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

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=== periods and generators ===

Earlier we mentioned the term “rank”. I warned you then that it wasn’t actually the same thing as ~~geometric~~ dimensionality, even though we could use dimensionality in the PTS to help differentiate rank-2 from rank-1 temperaments. Now it’s time to learn the true meaning of rank: it’s how many generators a temperament has.

Earlier we mentioned the term “rank”. I warned you then that it wasn’t actually the same thing as dimensionality, even though we could use dimensionality in the PTS to help differentiate rank-2 from rank-1 temperaments. Now it’s time to learn the true meaning of rank: it’s how many generators a temperament has. So, it ''is'' the dimensionality of the ''tempered'' lattice; but it's still important to stay clear about the fact that it's different from the dimensionality of the original system from which you are tempering.

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.

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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).

=== intersections and unions ===

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 === periods and generators ===
-Earlier we mentioned the term “rank”. I warned you then that it wasn’t actually the same thing as geometric dimensionality, even though we could use dimensionality in the PTS to help differentiate rank-2 from rank-1 temperaments. Now it’s time to learn the true meaning of rank: it’s how many generators a temperament has.
+Earlier we mentioned the term “rank”. I warned you then that it wasn’t actually the same thing as dimensionality, even though we could use dimensionality in the PTS to help differentiate rank-2 from rank-1 temperaments. Now it’s time to learn the true meaning of rank: it’s how many generators a temperament has. So, it ''is'' the dimensionality of the ''tempered'' lattice; but it's still important to stay clear about the fact that it's different from the dimensionality of the original system from which you are tempering.
 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.
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 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).
 === intersections and unions ===