article--> | ||