Fractal scale: Difference between revisions

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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, its ratio and its type. For example, the order-5 2~~:3:4~~ linear fractal scale is a 32-tone octave-repeating scale

A fractal scale can be uniquely identified by its order, its ratio and its type. For example, the order-5 1:2 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.

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=== Linear fractal scales ===

{| class="wikitable"

|+2:~~3:4~~ linear fractal scales

|+1:1 linear fractal scales

|-

! Order

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{| class="wikitable"

|+3:~~4:6~~ linear fractal scales

|+1:2 linear fractal scales

|-

! Order

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{| class="wikitable"

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

|+ Golden (1:φ) logarithmic fractal scales, as approximated by 55edo

! Order

! Number of steps

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|}

~~The fractal scale of <math>1:\sqrt{2}:2</math> is accurately consistent with edo to the power of 2 (e.g. [[16edo]]).~~

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

[[User:R-4981|R-4981]] calls the order-4 <math>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.

{| class="wikitable"

|+ ~~<math>~~1:2^{1~~/4}:2^{3/4}:2</math>~~ logarithmic fractal scales, represented in 64edo

|+ 1:2:1 logarithmic fractal scales, represented in 64edo

! Order

! Number of steps

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=== Truncated fractal scales ===

Since the division is applied to every interval in each step, some intervals will become very small compared to others. For example, if we divide the octave logarithmically by the ratio <math>1:2^a:2</math> with <math>1/2 < a ~~< 1~~</math> (like the golden logarithmic fractal scale above), then after ''N'' steps, the largest scale step will have size <math>2^{aN}</math> and the smallest scale step will have size <math>2^{(1-a)N}</math>. The ratio between the two is <math>2^{(2a-1)N} > 2^N</math> which grows exponentially as ''N'' grows linearly, so the scale will have very uneven steps.

Since the division is applied to every interval in each step, some intervals will become very small compared to others. For example, if we divide the octave logarithmically by the ratio <math>1:a</math> with <math>1 < a</math> (like the golden logarithmic fractal scale above), then after ''N'' steps, the largest scale step will have size <math>2^{aN}</math> and the smallest scale step will have size <math>2^{(1-a)N}</math>. The ratio between the two is <math>2^{(2a-1)N} > 2^N</math> which grows exponentially as ''N'' grows linearly, so the scale will have very uneven steps.

If we wish to make the scale steps more even, then we can choose some smallest '''threshold interval''' ''ε'' in the linear case (or 1+''ε'' in the logarithmic case). Here we consider the linear case. On each step, we divide an interval if it is larger than ''ε'', otherwise we leave it untouched if it is smaller than ''ε''. Since the divided intervals get smaller and smaller, we will eventually reach a point where the intervals become smaller than ''ε'', so this process will terminate after a finite amount of steps (and hence the scale is also finite). Here is an example:

{| class="wikitable"

|+3:~~4:6~~ truncated linear fractal scales, threshold ''ε'' = 1/16

|+1:2 truncated linear fractal scales, threshold ''ε'' = 1/16

!Order

!Number of steps

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|}

Alternatively, we can do '''pre-stopping''' -- we stop dividing the interval if any of the ''divided parts'' of the interval would be smaller than the threshold. ~~This has~~ the ~~advantage that no interval in any~~ of the ~~scales will be smaller than the threshold~~. Here is another example:

Alternatively, we can do '''pre-stopping''' -- we stop dividing the interval if any of the ''divided parts'' of the interval would be smaller than the threshold. The threshold now becomes a ''true lower bound'' for the size of the scale steps. Here is another example:

{| class="wikitable"

|+ Pre-stopped 3:~~4:6~~ truncated linear fractal scales, threshold ''ε'' = 1/16

|+ Pre-stopped 1:2 truncated linear fractal scales, threshold ''ε'' = 1/16

!Order

!Number of steps

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|none

|}

== 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. 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"

|+ Modes of 1:2:3 logarithmic fractal scales in 36edo

! Mode rotational order

! Division pattern

! [[FRACNAMS]]

! [[Step pattern]] (36edo)

! Scale [[degree]]s (36edo)

|-

| 0

| 1:2:3

| Rock

| 1 2 3 2 4 6 3 6 9

| 1 3 6 8 12 18 21 27 36

|-

| 1

| 2:3:1

| Paper

| 4 6 2 6 9 3 2 3 1

| 4 10 12 18 27 30 32 35 36

|-

| 2

| 3:1:2

| Scissors

| 9 3 6 3 1 2 6 2 4

| 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 ==

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ratio = [1, 2]

# the "shape" of the ratio, only input integers please.

# [a, b] corresponds to the ratio 1:~~2^{a/(a+~~b~~)}:2~~,

# [a, b] corresponds to the ratio a:b,

# [a, b, c] corresponds to the ratio 1:~~2^{a/(a+~~b~~+c)}~~:~~2^{(a+b)/(a+b+~~c~~)}:2~~, etc.

# [a, b, c] corresponds to the ratio a:b:c, etc.

c = ratio

@@ Line 4: / Line 4: @@
 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, its ratio and its type. For example, the order-5 2:3:4 linear fractal scale is a 32-tone octave-repeating scale
+A fractal scale can be uniquely identified by its order, its ratio and its type. For example, the order-5 1:2 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.
@@ Line 11: / Line 11: @@
 === Linear fractal scales ===
 {| class="wikitable"
-|+2:3:4 linear fractal scales
+|+1:1 linear fractal scales
 |-
 ! Order
@@ Line 35: / Line 35: @@
 {| class="wikitable"
-|+3:4:6 linear fractal scales
+|+1:2 linear fractal scales
 |-
 ! Order
@@ Line 63: / Line 63: @@
 {| class="wikitable"
-|+ Golden (<math>1:2^{\phi-1}:2</math>) logarithmic fractal scales, as approximated by 55edo
+|+ Golden (1:φ) logarithmic fractal scales, as approximated by 55edo
 ! Order
 ! Number of steps
@@ Line 101: / Line 101: @@
 |}
-The fractal scale of <math>1:\sqrt{2}:2</math> is accurately consistent with edo to the power of 2 (e.g. [[16edo]]).
+[[User:R-4981|R-4981]] calls the order-4 <math>1:\frac{1-\sqrt{3}}{\sqrt{3}}</math> fractal scale [[redbull]].
-[[User:R-4981|R-4981]] calls the order-4 <math>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.
 {| class="wikitable"
-|+ <math>1:2^{1/4}:2^{3/4}:2</math> logarithmic fractal scales, represented in 64edo
+|+ 1:2:1 logarithmic fractal scales, represented in 64edo
 ! Order
 ! Number of steps
@@ Line 144: / Line 142: @@
 === Truncated fractal scales ===
-Since the division is applied to every interval in each step, some intervals will become very small compared to others. For example, if we divide the octave logarithmically by the ratio <math>1:2^a:2</math> with <math>1/2 < a < 1</math> (like the golden logarithmic fractal scale above), then after ''N'' steps, the largest scale step will have size <math>2^{aN}</math> and the smallest scale step will have size <math>2^{(1-a)N}</math>. The ratio between the two is <math>2^{(2a-1)N} > 2^N</math> which grows exponentially as ''N'' grows linearly, so the scale will have very uneven steps.
+Since the division is applied to every interval in each step, some intervals will become very small compared to others. For example, if we divide the octave logarithmically by the ratio <math>1:a</math> with <math>1 < a</math> (like the golden logarithmic fractal scale above), then after ''N'' steps, the largest scale step will have size <math>2^{aN}</math> and the smallest scale step will have size <math>2^{(1-a)N}</math>. The ratio between the two is <math>2^{(2a-1)N} > 2^N</math> which grows exponentially as ''N'' grows linearly, so the scale will have very uneven steps.
 If we wish to make the scale steps more even, then we can choose some smallest '''threshold interval''' ''ε'' in the linear case (or 1+''ε'' in the logarithmic case). Here we consider the linear case. On each step, we divide an interval if it is larger than ''ε'', otherwise we leave it untouched if it is smaller than ''ε''. Since the divided intervals get smaller and smaller, we will eventually reach a point where the intervals become smaller than ''ε'', so this process will terminate after a finite amount of steps (and hence the scale is also finite). Here is an example:
 {| class="wikitable"
-|+3:4:6 truncated linear fractal scales, threshold ''ε'' = 1/16
+|+1:2 truncated linear fractal scales, threshold ''ε'' = 1/16
 !Order
 !Number of steps
@@ Line 208: / Line 206: @@
 |}
-Alternatively, we can do '''pre-stopping''' -- we stop dividing the interval if any of the ''divided parts'' of the interval would be smaller than the threshold. This has the advantage that no interval in any of the scales will be smaller than the threshold. Here is another example:
+Alternatively, we can do '''pre-stopping''' -- we stop dividing the interval if any of the ''divided parts'' of the interval would be smaller than the threshold. The threshold now becomes a ''true lower bound'' for the size of the scale steps. Here is another example:
 {| class="wikitable"
-|+ Pre-stopped 3:4:6 truncated linear fractal scales, threshold ''ε'' = 1/16
+|+ Pre-stopped 1:2 truncated linear fractal scales, threshold ''ε'' = 1/16
 !Order
 !Number of steps
@@ Line 256: / Line 254: @@
 |none
 |}
+== 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. 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"
+|+ Modes of 1:2:3 logarithmic fractal scales in 36edo
+! Mode rotational order
+! Division pattern
+! [[FRACNAMS]]
+! [[Step pattern]] (36edo)
+! Scale [[degree]]s (36edo)
+|-
+| 0
+| 1:2:3
+| Rock
+| 1 2 3 2 4 6 3 6 9
+| 1 3 6 8 12 18 21 27 36
+|-
+| 1
+| 2:3:1
+| Paper
+| 4 6 2 6 9 3 2 3 1
+| 4 10 12 18 27 30 32 35 36
+|-
+| 2
+| 3:1:2
+| Scissors
+| 9 3 6 3 1 2 6 2 4
+| 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 ==
@@ Line 266: / Line 366: @@
 ratio = [1, 2]
 # the "shape" of the ratio, only input integers please.
-# [a, b] corresponds to the ratio 1:2^{a/(a+b)}:2,
+# [a, b] corresponds to the ratio a:b,
-# [a, b, c] corresponds to the ratio 1:2^{a/(a+b+c)}:2^{(a+b)/(a+b+c)}:2, etc.
+# [a, b, c] corresponds to the ratio a:b:c, etc.
 c = ratio