Douglas Blumeyer's RTT How-To: Difference between revisions

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Let’s put it this way. When we look at 12-ET in terms of itself, rather than in terms of any particular rank-2 temperament, its generator is 1\12. That’s the simplest, smallest generator which if we iterate it 12 times will touch every pitch in 12-ET. But when we look at 12-ET not as the end goal, but rather as a foundation upon which we could work with a given temperament, things change. We don’t necessarily need to include every pitch in 12-ET to realize a temperament it supports.

For example, for meantone, even if I iterated the generator only four times, starting at step 0, touching steps 5, 10, 3 (it would be 15, but we octave-reduce here, subtracting 12 to stay within 12, landing back at 15 - 12 = 3), and 8, we’d realize meantone. That’s because fourths and fifths are octave-complements, and so in a sense they are equivalent. So, moving four fourths up like this is the same thing as moving four fifths down, and we can see that gets me to the same place as if I moved one major third down, which being 4 steps, would also take me to step 8.

For example, for meantone, even if I iterated the generator only four times, starting at step 0, touching steps 5, 10, 3 (it would be 15, but we octave-reduce here, subtracting 12 to stay within 12, landing back at 15 - 12 = 3), and 8, we’d realize meantone. That’s because fourths and fifths are octave-complements, and so in a sense they are equivalent. So, moving four fourths up like this is the same thing as moving four fifths down, and we can see that gets me to the same place as if I moved one major third down, which — being 4 steps — would also take me to step 8. That's the central idea of meantone temperament, and so this is what I mean by we've "realized" it.

If we continued to iterate this 12-ET meantone generator, we would happen to eventually touch every pitch in 12-ET, because 5 and 12 are coprime; we’d continue onward from 8 to 1 (13 - 12 = 1), then 6, 11, 4, 9, 2, 7, and circle back to 0. On the other hand, augmented temperament in 12-ET could never reach most of the pitches, because 4 is not coprime with 12; the 4\12 generator is essentially 1\3, and can only reach 0, 4, and 8. From augmented temperament’s perspective, that’s acceptable, though: this set of pitches still realizes the fact that three major thirds get you back where you started, which is its whole point.

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 Let’s put it this way. When we look at 12-ET in terms of itself, rather than in terms of any particular rank-2 temperament, its generator is 1\12. That’s the simplest, smallest generator which if we iterate it 12 times will touch every pitch in 12-ET. But when we look at 12-ET not as the end goal, but rather as a foundation upon which we could work with a given temperament, things change. We don’t necessarily need to include every pitch in 12-ET to realize a temperament it supports.
-For example, for meantone, even if I iterated the generator only four times, starting at step 0, touching steps 5, 10, 3 (it would be 15, but we octave-reduce here, subtracting 12 to stay within 12, landing back at 15 - 12 = 3), and 8, we’d realize meantone. That’s because fourths and fifths are octave-complements, and so in a sense they are equivalent. So, moving four fourths up like this is the same thing as moving four fifths down, and we can see that gets me to the same place as if I moved one major third down, which being 4 steps, would also take me to step 8.
+For example, for meantone, even if I iterated the generator only four times, starting at step 0, touching steps 5, 10, 3 (it would be 15, but we octave-reduce here, subtracting 12 to stay within 12, landing back at 15 - 12 = 3), and 8, we’d realize meantone. That’s because fourths and fifths are octave-complements, and so in a sense they are equivalent. So, moving four fourths up like this is the same thing as moving four fifths down, and we can see that gets me to the same place as if I moved one major third down, which — being 4 steps — would also take me to step 8. That's the central idea of meantone temperament, and so this is what I mean by we've "realized" it.
 If we continued to iterate this 12-ET meantone generator, we would happen to eventually touch every pitch in 12-ET, because 5 and 12 are coprime; we’d continue onward from 8 to 1 (13 - 12 = 1), then 6, 11, 4, 9, 2, 7, and circle back to 0. On the other hand, augmented temperament in 12-ET could never reach most of the pitches, because 4 is not coprime with 12; the 4\12 generator is essentially 1\3, and can only reach 0, 4, and 8. From augmented temperament’s perspective, that’s acceptable, though: this set of pitches still realizes the fact that three major thirds get you back where you started, which is its whole point.