Fractal scale: Difference between revisions

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A '''fractal scale''' is a [[scale]] obtained by dividing a starting [[interval]] in two or more parts, and then by dividing these parts recursively using the same ~~[[frequency~~ ratio]]. In practice, the starting interval ~~can be~~ used as the [[period]] of the scale, allowing to use fractal scales as [[periodic scale]]s.

A '''fractal scale''' is a [[scale]] obtained by dividing a starting [[interval]] in two or more parts, and then by dividing these parts recursively using the same ratio. In practice, the starting interval is generally used as the [[period]] of the scale, allowing to use fractal scales as [[periodic scale]]s. Fractal scales come in three main types, associated respectively with frequencies, pitch and length: '''linear fractal scales''', defined using a ratio of [[frequency ratio]]s; '''logarithmic fractal scales''', defined using a ratio of [[logarithmic ratio]]s (e.g. [[cent]]s); and '''inverse linear fractal scales''', defined using a ratio of length ratios.

The ''order'' of a fractal scale is the number of iterations of the division process used to obtain the scale. An order-0 fractal scale contains only the starting interval. An order-1 fractal scale contains the original ratio only once. An order-N scale with M steps in its division contains M<sup>N</sup> steps.

A fractal scale can be uniquely identified by its order and its ~~ratio (e.g~~. order-5 2:3:4 fractal scale~~). [[Interval size measure|Logarithmic ratio]]s expressed in [[cent]]s can be converted to frequency ratios using <math>ratio = 2^{\frac{cents}{1200}}</math>.~~

A fractal scale can be uniquely identified by its order, its ratio and its type. For example, the order-5 2:3:4 linear fractal scale is a 32-tone octave-repeating scale

Fractal scales provides a certain form of symmetry which is very different in nature than that of other scale families, such as [[MOS scale]]s or [[Regular temperament theory|regularly tempered scale]]s.

== Examples ==

=== ~~2-step divisions~~ ===

=== Linear fractal scales ===

{{todo|add examples|inline=1|comment=Create table with a simple example.}}

=== Logarithmic fractal scales ===

A series of [[octave]]-repeating fractal scales can be created using the [[golden ratio]] (here treated as [[logarithmic phi]]) and the octave. Various [[edo]]s approximate this series to a certain degree of precision. The example below uses the first nine terms of the Fibonacci sequence (1, 2, 3, 5, 8, 13, 21, 34, 55) to approximate golden fractal scales in [[55edo]].

{| class="wikitable"

|+ Golden (<math>1:2^{\phi-1}:2</math>) fractal scales, as approximated by 55edo

|+ Golden (<math>1:2^{\phi-1}:2</math>) logarithmic fractal scales, as approximated by 55edo

! Order

! Number of steps

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[[User:R-4981|R-4981]] calls the order-4 <math>1:2^{1/\sqrt{3}}:2</math> fractal scale [[redbull]].

~~=== 3-step divisions ===~~

The initial division may contain more than 2 intervals. Here is a simple example with 3 divisions.

{{todo|add examples|inline=1|comment=Create table with a simple example with a 3-step ratio.}}

=== Inverse linear fractal scales ===

{{todo|add examples|inline=1|comment=Create table with a simple example.}}

@@ Line 1: / Line 1: @@
 {{Wikipedia|Fractal}}
-A '''fractal scale''' is a [[scale]] obtained by dividing a starting [[interval]] in two or more parts, and then by dividing these parts recursively using the same [[frequency ratio]]. In practice, the starting interval can be used as the [[period]] of the scale, allowing to use fractal scales as [[periodic scale]]s.
+A '''fractal scale''' is a [[scale]] obtained by dividing a starting [[interval]] in two or more parts, and then by dividing these parts recursively using the same ratio. In practice, the starting interval is generally used as the [[period]] of the scale, allowing to use fractal scales as [[periodic scale]]s. Fractal scales come in three main types, associated respectively with frequencies, pitch and length: '''linear fractal scales''', defined using a ratio of [[frequency ratio]]s; '''logarithmic fractal scales''', defined using a ratio of [[logarithmic ratio]]s (e.g. [[cent]]s); and '''inverse linear fractal scales''', defined using a ratio of length ratios.
 The ''order'' of a fractal scale is the number of iterations of the division process used to obtain the scale. An order-0 fractal scale contains only the starting interval. An order-1 fractal scale contains the original ratio only once. An order-N scale with M steps in its division contains M<sup>N</sup> steps.
-A fractal scale can be uniquely identified by its order and its ratio (e.g. order-5 2:3:4 fractal scale). [[Interval size measure|Logarithmic ratio]]s expressed in [[cent]]s can be converted to frequency ratios using <math>ratio = 2^{\frac{cents}{1200}}</math>.
+A fractal scale can be uniquely identified by its order, its ratio and its type. For example, the order-5 2:3:4 linear fractal scale is a 32-tone octave-repeating scale
 Fractal scales provides a certain form of symmetry which is very different in nature than that of other scale families, such as [[MOS scale]]s or [[Regular temperament theory|regularly tempered scale]]s.
 == Examples ==
-=== 2-step divisions ===
+=== Linear fractal scales ===
+{{todo|add examples|inline=1|comment=Create table with a simple example.}}
+=== Logarithmic fractal scales ===
 A series of [[octave]]-repeating fractal scales can be created using the [[golden ratio]] (here treated as [[logarithmic phi]]) and the octave. Various [[edo]]s approximate this series to a certain degree of precision. The example below uses the first nine terms of the Fibonacci sequence (1, 2, 3, 5, 8, 13, 21, 34, 55) to approximate golden fractal scales in [[55edo]].
 {| class="wikitable"
-|+ Golden (<math>1:2^{\phi-1}:2</math>) fractal scales, as approximated by 55edo
+|+ Golden (<math>1:2^{\phi-1}:2</math>) logarithmic fractal scales, as approximated by 55edo
 ! Order
 ! Number of steps
@@ Line 55: / Line 58: @@
 [[User:R-4981|R-4981]] calls the order-4 <math>1:2^{1/\sqrt{3}}:2</math> fractal scale [[redbull]].
-=== 3-step divisions ===
 The initial division may contain more than 2 intervals. Here is a simple example with 3 divisions.
+{{todo|add examples|inline=1|comment=Create table with a simple example with a 3-step ratio.}}
+=== Inverse linear fractal scales ===
 {{todo|add examples|inline=1|comment=Create table with a simple example.}}