User:Akselai/FM scale: Difference between revisions

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<math>f(x) = f_1(f_2(\cdots f_n(t))).</math>

<math>f(x) = f_1(f_2(\cdots f_n(0))).</math>

We also define the '''[https://en.wikipedia.org/wiki/Sigmoid_function sigmoid function]''' to be

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{{todo|hint|inline=1|comment=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"~~.

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>\text{FM}(i) = \displaystyle \int_0^t f(x) dx</math>

<math>\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.

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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>A_i</math>-th spec, where <math>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>\text{FM}(x) = \displaystyle\sum_{1 \leq i \leq x}\sigma(x - A_i).</math>

The ''i''-th scale step will be mapped to the <math>A_i</math>-th spec, where <math>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>\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 ==

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== Properties ==

An FM scale is aperiodic if and only if some of the <math>a_i</math> is irrational.

An FM scale is aperiodic if and only if some of the <math>a_i</math> are irrational multiples of ''π''.

Since

@@ Line 6: / Line 6: @@
 <math>f_i(t) = a_i \sin(x + t) + c_i, 1 \leq i \leq n</math>
-<math>f(x) = f_1(f_2(\cdots f_n(t))).</math>
+<math>f(x) = f_1(f_2(\cdots f_n(0))).</math>
 We also define the '''[https://en.wikipedia.org/wiki/Sigmoid_function sigmoid function]''' to be
@@ Line 25: / Line 25: @@
 {{todo|hint|inline=1|comment=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".
+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>\text{FM}(i) = \displaystyle \int_0^t f(x) dx</math>
+<math>\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.
@@ Line 37: / Line 38: @@
 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>A_i</math>-th spec, where <math>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>\text{FM}(x) = \displaystyle\sum_{1 \leq i \leq x}\sigma(x - A_i).</math>
+The ''i''-th scale step will be mapped to the <math>A_i</math>-th spec, where <math>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>\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 ==
@@ Line 46: / Line 47: @@
 == Properties ==
-An FM scale is aperiodic if and only if some of the <math>a_i</math> is irrational.
+An FM scale is aperiodic if and only if some of the <math>a_i</math> are irrational multiples of ''π''.
 Since