Douglas Blumeyer's RTT How-To: Difference between revisions
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This is the reference I wish I had when I was learning RTT, or [[Regular temperament theory|Regular Temperament Theory]]. There are other great resources out there, but this is how I would have liked to have learned it myself. I might say these materials lean more visual and geometric than others I've seen, and focus on elementary computation and representation rather than theory. It's not really a big picture introduction, it doesn't explore musical applications, and its algorithms are for humans, not computers. In any case, I hope others are able to benefit from these tools and explanations. | This is the reference I wish I had when I was learning RTT, or [[Regular temperament theory|Regular Temperament Theory]]. There are other great resources out there, but this is how I would have liked to have learned it myself. I might say these materials lean more visual and geometric than others I've seen, and focus on elementary computation and representation rather than theory. It's not really a big picture introduction, it doesn't explore musical applications, and its algorithms are for humans, not computers. In any case, I hope others are able to benefit from these tools and explanations. | ||
== intro == | == intro == | ||
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In this first section, you will learn about maps — one of the basic building blocks of temperaments — and the effect maps have on musical intervals. | In this first section, you will learn about maps — one of the basic building blocks of temperaments — and the effect maps have on musical intervals. | ||
=== vectors and covectors === | === vectors and covectors === | ||
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In this section, we will be going into potentially excruciating detail about how to read the projective tuning space diagram featured prominently in Paul Erlich's Middle Path paper. For me personally, attaining total understanding of this diagram was critical before the linear algebra stuff (that we'll discuss afterwards) started to mean much to me. But other people might not work that way, and the extent of detail I go into in this section is not necessary to become competent with RTT (in fact, to my delight, one of the points I make in this section was news to Paul himself). So if you're already confident about reading the PTS diagram, you may try skipping ahead. | In this section, we will be going into potentially excruciating detail about how to read the projective tuning space diagram featured prominently in Paul Erlich's Middle Path paper. For me personally, attaining total understanding of this diagram was critical before the linear algebra stuff (that we'll discuss afterwards) started to mean much to me. But other people might not work that way, and the extent of detail I go into in this section is not necessary to become competent with RTT (in fact, to my delight, one of the points I make in this section was news to Paul himself). So if you're already confident about reading the PTS diagram, you may try skipping ahead. | ||
=== intro to PTS === | === intro to PTS === | ||
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And so it makes sense that {{map|17 27 40}} and {{map|17 27 39}} are aligned horizontally, because the only difference between their maps is in the 5-term, and the 5-axis is horizontal. | And so it makes sense that {{map|17 27 40}} and {{map|17 27 39}} are aligned horizontally, because the only difference between their maps is in the 5-term, and the 5-axis is horizontal. | ||
=== scaled axes === | === scaled axes === | ||
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We're about to take our first look at temperaments beyond mere equal temperaments. By the end of this section, you'll be able to explain the musical meaning of the patterns in the numerals along lines in PTS, the labels of these lines, as well as what's happening at their intersections and what their slopes mean. In other words, pretty much all of the major remaining visual elements on PTS should make sense to you. | We're about to take our first look at temperaments beyond mere equal temperaments. By the end of this section, you'll be able to explain the musical meaning of the patterns in the numerals along lines in PTS, the labels of these lines, as well as what's happening at their intersections and what their slopes mean. In other words, pretty much all of the major remaining visual elements on PTS should make sense to you. | ||
=== temperament lines === | === temperament lines === | ||
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And the related constraint for rank-1 from two rank-2 is that you can’t choose two temperaments whose names are printed smaller on the page than another temperament between them. More on that later. | And the related constraint for rank-1 from two rank-2 is that you can’t choose two temperaments whose names are printed smaller on the page than another temperament between them. More on that later. | ||
== matrices == | |||
From the PTS diagram, we can visually pick out rank-1 temperaments at the intersection of rank-2 temperaments as well as rank-2 temperaments as the unions of rank-1 temperaments. But we can also understand these results through covectors and vectors. And we're going to need to learn how, because PTS can only take us so far. 5-limit PTS is good for humans because we live in a physically 3-dimensional world (and spend a lot of time sitting in front of 2D pages on paper and on computer screens), but as soon as you want to start working in 7-limit harmony, which is 4D, visual analogies will begin to fail us, and if we’re not equipped with the necessary mathematical abstractions, we’ll no longer be able to effectively navigate. | |||
Don’t worry: we’re not going 4D just yet. We’ve still got plenty we can cover using only the 5-limit. But we may put away PTS for a couple sections. It’s matrix time. By the end of this section, you'll understand how to represent a temperament in matrix form, how to interpret them, notate them, and use them, as well as how to apply important transformations between different kinds of these matrices. | |||
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That looks like an identity matrix! Well, in this case the best interpretation can be found by checking its mapping of 2/1, 3/1, and 5/1, or in other words {{vector|1}}, {{vector|0 1}}, and {{vector|0 0 1}}. Each prime is generated by a different generator, independently. And if you think about the implications of that, you’ll realize that this is simply another way of expressing the idea of 5-limit JI! Because the three generators are entirely independent, we are capable of exactly generating literally any 5-limit interval. Which is another way of confirming our hypothesis that no commas are tempered out. | That looks like an identity matrix! Well, in this case the best interpretation can be found by checking its mapping of 2/1, 3/1, and 5/1, or in other words {{vector|1}}, {{vector|0 1}}, and {{vector|0 0 1}}. Each prime is generated by a different generator, independently. And if you think about the implications of that, you’ll realize that this is simply another way of expressing the idea of 5-limit JI! Because the three generators are entirely independent, we are capable of exactly generating literally any 5-limit interval. Which is another way of confirming our hypothesis that no commas are tempered out. | ||
=== tempered lattice === | === tempered lattice === | ||
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And so we can see that tempering has reduced the dimensionality of our lattice by 1. Or in other words, the dimensionality of our lattice was always the rank; it’s just that in JI, the rank was equal to the dimensionality. And what’s happened by reducing this rank is that we eliminated one of the primes in a sense, by making it so we can only express things in terms of it via combinations of the other remaining primes. | And so we can see that tempering has reduced the dimensionality of our lattice by 1. Or in other words, the dimensionality of our lattice was always the rank; it’s just that in JI, the rank was equal to the dimensionality. And what’s happened by reducing this rank is that we eliminated one of the primes in a sense, by making it so we can only express things in terms of it via combinations of the other remaining primes. | ||
=== rank and nullity === | === rank and nullity === | ||
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== multimaps & multicommas == | == multimaps & multicommas == | ||
=== multimaps === | === multimaps === | ||
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|monzo | |monzo | ||
|} | |} | ||
=== tuning === | === tuning === | ||