Generator-offset property: Difference between revisions
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# ''a''<sub>4</sub> − ''a''<sub>2</sub> = ''g''<sub>1</sub> − 2 ''g''<sub>2</sub> + ''g''<sub>3</sub> = (''g''<sub>3</sub> − ''g''<sub>2</sub>) + (''g''<sub>1</sub> − ''g''<sub>2</sub>) = (chroma ± ε) != 0 by choice of tuning. | # ''a''<sub>4</sub> − ''a''<sub>2</sub> = ''g''<sub>1</sub> − 2 ''g''<sub>2</sub> + ''g''<sub>3</sub> = (''g''<sub>3</sub> − ''g''<sub>2</sub>) + (''g''<sub>1</sub> − ''g''<sub>2</sub>) = (chroma ± ε) != 0 by choice of tuning. | ||
By applying this argument to 1-steps, we see that there must be 4 step sizes in some tuning, a contradiction. Thus ''g''<sub>1</sub> and ''g''<sub>2</sub> must themselves be step sizes. Thus we see that an even-cardinality, unconditionally MV3, AG scale must be of the form ''xy...xyxz''. But this pattern is not unconditionally MV3 if ''n'' ≥ 6, 3-steps come in 4 sizes: ''xyx'', ''yxy'', ''yxz'' and | By applying this argument to 1-steps, we see that there must be 4 step sizes in some tuning, a contradiction. Thus ''g''<sub>1</sub> and ''g''<sub>2</sub> must themselves be step sizes. Thus we see that an even-cardinality, unconditionally MV3, AG scale must be of the form ''xy...xyxz''. But this pattern is not unconditionally MV3 if ''n'' ≥ 6, since 3-steps come in 4 sizes: ''xyx'', ''yxy'', ''yxz'' and | ||
''xzx''. Thus ''n'' = 4 and the scale is ''xyxz''. | ''xzx''. Thus ''n'' = 4 and the scale is ''xyxz''. | ||