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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== 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 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