Projection: Difference between revisions

Line 812:

In the following subsections, we examine units-only analyses of some RTT objects~~. For~~ simpler of examples of these, see [[Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis#Units-only analyses]].

In the first several of the following subsections, we examine units-only analyses of some RTT objects; for simpler of examples of these, see [[Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis#Units-only analyses]]. Then the last few sections are more general units analyses.

===Generator embedding, mapping: projection matrix===

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\end{align}

</math>

===The JI mapping times the JI generator embedding===

This situation is a variation upon the situation [[Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis#Cancelation|described here]], where the projection matrix <math>P</math> is derived as the matrix product of the generator embedding matrix <math>G</math> and the temperament mapping matrix <math>M</math>. What we're going to do here is work through the variation where it's the ''JI'' embedding matrix <math>G_{\text{j}}</math> and the ''JI'' mapping matrix <math>M_{\text{j}}</math>.

The key difference here is that both of these matrices are [[identity matrix|identity matrices]]. Thus, upon multiplying them, the result is ''also'' an identity matrix.

As you'll recall, the units of a projection matrix are <math>\small 𝗽</math>/<math>\small 𝗽</math>, where these vectorized units do not cancel, because one increments along rows and the other along columns, so we get different combinations of primes in different entries. However, ''along the diagonal'', the indices match, and cancel out. Like so:

<math>

\left[ \begin{array} {c}

\frac{\color{ForestGreen}\mathsf{p_1}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{Plum}\mathsf{p_3}} \\[6pt]

\frac{\color{NavyBlue}\mathsf{p_2}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{Plum}\mathsf{p_3}} \\[6pt]

\frac{\color{Plum}\mathsf{p_3}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{Plum}\mathsf{p_3}}{\color{Plum}\mathsf{p_3}} \\[6pt]

\end{array} \right]

→

\left[ \begin{array} {c}

\frac{\cancel{\color{ForestGreen}\mathsf{p_1}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{Plum}\mathsf{p_3}} \\

\frac{\color{NavyBlue}\mathsf{p_2}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\cancel{\color{NavyBlue}\mathsf{p_2}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{Plum}\mathsf{p_3}} \\

\frac{\color{Plum}\mathsf{p_3}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\cancel{\color{Plum}\mathsf{p_3}}}{\cancel{\color{Plum}\mathsf{p_3}}} \\

\end{array} \right]

→

\left[ \begin{array} {c}

{\large\mathsf{𝟙}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{Plum}\mathsf{p_3}} \\[6pt]

\frac{\color{NavyBlue}\mathsf{p_2}}{\color{ForestGreen}\mathsf{p_1}} & {\large\mathsf{𝟙}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{Plum}\mathsf{p_3}} \\[6pt]

\frac{\color{Plum}\mathsf{p_3}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{NavyBlue}\mathsf{p_2}} & {\large\mathsf{𝟙}} \\[6pt]

\end{array} \right]

</math>

So we keep the <math>\small 𝗽</math>/<math>\small 𝗽</math> unit because most of the entries still relate one prime to another.

However, if this units pattern were applied ''to an identity matrix'', like so:

<math>

\left[ \begin{array} {c}

1\,{\large\mathsf{𝟙}} & 0\,\frac{\color{ForestGreen}\mathsf{p_1}}{\color{NavyBlue}\mathsf{p_2}} & 0\,\color{ForestGreen}\frac{\mathsf{p_1}}{\color{Plum}\mathsf{p_3}} \\[6pt]

0\,\frac{\color{NavyBlue}\mathsf{p_2}}{\color{ForestGreen}\mathsf{p_1}} & 1\,{\large\mathsf{𝟙}} & 0\,\frac{\color{NavyBlue}\mathsf{p_2}}{\color{Plum}\mathsf{p_3}} \\[6pt]

0\,\frac{\color{Plum}\mathsf{p_3}}{\color{ForestGreen}\mathsf{p_1}} & 0\,\frac{\color{Plum}\mathsf{p_3}}{\color{NavyBlue}\mathsf{p_2}} & 1\,{\large\mathsf{𝟙}} \\[6pt]

\end{array} \right]

</math>

And then we eliminate units for all the entries whose quantities are zero:

<math>

\left[ \begin{array} {c}

1\,{\large\mathsf{𝟙}} & 0 & 0 \\

0 & 1\,{\large\mathsf{𝟙}} & 0 \\

0 & 0 & 1\,{\large\mathsf{𝟙}} \\

\end{array} \right]

</math>

Having zeroed out all of the entries who had any dimensionful units, we can now rightly say that this matrix as a whole is unitless (has dimensionless units).

Now we can see why, despite the fact that <math>GM = P</math> in a meaningful sense, it is not the case that <math>G_{\text{j}}M_{\text{j}} = P_{\text{j}}</math> in any meaningful sense. Instead <math>G_{\text{j}}M_{\text{j}} = I</math>, a completely generic, unitless identity matrix.

===Derivation of generator embedding matrix from unchanged-interval basis and mapping===

The generator embedding matrix <math>G</math> is related to the mapping <math>M</math> and unchanged-interval basis <math>\mathrm{U}</math> by the following formula:

<math>

G=\mathrm{U}(M\mathrm{U})^{-1}

</math>

This formula is used, among other places, in the sum polytope method for computing miniaverage tunings of regular temperaments (see here: [[Generator embedding optimization#Convert to generators]]).

And it turns out this formula works fine when subjected to a units analysis. <math>\mathrm{U}</math> has units of primes. And <math>M</math> has units of generators per primes. So we have <math>\small 𝗽\!·\!((𝗴</math>/<math>\small 𝗽)\!·\!𝗽)^{-1}</math>. The <math>\small 𝗽</math>'s on the inside of the parens cancel, and we're left with <math>\small 𝗽𝗴^{-1}</math>, or in other words, <math>\small 𝗽</math>/<math>\small 𝗴</math>, which are indeed the units of <math>G</math>.

===Derivation of projection matrix from unrotated vector list and scaling factor matrix===

In a similar fashion we can show that the formula for the projection matrix <math>P</math> from the [[unrotated vector list]] <math>\mathrm{V}</math> and the [[scaling factor matrix]] <math>\textit{Λ}</math> holds up to units analysis. You can read about these objects and their relationships below: [[Projection#The unrotated vectors and scaling factors]]. The formula is:

<math>

\mathrm{V}\textit{Λ}\mathrm{V}^{-1}

</math>

With <math>\mathrm{V}</math> having units of <math>\small 𝗽</math> and <math>\textit{Λ}</math> having no units (having dimensionless units), we find: <math>{\small 𝗽}\!·\!\mathsf{𝟙}\!·\!{\small 𝗽^{-1}}</math>. This simplifies to <math>\small 𝗽</math>/<math>\small 𝗽</math>, which matches our projection matrix's units.

==Projection properties==

@@ Line 812: / Line 812: @@
-In the following subsections, we examine units-only analyses of some RTT objects. For simpler of examples of these, see [[Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis#Units-only analyses]].
+In the first several of the following subsections, we examine units-only analyses of some RTT objects; for simpler of examples of these, see [[Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis#Units-only analyses]]. Then the last few sections are more general units analyses.
 ===Generator embedding, mapping: projection matrix===
@@ Line 1,186: / Line 1,186: @@
 \end{align}
 </math>
+===The JI mapping times the JI generator embedding===
+This situation is a variation upon the situation [[Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis#Cancelation|described here]], where the projection matrix <math>P</math> is derived as the matrix product of the generator embedding matrix <math>G</math> and the temperament mapping matrix <math>M</math>. What we're going to do here is work through the variation where it's the ''JI'' embedding matrix <math>G_{\text{j}}</math> and the ''JI'' mapping matrix <math>M_{\text{j}}</math>.
+The key difference here is that both of these matrices are [[identity matrix|identity matrices]]. Thus, upon multiplying them, the result is ''also'' an identity matrix.
+As you'll recall, the units of a projection matrix are <math>\small 𝗽</math>/<math>\small 𝗽</math>, where these vectorized units do not cancel, because one increments along rows and the other along columns, so we get different combinations of primes in different entries. However, ''along the diagonal'', the indices match, and cancel out. Like so:
+<math>
+\left[ \begin{array} {c}
+\frac{\color{ForestGreen}\mathsf{p_1}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{Plum}\mathsf{p_3}} \\[6pt]
+\frac{\color{NavyBlue}\mathsf{p_2}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{Plum}\mathsf{p_3}} \\[6pt]
+\frac{\color{Plum}\mathsf{p_3}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{Plum}\mathsf{p_3}}{\color{Plum}\mathsf{p_3}} \\[6pt]
+\end{array} \right]
+→
+\left[ \begin{array} {c}
+\frac{\cancel{\color{ForestGreen}\mathsf{p_1}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{Plum}\mathsf{p_3}} \\
+\frac{\color{NavyBlue}\mathsf{p_2}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\cancel{\color{NavyBlue}\mathsf{p_2}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{Plum}\mathsf{p_3}} \\
+\frac{\color{Plum}\mathsf{p_3}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\cancel{\color{Plum}\mathsf{p_3}}}{\cancel{\color{Plum}\mathsf{p_3}}} \\
+\end{array} \right]
+→
+\left[ \begin{array} {c}
+{\large\mathsf{𝟙}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{Plum}\mathsf{p_3}} \\[6pt]
+\frac{\color{NavyBlue}\mathsf{p_2}}{\color{ForestGreen}\mathsf{p_1}} & {\large\mathsf{𝟙}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{Plum}\mathsf{p_3}} \\[6pt]
+\frac{\color{Plum}\mathsf{p_3}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{NavyBlue}\mathsf{p_2}} & {\large\mathsf{𝟙}} \\[6pt]
+\end{array} \right]
+</math>
+So we keep the <math>\small 𝗽</math>/<math>\small 𝗽</math> unit because most of the entries still relate one prime to another.
+However, if this units pattern were applied ''to an identity matrix'', like so:
+<math>
+\left[ \begin{array} {c}
+\,{\large\mathsf{𝟙}} & 0\,\frac{\color{ForestGreen}\mathsf{p_1}}{\color{NavyBlue}\mathsf{p_2}} & 0\,\color{ForestGreen}\frac{\mathsf{p_1}}{\color{Plum}\mathsf{p_3}} \\[6pt]
+\,\frac{\color{NavyBlue}\mathsf{p_2}}{\color{ForestGreen}\mathsf{p_1}} & 1\,{\large\mathsf{𝟙}} & 0\,\frac{\color{NavyBlue}\mathsf{p_2}}{\color{Plum}\mathsf{p_3}} \\[6pt]
+\,\frac{\color{Plum}\mathsf{p_3}}{\color{ForestGreen}\mathsf{p_1}} & 0\,\frac{\color{Plum}\mathsf{p_3}}{\color{NavyBlue}\mathsf{p_2}} & 1\,{\large\mathsf{𝟙}} \\[6pt]
+\end{array} \right]
+</math>
+And then we eliminate units for all the entries whose quantities are zero:
+<math>
+\left[ \begin{array} {c}
+\,{\large\mathsf{𝟙}} & 0 & 0 \\
+& 1\,{\large\mathsf{𝟙}} & 0 \\
+& 0 & 1\,{\large\mathsf{𝟙}} \\
+\end{array} \right]
+</math>
+Having zeroed out all of the entries who had any dimensionful units, we can now rightly say that this matrix as a whole is unitless (has dimensionless units).
+Now we can see why, despite the fact that <math>GM = P</math> in a meaningful sense, it is not the case that <math>G_{\text{j}}M_{\text{j}} = P_{\text{j}}</math> in any meaningful sense. Instead <math>G_{\text{j}}M_{\text{j}} = I</math>, a completely generic, unitless identity matrix.
+===Derivation of generator embedding matrix from unchanged-interval basis and mapping===
+The generator embedding matrix <math>G</math> is related to the mapping <math>M</math> and unchanged-interval basis <math>\mathrm{U}</math> by the following formula:
+<math>
+G=\mathrm{U}(M\mathrm{U})^{-1}
+</math>
+This formula is used, among other places, in the sum polytope method for computing miniaverage tunings of regular temperaments (see here: [[Generator embedding optimization#Convert to generators]]).
+And it turns out this formula works fine when subjected to a units analysis. <math>\mathrm{U}</math> has units of primes. And <math>M</math> has units of generators per primes. So we have <math>\small 𝗽\!·\!((𝗴</math>/<math>\small 𝗽)\!·\!𝗽)^{-1}</math>. The <math>\small 𝗽</math>'s on the inside of the parens cancel, and we're left with <math>\small 𝗽𝗴^{-1}</math>, or in other words, <math>\small 𝗽</math>/<math>\small 𝗴</math>, which are indeed the units of <math>G</math>.
+===Derivation of projection matrix from unrotated vector list and scaling factor matrix===
+In a similar fashion we can show that the formula for the projection matrix <math>P</math> from the [[unrotated vector list]] <math>\mathrm{V}</math> and the [[scaling factor matrix]] <math>\textit{Λ}</math> holds up to units analysis. You can read about these objects and their relationships below: [[Projection#The unrotated vectors and scaling factors]]. The formula is:
+<math>
+\mathrm{V}\textit{Λ}\mathrm{V}^{-1}
+</math>
+With <math>\mathrm{V}</math> having units of <math>\small 𝗽</math> and <math>\textit{Λ}</math> having no units (having dimensionless units), we find: <math>{\small 𝗽}\!·\!\mathsf{𝟙}\!·\!{\small 𝗽^{-1}}</math>. This simplifies to <math>\small 𝗽</math>/<math>\small 𝗽</math>, which matches our projection matrix's units.
 ==Projection properties==