Ternary scale theorems: Difference between revisions

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Let ''r'' be odd and ''r'' ≥ 3. Consider the following abstract sizes for the dyad class of ''k''-steps reached by stacking ''r'' generators:

# from '''g'''1 '''g'''2 ... '''g'''1, we get {{nowrap|''a''1 {{=}} (''r'' − 1)/2 * '''g''' + '''g'''1}} {{nowrap|{{=}} {{ceil|''r''/2}} '''g'''1 + {{floor|''r''/2}} '''g'''2}}

# from '''g'''1 '''g'''2 ... '''g'''1, we get {{nowrap|''a''1 {{=}} {{sfrac|''r'' − 1|2}} * '''g''' + '''g'''1}} {{nowrap|{{=}} {{ceil|{{frac|''r''|2}}}} '''g'''1 + {{floor|{{frac|''r''|2}}}} '''g'''2}}

# from '''g'''2 '''g'''1 ... '''g'''2, we get {{nowrap|''a''2 {{=}} (''r'' − 1)/2 * '''g''' + '''g'''2}} {{nowrap|{{=}} {{floor|''r''/2}} '''g'''1 + {{ceil|''r''/2}} '''g'''2}}

# from '''g'''2 '''g'''1 ... '''g'''2, we get {{nowrap|''a''2 {{=}} {{sfrac|''r'' − 1|2}} * '''g''' + '''g'''2}} {{nowrap|{{=}} {{floor|{{frac|''r''|2}}}} '''g'''1 + {{ceil|{{frac|''r''|2}}}} '''g'''2}}

# from '''g'''2 (...even # of gens...) '''g'''1 '''g'''3 '''g'''1 (...even # of gens...) '''g'''2, we get {{nowrap|''a''3 {{=}} (''r'' − 1)/2 '''g'''1 + (''r'' − 1)/2 '''g'''2 + '''g'''3}} {{nowrap|≡ (''r'' − ''n''/2 − 3/2)'''g'''1 + (''r'' − ''n''/2 − 1/2)'''g'''2 (mod '''e''')}}.

# from '''g'''2 (...even # of gens...) '''g'''1 '''g'''3 '''g'''1 (...even # of gens...) '''g'''2, we get {{nowrap|''a''3 {{=}} {{sfrac|''r'' − 1|2}} '''g'''1 + {{sfrac|''r'' − 1)|2}} '''g'''2 + '''g'''3}} {{nowrap|≡ {{sfrac|''r'' − ''n''|2}} − {{sfrac|3|2}})'''g'''1 + {{sfrac|''r'' − ''n''|2}} − {{sfrac|1|2}}'''g'''2 (mod '''e''')}}.

# from '''g'''1 (...odd # of gens...) '''g'''1 '''g'''3 '''g'''1 (...odd # of gens...) '''g'''1, we get {{nowrap|''a''4 {{=}} (''r'' + 1)/2 '''g'''1 + (''r'' − 3)/2 '''g'''2 + '''g'''3}} {{nowrap|≡ (''r'' − ''n''/2 − 1/2)'''g'''1 + (''r'' − ''n''/2 − 3/2)'''g'''2 (mod '''e''')}}.

# from '''g'''1 (...odd # of gens...) '''g'''1 '''g'''3 '''g'''1 (...odd # of gens...) '''g'''1, we get {{nowrap|''a''4 {{=}} {{sfrac|''r'' + 1)|2}} '''g'''1 + {{sfrac|''r'' − 3|2}} '''g'''2 + '''g'''3}} {{nowrap|≡ {{sfrac|''r'' − ''n''|2}} − {{sfrac|1|2}})'''g'''1 + {{sfrac|''r'' − ''n''|2}} − {{sfrac|3|2}}'''g'''2 (mod '''e''')}}.

Since {{nowrap|''n'' > 0}}, these are all distinct by ℤ-linear independence; hence there are at least 4 sizes for ''k''-steps. A 1-step must be reached by stacking an odd number of generators, thus by applying this argument to 1-steps, we see that there must be at least 4 step sizes in some tuning, a contradiction. Thus '''g'''1 and '''g'''2 must themselves be step sizes. Thus we see that an even-length SGA ternary scale must be of the form (xy)''r''xz. (Note that (xy)''r''xz is not SV3, since it has only two kinds of 2-steps, '''xy''' and '''xz'''.) This proves (1).

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 Let ''r'' be odd and ''r'' &ge; 3. Consider the following abstract sizes for the dyad class of ''k''-steps reached by stacking ''r'' generators:
-# from '''g'''<sub>1</sub> '''g'''<sub>2</sub> ... '''g'''<sub>1</sub>, we get {{nowrap|''a''<sub>1</sub> {{=}} (''r'' &minus; 1)/2 * '''g''' + '''g'''<sub>1</sub>}} {{nowrap|{{=}} {{ceil|''r''/2}} '''g'''<sub>1</sub> + {{floor|''r''/2}} '''g'''<sub>2</sub>}}
+# from '''g'''<sub>1</sub> '''g'''<sub>2</sub> ... '''g'''<sub>1</sub>, we get {{nowrap|''a''<sub>1</sub> {{=}} {{sfrac|''r'' &minus; 1|2}} * '''g''' + '''g'''<sub>1</sub>}} {{nowrap|{{=}} {{ceil|{{frac|''r''|2}}}} '''g'''<sub>1</sub> + {{floor|{{frac|''r''|2}}}} '''g'''<sub>2</sub>}}
-# from '''g'''<sub>2</sub> '''g'''<sub>1</sub> ... '''g'''<sub>2</sub>, we get {{nowrap|''a''<sub>2</sub> {{=}} (''r'' &minus; 1)/2 * '''g''' + '''g'''<sub>2</sub>}} {{nowrap|{{=}} {{floor|''r''/2}} '''g'''<sub>1</sub> + {{ceil|''r''/2}} '''g'''<sub>2</sub>}}
+# from '''g'''<sub>2</sub> '''g'''<sub>1</sub> ... '''g'''<sub>2</sub>, we get {{nowrap|''a''<sub>2</sub> {{=}} {{sfrac|''r'' &minus; 1|2}} * '''g''' + '''g'''<sub>2</sub>}} {{nowrap|{{=}} {{floor|{{frac|''r''|2}}}} '''g'''<sub>1</sub> + {{ceil|{{frac|''r''|2}}}} '''g'''<sub>2</sub>}}
-# from '''g'''<sub>2</sub> (...even # of gens...) '''g'''<sub>1</sub> '''g'''<sub>3</sub> '''g'''<sub>1</sub> (...even # of gens...) '''g'''<sub>2</sub>, we get {{nowrap|''a''<sub>3</sub> {{=}} (''r'' &minus; 1)/2 '''g'''<sub>1</sub> + (''r'' &minus; 1)/2 '''g'''<sub>2</sub> + '''g'''<sub>3</sub>}} {{nowrap|≡ (''r'' &minus; ''n''/2 &minus; 3/2)'''g'''<sub>1</sub> + (''r'' &minus; ''n''/2 &minus; 1/2)'''g'''<sub>2</sub> (mod '''e''')}}.
+# from '''g'''<sub>2</sub> (...even # of gens...) '''g'''<sub>1</sub> '''g'''<sub>3</sub> '''g'''<sub>1</sub> (...even # of gens...) '''g'''<sub>2</sub>, we get {{nowrap|''a''<sub>3</sub> {{=}} {{sfrac|''r'' &minus; 1|2}} '''g'''<sub>1</sub> + {{sfrac|''r'' &minus; 1)|2}} '''g'''<sub>2</sub> + '''g'''<sub>3</sub>}} {{nowrap|≡ {{sfrac|''r'' &minus; ''n''|2}} &minus; {{sfrac|3|2}})'''g'''<sub>1</sub> + {{sfrac|''r'' &minus; ''n''|2}} &minus; {{sfrac|1|2}}'''g'''<sub>2</sub> (mod '''e''')}}.
-# from '''g'''<sub>1</sub> (...odd # of gens...) '''g'''<sub>1</sub> '''g'''<sub>3</sub> '''g'''<sub>1</sub> (...odd # of gens...) '''g'''<sub>1</sub>, we get {{nowrap|''a''<sub>4</sub> {{=}} (''r'' + 1)/2 '''g'''<sub>1</sub> + (''r'' &minus; 3)/2 '''g'''<sub>2</sub> + '''g'''<sub>3</sub>}} {{nowrap|≡ (''r'' &minus; ''n''/2 &minus; 1/2)'''g'''<sub>1</sub> + (''r'' &minus; ''n''/2 &minus; 3/2)'''g'''<sub>2</sub> (mod '''e''')}}.
+# from '''g'''<sub>1</sub> (...odd # of gens...) '''g'''<sub>1</sub> '''g'''<sub>3</sub> '''g'''<sub>1</sub> (...odd # of gens...) '''g'''<sub>1</sub>, we get {{nowrap|''a''<sub>4</sub> {{=}} {{sfrac|''r'' + 1)|2}} '''g'''<sub>1</sub> + {{sfrac|''r'' &minus; 3|2}} '''g'''<sub>2</sub> + '''g'''<sub>3</sub>}} {{nowrap|≡ {{sfrac|''r'' &minus; ''n''|2}} &minus; {{sfrac|1|2}})'''g'''<sub>1</sub> + {{sfrac|''r'' &minus; ''n''|2}} &minus; {{sfrac|3|2}}'''g'''<sub>2</sub> (mod '''e''')}}.
 Since {{nowrap|''n'' &gt; 0}}, these are all distinct by ℤ-linear independence; hence there are at least 4 sizes for ''k''-steps. A 1-step must be reached by stacking an odd number of generators, thus by applying this argument to 1-steps, we see that there must be at least 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 SGA ternary scale must be of the form (xy)<sup>''r''</sup>xz. (Note that (xy)<sup>''r''</sup>xz is not SV3, since it has only two kinds of 2-steps, '''xy''' and '''xz'''.) This proves (1).