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

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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]]”; I think the association with "[[patent val]]" is confused, and "patent" isn't a good word for it in the first place, and I would prefer to characterize it as a “generatable map” myself.)

And why is this cool? Well, if 12~~-ET~~ approximates the harmonic building blocks well individually, then JI intervals built out of them, like 16/15, 5/4, 10/9, etc. should also be reasonably well-approximated overall, and thus recognizable as their JI counterparts in musical context. You could think of it like taking all the primes in a prime factorization and swapping in their approximations. For example, if 16/15 = 2⁴ × 3⁻¹ × 5⁻¹ ≈ 1.067, and in 12~~-ET~~ 2, 3, and 5 ~~are approximated~~ by 1.059¹² ≈ 1.998, 1.059¹⁹ ≈ 2.992, and 1.059²⁸ ≈ 5.029, respectively, then in 12~~-ET~~ 16/15 ~~maps~~ to 1.998⁴ × 2.992⁻¹ × 5.029⁻¹ ≈ 1.059, which is indeed pretty close to 1.067. Of course, we should also note that 1.059 is the same as our step of 12~~-ET~~, which checks out with our calculation we made in the previous section that the best approximation of 16/15 in 12~~-ET~~ would be 1 step.

And why is this cool? Well, if {{val|12 19 28}} approximates the harmonic building blocks well individually, then JI intervals built out of them, like 16/15, 5/4, 10/9, etc. should also be reasonably well-approximated overall, and thus recognizable as their JI counterparts in musical context. You could think of it like taking all the primes in a prime factorization and swapping in their approximations. For example, if 16/15 = 2⁴ × 3⁻¹ × 5⁻¹ ≈ 1.067, and {{val|12 19 28}} approximates 2, 3, and 5 by 1.059¹² ≈ 1.998, 1.059¹⁹ ≈ 2.992, and 1.059²⁸ ≈ 5.029, respectively, then {{val|12 19 28}} maps 16/15 to 1.998⁴ × 2.992⁻¹ × 5.029⁻¹ ≈ 1.059, which is indeed pretty close to 1.067. Of course, we should also note that 1.059 is the same as our step of {{val|12 19 28}}, which checks out with our calculation we made in the previous section that the best approximation of 16/15 in {{val|12 19 28}} would be 1 step.

=== tuning & pure octaves ===

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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 detune 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 12~~-ET~~ 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.

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 12~~-ET~~, the 5 is a bit sharp, and the 2 and 3 are each a tiny bit flat ''(as you can see in Figure 1c)''.

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 1c)''.

[[File:Why not just srhink every block.png|thumb|left|600px|'''Figure 1e.''' Visualization of pointlessness of tuning all primes sharp (or flat, as you could imagine)]]

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=== a multitude of maps ===

Suppose we want to experiment with 12~~-ET’s~~ map a bit. We’ll change one of the terms by 1, so now we have {{val|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.

Suppose we want to experiment with {{val|12 19 28}}’s map a bit. We’ll change one of the terms by 1, so now we have {{val|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 vinculum 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 vinculum and thus their damages compound, gets mapped to a whopping 4 steps, despite only being about 133¢ in JI.

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[[File:17-ET mistunings.png|thumb|600px|right|'''Figure 1f.''' Mistunings of various 17-ET maps, showing how the supposed "patent" val's total error can be improved upon by allowing flexible octaves and second-closest mappings of primes. Also shows that the pure octave 17c is improper insofar as all primes are mistuned in one direction.]]

So the case is cut-and-dry for 12-ET. 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]].

[[File:17-ET.png|thumb|400px|left|'''Figure 1g.''' Visualization of the 17-ETs on the GPV continuum, showing how the pure octave 17c is a "lie" insofar as there exists no generator that exactly reaches prime 2 in 17 steps while more closely approximating prime 5 as 40 steps than 39 steps.]]

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 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]]”; I think the association with "[[patent val]]" is confused, and "patent" isn't a good word for it in the first place, and I would prefer to characterize it as a “generatable map” myself.)
-And why is this cool? Well, if 12-ET approximates the harmonic building blocks well individually, then JI intervals built out of them, like 16/15, 5/4, 10/9, etc. should also be reasonably well-approximated overall, and thus recognizable as their JI counterparts in musical context. You could think of it like taking all the primes in a prime factorization and swapping in their approximations. For example, if 16/15 = 2⁴ × 3⁻¹ × 5⁻¹ ≈ 1.067, and in 12-ET 2, 3, and 5 are approximated by 1.059¹² ≈ 1.998, 1.059¹⁹ ≈ 2.992, and 1.059²⁸ ≈ 5.029, respectively, then in 12-ET 16/15 maps to 1.998⁴ × 2.992⁻¹ × 5.029⁻¹ ≈ 1.059, which is indeed pretty close to 1.067. Of course, we should also note that 1.059 is the same as our step of 12-ET, which checks out with our calculation we made in the previous section that the best approximation of 16/15 in 12-ET would be 1 step.
+And why is this cool? Well, if {{val|12 19 28}} approximates the harmonic building blocks well individually, then JI intervals built out of them, like 16/15, 5/4, 10/9, etc. should also be reasonably well-approximated overall, and thus recognizable as their JI counterparts in musical context. You could think of it like taking all the primes in a prime factorization and swapping in their approximations. For example, if 16/15 = 2⁴ × 3⁻¹ × 5⁻¹ ≈ 1.067, and {{val|12 19 28}} approximates 2, 3, and 5 by 1.059¹² ≈ 1.998, 1.059¹⁹ ≈ 2.992, and 1.059²⁸ ≈ 5.029, respectively, then {{val|12 19 28}} maps 16/15 to 1.998⁴ × 2.992⁻¹ × 5.029⁻¹ ≈ 1.059, which is indeed pretty close to 1.067. Of course, we should also note that 1.059 is the same as our step of {{val|12 19 28}}, which checks out with our calculation we made in the previous section that the best approximation of 16/15 in {{val|12 19 28}} would be 1 step.
 === tuning & pure octaves ===
@@ Line 116: / Line 116: @@
 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 detune 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 12-ET 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.
-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 12-ET, the 5 is a bit sharp, and the 2 and 3 are each a tiny bit flat ''(as you can see in Figure 1c)''.
+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 1c)''.
 [[File:Why not just srhink every block.png|thumb|left|600px|'''Figure 1e.''' Visualization of pointlessness of tuning all primes sharp (or flat, as you could imagine)]]
@@ Line 129: / Line 129: @@
 === a multitude of maps ===
-Suppose we want to experiment with 12-ET’s map a bit. We’ll change one of the terms by 1, so now we have {{val|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.
+Suppose we want to experiment with {{val|12 19 28}}’s map a bit. We’ll change one of the terms by 1, so now we have {{val|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 vinculum 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 vinculum and thus their damages compound, gets mapped to a whopping 4 steps, despite only being about 133¢ in JI.
@@ Line 137: / Line 137: @@
 [[File:17-ET mistunings.png|thumb|600px|right|'''Figure 1f.''' Mistunings of various 17-ET maps, showing how the supposed "patent" val's total error can be improved upon by allowing flexible octaves and second-closest mappings of primes. Also shows that the pure octave 17c is improper insofar as all primes are mistuned in one direction.]]
-So the case is cut-and-dry for 12-ET. 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]].
 [[File:17-ET.png|thumb|400px|left|'''Figure 1g.''' Visualization of the 17-ETs on the GPV continuum, showing how the pure octave 17c is a "lie" insofar as there exists no generator that exactly reaches prime 2 in 17 steps while more closely approximating prime 5 as 40 steps than 39 steps.]]