User:Frostburn/Theory From First Principles: Difference between revisions

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In combination with implicit scalar multiplication the similarity with Scale Workshop's N-of-EDO notation is unmistakable e.g. <math>7 \backslash 12 = 700 ¢</math> .

Absolute pitch can be incorporated when we introduce a basis vector <math>e_0</math> and associate it with logarithmic frequency

<math>\overrightarrow{Hz} \mapsto e_0</math>

It represents a single point of origin in projective geometry (remember how we chose to ignore that extra scalar in front when summing absolute pitches).

In practice it's handy to have a reference frequency that humans can hear so we might use the shifted origin

<math>\tilde{e_0} := \overrightarrow{440Hz} = e_0 + 3 e_2 + e_5 + e_{11}</math>.

When working with absolute pitch, the inverse of the arrow function is called <math>\mathrm{freq}</math> instead of <math>\mathrm{ratio}</math>

<math>\mathrm{freq}(\overrightarrow{\frac{p}{q} Hz}) = \frac{p}{q} Hz</math> .

Care must be taken when the multiplier of the projective origin is not 1. e.g. Let's consider 1.5 Hz which is a perfect fifth above 1 Hz. It's represented as <math>e_0 - e_2 + e_3</math>.

If some other calculation gave us the result <math>2 e_0 - e_2 + e_3</math> it wouldn't represent 1.5Hz it's

<math>\mathrm{freq}(2 e_0 - e_2 + e_3) = 1.5 Hz^2 \sim \sqrt{\frac{3}{2}} Hz \approx 1.22 Hz</math>

instead.

== Expanding geometry ==

It's instructive to construct <math>\mathrm{ratio}</math> more explicitly by considering the inverses of the basis vectors

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Note that above I've implicitly used a convenient metric to carry out the calculations, which is fine due to the new basis still being orhogonal. Explicitly we'd have

<math>\overrightarrow{13/5}^{-1} = (w_{13} \hat{m} - w_5 \hat{k}) / (w_5^2 + w_{13}^2)</math>, where the weights decide how much 13/5 "leans" towards 5/1 or 13/1.

We make the projective origin <math>e_0</math> non-invertible by enforcing a null metric weight <math>e_0 \cdot e_0 = 0</math> which can be handy in some calculations and often does the right thing.

Most of the theory developed here deals with relative pitch. It's always possible to ground a result on an origin e.g.

<math>

\mathrm{freq}(\tilde{e_0} + \overleftarrow{12} \cdot \overrightarrow{15/8} \backslash 12) = \mathrm{freq}(e_0 + \frac{47}{12} e_2 + e_5 + e_{11}) \approx 830.6 Hz

</math>

== On units ==

Scalars do not have units. That's what makes them scalars. Do pitches have units? Maybe they're like radians, unitless but it makes no sense to add them to other kinds of objects. Whatever the case may be, prime count vectors (i.e. monzos) have inverse units to vals. This ~~should be~~ enough to distinguish them during SW3 runtime and prevent vals from being interpreted as pitch or turned into frequencies.

Scalars do not have units. That's what makes them scalars. Do relative pitches have units? Maybe they're like radians, unitless but it makes no sense to add them to other kinds of objects. Whatever the case may be, prime count vectors (i.e. monzos) have inverse units to vals. This is enough to distinguish them during SW3 runtime and prevent vals from being interpreted as pitch or turned into frequencies.

Taking these considerations more seriously and remembering that cents are a vector quantity we can try to figure out what units vals have: One cent is one hundreth of a semitone and one octave consists of twelve of these semitones. All vector quantities. Let's call the dimensioneless version of a semitone a demitone. To re-iterate: A cent is 1/100 demitones in the direction of <math>e_2</math>. Let's call <math>\hat{i}</math> the direction of <math>e_2</math> i.e. <math>e_2 = w_2 \hat{i} = 12 d \hat{i}</math>, where <math>d</math> is the metric weight of a demitone. The basis vector itself has unit metric <math>\hat{i} \cdot \hat{i} = 1</math>.

Taking these considerations more seriously and remembering that cents are a vector quantity we can try to figure out what units vals such as the ''jorp'' (€) have: One cent is one hundreth of a semitone and one octave consists of twelve of these semitones. All vector quantities. Let's call the dimensioneless version of a semitone a demitone. To re-iterate: A cent is 1/100 demitones in the direction of <math>e_2</math>. Let's call <math>\hat{i}</math> the direction of <math>e_2</math> i.e. <math>e_2 = w_2 \hat{i} = 12 d \hat{i}</math>, where <math>d</math> is the metric weight of a demitone. The basis vector itself has unit metric <math>\hat{i} \cdot \hat{i} = 1</math>.

A reciprocal cent satisfies <math>¢^{-1} \cdot ¢ = 1</math> so as per the usual definition of the geometric inverse of a vector we have <math>¢^{-1} = ¢ / (¢ \cdot ¢) = \frac{1}{1200}e_2 / (\frac{1}{1200}^2 e_2 \cdot e_2) = 1200 w_2 \hat{i} / (w_2^2 \hat{i} \cdot \hat{i}) = \frac{1200}{w_2}\hat{i}</math>.

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which strictly speaking consists of only a single element, namely the unison 1/1. However all commas lie close to this plane so in 3D subspaces of <math>\mathbb{S}</math> it's informative to face the unison plane head-on to get a 2D view of all commas of interest. The null-space of all reasonable temperaments (i.e. the set of all commas and their products that vanish in the temperament) are also closely aligned with this plane.

~~== The nature of 1 and improving representation ==~~

~~We've purposefully avoided using <math>e_1</math> i.e. <math>\overrightarrow{1} = 0</math>. This is because it introduces ambiguity into the logarithm formula. e.g.~~

~~<math>\log(6) = \log(2\cdot 3) = \log(1 \cdot 1 \cdot 2 \cdot 3) = \log(1) + \log(1) + \log(2) + log(3) \mapsto^? 2 e_1 + e_2 + e_3</math>.~~

~~If we stipulate that the logarithm of unity is only added once and that <math>e_1 \cdot e_1 = 0</math> we might get a useful bookkeeping tool.~~

~~The right-facing arrow function should be redefined:~~

~~<math>\overrightarrow{1\cdot 2^x\cdot 3^y\cdot\ldots} \mapsto e_1 + x e_2 + y e_3 + \ldots</math>~~

The point is to differentiate intervals from interval classes using projective geometric algebra. A ratio and it's square are different intervals, but any rational multiple of a val represents the same equal temperament and the same goes for temperaments (wedges). I've forgotten how to do the projective stuff to Make Things Work™, so I'm just leaving this chapter here to remind myself to read up on the relevant literature...

@@ Line 51: / Line 51: @@
 In combination with implicit scalar multiplication the similarity with Scale Workshop's N-of-EDO notation is unmistakable e.g. <math>7 \backslash 12 = 700 ¢</math> .
+Absolute pitch can be incorporated when we introduce a basis vector <math>e_0</math> and associate it with logarithmic frequency
+<math>\overrightarrow{Hz} \mapsto e_0</math>
+It represents a single point of origin in projective geometry (remember how we chose to ignore that extra scalar in front when summing absolute pitches).
+In practice it's handy to have a reference frequency that humans can hear so we might use the shifted origin
+<math>\tilde{e_0} := \overrightarrow{440Hz} = e_0 + 3 e_2 + e_5 + e_{11}</math>.
+When working with absolute pitch, the inverse of the arrow function is called <math>\mathrm{freq}</math> instead of <math>\mathrm{ratio}</math>
+<math>\mathrm{freq}(\overrightarrow{\frac{p}{q} Hz}) = \frac{p}{q} Hz</math> .
+Care must be taken when the multiplier of the projective origin is not 1. e.g. Let's consider 1.5 Hz which is a perfect fifth above 1 Hz. It's represented as <math>e_0 - e_2 + e_3</math>.
+If some other calculation gave us the result <math>2 e_0 - e_2 + e_3</math> it wouldn't represent 1.5Hz it's
+<math>\mathrm{freq}(2 e_0 - e_2 + e_3) = 1.5 Hz^2 \sim \sqrt{\frac{3}{2}} Hz \approx 1.22 Hz</math>
+instead.
 == Expanding geometry ==
 It's instructive to construct <math>\mathrm{ratio}</math> more explicitly by considering the inverses of the basis vectors
@@ Line 110: / Line 133: @@
 Note that above I've implicitly used a convenient metric to carry out the calculations, which is fine due to the new basis still being orhogonal. Explicitly we'd have
 <math>\overrightarrow{13/5}^{-1} = (w_{13} \hat{m} - w_5 \hat{k}) / (w_5^2 + w_{13}^2)</math>, where the weights decide how much 13/5 "leans" towards 5/1 or 13/1.
+We make the projective origin <math>e_0</math> non-invertible by enforcing a null metric weight <math>e_0 \cdot e_0 = 0</math> which can be handy in some calculations and often does the right thing.
+Most of the theory developed here deals with relative pitch. It's always possible to ground a result on an origin e.g.
+<math>
+\mathrm{freq}(\tilde{e_0} + \overleftarrow{12} \cdot \overrightarrow{15/8} \backslash 12) = \mathrm{freq}(e_0 + \frac{47}{12} e_2 + e_5 + e_{11}) \approx 830.6 Hz
+</math>
 == On units ==
-Scalars do not have units. That's what makes them scalars. Do pitches have units? Maybe they're like radians, unitless but it makes no sense to add them to other kinds of objects. Whatever the case may be, prime count vectors (i.e. monzos) have inverse units to vals. This should be enough to distinguish them during SW3 runtime and prevent vals from being interpreted as pitch or turned into frequencies.
+Scalars do not have units. That's what makes them scalars. Do relative pitches have units? Maybe they're like radians, unitless but it makes no sense to add them to other kinds of objects. Whatever the case may be, prime count vectors (i.e. monzos) have inverse units to vals. This is enough to distinguish them during SW3 runtime and prevent vals from being interpreted as pitch or turned into frequencies.
-Taking these considerations more seriously and remembering that cents are a vector quantity we can try to figure out what units vals have: One cent is one hundreth of a semitone and one octave consists of twelve of these semitones. All vector quantities. Let's call the dimensioneless version of a semitone a demitone. To re-iterate: A cent is 1/100 demitones in the direction of <math>e_2</math>. Let's call <math>\hat{i}</math> the direction of <math>e_2</math> i.e. <math>e_2 = w_2 \hat{i} = 12 d \hat{i}</math>, where <math>d</math> is the metric weight of a demitone. The basis vector itself has unit metric <math>\hat{i} \cdot \hat{i} = 1</math>.
+Taking these considerations more seriously and remembering that cents are a vector quantity we can try to figure out what units vals such as the ''jorp'' (€) have: One cent is one hundreth of a semitone and one octave consists of twelve of these semitones. All vector quantities. Let's call the dimensioneless version of a semitone a demitone. To re-iterate: A cent is 1/100 demitones in the direction of <math>e_2</math>. Let's call <math>\hat{i}</math> the direction of <math>e_2</math> i.e. <math>e_2 = w_2 \hat{i} = 12 d \hat{i}</math>, where <math>d</math> is the metric weight of a demitone. The basis vector itself has unit metric <math>\hat{i} \cdot \hat{i} = 1</math>.
 A reciprocal cent satisfies <math>¢^{-1} \cdot ¢ = 1</math> so as per the usual definition of the geometric inverse of a vector we have <math>¢^{-1} = ¢ / (¢ \cdot ¢) = \frac{1}{1200}e_2 / (\frac{1}{1200}^2 e_2 \cdot e_2) = 1200 w_2 \hat{i} / (w_2^2 \hat{i} \cdot \hat{i}) = \frac{1200}{w_2}\hat{i}</math>.
@@ Line 179: / Line 210: @@
 which strictly speaking consists of only a single element, namely the unison 1/1. However all commas lie close to this plane so in 3D subspaces of <math>\mathbb{S}</math> it's informative to face the unison plane head-on to get a 2D view of all commas of interest. The null-space of all reasonable temperaments (i.e. the set of all commas and their products that vanish in the temperament) are also closely aligned with this plane.
-== The nature of 1 and improving representation ==
-We've purposefully avoided using <math>e_1</math> i.e. <math>\overrightarrow{1} = 0</math>. This is because it introduces ambiguity into the logarithm formula. e.g.
-<math>\log(6) = \log(2\cdot 3) = \log(1 \cdot 1 \cdot 2 \cdot 3) = \log(1) + \log(1) + \log(2) + log(3) \mapsto^? 2 e_1 + e_2 + e_3</math>.
-If we stipulate that the logarithm of unity is only added once and that <math>e_1 \cdot e_1 = 0</math> we might get a useful bookkeeping tool.
-The right-facing arrow function should be redefined:
-<math>\overrightarrow{1\cdot 2^x\cdot 3^y\cdot\ldots} \mapsto e_1 + x e_2 + y e_3 + \ldots</math>
-The point is to differentiate intervals from interval classes using projective geometric algebra. A ratio and it's square are different intervals, but any rational multiple of a val represents the same equal temperament and the same goes for temperaments (wedges). I've forgotten how to do the projective stuff to Make Things Work™, so I'm just leaving this chapter here to remind myself to read up on the relevant literature...