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

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Why did I do this to you? Well, I decided map space was conceptually easier to introduce than tuning space. Paul himself prefers to think of this diagram as a projection of tuning space, however, so I don’t want to leave this material before clarifying the difference. Also, there are different helpful insights you can get from thinking of PTS as tuning space. Let’s consider those now.

The first key difference to notice is that we can ~~normalize~~ coordinates in tuning space, so that the first term of every coordinate is the same, namely, one octave, or 1200 cents. For example, note that while in map space, {{map|3 5 7}} is located physically in front of {{map|6 10 14}}, in tuning space, these two points collapse to literally the same point, {{map|1200 2000 2800}}. This can be helpful in a similar way to how the scaled axes of PTS help us visually compare maps’ proximity to the central JI spoke: they are now expressed closer to in terms of their deviation from JI, so we can more immediately compare maps to each other, as well as individually directly to the pure JI primes, as long as we memorize the cents values of those (they’re 1200, 1901.955, and 2786.314). For example, in map space, it may not be immediately obvious that {{map|6 9 14}} is halfway between {{map|3 5 7}} and {{map|3 4 7}}, but in tuning space it is immediately obvious that {{map|1200 1800 2800}} is halfway between {{map|1200 2000 2800}} and {{map|1200 1600 2800}}.

The first key difference to notice is that we can standardize coordinates in tuning space, so that the first term of every coordinate is the same, namely, one octave, or 1200 cents. For example, note that while in map space, {{map|3 5 7}} is located physically in front of {{map|6 10 14}}, in tuning space, these two points collapse to literally the same point, {{map|1200 2000 2800}}. This can be helpful in a similar way to how the scaled axes of PTS help us visually compare maps’ proximity to the central JI spoke: they are now expressed closer to in terms of their deviation from JI, so we can more immediately compare maps to each other, as well as individually directly to the pure JI primes, as long as we memorize the cents values of those (they’re 1200, 1901.955, and 2786.314). For example, in map space, it may not be immediately obvious that {{map|6 9 14}} is halfway between {{map|3 5 7}} and {{map|3 4 7}}, but in tuning space it is immediately obvious that {{map|1200 1800 2800}} is halfway between {{map|1200 2000 2800}} and {{map|1200 1600 2800}}.

So if we take a look at a cross-section of projection again, but in terms of tuning space now ''(see Figure 3k)'', we can see how every point is about the same distance from us.

[[File:Tuning space version.png|400px|thumb|right|'''Figure 3k.''' Demonstration of projection in terms of ''tuning'' space (compare with Figure 3i, which shows projection in terms of ''map'' space). As you can see here, all the points are in about the same region of space, since tuning space ~~normalizes itself nearby to~~ JI.]]

[[File:Tuning space version.png|400px|thumb|right|'''Figure 3k.''' Demonstration of projection in terms of ''tuning'' space (compare with Figure 3i, which shows projection in terms of ''map'' space). As you can see here, all the points are in about the same region of space, since tuning space tends toward JI.]]

The other major difference is that tuning space is continuous, where map space is discrete. In other words, to find a map between {{map|6 10 14}} and {{map|6 9 14}}, you’re subdividing it by 2 or 3 and picking a point in between, that sort of thing. But between {{map|1200 2000 2800}} and {{map|1200 1800 2800}} you’ve got an infinitude of choices smoothly transitioning between each other; you’ve basically got knobs you can turn on the proportions of the tuning of 2, 3, and 5. Everything from from {{map|1200 1999.999 2800}} to {{map|1200 1901.955 2800}} to {{map|1200 1817.643 2800}} is along the way.

@@ Line 286: / Line 286: @@
 Why did I do this to you? Well, I decided map space was conceptually easier to introduce than tuning space. Paul himself prefers to think of this diagram as a projection of tuning space, however, so I don’t want to leave this material before clarifying the difference. Also, there are different helpful insights you can get from thinking of PTS as tuning space. Let’s consider those now.
-The first key difference to notice is that we can normalize coordinates in tuning space, so that the first term of every coordinate is the same, namely, one octave, or 1200 cents. For example, note that while in map space, {{map|3 5 7}} is located physically in front of {{map|6 10 14}}, in tuning space, these two points collapse to literally the same point, {{map|1200 2000 2800}}. This can be helpful in a similar way to how the scaled axes of PTS help us visually compare maps’ proximity to the central JI spoke: they are now expressed closer to in terms of their deviation from JI, so we can more immediately compare maps to each other, as well as individually directly to the pure JI primes, as long as we memorize the cents values of those (they’re 1200, 1901.955, and 2786.314). For example, in map space, it may not be immediately obvious that {{map|6 9 14}} is halfway between {{map|3 5 7}} and {{map|3 4 7}}, but in tuning space it is immediately obvious that {{map|1200 1800 2800}} is halfway between {{map|1200 2000 2800}} and {{map|1200 1600 2800}}.
+The first key difference to notice is that we can standardize coordinates in tuning space, so that the first term of every coordinate is the same, namely, one octave, or 1200 cents. For example, note that while in map space, {{map|3 5 7}} is located physically in front of {{map|6 10 14}}, in tuning space, these two points collapse to literally the same point, {{map|1200 2000 2800}}. This can be helpful in a similar way to how the scaled axes of PTS help us visually compare maps’ proximity to the central JI spoke: they are now expressed closer to in terms of their deviation from JI, so we can more immediately compare maps to each other, as well as individually directly to the pure JI primes, as long as we memorize the cents values of those (they’re 1200, 1901.955, and 2786.314). For example, in map space, it may not be immediately obvious that {{map|6 9 14}} is halfway between {{map|3 5 7}} and {{map|3 4 7}}, but in tuning space it is immediately obvious that {{map|1200 1800 2800}} is halfway between {{map|1200 2000 2800}} and {{map|1200 1600 2800}}.
 So if we take a look at a cross-section of projection again, but in terms of tuning space now ''(see Figure 3k)'', we can see how every point is about the same distance from us.
-[[File:Tuning space version.png|400px|thumb|right|'''Figure 3k.''' Demonstration of projection in terms of ''tuning'' space (compare with Figure 3i, which shows projection in terms of ''map'' space). As you can see here, all the points are in about the same region of space, since tuning space normalizes itself nearby to JI.]]
+[[File:Tuning space version.png|400px|thumb|right|'''Figure 3k.''' Demonstration of projection in terms of ''tuning'' space (compare with Figure 3i, which shows projection in terms of ''map'' space). As you can see here, all the points are in about the same region of space, since tuning space tends toward JI.]]
 The other major difference is that tuning space is continuous, where map space is discrete. In other words, to find a map between {{map|6 10 14}} and {{map|6 9 14}}, you’re subdividing it by 2 or 3 and picking a point in between, that sort of thing. But between {{map|1200 2000 2800}} and {{map|1200 1800 2800}} you’ve got an infinitude of choices smoothly transitioning between each other; you’ve basically got knobs you can turn on the proportions of the tuning of 2, 3, and 5. Everything from from {{map|1200 1999.999 2800}} to {{map|1200 1901.955 2800}} to {{map|1200 1817.643 2800}} is along the way.