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(t))). }[/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
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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. The x-th "scale step" in such a scale is called the x-th spec (pl. specs), which comes from the phrase "tone spectrum". It is defined as the integral of the FM function: [math]\displaystyle{ \text{FM}(i) = \displaystyle \int_0^t f(t) }[/math]
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 k \leq i} 1+f(k) }[/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
Properties
An FM scale is aperiodic if and only if some of the [math]\displaystyle{ a_i }[/math] is irrational.
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.
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Todo: elaborate this
Can we elaborate and verify these? |