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

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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 {{map|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 {{map|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 error on 5 which is lamentable.

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 {{map|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 temperings may be proportional and thus ~~damage~~ cancels out rather than compounds. But in general, this diagram sends the right message.]]

[[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 error 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)''

This matter of choosing the exact generator for a map is called '''tuning''', and if you’ll believe it, we won’t actually talk about that in detail again until much later. Temperament — the second ‘T’ in “RTT” — is the discipline concerned with choosing an interesting map, and tuning can remain largely independent from it. The temperament is only concerned with the fact that — no matter what exact size you ultimately make the generator — it is the case e.g. that 12 of them make a 2, 19 of them make a 3, and 28 of them make a 5. So, for now, whenever we show a value for g, assume we’ve given a computer a formula for optimizing the tuning to approximate all three primes equally well. As for us humans, let’s stay focused on tempering.

Damage, by the way, is a technical term. That refers to the delta in cents of the tempering for a prime (which is known as the error, by the way) but divided by log₂ of that prime. So for octaves, damage is the same as error. So if prime 3 was tuned 4.1 cents flat, that's its error, but if you want to know damage, you need 4.1/log₂3 = 2.587. We typically use damage instead of error when comparing across primes, because damage tells us how much a prime has been impacted relative to its complexity; we care much more about error to lower primes like 2, 3, and 5 than we do really high up and obscure building blocks like 37 and 41.

=== a multitude of maps ===

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Suppose we want to experiment with {{map|12 19 28}}’s map a bit. We’ll change one of the terms by 1, so now we have {{map|12 20 28}}. Because the previous map did such a great job of approximating the 5-limit (i.e. log(2:3:5)), though, it should be unsurprising that this new map cannot achieve that feat. The proportions, 12:20:28, should now be about as out of whack as they can get. The best generator we can do here is about 1.0583 (getting a little more precise now), and 1.0583¹² ≈ 1.9738 which isn’t so bad, but 1.0583¹⁹ = 3.1058 and 1.0583²⁸ = 4.8870 which are both way off! And they’re way off in the opposite direction — 3.1058 is too big and 4.8870 is too small — which is why our tuning formula for g, which is designed to make the approximation good for every prime at once, can’t improve the situation: either sharpening or flattening helps one but hurts the other.

The results of such inaccurate approximation are a bit chaotic. A ratio like 16/15 — where the factors of 3 and 5 are on the same side of the fraction bar and therefore cancel out each other’s ~~damage~~ — fares relatively alright, if by “alright” we mean it gets tempered out despite being about 112¢ in JI. On the other hand, an interval like 27/25 where the factors of 3 and 5 are on opposite sides of the fraction bar and thus their ~~damages~~ compound, gets mapped to a whopping 4 steps, despite only being about 133¢ in JI.

The results of such inaccurate approximation are a bit chaotic. A ratio like 16/15 — where the factors of 3 and 5 are on the same side of the fraction bar and therefore cancel out each other’s error — fares relatively alright, if by “alright” we mean it gets tempered out despite being about 112¢ in JI. On the other hand, an interval like 27/25 where the factors of 3 and 5 are on opposite sides of the fraction bar and thus their errors compound, gets mapped to a whopping 4 steps, despite only being about 133¢ in JI.

If your goal is to evoke JI-like harmony, then, {{map|12 20 28}} is not your friend. Feel free to work out some other variations on {{map|12 19 28}} if you like, such as {{map|12 19 29}} maybe, but I guarantee you won’t find a better one that starts with 12 than {{map|12 19 28}}.

[[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<ref>Yes, this diagram is showing error, not damage. If it showed damage, the difference would be even more dramatic. And most people care more about damage than error. But damage is simpler to convey, so that's why I went with it.</ref> 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.]]

[[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 {{map|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 {{map|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 {{map|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 {{map|17 27 39}}, it’s way small, while for {{map|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:Tuning projection.png|400px|thumb|left|'''Figure 3l.''' Demonstration of tuning projection. As long as the tunings change in a fixed proportion, the tuning will project to the same point on PTS.]]

But perhaps even more interesting than this continuous tuning space that appears in PTS between points is the continuous tuning space that does not appear in PTS because it exists within each point, that is, exactly out from and deeper into the page at each point. In tuning space, as we’ve just established, there are no maps in front of or behind each other that get collapsed to a single point. But there are still many things that get collapsed to a single point like this, but in tuning space they are different tunings ''(see Figure 3l)''. For example, {{map|1200 1900 2800}} is the way we’d write 12-ET in tuning space. But there are other tunings represented by this same point in PTS, such as {{map|1200.12 1900.19 2800.28}} (note that in order to remain at the same point, we’ve maintained the exact proportions of all the prime tunings). That tuning might not be of particular interest. I just used it as a simple example to illustrate the point. A more useful example would be {{map|1198.440 1897.531 2796.361}}, which by some algorithm is the optimal tuning for 12-ET (minimizes ~~damage~~ across primes or intervals); it may not be as obvious from looking at that one, but if you check the proportions of those terms with each other, you will find they are still exactly 12:19:28.

But perhaps even more interesting than this continuous tuning space that appears in PTS between points is the continuous tuning space that does not appear in PTS because it exists within each point, that is, exactly out from and deeper into the page at each point. In tuning space, as we’ve just established, there are no maps in front of or behind each other that get collapsed to a single point. But there are still many things that get collapsed to a single point like this, but in tuning space they are different tunings ''(see Figure 3l)''. For example, {{map|1200 1900 2800}} is the way we’d write 12-ET in tuning space. But there are other tunings represented by this same point in PTS, such as {{map|1200.12 1900.19 2800.28}} (note that in order to remain at the same point, we’ve maintained the exact proportions of all the prime tunings). That tuning might not be of particular interest. I just used it as a simple example to illustrate the point. A more useful example would be {{map|1198.440 1897.531 2796.361}}, which by some algorithm is the optimal tuning for 12-ET (minimizes error across primes or intervals); it may not be as obvious from looking at that one, but if you check the proportions of those terms with each other, you will find they are still exactly 12:19:28.

The key point here is that, as we mentioned before, the problems of tuning and tempering are largely separate. PTS projects all tunings of the same temperament to the same point. This way, issues of tuning are completely hidden and ignored on PTS, so we can focus instead on tempering.

@@ Line 128: / Line 128: @@
 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 {{map|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 {{map|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 error on 5 which is lamentable.
 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 {{map|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 temperings may be proportional and thus damage cancels out rather than compounds. But in general, this diagram sends the right message.]]
+[[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 error 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)''
 This matter of choosing the exact generator for a map is called '''tuning''', and if you’ll believe it, we won’t actually talk about that in detail again until much later. Temperament — the second ‘T’ in “RTT” — is the discipline concerned with choosing an interesting map, and tuning can remain largely independent from it. The temperament is only concerned with the fact that — no matter what exact size you ultimately make the generator — it is the case e.g. that 12 of them make a 2, 19 of them make a 3, and 28 of them make a 5. So, for now, whenever we show a value for g, assume we’ve given a computer a formula for optimizing the tuning to approximate all three primes equally well. As for us humans, let’s stay focused on tempering.
-Damage, by the way, is a technical term. That refers to the delta in cents of the tempering for a prime (which is known as the error, by the way) but divided by log₂ of that prime. So for octaves, damage is the same as error. So if prime 3 was tuned 4.1 cents flat, that's its error, but if you want to know damage, you need 4.1/log₂3 = 2.587. We typically use damage instead of error when comparing across primes, because damage tells us how much a prime has been impacted relative to its complexity; we care much more about error to lower primes like 2, 3, and 5 than we do really high up and obscure building blocks like 37 and 41.
 === a multitude of maps ===
@@ Line 144: / Line 142: @@
 Suppose we want to experiment with {{map|12 19 28}}’s map a bit. We’ll change one of the terms by 1, so now we have {{map|12 20 28}}. Because the previous map did such a great job of approximating the 5-limit (i.e. log(2:3:5)), though, it should be unsurprising that this new map cannot achieve that feat. The proportions, 12:20:28, should now be about as out of whack as they can get. The best generator we can do here is about 1.0583 (getting a little more precise now), and 1.0583¹² ≈ 1.9738 which isn’t so bad, but 1.0583¹⁹ = 3.1058 and 1.0583²⁸ = 4.8870 which are both way off! And they’re way off in the opposite direction — 3.1058 is too big and 4.8870 is too small — which is why our tuning formula for g, which is designed to make the approximation good for every prime at once, can’t improve the situation: either sharpening or flattening helps one but hurts the other.
-The results of such inaccurate approximation are a bit chaotic. A ratio like 16/15 — where the factors of 3 and 5 are on the same side of the fraction bar and therefore cancel out each other’s damage — fares relatively alright, if by “alright” we mean it gets tempered out despite being about 112¢ in JI. On the other hand, an interval like 27/25 where the factors of 3 and 5 are on opposite sides of the fraction bar and thus their damages compound, gets mapped to a whopping 4 steps, despite only being about 133¢ in JI.
+The results of such inaccurate approximation are a bit chaotic. A ratio like 16/15 — where the factors of 3 and 5 are on the same side of the fraction bar and therefore cancel out each other’s error — fares relatively alright, if by “alright” we mean it gets tempered out despite being about 112¢ in JI. On the other hand, an interval like 27/25 where the factors of 3 and 5 are on opposite sides of the fraction bar and thus their errors compound, gets mapped to a whopping 4 steps, despite only being about 133¢ in JI.
 If your goal is to evoke JI-like harmony, then, {{map|12 20 28}} is not your friend. Feel free to work out some other variations on {{map|12 19 28}} if you like, such as {{map|12 19 29}} maybe, but I guarantee you won’t find a better one that starts with 12 than {{map|12 19 28}}.
-[[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<ref>Yes, this diagram is showing error, not damage. If it showed damage, the difference would be even more dramatic. And most people care more about damage than error. But damage is simpler to convey, so that's why I went with it.</ref> 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.]]
+[[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 {{map|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 {{map|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 {{map|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 {{map|17 27 39}}, it’s way small, while for {{map|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:Tuning projection.png|400px|thumb|left|'''Figure 3l.''' Demonstration of tuning projection. As long as the tunings change in a fixed proportion, the tuning will project to the same point on PTS.]]
-But perhaps even more interesting than this continuous tuning space that appears in PTS between points is the continuous tuning space that does not appear in PTS because it exists within each point, that is, exactly out from and deeper into the page at each point. In tuning space, as we’ve just established, there are no maps in front of or behind each other that get collapsed to a single point. But there are still many things that get collapsed to a single point like this, but in tuning space they are different tunings ''(see Figure 3l)''. For example, {{map|1200 1900 2800}} is the way we’d write 12-ET in tuning space. But there are other tunings represented by this same point in PTS, such as {{map|1200.12 1900.19 2800.28}} (note that in order to remain at the same point, we’ve maintained the exact proportions of all the prime tunings). That tuning might not be of particular interest. I just used it as a simple example to illustrate the point. A more useful example would be {{map|1198.440 1897.531 2796.361}}, which by some algorithm is the optimal tuning for 12-ET (minimizes damage across primes or intervals); it may not be as obvious from looking at that one, but if you check the proportions of those terms with each other, you will find they are still exactly 12:19:28.
+But perhaps even more interesting than this continuous tuning space that appears in PTS between points is the continuous tuning space that does not appear in PTS because it exists within each point, that is, exactly out from and deeper into the page at each point. In tuning space, as we’ve just established, there are no maps in front of or behind each other that get collapsed to a single point. But there are still many things that get collapsed to a single point like this, but in tuning space they are different tunings ''(see Figure 3l)''. For example, {{map|1200 1900 2800}} is the way we’d write 12-ET in tuning space. But there are other tunings represented by this same point in PTS, such as {{map|1200.12 1900.19 2800.28}} (note that in order to remain at the same point, we’ve maintained the exact proportions of all the prime tunings). That tuning might not be of particular interest. I just used it as a simple example to illustrate the point. A more useful example would be {{map|1198.440 1897.531 2796.361}}, which by some algorithm is the optimal tuning for 12-ET (minimizes error across primes or intervals); it may not be as obvious from looking at that one, but if you check the proportions of those terms with each other, you will find they are still exactly 12:19:28.
 The key point here is that, as we mentioned before, the problems of tuning and tempering are largely separate. PTS projects all tunings of the same temperament to the same point. This way, issues of tuning are completely hidden and ignored on PTS, so we can focus instead on tempering.