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

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So if we take a look at a cross-section of projection again, but in terms of tuning space now ''(see Figure 2k)'', we can see how every point is about the same distance from us.

[[File:Tuning space version.png|~~500px~~|thumb|'''Figure 2k.''' Demonstration of projection in terms of ''tuning'' space (compare with Figure 2i).]]

[[File:Tuning space version.png|300px|thumb|right|'''Figure 2k.''' Demonstration of projection in terms of ''tuning'' space (compare with Figure 2i).]]

The other major difference is that tuning space is continuous, where map space is discrete. In other words, to find a map between {{val|6 10 14}} and {{val|6 9 14}}, you’re subdividing it by 2 or 3 and picking a point in between, that sort of thing. But between {{val|1200 2000 2800}} and {{val|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 {{val|1200 1999.999 2800}} to {{val|1200 1901.955 2800}} to {{val|1200 1817.643 2800}} is along the way.

But perhaps even more interesting than this continuous tuning space that appears in PTS between points is the continuous tuning space that does not appear in PTS because it exists within each point, that is, exactly out from and deeper into the page at each point. In tuning space, as we’ve just established, there are no maps in front of or behind each other that get collapsed to a single point. But there are still many things that get collapsed to a single point like this, but in tuning space they are different tunings. For example, {{val|~~1200 1900 2800}} is the way we’d write 12-ET in tuning space. But there are other tunings represented by this same point in PTS, such as {{val~~|~~1200.12 1900.19 2800~~.~~28}} (note that in order to remain at the same point, we’ve maintained the exact proportions~~ of ~~all the prime tunings). That~~ tuning might not be of particular interest. I just used it as a simple example to illustrate the point. A more useful example would be {{val|1198.440 1897.531 2796.361}}, which by some algorithm is the optimal tuning for 12-ET (minimizes damage across primes or intervals); it may not be as obvious from looking at that one, but if you check the proportions of those terms with each other, you will find they are still exactly 12:19:28.

[[File:Tuning projection.png|300px|thumb|left|'''Figure 2l.''' Demonstrating of tuning projection.]]

But perhaps even more interesting than this continuous tuning space that appears in PTS between points is the continuous tuning space that does not appear in PTS because it exists within each point, that is, exactly out from and deeper into the page at each point. In tuning space, as we’ve just established, there are no maps in front of or behind each other that get collapsed to a single point. But there are still many things that get collapsed to a single point like this, but in tuning space they are different tunings ''(see Figure 2l)''. For example, {{val|1200 1900 2800}} is the way we’d write 12-ET in tuning space. But there are other tunings represented by this same point in PTS, such as {{val|1200.12 1900.19 2800.28}} (note that in order to remain at the same point, we’ve maintained the exact proportions of all the prime tunings). That tuning might not be of particular interest. I just used it as a simple example to illustrate the point. A more useful example would be {{val|1198.440 1897.531 2796.361}}, which by some algorithm is the optimal tuning for 12-ET (minimizes damage across primes or intervals); it may not be as obvious from looking at that one, but if you check the proportions of those terms with each other, you will find they are still exactly 12:19:28.

=== regions ===

@@ Line 272: / Line 272: @@
 So if we take a look at a cross-section of projection again, but in terms of tuning space now ''(see Figure 2k)'', we can see how every point is about the same distance from us.
-[[File:Tuning space version.png|500px|thumb|'''Figure 2k.''' Demonstration of projection in terms of ''tuning'' space (compare with Figure 2i).]]
+[[File:Tuning space version.png|300px|thumb|right|'''Figure 2k.''' Demonstration of projection in terms of ''tuning'' space (compare with Figure 2i).]]
 The other major difference is that tuning space is continuous, where map space is discrete. In other words, to find a map between {{val|6 10 14}} and {{val|6 9 14}}, you’re subdividing it by 2 or 3 and picking a point in between, that sort of thing. But between {{val|1200 2000 2800}} and {{val|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 {{val|1200 1999.999 2800}} to {{val|1200 1901.955 2800}} to {{val|1200 1817.643 2800}} is along the way.
-But perhaps even more interesting than this continuous tuning space that appears in PTS between points is the continuous tuning space that does not appear in PTS because it exists within each point, that is, exactly out from and deeper into the page at each point. In tuning space, as we’ve just established, there are no maps in front of or behind each other that get collapsed to a single point. But there are still many things that get collapsed to a single point like this, but in tuning space they are different tunings. For example, {{val|1200 1900 2800}} is the way we’d write 12-ET in tuning space. But there are other tunings represented by this same point in PTS, such as {{val|1200.12 1900.19 2800.28}} (note that in order to remain at the same point, we’ve maintained the exact proportions of all the prime tunings). That tuning might not be of particular interest. I just used it as a simple example to illustrate the point. A more useful example would be {{val|1198.440 1897.531 2796.361}}, which by some algorithm is the optimal tuning for 12-ET (minimizes damage across primes or intervals); it may not be as obvious from looking at that one, but if you check the proportions of those terms with each other, you will find they are still exactly 12:19:28.
+[[File:Tuning projection.png|300px|thumb|left|'''Figure 2l.''' Demonstrating of tuning projection.]]
+But perhaps even more interesting than this continuous tuning space that appears in PTS between points is the continuous tuning space that does not appear in PTS because it exists within each point, that is, exactly out from and deeper into the page at each point. In tuning space, as we’ve just established, there are no maps in front of or behind each other that get collapsed to a single point. But there are still many things that get collapsed to a single point like this, but in tuning space they are different tunings ''(see Figure 2l)''. For example, {{val|1200 1900 2800}} is the way we’d write 12-ET in tuning space. But there are other tunings represented by this same point in PTS, such as {{val|1200.12 1900.19 2800.28}} (note that in order to remain at the same point, we’ve maintained the exact proportions of all the prime tunings). That tuning might not be of particular interest. I just used it as a simple example to illustrate the point. A more useful example would be {{val|1198.440 1897.531 2796.361}}, which by some algorithm is the optimal tuning for 12-ET (minimizes damage across primes or intervals); it may not be as obvious from looking at that one, but if you check the proportions of those terms with each other, you will find they are still exactly 12:19:28.
 === regions ===