Ternary scale theorems: Difference between revisions
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==== Statement (3) ==== | ==== Statement (3) ==== | ||
For (3), we now only need to see that if len(''S'') is odd and ''S'' is SGA, ''S'' is abstractly SV3. But the argument in case 2 above works when you substitute any interval | For (3), we now only need to see that if len(''S'') is odd and ''S'' is SGA, ''S'' is abstractly SV3. But the argument in case 2 above works when you substitute any odd-step interval classes in ''S'' instead of a 1-step (abstract SV3 wasn't used). To get even-step interval classes, we can take octave complements. Hence any interval class in such a scale comes in (abstractly) exactly 3 sizes. | ||
==== Statement (4) ==== | ==== Statement (4) ==== | ||
For (4), assume ''S'' is ''a''X ''b''Y ''b''Z, ''a'' odd. If ''b'' = 1, there's nothing to prove, so assume ''b'' > 1. | For (4), assume ''S'' is ''a''X ''b''Y ''b''Z, ''a'' odd. If ''b'' = 1, there's nothing to prove, so assume ''b'' > 1. | ||