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

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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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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 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<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.]]

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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 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 146: / Line 148: @@
 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 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<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.]]
 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]].