User:Inthar/MV3: Difference between revisions
| Line 97: | Line 97: | ||
* k g0 | * k g0 | ||
* (k-1) g0 + g1 | * (k-1) g0 + g1 | ||
* (k- | * (k-1) g0 + g2. | ||
It is clear that the last two sizes must occur the same number of times. | |||
In case 2, let (2,1)-(1,1) = g1, (1,2)-(2,1) = g2 be the two alternating generators. Let g3 be the leftover generator after stacking alternating g1 and g2. Then the generator circle looks like g1 g2 g1 g2 ... g1 g2 g3. Then the generators corresponding to a step are: | In case 2, let (2,1)-(1,1) = g1, (1,2)-(2,1) = g2 be the two alternating generators. Let g3 be the leftover generator after stacking alternating g1 and g2. Then the generator circle looks like g1 g2 g1 g2 ... g1 g2 g3. Then the generators corresponding to a step are: | ||
| Line 103: | Line 104: | ||
* (k-1) g1 + k g2 | * (k-1) g1 + k g2 | ||
* (k-1) g1 + (k-1) g2 + g3 | * (k-1) g1 + (k-1) g2 + g3 | ||
if a step is an odd number of generators (can always ensure this by taking octave complements of all the generators). QED. | if a step is an odd number of generators (can always ensure this by taking octave complements of all the generators). The first two sizes must occur the same number of times. QED. | ||
== MV3 Theorem 2 == | == MV3 Theorem 2 == | ||
''Once you have chosen a rank-3 temperament and a specific generator interval, there is a mechanical procedure to generate all max-variety-3 scales of a certain size (of which there are, however, infinitely many).'' | ''Once you have chosen a rank-3 temperament and a specific generator interval, there is a mechanical procedure to generate all max-variety-3 scales of a certain size (of which there are, however, infinitely many).'' | ||