Ternary scale theorems: Difference between revisions

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==== Statement (1) ====

In case 1, let {{nowrap|'''g'''1 {{=}} (2, 1) − (1, 1)|'''g'''2 {{=}} (1, 2) − (2, 1)}}, and {{nowrap|'''g'''3 {{=}} (1, 1) − ({{~~sfrac~~|''n''|2}}, 2)}} {{nowrap|{{=}} ((−{{~~sfrac~~|''n''|2}} − 1)*'''g'''1 − {{~~sfrac~~|''n''|2}}*'''g'''2) (mod '''e''')}}. We assume that '''g'''1, '''g'''2 and '''e''' are ℤ-linearly independent. We have the chain '''g'''1 '''g'''2 '''g'''1 '''g'''2 ... '''g'''1 '''g'''3 which visits every note in ''s''.

In case 1, let {{nowrap|'''g'''1 {{=}} (2, 1) − (1, 1)|'''g'''2 {{=}} (1, 2) − (2, 1)}}, and {{nowrap|'''g'''3 {{=}} (1, 1) − ({{frac|''n''|2}}, 2)}} {{nowrap|{{=}} ((−{{frac|''n''|2}} − 1)*'''g'''1 − {{frac|''n''|2}}*'''g'''2) (mod '''e''')}}. We assume that '''g'''1, '''g'''2 and '''e''' are ℤ-linearly independent. We have the chain '''g'''1 '''g'''2 '''g'''1 '''g'''2 ... '''g'''1 '''g'''3 which visits every note in ''s''.

Since ''s'' is generator-offset it is well-formed with respect to the aggregate generator {{nowrap|'''g''' {{=}} ('''g'''2 + '''g'''1)}}. Since '''g'''1 and '''g'''2 subtend the same number of steps by the SGA assumption, each is an odd-step. All multiples of the aggregate generator '''g''' must be even-steps, and those dyads that are "offset" by '''g'''1 must be odd-steps. Letting ''M'' be the subset consisting of all even-numbered notes (which are generated by '''g''') and considering ''M'' as a scale by dividing degree indices in ''M'' by two, ''M'' is well-formed with respect to '''g''', thus ''M'' (and its offset) must be a MOS subset. Hence {{nowrap|('''g'''3 + '''g'''1)}}, the imperfect generator of the MOS generated by '''g''', subtends the same number of steps as '''g'''. Thus '''g'''2 and '''g'''3 subtend the same number of steps, a fact we need in order to be able to substitute one instance of '''g'''2 with '''g'''3 in the next part.

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 ==== Statement (1) ====
-In case 1, let {{nowrap|'''g'''<sub>1</sub> {{=}} (2, 1) &minus; (1, 1)|'''g'''<sub>2</sub> {{=}} (1, 2) &minus; (2, 1)}}, and {{nowrap|'''g'''<sub>3</sub> {{=}} (1, 1) &minus; ({{sfrac|''n''|2}}, 2)}} {{nowrap|{{=}} ((&minus;{{sfrac|''n''|2}} &minus; 1)*'''g'''<sub>1</sub> &minus; {{sfrac|''n''|2}}*'''g'''<sub>2</sub>) (mod '''e''')}}. We assume that '''g'''<sub>1</sub>, '''g'''<sub>2</sub> and '''e''' are ℤ-linearly independent. We have the chain '''g'''<sub>1</sub> '''g'''<sub>2</sub> '''g'''<sub>1</sub> '''g'''<sub>2</sub> ... '''g'''<sub>1</sub> '''g'''<sub>3</sub> which visits every note in ''s''.
+In case 1, let {{nowrap|'''g'''<sub>1</sub> {{=}} (2, 1) &minus; (1, 1)|'''g'''<sub>2</sub> {{=}} (1, 2) &minus; (2, 1)}}, and {{nowrap|'''g'''<sub>3</sub> {{=}} (1, 1) &minus; ({{frac|''n''|2}}, 2)}} {{nowrap|{{=}} ((&minus;{{frac|''n''|2}} &minus; 1)*'''g'''<sub>1</sub> &minus; {{frac|''n''|2}}*'''g'''<sub>2</sub>) (mod '''e''')}}. We assume that '''g'''<sub>1</sub>, '''g'''<sub>2</sub> and '''e''' are ℤ-linearly independent. We have the chain '''g'''<sub>1</sub> '''g'''<sub>2</sub> '''g'''<sub>1</sub> '''g'''<sub>2</sub> ... '''g'''<sub>1</sub> '''g'''<sub>3</sub> which visits every note in ''s''.
 Since ''s'' is generator-offset it is well-formed with respect to the aggregate generator {{nowrap|'''g''' {{=}} ('''g'''<sub>2</sub> + '''g'''<sub>1</sub>)}}. Since '''g'''<sub>1</sub> and '''g'''<sub>2</sub> subtend the same number of steps by the SGA assumption, each is an odd-step. All multiples of the aggregate generator '''g''' must be even-steps, and those dyads that are "offset" by '''g'''<sub>1</sub> must be odd-steps. Letting ''M'' be the subset consisting of all even-numbered notes (which are generated by '''g''') and considering ''M'' as a scale by dividing degree indices in ''M'' by two, ''M'' is well-formed with respect to '''g''', thus ''M'' (and its offset) must be a MOS subset. Hence {{nowrap|('''g'''<sub>3</sub> + '''g'''<sub>1</sub>)}}, the imperfect generator of the MOS generated by '''g''', subtends the same number of steps as '''g'''. Thus '''g'''<sub>2</sub> and '''g'''<sub>3</sub> subtend the same number of steps, a fact we need in order to be able to substitute one instance of '''g'''<sub>2</sub> with '''g'''<sub>3</sub> in the next part.