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

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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}}.

[[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.]]

[[File:17-ET mistunings.png|thumb|600px|right|'''Figure 1f.''' 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.]]

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 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.]]

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 ~~mistune~~ 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.

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 detune 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.

@@ Line 138: / Line 138: @@
 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 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.]]
+[[File:17-ET mistunings.png|thumb|600px|right|'''Figure 1f.''' 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.]]
 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 144: / Line 144: @@
 [[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.]]
-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 mistune 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.
+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 detune 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.