Fractal scale
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 scales. Fractal scales come in three main types, associated respectively with frequencies, pitch and length: linear fractal scales, defined using a ratio of frequency ratios; logarithmic fractal scales, defined using a ratio of logarithmic ratios (e.g. cents); 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 MN steps.
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 scales or regularly tempered scales.
Examples
Linear fractal scales
| Order | Number of steps | Chord |
|---|---|---|
| 0 | 1 | 1:2 |
| 1 | 2 | 2:3:4 (2afdo) |
| 2 | 4 | 4:5:6:7:8 (4afdo) |
| 3 | 8 | 8:9:10:11:12:13:14:15:16 (8afdo) |
| Order | Number of steps | Chord |
|---|---|---|
| 0 | 1 | 1:2 |
| 1 | 2 | 3:4:6 |
| 2 | 4 | 9:10:12:14:18 |
| 3 | 8 | 27:28:30:32:36:38:42:46:54 |
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 edos 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.
| Order | Number of steps | Step visualization | Step pattern (55edo) | Scale degrees (55edo) |
|---|---|---|---|---|
| 0 | 1 | ├──────────────────────────────────────────────────────┤ | 55 | 55 |
| 1 | 2 | ├─────────────────────────────────┼────────────────────┤ | 34 21 | 34 55 |
| 2 | 4 | ├────────────────────┼────────────┼────────────┼───────┤ | 21 13 13 8 | 21 34 47 55 |
| 3 | 8 | ├────────────┼───────┼───────┼────┼───────┼────┼────┼──┤ | 13 8 8 5 8 5 5 3 | 13 21 29 34 42 47 52 55 |
| 4 | 16 | ├───────┼────┼────┼──┼────┼──┼──┼─┼────┼──┼──┼─┼──┼─┼─┼┤ | 8 5 5 3 5 3 3 2 5 3 3 2 3 2 2 1 | 8 13 18 21 26 29 32 34 39 42 45 47 50 52 54 55 |
The fractal scale of [math]\displaystyle{ 1:\sqrt{2}:2 }[/math] is accurately consistent with edo to the power of 2 (e.g. 16edo).
R-4981 calls the order-4 [math]\displaystyle{ 1:2^{1/\sqrt{3}}:2 }[/math] fractal scale redbull.
The initial division may contain more than 2 intervals. Here is a simple example with 3 divisions.
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Todo: add examples Create table with a simple example with a 3-step ratio. |
Inverse linear fractal scales
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Todo: add examples Create table with a simple example. |
Formulas
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Todo: expand Add formulas to calculate the steps and degrees of a given fractal scale, provided the order and the ratio. |