Projection: Difference between revisions

Line 950:

Here's one way to think about it. When we look at a mapping matrix like meantone's {{ket|{{bra|1 1 0}} {{bra|0 1 4}}}} above, we're basically plotting vectors in a brand-new 2D space, that is, one completely distinct from the 3D space we started with in JI. So while the <math>x</math>, <math>y</math>, and <math>z</math> axes for JI corresponded to our primes <math>\text{p}_1</math>, <math>\text{p}_2</math>, and <math>\text{p}_3</math>, our <math>x</math> and <math>y</math> axes (no <math>z</math> here!) of our new meantone space correspond to its generators <math>\text{g}_1</math> and <math>\text{p}_2</math>. That's how ''its'' 2D plane exists. However, as for a tuning of meantone's projection matrix, such as quarter-comma's, it remains in the original 3D space with the <math>\text{p}_1</math>, <math>\text{p}_2</math>, and <math>\text{p}_3</math> axes. What this means is that everything in that space gets projected — or smooshed down, you could think of it — into a single plane. So this plane is 2D, certainly, but importantly, it still requires three coordinates to describe ''because it exists titled at some angle through the original JI space''. It is a 2D object occupying 3D space, just like any 2D document you might have in your real-life 3D physical space sitting up on a bookshelf at some slight angle.

TODO: diagram, at least one diagram here would be really valuable; perhaps this will help https://app.asana.com/0/1200134195404323/1201659347245292. also plane always flat at bottom projection diagram - might be easier to discern how least-squares points straight up and others are at an angle - as opposed to having the x/y/z axes always in the same position and then try to figure out what angle is this plane oriented through the space. this could help too: https://stackoverflow.com/questions/42893410/how-to-get-the-tilt-angle-of-a-plane-defined-by-three-coordinates-in-python

===Visualization of simpler problem ===

Line 994:

Line 992:

For comparison's sake, here's the tuning of 5-ET we get by taking the pseudoinverse of its mapping (i.e. the [[minimax-E-copfr-S]] tuning). Note that here, the projection lines are exactly perpendicular to the unchanged-interval line.

[[File:Projection demo - pseudoinverse case.png|none|thumb|900x900px|The projection line from the blackwood comma to the origin (its vanishing) is identical to the previous diagram, and we're showing the same set of JI pitches projecting to the generator steps of 5-ET up to the octave, and all projection lines are still parallel. What's different in this diagram is the angle of the unchanged-interval line, which is now exactly perpendicular to the projection lines (note the 90° square angle indicators). The reason why the pseudoinverse gives this perpendicular line is that in a sense this tuning is the "best fit line" to this temperament; it minimizes the overall lengths of all projection lines. Tuning tempered intervals is much trickier than this, however; this simplistic approach is deceptive in that it doesn't how 1) harmonic space isn't based on Euclidean distances and 2) primes 2 and 3 are not necessarily of equal importance.]]

And so from here we can try to generalize these insights to higher dimensions. Meantone temperament, again, would exist in 3D. The meantone comma at the point {{vector|-4 4 -1}} would make a blue line to unison at the origin {{vector|0 0 0}}. Also through the origin we'd have an green "unchanged-interval plane". This could be tilted at any angle; the tilt would depend only the tuning of the temperament. No matter the tuning, every interval will have a parallel blue line through the space projecting it onto this plane. And this plane would also therefore represent the tempered lattice, and would have a regularly-spaced 2D grid of points on it.

===Tunings and commas===

@@ Line 950: / Line 950: @@
 Here's one way to think about it. When we look at a mapping matrix like meantone's {{ket|{{bra|1 1 0}} {{bra|0 1 4}}}} above, we're basically plotting vectors in a brand-new 2D space, that is, one completely distinct from the 3D space we started with in JI. So while the <math>x</math>, <math>y</math>, and <math>z</math> axes for JI corresponded to our primes <math>\text{p}_1</math>, <math>\text{p}_2</math>, and <math>\text{p}_3</math>, our <math>x</math> and <math>y</math> axes (no <math>z</math> here!) of our new meantone space correspond to its generators <math>\text{g}_1</math> and <math>\text{p}_2</math>. That's how ''its'' 2D plane exists. However, as for a tuning of meantone's projection matrix, such as quarter-comma's, it remains in the original 3D space with the <math>\text{p}_1</math>, <math>\text{p}_2</math>, and <math>\text{p}_3</math> axes. What this means is that everything in that space gets projected — or smooshed down, you could think of it — into a single plane. So this plane is 2D, certainly, but importantly, it still requires three coordinates to describe ''because it exists titled at some angle through the original JI space''. It is a 2D object occupying 3D space, just like any 2D document you might have in your real-life 3D physical space sitting up on a bookshelf at some slight angle.
- TODO: diagram, at least one diagram here would be really valuable; perhaps this will help https://app.asana.com/0/1200134195404323/1201659347245292. also plane always flat at bottom projection diagram - might be easier to discern how least-squares points straight up and others are at an angle - as opposed to having the x/y/z axes always in the same position and then try to figure out what angle is this plane oriented through the space. this could help too: https://stackoverflow.com/questions/42893410/how-to-get-the-tilt-angle-of-a-plane-defined-by-three-coordinates-in-python
 ===Visualization of simpler problem ===
@@ Line 994: / Line 992: @@
 For comparison's sake, here's the tuning of 5-ET we get by taking the pseudoinverse of its mapping (i.e. the [[minimax-E-copfr-S]] tuning). Note that here, the projection lines are exactly perpendicular to the unchanged-interval line.
 [[File:Projection demo - pseudoinverse case.png|none|thumb|900x900px|The projection line from the blackwood comma to the origin (its vanishing) is identical to the previous diagram, and we're showing the same set of JI pitches projecting to the generator steps of 5-ET up to the octave, and all projection lines are still parallel. What's different in this diagram is the angle of the unchanged-interval line, which is now exactly perpendicular to the projection lines (note the 90° square angle indicators). The reason why the pseudoinverse gives this perpendicular line is that in a sense this tuning is the "best fit line" to this temperament; it minimizes the overall lengths of all projection lines. Tuning tempered intervals is much trickier than this, however; this simplistic approach is deceptive in that it doesn't how 1) harmonic space isn't based on Euclidean distances and 2) primes 2 and 3 are not necessarily of equal importance.]]
+And so from here we can try to generalize these insights to higher dimensions. Meantone temperament, again, would exist in 3D. The meantone comma at the point {{vector|-4 4 -1}} would make a blue line to unison at the origin {{vector|0 0 0}}. Also through the origin we'd have an green "unchanged-interval plane". This could be tilted at any angle; the tilt would depend only the tuning of the temperament. No matter the tuning, every interval will have a parallel blue line through the space projecting it onto this plane. And this plane would also therefore represent the tempered lattice, and would have a regularly-spaced 2D grid of points on it.
 ===Tunings and commas===