Douglas Blumeyer's RTT How-To: Difference between revisions
Cmloegcmluin (talk | contribs) →a multitude of maps: remove "concerning" on Dave Keenan's recommendation |
Cmloegcmluin (talk | contribs) detune → deviate from JI, or temper (Dave Keenan's suggestion) |
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Why is this rare? Well, it’s like a game of trying to get these numbers to line up ''(see Figure 2d)'': | Why is this rare? Well, it’s like a game of trying to get these numbers to line up ''(see Figure 2d)'': | ||
[[File:Near linings up rare2.png|600px|thumb|right|'''Figure 2d.''' Texture of ETs approximating prime harmonics. Where the ''numerals'' (and tick marks) line up, all primes are well-approximated by a single step size (the boundaries between cells are midpoints between perfect approximations, or in other words, the point where the closest approximation switches over from one generator count to the next) (the numerals and tick marks are meant to be centered in each cell). Nudging one of the maps' vertical lines to the right would mean decreasing the generator size, flattening the tunings of all the primes, and vice versa, nudging it to the left would mean increasing the generator size, sharpening the tunings of all the primes. You can visualize this on Figure 2c. as shrinking or growing the height of the rectangular bricks. The positions of each map's vertical line, or in other words the tuning of its generator, has been optimized using some formula to distribute the | [[File:Near linings up rare2.png|600px|thumb|right|'''Figure 2d.''' Texture of ETs approximating prime harmonics. Where the ''numerals'' (and tick marks) line up, all primes are well-approximated by a single step size (the boundaries between cells are midpoints between perfect approximations, or in other words, the point where the closest approximation switches over from one generator count to the next) (the numerals and tick marks are meant to be centered in each cell). Nudging one of the maps' vertical lines to the right would mean decreasing the generator size, flattening the tunings of all the primes, and vice versa, nudging it to the left would mean increasing the generator size, sharpening the tunings of all the primes. You can visualize this on Figure 2c. as shrinking or growing the height of the rectangular bricks. The positions of each map's vertical line, or in other words the tuning of its generator, has been optimized using some formula to distribute the deviations amongst the three primes; that's why you do not see any vertical line here for which the closest step counts for each prime are all on one side of it.]] | ||
If the distance between entries in the row for 2 are defined as 1 unit apart, then the distance between entries in the row for prime 3 are 1/log₂3 units apart, and 1/log₂5 units apart for the prime 5. So, near-linings up don’t happen all that often!<ref>For more information, see: [[The_Riemann_zeta_function_and_tuning|The Riemann zeta function and tuning]].</ref> (By the way, any vertical line drawn through a chart like this is called a GPV, or “[[generalized patent val]]”.) | If the distance between entries in the row for 2 are defined as 1 unit apart, then the distance between entries in the row for prime 3 are 1/log₂3 units apart, and 1/log₂5 units apart for the prime 5. So, near-linings up don’t happen all that often!<ref>For more information, see: [[The_Riemann_zeta_function_and_tuning|The Riemann zeta function and tuning]].</ref> (By the way, any vertical line drawn through a chart like this is called a GPV, or “[[generalized patent val]]”.) | ||
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=== tuning & pure octaves === | === tuning & pure octaves === | ||
Now, because the octave is the [[interval of equivalence]] in terms of human pitch perception, it’s a major convenience to enforce pure octaves, and so many people prefer the first term to be exact. In fact, I’ll bet many readers have never even heard of or imagined impure octaves, if my own anecdotal experience is any indicator; the idea that I could | Now, because the octave is the [[interval of equivalence]] in terms of human pitch perception, it’s a major convenience to enforce pure octaves, and so many people prefer the first term to be exact. In fact, I’ll bet many readers have never even heard of or imagined impure octaves, if my own anecdotal experience is any indicator; the idea that I could temper octaves to optimize tunings came rather late to me. | ||
Well, you’ll notice that in the previous section, we did approximate the octave, using 1.998 instead of 2. But another thing {{val|12 19 28}} has going for it is that it excels at approximating 5-limit JI even if we constrain ourselves to pure octaves, locking g¹² to exactly 2: (¹²√2)¹⁹ ≈ 2.997 and (¹²√2)²⁸ ≈ 5.040. You can see that actually the approximation of 3 is even better here, marginally; it’s the damage to 5 which is lamentable. | Well, you’ll notice that in the previous section, we did approximate the octave, using 1.998 instead of 2. But another thing {{val|12 19 28}} has going for it is that it excels at approximating 5-limit JI even if we constrain ourselves to pure octaves, locking g¹² to exactly 2: (¹²√2)¹⁹ ≈ 2.997 and (¹²√2)²⁸ ≈ 5.040. You can see that actually the approximation of 3 is even better here, marginally; it’s the damage to 5 which is lamentable. | ||
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When we don’t enforce pure octaves, tuning becomes a more interesting problem. Approximating all three primes at once with the same generator is a balancing act. At least one of the primes will be tuned a bit sharp while at least one of them will be tuned a bit flat. In the case of {{val|12 19 28}}, the 5 is a bit sharp, and the 2 and 3 are each a tiny bit flat ''(as you can see in Figure 2c)''. | When we don’t enforce pure octaves, tuning becomes a more interesting problem. Approximating all three primes at once with the same generator is a balancing act. At least one of the primes will be tuned a bit sharp while at least one of them will be tuned a bit flat. In the case of {{val|12 19 28}}, the 5 is a bit sharp, and the 2 and 3 are each a tiny bit flat ''(as you can see in Figure 2c)''. | ||
[[File:Why not just srhink every block.png|thumb|left|600px|'''Figure 2e.''' Visualization of pointlessness of tuning all primes sharp (you should be able to imagine the opposite case, where all primes are tuned flat). To be completely accurate, depending on your actual scale, there maybe cases where tuning all the primes sharp (or pure) may not be pointless, depending on which combinations of primes you use in your pitches and in particular which sides of the fraction bar they're on i.e. if they are on opposite sides then their | [[File:Why not just srhink every block.png|thumb|left|600px|'''Figure 2e.''' Visualization of pointlessness of tuning all primes sharp (you should be able to imagine the opposite case, where all primes are tuned flat). To be completely accurate, depending on your actual scale, there maybe cases where tuning all the primes sharp (or pure) may not be pointless, depending on which combinations of primes you use in your pitches and in particular which sides of the fraction bar they're on i.e. if they are on opposite sides then their temperings may be proportional and thus damage cancels out rather than compounds. But in general, this diagram sends the right message.]] | ||
If you think about it, you would never want to tune all the primes sharp at the same time, or all of them flat; if you care about this particular proportion of their tunings, why wouldn’t you shift them all in the same direction, toward accuracy, while maintaining that proportion? ''(see Figure 2e)'' | If you think about it, you would never want to tune all the primes sharp at the same time, or all of them flat; if you care about this particular proportion of their tunings, why wouldn’t you shift them all in the same direction, toward accuracy, while maintaining that proportion? ''(see Figure 2e)'' | ||
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If your goal is to evoke JI-like harmony, then, {{val|12 20 28}} is not your friend. Feel free to work out some other variations on {{val|12 19 28}} if you like, such as {{val|12 19 29}} maybe, but I guarantee you won’t find a better one that starts with 12 than {{val|12 19 28}}. | If your goal is to evoke JI-like harmony, then, {{val|12 20 28}} is not your friend. Feel free to work out some other variations on {{val|12 19 28}} if you like, such as {{val|12 19 29}} maybe, but I guarantee you won’t find a better one that starts with 12 than {{val|12 19 28}}. | ||
[[File:17-ET mistunings.png|thumb|600px|right|'''Figure 2f.''' | [[File:17-ET mistunings.png|thumb|600px|right|'''Figure 2f.''' Deviations from JI for various 17-ET maps, showing how the supposed "patent" val's total error can be improved upon by allowing tempered octaves and second-closest mappings of primes. It also shows how pure octave 17c has no primes tuned flat.]] | ||
So the case is cut-and-dry for {{val|12 19 28}}, and therefore from now on I'm simply going to refer to this ET by "12-ET" rather than spelling out its map. But other ETs find themselves in trickier situations. Consider [[17edo|17-ET]]. One option we have is the map {{val|17 27 39}}, with a generator of about 1.0418, and prime approximations of 2.0045, 3.0177, 4.9302. But we have a second reasonable option here, too, where {{val|17 27 40}} gives us a generator of 1.0414, and prime approximations of 1.9929, 2.9898, and 5.0659. In either case, the approximations of 2 and 3 are close, but the approximation of 5 is way off. For {{val|17 27 39}}, it’s way small, while for {{val|17 27 40}} it’s way big. The conundrum could be described like this: any generator we could find that divides 2 into about 17 equal steps can do a good job dividing 3 into about 27 equal steps, too, but it will not do a good job of dividing 5 into equal steps; 5 is going to land, unfortunately, right about in the middle between the 39th and 40th steps, as far as possible from either of these two nearest approximations. To do a good job approximating prime 5, we’d really just want to subdivide each of these steps in half, or in other words, we’d want [[34edo|34-ET]]. | So the case is cut-and-dry for {{val|12 19 28}}, and therefore from now on I'm simply going to refer to this ET by "12-ET" rather than spelling out its map. But other ETs find themselves in trickier situations. Consider [[17edo|17-ET]]. One option we have is the map {{val|17 27 39}}, with a generator of about 1.0418, and prime approximations of 2.0045, 3.0177, 4.9302. But we have a second reasonable option here, too, where {{val|17 27 40}} gives us a generator of 1.0414, and prime approximations of 1.9929, 2.9898, and 5.0659. In either case, the approximations of 2 and 3 are close, but the approximation of 5 is way off. For {{val|17 27 39}}, it’s way small, while for {{val|17 27 40}} it’s way big. The conundrum could be described like this: any generator we could find that divides 2 into about 17 equal steps can do a good job dividing 3 into about 27 equal steps, too, but it will not do a good job of dividing 5 into equal steps; 5 is going to land, unfortunately, right about in the middle between the 39th and 40th steps, as far as possible from either of these two nearest approximations. To do a good job approximating prime 5, we’d really just want to subdivide each of these steps in half, or in other words, we’d want [[34edo|34-ET]]. | ||
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[[File:17-ET.png|thumb|400px|left|'''Figure 2g.''' Visualization of the 17-ETs on the GPV continuum, showing how for the pure octave 17c there exists no generator that exactly reaches prime 2 in 17 steps while more closely approximating prime 5 as 40 steps than 39 steps. (If this diagram is unclear, please refer back to Figure 2d., which has the same type of information but with more thorough labelling.)]] | [[File:17-ET.png|thumb|400px|left|'''Figure 2g.''' Visualization of the 17-ETs on the GPV continuum, showing how for the pure octave 17c there exists no generator that exactly reaches prime 2 in 17 steps while more closely approximating prime 5 as 40 steps than 39 steps. (If this diagram is unclear, please refer back to Figure 2d., which has the same type of information but with more thorough labelling.)]] | ||
Curiously, {{val|17 27 39}} is the map for which each prime individually is as closely approximated as possible when prime 2 is exact, so it is in a sense the naively best map for 17-ET, however, if that constraint is lifted, and we’re allowed to either | Curiously, {{val|17 27 39}} is the map for which each prime individually is as closely approximated as possible when prime 2 is exact, so it is in a sense the naively best map for 17-ET, however, if that constraint is lifted, and we’re allowed to either temper prime 2 and/or choose the next-closest approximations for prime 5, the overall approximation can be improved; in other words, even though 39 steps can take you just a tiny bit closer to prime 5 than 40 steps can, the tiny amount by which it is closer is less than the improvements to the tuning of primes 2 and 3 you can get by using {{val|17 27 40}}. So again, the choice is not always cut-and-dry; there’s still a lot of personal preference going on in the tempering process. | ||
So some musicians may conclude “17-ET is clearly not cut out for 5-limit music,” and move on to another ET. Other musicians may snicker maniacally, and choose one or the other map, and begin exploiting the profound and unusual 5-limit harmonic mechanisms it affords. {{val|17 27 40}}, like {{val|12 19 28}}, tempers out the meantone comma {{monzo|-4 4 -1}}, so even though fifths and major thirds are different sizes in these two ETs, the relationship that four fifths equals one major third is shared. {{val|17 27 39}}, on the other hand, does not work like that, but what it does do is temper out 25/24, {{monzo|-3 -1 2}}, or in other words, it equates one fifth with two major thirds. | So some musicians may conclude “17-ET is clearly not cut out for 5-limit music,” and move on to another ET. Other musicians may snicker maniacally, and choose one or the other map, and begin exploiting the profound and unusual 5-limit harmonic mechanisms it affords. {{val|17 27 40}}, like {{val|12 19 28}}, tempers out the meantone comma {{monzo|-4 4 -1}}, so even though fifths and major thirds are different sizes in these two ETs, the relationship that four fifths equals one major third is shared. {{val|17 27 39}}, on the other hand, does not work like that, but what it does do is temper out 25/24, {{monzo|-3 -1 2}}, or in other words, it equates one fifth with two major thirds. | ||
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[[File:Understanding projection.png|600px|thumb|left|'''Figure 3h.''' Visualization of the projection process. (In real life, the cube is infinite in size. I made it smaller here to help make the shape clearer.)]] | [[File:Understanding projection.png|600px|thumb|left|'''Figure 3h.''' Visualization of the projection process. (In real life, the cube is infinite in size. I made it smaller here to help make the shape clearer.)]] | ||
PTS doesn’t show the entire tuning cube. You can see evidence of this in the fact that some numerals have been cut off on its edges. We’ve cropped things around the central region of information, which is where the ETs best approximating JI are found (note how close 53-ET is to the center!). Paul added some concentric hexagons to the center of his diagram, which you could think of as concentric around that interior diagonal, or in other words, are defined by gradually increasing thresholds of deviations from | PTS doesn’t show the entire tuning cube. You can see evidence of this in the fact that some numerals have been cut off on its edges. We’ve cropped things around the central region of information, which is where the ETs best approximating JI are found (note how close 53-ET is to the center!). Paul added some concentric hexagons to the center of his diagram, which you could think of as concentric around that interior diagonal, or in other words, are defined by gradually increasing thresholds of deviations from JI for any one prime at a time. | ||
No maps past [[99edo|99-ET]] are drawn on this diagram. ETs with that many steps are considered too complex (read: big numbers, impractical) to bother cluttering the diagram with. Better to leave the more useful information easier to read. | No maps past [[99edo|99-ET]] are drawn on this diagram. ETs with that many steps are considered too complex (read: big numbers, impractical) to bother cluttering the diagram with. Better to leave the more useful information easier to read. | ||
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So: we now know which point is {{val|12 19 28}}, and we know a couple of 17’s, 40’s and a 41. But can we answer in the general case? Given an arbitrary map, like {{val|7 11 16}}, can we find it on the diagram? Well, you may look to the first term, 7, which tells you it’s [[7edo|7-ET]]. There’s only one big 7 on this diagram, so it’s probably that. (You’re right). But that one’s easy. The 7 is huge. | So: we now know which point is {{val|12 19 28}}, and we know a couple of 17’s, 40’s and a 41. But can we answer in the general case? Given an arbitrary map, like {{val|7 11 16}}, can we find it on the diagram? Well, you may look to the first term, 7, which tells you it’s [[7edo|7-ET]]. There’s only one big 7 on this diagram, so it’s probably that. (You’re right). But that one’s easy. The 7 is huge. | ||
What if I gave you {{val|43 68 100}}. Where’s [[43edo|43-ET]]? I’ll bet you’re still complaining: the map expresses the | What if I gave you {{val|43 68 100}}. Where’s [[43edo|43-ET]]? I’ll bet you’re still complaining: the map expresses the tempering of 2, 3, and 5 in terms of their shared generator, but doesn’t tell us directly which primes are sharp, and which primes are flat, so how could we know in which region to look for this ET? | ||
The answer to that is, unfortunately: that’s just how it is. It can be a bit of a hunt sometimes. But the chances are, in the real world, if you’re looking for a map or thinking about it, then you probably already have at least some other information about it to help you find it, whether it’s memorized in your head, or you’re reading it off the results page for an automatic temperament search tool. | The answer to that is, unfortunately: that’s just how it is. It can be a bit of a hunt sometimes. But the chances are, in the real world, if you’re looking for a map or thinking about it, then you probably already have at least some other information about it to help you find it, whether it’s memorized in your head, or you’re reading it off the results page for an automatic temperament search tool. | ||
Probably you have the information about the primes’ | Probably you have the information about the primes’ tempering; maybe you get lucky and a 43 jumps out at you but it’s not the one you’re looking for, but you can use what you know about the perspectival scaling and axis directions and log-of-prime scaling to find other 43’s relative to it. | ||
Or maybe you know which commas {{val|43 68 100}} tempers out, so you can find it along the line for that comma’s temperament. | Or maybe you know which commas {{val|43 68 100}} tempers out, so you can find it along the line for that comma’s temperament. | ||