Generator-offset property: Difference between revisions
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Now we only need to see that SGA + 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. <math>\square</math> | Now we only need to see that SGA + 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. <math>\square</math> | ||
== | == Open conjectures == | ||
=== Conjecture 2 === | === Conjecture 2 === | ||
If a non-multiperiod 3-step size scale word is | If a non-multiperiod 3-step size scale word is | ||
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=== Conjecture 3 === | === Conjecture 3 === | ||
An SGA scale is always pairwise-well-formed. That is, the result of identifying any two step sizes of an SGA scale is always a non-multiperiod mos. | An SGA scale is always pairwise-well-formed. That is, the result of identifying any two step sizes of an SGA scale is always a non-multiperiod mos. | ||
== Falsified conjectures == | |||
=== Conjecture -1 === | |||
An odd generator-offset scale is SGA. | |||
==== Counterexamples ==== | |||
This is false for the 5-note scale 0 250 300 700 750 1200 with alternants 700 and 50: the 1-steps form the chain 250 50 400 50 450 and the 2-steps form the chain 300 450 700 250 500. | |||
A 7-note counterexample: 0 210 220 700 710 920 930 1200 | |||
[[Category:Theory]] | [[Category:Theory]] | ||
[[Category:AG scales| ]]<!--Main article--> | [[Category:AG scales| ]]<!--Main article--> | ||