Projection: Difference between revisions

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And since we have only a single comma, we know the angle at which every point in space that's off the unchanged-interval line will be projected onto it. We can figure it out by drawing a line from our comma, {{Vector|-8 5}}, to the origin, {{Vector|0 0}}. Every other projection line will be parallel to this line.

[[File:Projection demo - arbitrary tuning case.png|none|thumb|900x900px|The purple grid is the original 2D JI lattice. The green line is the 1D tempered lattice for this unchanged-3/2 tuning of 5-ET, which we can also think of as the unchanged-interval line. We've only shown 5 points on it (other than the origin), i.e. we iterated the generator until we reached the next octave. For each of these green points on the tempered lattice, we show one purple point on the JI lattice which projects to it, via a blue arrow. In particular, we show the blackwood comma, the vanishing comma of this temperament, projecting to the unison [0}. Note that each next iteration of the generator adds a new green dot 1/3 of the way to the next JI lattice node which is diagonally up and to the left; this corresponds to the fact that 3/2 is our unchanged-interval, and 3/2 [-1 1⟩ maps to ~3/2 [3} in 5-ET. In other words, each generator is 2^(-1/3)·3^(1/3) so that (2^(-1/3)·3^(1/3))^3 = 2^(-1)·3^(1) = 3/2. Notice that all of the blue projection lines are exactly parallel, but that the green unchanged-interval line is not parallel or perpendicular to them.]]

[[File:Projection demo - arbitrary tuning case.png|none|thumb|900x900px|The purple grid is the original 2D JI lattice. The green line is the 1D tempered lattice for this unchanged-3/2 tuning of 5-ET, which we can also think of as the unchanged-interval line. We've only shown 5 points on it (other than the origin), i.e. we iterated the generator until we reached the next octave. For each of these green points on the tempered lattice, we show one purple point on the JI lattice which projects to it, via a blue arrow.

In particular, we show the blackwood comma, the vanishing comma of this temperament, projecting to the unison [0}.

Note that each next iteration of the generator adds a new green dot 1/3 of the way to the next JI lattice node which is diagonally up and to the left; this corresponds to the fact that 3/2 is our unchanged-interval, and 3/2 [-1 1⟩ maps to ~3/2 [3} in 5-ET. In other words, each generator is 2^(-1/3)·3^(1/3) so that (2^(-1/3)·3^(1/3))^3 = 2^(-1)·3^(1) = 3/2.

Notice that all of the blue projection lines are exactly parallel, but that the green unchanged-interval line is not parallel or perpendicular to them.]]

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.]]

[[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.

The green dots are much more closely spaced here than in the previous diagram, but the generator size (in cents) is very similar; it's just that depending on how you're angled in this space, the combination of powers of 2 and 3 that define the generator size will be different, which can manifest as very different spacings visually.]]

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.

@@ Line 989: / Line 989: @@
 And since we have only a single comma, we know the angle at which every point in space that's off the unchanged-interval line will be projected onto it. We can figure it out by drawing a line from our comma, {{Vector|-8 5}}, to the origin, {{Vector|0 0}}. Every other projection line will be parallel to this line.
-[[File:Projection demo - arbitrary tuning case.png|none|thumb|900x900px|The purple grid is the original 2D JI lattice. The green line is the 1D tempered lattice for this unchanged-3/2 tuning of 5-ET, which we can also think of as the unchanged-interval line. We've only shown 5 points on it (other than the origin), i.e. we iterated the generator until we reached the next octave. For each of these green points on the tempered lattice, we show one purple point on the JI lattice which projects to it, via a blue arrow. In particular, we show the blackwood comma, the vanishing comma of this temperament, projecting to the unison [0}. Note that each next iteration of the generator adds a new green dot 1/3 of the way to the next JI lattice node which is diagonally up and to the left; this corresponds to the fact that 3/2 is our unchanged-interval, and 3/2 [-1 1⟩ maps to ~3/2 [3} in 5-ET. In other words, each generator is 2^(-1/3)·3^(1/3) so that (2^(-1/3)·3^(1/3))^3 = 2^(-1)·3^(1) = 3/2. Notice that all of the blue projection lines are exactly parallel, but that the green unchanged-interval line is not parallel or perpendicular to them.]]
+[[File:Projection demo - arbitrary tuning case.png|none|thumb|900x900px|The purple grid is the original 2D JI lattice. The green line is the 1D tempered lattice for this unchanged-3/2 tuning of 5-ET, which we can also think of as the unchanged-interval line. We've only shown 5 points on it (other than the origin), i.e. we iterated the generator until we reached the next octave. For each of these green points on the tempered lattice, we show one purple point on the JI lattice which projects to it, via a blue arrow.
+In particular, we show the blackwood comma, the vanishing comma of this temperament, projecting to the unison [0}.
+Note that each next iteration of the generator adds a new green dot 1/3 of the way to the next JI lattice node which is diagonally up and to the left; this corresponds to the fact that 3/2 is our unchanged-interval, and 3/2 [-1 1⟩ maps to ~3/2 [3} in 5-ET. In other words, each generator is 2^(-1/3)·3^(1/3) so that (2^(-1/3)·3^(1/3))^3 = 2^(-1)·3^(1) = 3/2.
+Notice that all of the blue projection lines are exactly parallel, but that the green unchanged-interval line is not parallel or perpendicular to them.]]
 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.]]
+[[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.
+The green dots are much more closely spaced here than in the previous diagram, but the generator size (in cents) is very similar; it's just that depending on how you're angled in this space, the combination of powers of 2 and 3 that define the generator size will be different, which can manifest as very different spacings visually.]]
 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.