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

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When we spoke of the generator for a rank-2 temperament such as meantone, we were taking advantage of the fact that the other generator is generally assumed to be the octave, and it gets its own special name: the period. It’s technically a generator too, but when we say “the” generator of a rank-2 temperament, we mean the one that’s not the period.

In rank-2 temperaments, the period usually serves as the interval of repetition. Rank-1 temperaments have only one generator, but by definition it’s some integer fraction of the interval of repetition. So, in an ET, the period is not literally a separate generator, but it may still make sense in context to refer to its interval of repetition — octave or otherwise — as the period, especially when comparing the ET with a related rank-2 temperament.

In rank-2 temperaments, the period usually serves as the [[interval of repetition]]. Rank-1 temperaments have only one generator, but by definition it’s some integer fraction of the interval of repetition. So, in an ET, the period is not literally a separate generator, but it may still make sense in context to refer to its interval of repetition — octave or otherwise — as the period, especially when comparing the ET with a related rank-2 temperament.

As we’ll soon see, there’s more than one way to generate a given rank-2 temperament. For example, meantone can be generated by an octave and a fourth. But it could equivalently be generated by an octave and a fifth. And we’ve seen it can be generated by an octave and an augmented prime in 12-ET. It could even be generated by cycling a fourth against a fifth. And so on.

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 When we spoke of the generator for a rank-2 temperament such as meantone, we were taking advantage of the fact that the other generator is generally assumed to be the octave, and it gets its own special name: the period. It’s technically a generator too, but when we say “the” generator of a rank-2 temperament, we mean the one that’s not the period.
-In rank-2 temperaments, the period usually serves as the interval of repetition. Rank-1 temperaments have only one generator, but by definition it’s some integer fraction of the interval of repetition. So, in an ET, the period is not literally a separate generator, but it may still make sense in context to refer to its interval of repetition — octave or otherwise — as the period, especially when comparing the ET with a related rank-2 temperament.
+In rank-2 temperaments, the period usually serves as the [[interval of repetition]]. Rank-1 temperaments have only one generator, but by definition it’s some integer fraction of the interval of repetition. So, in an ET, the period is not literally a separate generator, but it may still make sense in context to refer to its interval of repetition — octave or otherwise — as the period, especially when comparing the ET with a related rank-2 temperament.
 As we’ll soon see, there’s more than one way to generate a given rank-2 temperament. For example, meantone can be generated by an octave and a fourth. But it could equivalently be generated by an octave and a fifth. And we’ve seen it can be generated by an octave and an augmented prime in 12-ET. It could even be generated by cycling a fourth against a fifth. And so on.