User:Frostburn/Theory From First Principles

Just using Xen Wiki as a notepad, don't mind me.

I'm currently working on the grammar for Scale Workshop 3. It will naturally include monzos and I'm debating if I want to include vals so I'm writing stuff down as a thinking aid.

Time Domain

We begin our journey in the time domain where one second (1 s) passes for every 9192631770 oscillations of the radiation emited by caesium 133 during the unperturbed ground-state hyperfine transition.

Frequency Domain

We invert time to arrive in the frequency domain where oscillations are measured in repetitions per second i.e. Hertz (Hz = s^-1).

Scalar Domain

Frequencies are scalar multiples of each other and especially the positive rational scalars are of special interest in music.

Pitch Domain

When we take the logarithm of a positive rational scalar its factors separate into a sum e.g. [math]\displaystyle{ \log(15/8) = \log(3) + \log(5) - 3\log(2) }[/math].

Adding Geometry

By the fundamental theorem of arithmetic logarithms of primes are linearly independent over [math]\displaystyle{ \mathbb{Q} }[/math], so we can interprete [math]\displaystyle{ \log(2), \log(3), \ldots }[/math] as basis vectors. We write [math]\displaystyle{ e_p }[/math] in place of [math]\displaystyle{ \log(p) }[/math].

To make things slightly more formal we define the right facing arrow function

[math]\displaystyle{ \overrightarrow{2^x 3^y 5^z \ldots} := x e_2 + y e_3 + z e_5 \ldots, x, y, z \in \mathbb{Q} }[/math]

which takes objects from the scalar domain to the geometric pitch domain.

We denote the inverse of the arrow function with [math]\displaystyle{ \mathrm{ratio} }[/math] i.e. it turns prime count vectors into ratios:

[math]\displaystyle{ \mathrm{ratio}(\overrightarrow{p/q}) = p/q }[/math]

Pitch is measured in cents (¢) which we define to be the vector quantity [math]\displaystyle{ ¢ := e_2 / 1200 }[/math] i.e. [math]\displaystyle{ \mathrm{ratio}(¢) = 2^{\frac{1}{1200}} \approx 1.0005777895 }[/math] .

Expanding geometry

TODO