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

By the [[Wikipedia:Fundamental theorem of arithmetic|fundamental theorem of arithmetic]] logarithms of primes are linearly independent over <math>\mathbb{Q}</math>, so we can interprete <math>\log(2), \log(3), \ldots</math> as orthogonal basis vectors. We write <math>e_p</math> in place of <math>\log(p)</math> and enforce orthogonality

<math>\overrightarrow{2^x 3^y 5^z \ldots} \mapsto x e_2 + y e_3 + z e_5 \ldots, x, y, z \in \mathbb{Q}</math>

i.e. the superscript vectors act as measuring sticks telling us how much of each prime there is in a vector.

One conceptually important measuring stick is the one constructed from the logarithms of the primes:

However rational approximations to the <math>\mathsf{JIP}</math> do belong in our space. Equal temperaments can be represented by such approximations called [[Val|vals]] which we define as

<math>\mathrm{val}(n; a, b, \ldots, z) := n \overrightarrow{a}^{-1} + \lfloor n \log_a(b) + \frac{1}{2}\rfloor \overrightarrow{b}^{-1} + \ldots + \lfloor n \log_a(z) + \frac{1}{2}\rfloor \overrightarrow{z}^{-1}</math> ,

<math>\overleftarrow{12} := \mathrm{val}(12; 2, 3, 5) = 12 e^2 + 19 e^3 + 28 e^5 =: \langle 12, 19, 28 \rbrack</math>

<math>\overleftarrow{5} := \mathrm{val}(5; 2, 3, 13/5) = 5 \cdot \overrightarrow{2}^{-1} + 8 \cdot \overrightarrow{3}^{-1} + 7 \cdot \overrightarrow{13/5}^{-1} = 5 e^2 + 8 e^3 - \frac{7}{2}e^5 + \frac{7}{2}e^{13}</math>

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" toward 5/1 or 13/1.

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>.

Now 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.

Therefore so does any linear combination of them e. g. <math>2 \cdot \overleftarrow{12} + \overleftarrow{7} = \overleftarrow{31}</math>

Defining the vee operator <math>a \vee b := \overline{\overline{b} \wedge \overline{a}}</math>, where the inner overlines are the inverses of the dual (always obvious from context). Wedges combine vals into temperaments that are closer to just intonation while vees do progressive damage to just intonation. As an exercise let's calculate the damage done by combining Meantone with Augmented:

To explore our space a bit longer we can consider <math>e_2</math> as an interval class consiting of all multiples of the octave. After all we have <math>\overrightarrow{8} = 3 \cdot \overrightarrow{2} = 3 e_2</math>. Wedging gives us larger classes. <math>e_2 \wedge e_3</math> can be seen as representing Pythagorean just intonation. Unfortunately the wedges are not powerful enough to distinguish subflavors <math>\overrightarrow{2} \wedge \overrightarrow{9} = 2 \cdot \overrightarrow{2} \wedge \overrightarrow{3} = \overrightarrow{4} \wedge \overrightarrow{3}</math> but at least the "orientation" is correct. Previously we used the term "just intonation" to refer to lack of tempering but it's also used to refer to an interval class such as 5-limit just intonation representable by <math>e_2 \wedge e_3 \wedge e_5</math>.

which strictly speaking consists of only a single element, namely the unison 1/1. However all commas lie close to this plane so it's informative to look at 3D subspaces of <math>\mathbb{S}</math> facing the unison plane to get a 2D view of all commas of interest. The null-space of all reasonable temperaments are also closely aligned with this plane.