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

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Moreover, there’s a special relationship between the positions of n-ETs and 2n-ETs, and indeed between n-ETs and 3n-ETs, 4n-ETs, etc. To understand why, it’s instructive to plot it out ''(see Figure 3i)''.

[[File:Hiding vals.png|500px|thumb|right|'''Figure 3i.''' ~~redundant~~ maps hiding behind their simpler counterparts]]

[[File:Hiding vals.png|500px|thumb|right|'''Figure 3i.''' Redundant maps hiding behind their simpler counterparts. The eye is the origin; the same as in Figure 3h. Projective tuning space is the plane resting at the bottom that we are projecting onto. The portion we see in the Middle Path version is only a tiny part right in the middle. The dotted lines just above where the PTS plane is drawn are there to indicate the elision of an infinitude of space; potentially you could go way up to insanely large ETs and they would all be between the origin-eye and this projective plane.]]

For simplicity, we’re looking at the octant cube here from the angle straight on to the 2-axis, so changes to the 2-terms don’t matter here. In the ~~bottom left~~ is the origin; that’s the point at the center of PTS. Close-by, we can see the map {{val|3 5 7}}, and two closely related maps {{val|3 4 7}} and {{val|3 5 8}}. Colored lines have been drawn from the origin through these points to the black line in the top-right, which represents the page; this is portraying how if our eye is at that origin, where on the page these points would appear to be.

For simplicity, we’re looking at the octant cube here from the angle straight on to the 2-axis, so changes to the 2-terms don’t matter here. At the top is the origin; that’s the point at the center of PTS. Close-by, we can see the map {{val|3 5 7}}, and two closely related maps {{val|3 4 7}} and {{val|3 5 8}}. Colored lines have been drawn from the origin through these points to the black line in the top-right, which represents the page; this is portraying how if our eye is at that origin, where on the page these points would appear to be.

In between where the colored lines touch the maps themselves and the page, we see a cluster of more maps, each of which starts with 6. In other words, these maps are about twice as far away from us as the others. Let’s consider {{val|6 10 14}} first. Notice that each of its terms is exactly 2x the corresponding term in {{val|3 5 7}}. In effect, {{val|6 10 14}} is redundant with {{val|3 5 7}}. If you imagine doing a mapping calculation or two, you can easily convince yourself that you’ll get the same answer as if you’d just done it with {{val|3 5 7}} instead and then simply divided by 2 one time at the end. It behaves in the exact same way as {{val|3 5 7}} in terms of the relationships between the intervals it maps, the only difference being that it needlessly includes twice as many steps to do so, never using every other one. So we don’t really care about {{val|6 10 14}}. Which is great, because it’s hidden exactly behind {{val|3 5 7}} from where we’re looking.

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Just as there are 2n-ETs halfway between n-ETs, there are 3n-ETs a third of the way between n-ETs. Look at these two [[29edo|29-ET]]s here. The [[58edo|58-ET]] is here halfway between them, and two [[87edo|87-ET]]s are here each a third of the way between.

=== map space vs. tuning space ===

@@ Line 263: / Line 263: @@
 Moreover, there’s a special relationship between the positions of n-ETs and 2n-ETs, and indeed between n-ETs and 3n-ETs, 4n-ETs, etc. To understand why, it’s instructive to plot it out ''(see Figure 3i)''.
-[[File:Hiding vals.png|500px|thumb|right|'''Figure 3i.''' redundant maps hiding behind their simpler counterparts]]
+[[File:Hiding vals.png|500px|thumb|right|'''Figure 3i.''' Redundant maps hiding behind their simpler counterparts. The eye is the origin; the same as in Figure 3h. Projective tuning space is the plane resting at the bottom that we are projecting onto. The portion we see in the Middle Path version is only a tiny part right in the middle. The dotted lines just above where the PTS plane is drawn are there to indicate the elision of an infinitude of space; potentially you could go way up to insanely large ETs and they would all be between the origin-eye and this projective plane.]]
-For simplicity, we’re looking at the octant cube here from the angle straight on to the 2-axis, so changes to the 2-terms don’t matter here. In the bottom left is the origin; that’s the point at the center of PTS. Close-by, we can see the map {{val|3 5 7}}, and two closely related maps {{val|3 4 7}} and {{val|3 5 8}}. Colored lines have been drawn from the origin through these points to the black line in the top-right, which represents the page; this is portraying how if our eye is at that origin, where on the page these points would appear to be.
+For simplicity, we’re looking at the octant cube here from the angle straight on to the 2-axis, so changes to the 2-terms don’t matter here. At the top is the origin; that’s the point at the center of PTS. Close-by, we can see the map {{val|3 5 7}}, and two closely related maps {{val|3 4 7}} and {{val|3 5 8}}. Colored lines have been drawn from the origin through these points to the black line in the top-right, which represents the page; this is portraying how if our eye is at that origin, where on the page these points would appear to be.
 In between where the colored lines touch the maps themselves and the page, we see a cluster of more maps, each of which starts with 6. In other words, these maps are about twice as far away from us as the others. Let’s consider {{val|6 10 14}} first. Notice that each of its terms is exactly 2x the corresponding term in {{val|3 5 7}}. In effect, {{val|6 10 14}} is redundant with {{val|3 5 7}}. If you imagine doing a mapping calculation or two, you can easily convince yourself that you’ll get the same answer as if you’d just done it with {{val|3 5 7}} instead and then simply divided by 2 one time at the end. It behaves in the exact same way as {{val|3 5 7}} in terms of the relationships between the intervals it maps, the only difference being that it needlessly includes twice as many steps to do so, never using every other one. So we don’t really care about {{val|6 10 14}}. Which is great, because it’s hidden exactly behind {{val|3 5 7}} from where we’re looking.
@@ Line 280: / Line 280: @@
 Just as there are 2n-ETs halfway between n-ETs, there are 3n-ETs a third of the way between n-ETs. Look at these two [[29edo|29-ET]]s here. The [[58edo|58-ET]] is here halfway between them, and two [[87edo|87-ET]]s are here each a third of the way between.
 === map space vs. tuning space ===