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

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<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 ==

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== 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 65: / Line 65: @@
 <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 ==
@@ Line 135: / Line 143: @@
 == 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 202: / 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...