Douglas Blumeyer's RTT How-To: Difference between revisions
Cmloegcmluin (talk | contribs) m null-space improvements, per Dave's recommendations (thanks Dave) |
Cmloegcmluin (talk | contribs) assorted finesse courtesy of Dave, and some tweaks I thought up last night |
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(to be broken out to pages named "Douglas Blumeyer's RTT How-To") | |||
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. In any case, I hope they help someone else. | 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. In any case, I hope they help someone else. | ||
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Determining desirable tunings is a whole other beast. Perhaps contrary to popular belief, xenharmonic musicians — composers and performers alike — can mostly insulate themselves from this stuff if they like. It’s fine to nab a popular and well-reviewed tuning off the shelf, without deeply understanding how or why it’s there, and just pump, jam, or riff away. There's a good chance you could naturally pick up what's cool about a tuning without ever learning the definition of "temper out" or "generator". But if you do want to be deliberate about it, to mod something, rifle through the obscure section, or even discover your own tuning, then you must prepare to delve deeper into the xenharmonic fold. That’s why this resource is here, for RTT. | Determining desirable tunings is a whole other beast. Perhaps contrary to popular belief, xenharmonic musicians — composers and performers alike — can mostly insulate themselves from this stuff if they like. It’s fine to nab a popular and well-reviewed tuning off the shelf, without deeply understanding how or why it’s there, and just pump, jam, or riff away. There's a good chance you could naturally pick up what's cool about a tuning without ever learning the definition of "temper out" or "generator". But if you do want to be deliberate about it, to mod something, rifle through the obscure section, or even discover your own tuning, then you must prepare to delve deeper into the xenharmonic fold. That’s why this resource is here, for RTT. | ||
As for whether ''determining'' | As for whether ''determining'' a middle path tuning is any harder than determining an ED or JI tuning, I think it would be fair to say that in the exact same way that a middle path tuning — once attained — combines the strengths of ED and of JI, determining a middle path tuning combines the challenges of determining good ED tunings and of determining good JI tunings. You have been warned. | ||
== maps == | == maps == | ||
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If you’ve previously worked with JI, you may already be familiar with vectors. Vectors are a compact way to express JI intervals in terms of their prime factorization, or in other words, their harmonic building blocks. In JI, and in most contexts in RTT, vectors simply list the count of each prime, in order. For example, 16/15 is {{monzo|4 -1 -1}} because it has four 2’s in its numerator, one 3 in its denominator, and also one 5 in its denominator. You can look at each term as an exponent: 2⁴ × 3⁻¹ × 5⁻¹ = 16/15. | If you’ve previously worked with JI, you may already be familiar with vectors. Vectors are a compact way to express JI intervals in terms of their prime factorization, or in other words, their harmonic building blocks. In JI, and in most contexts in RTT, vectors simply list the count of each prime, in order. For example, 16/15 is {{monzo|4 -1 -1}} because it has four 2’s in its numerator, one 3 in its denominator, and also one 5 in its denominator. You can look at each term as an exponent: 2⁴ × 3⁻¹ × 5⁻¹ = 16/15. | ||
And if you’ve previously worked with EDOs, you may already be familiar with covectors. Covectors are a compact way to express EDOs in terms of the count of its steps it takes to reach its approximation of each prime harmonic, in order. For example, 12-EDO is {{val|12 19 28}}. The first term is the same as the name of the EDO, because the first prime harmonic is 2/1, or in other words: the octave. So this covector tells us that it takes 12 steps to reach the octave, 19 steps get you to about 3/1 (the [[tritave]]), and 28 steps get you to about 5/1 (the | And if you’ve previously worked with EDOs, you may already be familiar with covectors. Covectors are a compact way to express EDOs in terms of the count of its steps it takes to reach its approximation of each prime harmonic, in order. For example, 12-EDO is {{val|12 19 28}}. The first term is the same as the name of the EDO, because the first prime harmonic is 2/1, or in other words: the octave. So this covector tells us that it takes 12 steps to reach the octave, 19 steps get you to about 3/1 (the [[tritave]]), and 28 steps get you to about 5/1 (the [[pentave]]). | ||
If the musical structure that the mathematical structure called a vector represents is an '''interval''', the musical structure that the mathematical structure called a covector represents is called a '''map'''. | If the musical structure that the mathematical structure called a vector represents is an '''interval''', the musical structure that the mathematical structure called a covector represents is called a '''map'''. | ||
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The immediate conclusion is that 12-EDO is not equipped to approximate the meantone comma directly as a melodic or harmonic interval, and this shouldn’t be surprising because 81/80 is only around 20¢, while the (smallest) step in 12-EDO is five times that. | The immediate conclusion is that 12-EDO is not equipped to approximate the meantone comma directly as a melodic or harmonic interval, and this shouldn’t be surprising because 81/80 is only around 20¢, while the (smallest) step in 12-EDO is five times that. | ||
But a more interesting way to think about this result involves treating {{monzo|-4 4 -1}} not as a single interval, but as the end result of moving by a combination of intervals. For example, moving up four fifths, 4 × {{monzo|-1 1 0}} = {{monzo|-4 4 0}}, and then moving down one | But a more interesting way to think about this result involves treating {{monzo|-4 4 -1}} not as a single interval, but as the end result of moving by a combination of intervals. For example, moving up four fifths, 4 × {{monzo|-1 1 0}} = {{monzo|-4 4 0}}, and then moving down one pentave {{monzo|0 0 -1}}, gets you right back where you started in 12-EDO. Or, in other words, moving by one pentave is the same thing as moving by four fifths ''(see Figure 1b)''. One can make compelling music that [[Keenan's comma pump page|exploits such harmonic mechanisms]]. | ||
From this perspective, the disappearance of 81/80 is not a shortcoming, but a fascinating feature of 12-EDO; we say that 12-EDO '''supports''' the meantone temperament. And 81/80 in 12-EDO is only the beginning of that journey. For many people, tempering commas is one of the biggest draws to RTT. | From this perspective, the disappearance of 81/80 is not a shortcoming, but a fascinating feature of 12-EDO; we say that 12-EDO '''supports''' the meantone temperament. And 81/80 in 12-EDO is only the beginning of that journey. For many people, tempering commas is one of the biggest draws to RTT. | ||
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== outro == | == outro == | ||
You’ve made it to the end. This is pretty much everything that I understand about RTT at this point. This took me a little over a month of full-time funemployed exploration to learn and put together; at the onset, while having considered myself a xenharmonicist for 15 years, I only understood a few scattered things about RTT, such as ET maps and commas and the overlapping MOS concepts. I hope that equipped with my results here it shouldn’t take a newcomer nearly that much time to get going. And of course I hope to continue learning more myself and perhaps even contributing to the theory one day, since there’s still plenty of frontier here. | You’ve made it to the end. This is pretty much everything that I understand about RTT at this point (May 2021). This took me a little over a month of full-time funemployed exploration to learn and put together; at the onset, while having considered myself a xenharmonicist for 15 years, I only understood a few scattered things about RTT, such as ET maps and commas and the overlapping MOS concepts. I hope that equipped with my results here it shouldn’t take a newcomer nearly that much time to get going. And of course I hope to continue learning more myself and perhaps even contributing to the theory one day, since there’s still plenty of frontier here. | ||
I couldn’t have put this together without the help of: | I couldn’t have put this together without the help of: | ||
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* [[Stephen Weigel]] | * [[Stephen Weigel]] | ||
plus many many more. And of course I owe a big debt to [[Gene Ward Smith]]. | plus many many more. And of course I owe a big debt to [[Gene Ward Smith]]. | ||
I take full responsibility for any errors or shortcomings of this work. Please feel free to edit this stuff yourself if you have something you'd like to correct, revise, or contribute. | |||
Happy tempering! | Happy tempering! | ||
<references/> | <references/> | ||