Fractal scale: Difference between revisions
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{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ 1:2:1 logarithmic fractal scales, represented in 64edo | ||
! Order | ! Order | ||
! Number of steps | ! Number of steps | ||
| Line 256: | Line 256: | ||
== Modes == | == Modes == | ||
It is possible for a fractal scale to have various modes. However, unlike [[MOS scales]], the modes of a fractal scale are not ordinary [[rotation]]s of a scale. Instead, modes correspond to the rotations of the ratio used to divide the scale. For example, here are the modes of an order-2 logarithmic fractal scale with scale steps divided into 1:2:3: | It is possible for a fractal scale to have various modes. However, unlike [[MOS scales]], the modes of a fractal scale are not ordinary [[rotation]]s of a scale. Instead, modes correspond to the rotations of the ratio used to divide the scale. This results in ''M'' modes for each order-''N'' fractal scale. For example, here are the modes of an order-2 logarithmic fractal scale with scale steps divided into 1:2:3: | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 265: | Line 262: | ||
! Mode rotational order | ! Mode rotational order | ||
! Division pattern | ! Division pattern | ||
! FRACNAMS | ! [[FRACNAMS]] | ||
! [[Step pattern]] (36edo) | ! [[Step pattern]] (36edo) | ||
! Scale [[degree]]s (36edo) | ! Scale [[degree]]s (36edo) | ||
| Line 271: | Line 268: | ||
| 0 | | 0 | ||
| 1:2:3 | | 1:2:3 | ||
| | | Rock | ||
| 1 2 3 2 4 6 3 6 9 | | 1 2 3 2 4 6 3 6 9 | ||
| 1 3 6 8 12 18 21 27 36 | | 1 3 6 8 12 18 21 27 36 | ||
| Line 277: | Line 274: | ||
| 1 | | 1 | ||
| 2:3:1 | | 2:3:1 | ||
| | | Paper | ||
| 4 6 2 6 9 3 2 3 1 | | 4 6 2 6 9 3 2 3 1 | ||
| 4 10 12 18 27 30 32 35 36 | | 4 10 12 18 27 30 32 35 36 | ||
| Line 283: | Line 280: | ||
| 2 | | 2 | ||
| 3:1:2 | | 3:1:2 | ||
| | | Scissors | ||
| 9 3 6 3 1 2 6 2 4 | | 9 3 6 3 1 2 6 2 4 | ||
| 9 12 18 21 22 24 30 32 36 | | 9 12 18 21 22 24 30 32 36 | ||
|} | |} | ||
Alternatively, it is possible to choose which rotation of the division pattern to use in the next step. This results in ''M''<sup>''N''</sup> modes for an order-''N'' fractal scale, which are as many modes are there are scale steps. | |||
Here is the example for the order-3 golden ratio (φ) fractal scale: | |||
{| class="wikitable" | |||
|+ Modes of 1:φ logarithmic fractal scales in 34edo | |||
! Mode rotational order | |||
! Mode binary numbering | |||
! Division pattern | |||
! [[FRACNAMS]] | |||
! [[Step pattern]] (34edo) | |||
! Scale [[degree]]s (34edo) | |||
|- | |||
| 0,0,0 | |||
| 0 | |||
| 1:φ, 1:φ, 1:φ | |||
| Brazilian | |||
| 8 5 5 3 5 3 3 2 | |||
| 8 13 18 21 26 29 32 34 | |||
|- | |||
| 0,0,1 | |||
| 1 | |||
| 1:φ, 1:φ, φ:1 | |||
| Carolina | |||
| 5 8 3 5 3 5 2 3 | |||
| 5 13 16 21 24 29 31 34 | |||
|- | |||
| 0,1,0 | |||
| 2 | |||
| 1:φ, φ:1, 1:φ | |||
| Georgia | |||
| 5 3 8 5 3 2 5 3 | |||
| 5 8 16 21 24 26 31 34 | |||
|- | |||
| 0,1,1 | |||
| 3 | |||
| 1:φ, φ:1, φ:1 | |||
| California | |||
| 3 5 5 8 2 3 3 5 | |||
| 3 8 13 21 23 26 29 34 | |||
|- | |||
| 1,0,0 | |||
| 4 | |||
| φ:1, 1:φ, 1:φ | |||
| Cariboo | |||
| 5 3 3 2 8 5 5 3 | |||
| 5 8 11 13 21 26 31 34 | |||
|- | |||
| 1,0,1 | |||
| 5 | |||
| φ:1, 1:φ, φ:1 | |||
| Montana | |||
| 3 5 2 3 5 8 3 5 | |||
| 3 8 10 13 18 26 29 34 | |||
|- | |||
| 1,1,0 | |||
| 6 | |||
| φ:1, φ:1, 1:φ | |||
| Bigbend | |||
| 3 2 5 3 5 3 8 5 | |||
| 3 5 10 13 18 21 29 34 | |||
|- | |||
| 1,1,1 | |||
| 7 | |||
| φ:1, φ:1, φ:1 | |||
| Omineca | |||
| 2 3 3 5 3 5 5 8 | |||
| 2 5 8 13 16 21 26 34 | |||
|} | |||
{{todo|improve definition|complete section|inline=1|comment=Add specific and detailed definitions.}} | |||
== Formulas == | == Formulas == | ||