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 frequency ratio. In practice, the starting interval can be used as the period of the scale, allowing to use fractal scales as periodic scales.
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 and its ratio (e.g. order-5 2:3:4 fractal scale). Logarithmic ratios expressed in cents can be converted to frequency ratios using [math]\displaystyle{ ratio = 2^{\frac{cents}{1200}} }[/math].
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
2-step divisions
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.
3-step divisions
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. |
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. |