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

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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).

Let’s bring up MOS theory again. We mentioned earlier that you might have been familiar with the scale tree if you’d worked with MOS scales before, and if so, the connection was scale cardinalities, or in other words, how many notes are in the resultant scales when you continuously iterate the generator until you reach points where there are only two scale step sizes. At these points scales tend to sound relatively good, and this is in fact the definition of a MOS scale. There’s a mathematical explanation for how to know, given a ratio between the size of your generator and period, the cardinalities of scales possible; we won’t re-explain it here. The point is that the scale tree can show you that pattern visually. And so if each temperament line in PTS is its own segment of the scale tree, then we can use it in a similar way.

For example, if we pick a point along the meantone line between 46 and 29, the cardinalities will be 5, 12, 17, 29, 46, etc. If we chose exactly the point at 29, then the cardinality pattern would terminate there, or in other words, eventually we’ll hit a scale with 29 notes and instead of two different step sizes there would only be one, and there’s no place else to go from there. The system has circled back around to its starting point, so it’s a closed system. Further generator iterations will only retread notes you’ve already touched. The same would be true if you chose exactly the point at 46, except that’s where you’d hit an ET instead.

Between ETs, in the stretches of rank-2 temperament lines where the generator is not a rational fraction of the octave, theoretically those temperaments could have infinite pitches; you could continuously iterate the generator and you’d never exactly circle back to the point where you started. If bigger numbers were shown on PTS, you could continue to use those numbers to guide your cardinalities forever.

The structure when you stop iterating the meantone generator with five notes is called meantone[5]. If you were to use the entirety of 12-ET as meantone then that’d be meantone[12]. But you can also realize meantone[12] in 19-ET; in the former you have only one step size, but in the latter you have two. You can’t realize meantone[19] in 12-ET, but you could also realize it in 31-ET.

=== meet and join ===

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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).
+Let’s bring up MOS theory again. We mentioned earlier that you might have been familiar with the scale tree if you’d worked with MOS scales before, and if so, the connection was scale cardinalities, or in other words, how many notes are in the resultant scales when you continuously iterate the generator until you reach points where there are only two scale step sizes. At these points scales tend to sound relatively good, and this is in fact the definition of a MOS scale. There’s a mathematical explanation for how to know, given a ratio between the size of your generator and period, the cardinalities of scales possible; we won’t re-explain it here. The point is that the scale tree can show you that pattern visually. And so if each temperament line in PTS is its own segment of the scale tree, then we can use it in a similar way.
+For example, if we pick a point along the meantone line between 46 and 29, the cardinalities will be 5, 12, 17, 29, 46, etc. If we chose exactly the point at 29, then the cardinality pattern would terminate there, or in other words, eventually we’ll hit a scale with 29 notes and instead of two different step sizes there would only be one, and there’s no place else to go from there. The system has circled back around to its starting point, so it’s a closed system. Further generator iterations will only retread notes you’ve already touched. The same would be true if you chose exactly the point at 46, except that’s where you’d hit an ET instead.
+Between ETs, in the stretches of rank-2 temperament lines where the generator is not a rational fraction of the octave, theoretically those temperaments could have infinite pitches; you could continuously iterate the generator and you’d never exactly circle back to the point where you started. If bigger numbers were shown on PTS, you could continue to use those numbers to guide your cardinalities forever.
+The structure when you stop iterating the meantone generator with five notes is called meantone[5]. If you were to use the entirety of 12-ET as meantone then that’d be meantone[12]. But you can also realize meantone[12] in 19-ET; in the former you have only one step size, but in the latter you have two. You can’t realize meantone[19] in 12-ET, but you could also realize it in 31-ET.
 === meet and join ===