Generator-offset property: Difference between revisions
m →Proposition 1 (Properties of SGA scales): xyxz is not SV3 |
m →Proposition 1 (Properties of SGA scales): correcting for xyxz |
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=== Proposition 1 (Properties of SGA scales) === | === Proposition 1 (Properties of SGA scales) === | ||
Let ''S'' be a 3-step-size scale word in L, M, and s of length ''n'', and suppose ''S'' is SGA. Then: | Let ''S'' be a 3-step-size scale word in L, M, and s of length ''n'', and suppose ''S'' is SGA. Then: | ||
# ''S'' is abstractly SV3 (i.e. SV3 for almost all tunings). | # If ''n'' is odd, ''S'' is abstractly SV3 (i.e. SV3 for almost all tunings). | ||
# ''S'' is of the form ''a''x ''b''y ''b''z for some permutation (x, y, z) of (L, M, s). | # ''S'' is of the form ''a''x ''b''y ''b''z for some permutation (x, y, z) of (L, M, s). | ||
# The length of ''S'' is odd. | # The length of ''S'' is odd, or ''S'' is equivalent to xyxz. | ||
# ''S'' = ''a''X ''b''Y ''b''Z is obtained from some mode of the (single-period) mos ''a''X 2''b''W by replacing all the W's successively with alternating Y's and Z's (or alternating Z's and Y's for the other chirality, fixing the mode of ''a''X 2''b''W). | # ''S'' = ''a''X ''b''Y ''b''Z is obtained from some mode of the (single-period) mos ''a''X 2''b''W by replacing all the W's successively with alternating Y's and Z's (or alternating Z's and Y's for the other chirality, fixing the mode of ''a''X 2''b''W). | ||
# The two alternants differ by replacing one Y with a Z. | # The two alternants differ by replacing one Y with a Z. | ||
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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-length, unconditionally SV3, GO scale must be of the form xy...xyxz. But this pattern is not abstractly SV3 if ''n'' ≥ 6, since 3-steps come in 4 sizes: xyx, yxy, yxz and xzx. Thus ''n'' = 4 and the scale is xyxz. | 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-length, unconditionally SV3, GO scale must be of the form xy...xyxz. But this pattern is not abstractly SV3 if ''n'' ≥ 6, since 3-steps come in 4 sizes: xyx, yxy, yxz and xzx. Thus ''n'' = 4 and the scale is xyxz. (Note that xyxz is not SV3, since it has only two kinds of 2-steps, xy and xz.) This proves (3). | ||
In case 2, let (2, 1) − (1, 1) = g<sub>1</sub>, (1, 2) − (2, 1) = g<sub>2</sub> be the two alternants. Let g<sub>3</sub> be the leftover generator after stacking alternating g<sub>1</sub> and g<sub>2</sub>. Then the generator circle looks like g<sub>1</sub> g<sub>2</sub> g<sub>1</sub> g<sub>2</sub> ... g<sub>1</sub> g<sub>2</sub> g<sub>3</sub>. Then the combinations of alternants corresponding to a step come in exactly 3 sizes: | In case 2, let (2, 1) − (1, 1) = g<sub>1</sub>, (1, 2) − (2, 1) = g<sub>2</sub> be the two alternants. Let g<sub>3</sub> be the leftover generator after stacking alternating g<sub>1</sub> and g<sub>2</sub>. Then the generator circle looks like g<sub>1</sub> g<sub>2</sub> g<sub>1</sub> g<sub>2</sub> ... g<sub>1</sub> g<sub>2</sub> g<sub>3</sub>. Then the combinations of alternants corresponding to a step come in exactly 3 sizes: | ||