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

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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.

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. Or an octave and an [https://en.wikipedia.org/wiki/Augmented_unison augmented unison]. It could even be generated by cycling a fourth against a fifth. And so on.

And so it’s good to have a standard form for the generators of a rank-2 temperament. One excellent standard is to set the period to an octave and the generator set to anything less than half the size of the period, as we did earlier, and again, when in this form, we call the temperament a linear temperament (not all rank-2 temperaments can be linear, e.g. if they repeat multiple times per octave, such as blackwood 5x or augmented 3x).

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 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.
+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. Or an octave and an [https://en.wikipedia.org/wiki/Augmented_unison augmented unison]. It could even be generated by cycling a fourth against a fifth. And so on.
 And so it’s good to have a standard form for the generators of a rank-2 temperament. One excellent standard is to set the period to an octave and the generator set to anything less than half the size of the period, as we did earlier, and again, when in this form, we call the temperament a linear temperament (not all rank-2 temperaments can be linear, e.g. if they repeat multiple times per octave, such as blackwood 5x or augmented 3x).