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

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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 {{map|6 10 14}} first. Notice that each of its terms is exactly 2x the corresponding term in {{map|3 5 7}}. In effect, {{map|6 10 14}} is redundant with {{map|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 {{map|3 5 7}} instead and then simply divided by 2 one time at the end. It behaves in the exact same way as {{map|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 {{map|6 10 14}}. Which is great, because it’s hidden exactly behind {{map|3 5 7}} from where we’re looking.

The same is true of the map pair {{map|3 4 7}} and {{map|6 8 14}}, as well as of {{map|3 5 8}} and {{map|6 10 16}}. Any map whose terms have a common factor other than 1 is going to be redundant in this sense, and therefore hidden. You can imagine that even further past {{map|3 5 7}} you’ll find {{map|9 15 21}}, {{map|12 20 28}}, and so on, and these we could call “enfactored” maps<ref>~~”Enfactored” is not yet an established term; elsewhere~~ you may see ~~enfactored maps~~ called ~~“contorted”~~, but ~~I think that name~~ is ~~strange and unhelpful~~.</ref><ref>On some versions of PTS which Paul prepared, these enfactored ETs are actually printed on the page.</ref>. More on those later. What’s important to realize here is that Paul found a way to collapse 3 dimensions worth of information down to 2 dimensions without losing anything important. Each of these lines connecting redundant ETs have been [https://en.wikipedia.org/wiki/Projection_(mathematics) projected] onto the page as a single point. That’s why the diagram is called "projective" tuning space.

The same is true of the map pair {{map|3 4 7}} and {{map|6 8 14}}, as well as of {{map|3 5 8}} and {{map|6 10 16}}. Any map whose terms have a common factor other than 1 is going to be redundant in this sense, and therefore hidden. You can imagine that even further past {{map|3 5 7}} you’ll find {{map|9 15 21}}, {{map|12 20 28}}, and so on, and these we could call “enfactored” maps.<ref>Elsewhere you may see these called "contorted", but as you can read on the page [[defactoring]], this is not technically correct, but has historically been frequently confused.</ref><ref>On some versions of PTS which Paul prepared, these enfactored ETs are actually printed on the page.</ref>. More on those later. What’s important to realize here is that Paul found a way to collapse 3 dimensions worth of information down to 2 dimensions without losing anything important. Each of these lines connecting redundant ETs have been [https://en.wikipedia.org/wiki/Projection_(mathematics) projected] onto the page as a single point. That’s why the diagram is called "projective" tuning space.

Now, to find a 6-ET with anything new to bring to the table, we’ll need to find one whose terms don’t share a common factor. That’s not hard. We’ll just take one of the ones halfway between the ones we just looked at. How about {{map|6 11 14}}, which is halfway between {{map|6 10 14}} and {{map|6 12 14}}. Notice that the purple line that runs through it lands halfway between the red and blue lines on the page. Similarly, {{map|6 10 15}} is halfway between {{map|6 10 14}} and {{map|6 10 16}}, and its yellow line appears halfway between the red and green lines on the page. What this is demonstrating is that halfway between any pair of n-ETs on the diagram, whether this pair is separated along the 3-axis or 5-axis, you will find a 2n-ET. We can’t really demonstrate this with 3-ET and 6-ET on the diagram, because those ETs are too inaccurate; they’ve been cropped off. But if we return to our 40-ET example, that will work just fine.

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 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 {{map|6 10 14}} first. Notice that each of its terms is exactly 2x the corresponding term in {{map|3 5 7}}. In effect, {{map|6 10 14}} is redundant with {{map|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 {{map|3 5 7}} instead and then simply divided by 2 one time at the end. It behaves in the exact same way as {{map|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 {{map|6 10 14}}. Which is great, because it’s hidden exactly behind {{map|3 5 7}} from where we’re looking.
-The same is true of the map pair {{map|3 4 7}} and {{map|6 8 14}}, as well as of {{map|3 5 8}} and {{map|6 10 16}}. Any map whose terms have a common factor other than 1 is going to be redundant in this sense, and therefore hidden. You can imagine that even further past {{map|3 5 7}} you’ll find {{map|9 15 21}}, {{map|12 20 28}}, and so on, and these we could call “enfactored” maps<ref>”Enfactored” is not yet an established term; elsewhere you may see enfactored maps called “contorted”, but I think that name is strange and unhelpful.</ref><ref>On some versions of PTS which Paul prepared, these enfactored ETs are actually printed on the page.</ref>. More on those later. What’s important to realize here is that Paul found a way to collapse 3 dimensions worth of information down to 2 dimensions without losing anything important. Each of these lines connecting redundant ETs have been [https://en.wikipedia.org/wiki/Projection_(mathematics) projected] onto the page as a single point. That’s why the diagram is called "projective" tuning space.
+The same is true of the map pair {{map|3 4 7}} and {{map|6 8 14}}, as well as of {{map|3 5 8}} and {{map|6 10 16}}. Any map whose terms have a common factor other than 1 is going to be redundant in this sense, and therefore hidden. You can imagine that even further past {{map|3 5 7}} you’ll find {{map|9 15 21}}, {{map|12 20 28}}, and so on, and these we could call “enfactored” maps.<ref>Elsewhere you may see these called "contorted", but as you can read on the page [[defactoring]], this is not technically correct, but has historically been frequently confused.</ref><ref>On some versions of PTS which Paul prepared, these enfactored ETs are actually printed on the page.</ref>. More on those later. What’s important to realize here is that Paul found a way to collapse 3 dimensions worth of information down to 2 dimensions without losing anything important. Each of these lines connecting redundant ETs have been [https://en.wikipedia.org/wiki/Projection_(mathematics) projected] onto the page as a single point. That’s why the diagram is called "projective" tuning space.
 Now, to find a 6-ET with anything new to bring to the table, we’ll need to find one whose terms don’t share a common factor. That’s not hard. We’ll just take one of the ones halfway between the ones we just looked at. How about {{map|6 11 14}}, which is halfway between {{map|6 10 14}} and {{map|6 12 14}}. Notice that the purple line that runs through it lands halfway between the red and blue lines on the page. Similarly, {{map|6 10 15}} is halfway between {{map|6 10 14}} and {{map|6 10 16}}, and its yellow line appears halfway between the red and green lines on the page. What this is demonstrating is that halfway between any pair of n-ETs on the diagram, whether this pair is separated along the 3-axis or 5-axis, you will find a 2n-ET. We can’t really demonstrate this with 3-ET and 6-ET on the diagram, because those ETs are too inaccurate; they’ve been cropped off. But if we return to our 40-ET example, that will work just fine.