User:Akselai/FM scale

A frequency-modulated scale (FM scale) is a scale with steps generated by a frequency modulation function. Such a scale is usually aperiodic, but is periodic under some conditions.

Definitions

A frequency modulation function is a function [math]\displaystyle{ f : \mathbb{R} \rightarrow \mathbb{R} }[/math] defined by

[math]\displaystyle{ f_i(t) = a_i \sin(x + t) + c_i, 1 \leq i \leq n }[/math]

[math]\displaystyle{ f(x) = f_1(f_2(\cdots f_n(0))). }[/math]

We also define the sigmoid function to be

[math]\displaystyle{ \displaystyle \sigma(x) = \frac{1}{1 + e^{-kx}} }[/math] for a parameter k. This will act as a interpolation function. The smaller the value of k, the smoother the function.

There are three types of FM scales. The first one is a scale in the strict sense, the other two are more general objects. Here, a value of 1 produced by the formulas correspond to one logarithmic unit of pitch, such as cent, semitone or octave.

First Definition

An FM scale of the first kind is a scale with step size [math]\displaystyle{ p + f(i) }[/math] between the i-th and (i+1)-th scale degree. Here, p is the ascension interval of the scale, analogous to the quasiperiod of a periodic scale. In other words, p is the average step size of the scale. Formally, [math]\displaystyle{ \text{FM}(i) = \displaystyle ip + \sum_{1 \leq j \leq i} f(j). }[/math]

Setting p = 0 gives the undulation scale.

Second Definition

Todo: hint It may be better to have "continuous scales" be a separate article.

An FM scale of the second kind is a function from the real numbers to musical intervals. Since these are usually continuous functions, it is meaningless to talk about scale steps of an FM scale. It is defined as the integral of the FM function: [math]\displaystyle{ \text{FM}(t) = \displaystyle \int_0^t f(x) dx }[/math]

The x-th "scale step" in such a scale is called the x-th spec (pl. specs), which comes from the phrase "tone spectrum". Unlike a usual scale, which is mathematically a function from the integers to musical intervals, here we have a scale with continuous scale degrees. So, we cannot put such a scale into Scala or a usual DAW retuning plugin; however, audio synthesis software such as Csound and SuperCollider provide good environments for continuous scales.

Third Definition

An FM scale of the third kind is a function from the real numbers to musical intervals again.

Suppose for each value i we provide an FM mapping, that is, we perturb the scale step indices from integers to nearby real numbers. This is done using a cumulative sum over the FM function.

The i-th scale step will be mapped to the [math]\displaystyle{ A_i }[/math]-th spec, where [math]\displaystyle{ A_i = \displaystyle \sum_{1 \leq j \leq i} 1+f(j) }[/math]. By smoothing out the scale steps using the sigmoid function, the FM scale becomes [math]\displaystyle{ \text{FM}(x) = \displaystyle\sum_{1 \leq i \leq x}\sigma(x - A_i). }[/math]

Setting k = ∞ gives a discrete scale, with unequal spec ranges corresponding to equal steps.

Examples

The plot of the undulation scale given by the FM function f(x) = sin(x + 2 sin x). 200 scale steps are shown.

Properties

An FM scale is aperiodic if and only if some of the [math]\displaystyle{ a_i }[/math] are irrational multiples of π.

Since [math]\displaystyle{ \displaystyle \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{0 \leq i \leq n} f(i) = 0, }[/math] p is the interval which is the average step size of the scale.

Todo: elaborate this Can we elaborate and verify these?