User:Akselai/FM scale - Revision history

Akselai: /* Properties */

2024-01-30T13:30:31Z

Properties

← Older revision		Revision as of 13:30, 30 January 2024
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	== Properties ==		== 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		Since

Akselai: /* Second Definition */

2024-01-30T13:20:05Z

Second Definition

← Older revision		Revision as of 13:20, 30 January 2024
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	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.		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:		It is defined as the integral of the FM function:
	<math>\text{FM}(x) = \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".		The x-th "scale step" in such a scale is called the x-th '''spec''' (pl. '''specs'''), which comes from the phrase "tone spectrum".

Akselai: /* Second Definition */

2024-01-30T13:19:45Z

Second Definition

← Older revision		Revision as of 13:19, 30 January 2024
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	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.		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:		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}(x) = \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".		The x-th "scale step" in such a scale is called the x-th '''spec''' (pl. '''specs'''), which comes from the phrase "tone spectrum".

Akselai: /* Second Definition */

2024-01-30T13:18:26Z

Second Definition

← Older revision		Revision as of 13:18, 30 January 2024
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	{{todo\|hint\|inline=1\|comment=It may be better to have "continuous scales" be a separate article.}}		{{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:		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}(i) = \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.		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.

Akselai: /* Definitions */

2024-01-30T13:13:43Z

Definitions

← Older revision		Revision as of 13:13, 30 January 2024
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	<math>f_i(t) = a_i \sin(x + t) + c_i, 1 \leq i \leq n</math>		<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		We also define the '''[https://en.wikipedia.org/wiki/Sigmoid_function sigmoid function]''' to be

Akselai: /* Third Definition */

2024-01-30T13:12:46Z

Third Definition

← Older revision		Revision as of 13:12, 30 January 2024
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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.		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.		Setting ''k'' = ∞ gives a discrete scale, with unequal spec ranges corresponding to equal steps.

	== Examples ==		== Examples ==

Akselai: /* Second Definition */

2024-01-30T13:08:09Z

Second Definition

← Older revision		Revision as of 13:08, 30 January 2024
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	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. 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:		It is defined as the integral of the FM function:
	<math>\text{FM}(i) = \displaystyle \int_0^t f(t)</math>		<math>\text{FM}(i) = \displaystyle \int_0^t f(x) dx</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.		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.

Akselai: Created page with "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..."

2024-01-30T13:07:42Z

Created page with "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..."

New page

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>f : \mathbb{R} \rightarrow \mathbb{R} </math> defined by

<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>

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

<math>\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>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>\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|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".
It is defined as the integral of the FM function:
<math>\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>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>

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

== Examples ==
[[File:Undulation.png|thumb|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>a_i</math> is irrational.

Since
<math>\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|inline=1|comment=Can we elaborate and verify these?}}