Ternary scale theorems: Difference between revisions
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# (''k'' − 1)g<sub>1</sub> + ''k''g<sub>2</sub> | # (''k'' − 1)g<sub>1</sub> + ''k''g<sub>2</sub> | ||
# (''k'' − 1)g<sub>1</sub> + (''k'' − 1) g<sub>2</sub> + g<sub>3</sub> | # (''k'' − 1)g<sub>1</sub> + (''k'' − 1) g<sub>2</sub> + g<sub>3</sub> | ||
(since the scale size is odd, we can always ensure this by taking octave complements of all the generators). By counting the length-k subwords of the word (g<sub>1</sub> g<sub>2</sub>)<sup>floor(''n''/2)</sup>, we see that the first two sizes must occur | (since the scale size is odd, we can always ensure this by taking octave complements of all the generators). By counting the length-k subwords of the (linear) word (g<sub>1</sub> g<sub>2</sub>)<sup>floor(''n''/2)</sup>, we see that the first two sizes must both occur (''n'' − ''k'') times. This proves (2). | ||
(The above holds for any odd ''n'' ≥ 3.) | (The above holds for any odd ''n'' ≥ 3.) | ||