Generator-offset property: Difference between revisions

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In case 1, let ''g''1 = (2,1) − (1,1) and ''g''2 = (1,2) − (2,1). We have the chain ''g''1 ''g''2 ''g''1 ''g''2 ... ''g''1 ''g''3.

Let ''r'' be odd and ''r'' >= 3. Consider the following abstract sizes for the interval class reached by stacking ''r'' generators:

Let ''r'' be odd and ''r'' ≥ 3. Consider the following abstract sizes for the interval class reached by stacking ''r'' generators:

# from ''g''1 ... ''g''1, we get ''a''1 = (''r'' − 1)/2*''g''0 + ''g''1 = (''r'' + 1)/2 ''g''1 + (''r'' − 1)/2 ''g''2

# from ''g''2 ... ''g''2, we get ''a''2 = (''r'' − 1)/2*''g''0 + ''g''2 = (''r'' − 1)/2 ''g''1 + (''r'' + 1)/2 ''g''2

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# ''a''4 − ''a''2 = ''g''1 − 2 ''g''2 + ''g''3 = (''g''3 − ''g''2) + (''g''1 − ''g''2) = (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''1 and ''g''2 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''1 and ''g''2 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

''xzx''. Thus ''n'' = 4 and the scale is ''xyxz''.

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if a step is an odd number of generators (since the scale size is odd, we can always ensure this by taking octave complements of all the generators). The first two sizes must occur the same number of times.

(The above holds for any odd ''n'' >= 3.)

(The above holds for any odd ''n'' ≥ 3.)

Now we only need to see that AG + odd cardinality => unconditionally MV3. But the argument in case 2 above works for any interval class (unconditional MV3 wasn't used), hence any interval class comes in at most 3 sizes regardless of tuning.

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 In case 1, let ''g''<sub>1</sub> = (2,1) &minus; (1,1) and ''g''<sub>2</sub> = (1,2) &minus; (2,1). 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>.
-Let ''r'' be odd and ''r'' >= 3. Consider the following abstract sizes for the interval class reached by stacking ''r'' generators:
+Let ''r'' be odd and ''r'' ≥ 3. Consider the following abstract sizes for the interval class reached by stacking ''r'' generators:
 # from ''g''<sub>1</sub> ... ''g''<sub>1</sub>, we get ''a''<sub>1</sub> = (''r'' &minus; 1)/2*''g''<sub>0</sub> + ''g''<sub>1</sub> = (''r'' + 1)/2 ''g''<sub>1</sub> + (''r'' &minus; 1)/2 ''g''<sub>2</sub>
 # from ''g''<sub>2</sub> ... ''g''<sub>2</sub>, we get ''a''<sub>2</sub> = (''r'' &minus; 1)/2*''g''<sub>0</sub> + ''g''<sub>2</sub> = (''r'' &minus; 1)/2 ''g''<sub>1</sub> + (''r'' + 1)/2 ''g''<sub>2</sub>
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 # ''a''<sub>4</sub> &minus; ''a''<sub>2</sub> = ''g''<sub>1</sub> &minus; 2 ''g''<sub>2</sub> + ''g''<sub>3</sub> = (''g''<sub>3</sub> &minus; ''g''<sub>2</sub>) + (''g''<sub>1</sub> &minus; ''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, 3-steps come in 4 sizes: ''xyx'', ''yxy'', ''yxz'' and
 ''xzx''. Thus ''n'' = 4 and the scale is ''xyxz''.
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 if a step is an odd number of generators (since the scale size is odd, we can always ensure this by taking octave complements of all the generators). The first two sizes must occur the same number of times.
-(The above holds for any odd ''n'' >= 3.)
+(The above holds for any odd ''n'' ≥ 3.)
 Now we only need to see that AG + odd cardinality => unconditionally MV3. But the argument in case 2 above works for any interval class (unconditional MV3 wasn't used), hence any interval class comes in at most 3 sizes regardless of tuning.