Projection: Difference between revisions

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In particular, this is the tuning where <math>\frac32</math> is unchanged (as are all of its multiples). So our unchanged-interval basis contains a single column vector. This describes a line we could draw across the 3-limit lattice, which we could call our "unchanged-interval line". The idea is that every pitch (i.e., every point in this 2D 3-limit space) will get projected onto this line. Every pitch that is already on this line therefore won't be moved by this tuning; that's why it's unchanged!

Note that this unchanged-interval line is also our tempered lattice. Normally, when we draw the tempered lattice separately from the JI lattice, we wouldn't draw it at an angle like this. But it's important that it's at this angle here, since that's the angle at which it has been re-embedded into our original JI space.

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.

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 In particular, this is the tuning where <math>\frac32</math> is unchanged (as are all of its multiples). So our unchanged-interval basis contains a single column vector. This describes a line we could draw across the 3-limit lattice, which we could call our "unchanged-interval line". The idea is that every pitch (i.e., every point in this 2D 3-limit space) will get projected onto this line. Every pitch that is already on this line therefore won't be moved by this tuning; that's why it's unchanged!
+Note that this unchanged-interval line is also our tempered lattice. Normally, when we draw the tempered lattice separately from the JI lattice, we wouldn't draw it at an angle like this. But it's important that it's at this angle here, since that's the angle at which it has been re-embedded into our original JI space.
 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.