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

Line 63:

Scalars do not have units. That's what makes them scalars. Do pitches have units? Maybe they are 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.

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>.

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>.

Let's define superscript basis vectors as

i.e. we have <math>e^2 = e_2 / w_2^2 = \hat{i} / w_2</math>. A reciprocal cent can now be expressed as <math>¢^{-1} = 1200 e^2 = 100 d^{-1} \hat{i}</math> or 100 reciprocal demitones in the <math>\hat{i}</math> direction.

== Exterior algebra nonsense ==

@@ Line 63: / Line 63: @@
 Scalars do not have units. That's what makes them scalars. Do pitches have units? Maybe they are 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.
+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>.
+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>.
+Let's define superscript basis vectors as
+<math>e^p := e_p ^ {-1}</math>
+i.e. we have <math>e^2 = e_2 / w_2^2 = \hat{i} / w_2</math>. A reciprocal cent can now be expressed as <math>¢^{-1} = 1200 e^2 = 100 d^{-1} \hat{i}</math> or 100 reciprocal demitones in the <math>\hat{i}</math> direction.
 == Exterior algebra nonsense ==