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

Line 132:

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 (or ~~flat~~, as you ~~could imagine)~~]]

[[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 detunings 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)''

Line 146:

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.''' Detunings 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 detuned in one direction~~.]]

[[File:17-ET mistunings.png|thumb|600px|right|'''Figure 2f.''' Detunings 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. It also shows the concerning case that 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]].

@@ Line 132: / Line 132: @@
 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 (or flat, as you could imagine)]]
+[[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 detunings 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)''
@@ Line 146: / Line 146: @@
 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.''' Detunings 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 detuned in one direction.]]
+[[File:17-ET mistunings.png|thumb|600px|right|'''Figure 2f.''' Detunings 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. It also shows the concerning case that 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]].