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 (or a unit fraction of 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.

The period, usually being the largest generator, serves as the [[interval of repetition]]. Rank-1 temperaments have only one generator, so that generator technically is the period, though we usually don’t think of it that way; that’s because the word “period” is designed to convey the way in which the other smaller generators can be understood to cycle within it, and so without any smaller generators doing that, it sort of loses its meaning. Sometimes the octave (or a fraction thereof) is treated as the “period” of an ET, especially when comparing the ET with a related rank-2 temperament; that may make sense in context, though technically this interval could only be the [[interval of equivalence]], not also the interval of repetition.

The period, usually being the largest generator, serves as the [[interval of repetition]]. Rank-1 temperaments have only one generator, so that generator technically is the period, though we usually don’t think of it that way; that’s because the word “period” is designed to convey the way in which the other smaller generators can be understood to cycle within it, and so without any smaller generators doing that ''(see Figure 4c)'', it sort of loses its meaning. Sometimes the octave (or a fraction thereof) is treated as the “period” of an ET, especially when comparing the ET with a related rank-2 temperament; that may make sense in context, though technically this interval could only be the [[interval of equivalence]], not also the interval of repetition ''(see Figure 4d)''.

[[File:Period vs generator wrt interval of repetition.png|400px|thumb|left|'''Figure 4c.''' The period is the interval of repetition.]]

[[File:Interval of repetition vs interval of equivalence.png|200px|thumb|'''Figure 4d.''' Basically, the octave is usually the interval of equivalence, except in cases like tritave-based systems such as [[Bohlen-Pierce]]. And the period is always the interval of repetition. So when the period is the octave, such as with linear temperaments, then it is both the interval of repetition and of equivalence.]]

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.

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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 (or a unit fraction of 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.
-The period, usually being the largest generator, serves as the [[interval of repetition]]. Rank-1 temperaments have only one generator, so that generator technically is the period, though we usually don’t think of it that way; that’s because the word “period” is designed to convey the way in which the other smaller generators can be understood to cycle within it, and so without any smaller generators doing that, it sort of loses its meaning. Sometimes the octave (or a fraction thereof) is treated as the “period” of an ET, especially when comparing the ET with a related rank-2 temperament; that may make sense in context, though technically this interval could only be the [[interval of equivalence]], not also the interval of repetition.
+The period, usually being the largest generator, serves as the [[interval of repetition]]. Rank-1 temperaments have only one generator, so that generator technically is the period, though we usually don’t think of it that way; that’s because the word “period” is designed to convey the way in which the other smaller generators can be understood to cycle within it, and so without any smaller generators doing that ''(see Figure 4c)'', it sort of loses its meaning. Sometimes the octave (or a fraction thereof) is treated as the “period” of an ET, especially when comparing the ET with a related rank-2 temperament; that may make sense in context, though technically this interval could only be the [[interval of equivalence]], not also the interval of repetition ''(see Figure 4d)''.
+[[File:Period vs generator wrt interval of repetition.png|400px|thumb|left|'''Figure 4c.''' The period is the interval of repetition.]]
+[[File:Interval of repetition vs interval of equivalence.png|200px|thumb|'''Figure 4d.''' Basically, the octave is usually the interval of equivalence, except in cases like tritave-based systems such as [[Bohlen-Pierce]]. And the period is always the interval of repetition. So when the period is the octave, such as with linear temperaments, then it is both the interval of repetition and of equivalence.]]
 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.