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

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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.

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.

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}}.

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 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.
+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.
 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}}.