Muddle: Difference between revisions

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== Construction ==

Let the target ~~scale~~ T be a sequence of steps [ t1, t2, t3, ... , tm ], the parent scale P be a sequence of steps [ p1, p2, p3, ... , pn ], and the resulting muddle scale S be a sequence of steps [ s1, s2, s3, ... , sm ]. Note that the number of steps in P must be equal to the sum of all ti from T. Also note that both ti and pi are both numeric values, as with si.

Let the target shape T be a sequence of steps [ t1, t2, t3, ... , tm ], the parent scale P be a sequence of steps [ p1, p2, p3, ... , pn ], and the resulting muddle scale S be a sequence of steps [ s1, s2, s3, ... , sm ]. Note that the number of steps in P must be equal to the sum of all ti from T. Also note that both ti and pi are both numeric values, as with si.

The first step s1 of the muddle scale is the sum of the first t1 steps from P, the next step s2 is the sum of the next t2 steps after that (after the previous t1 steps), the next step s3 is the sum of the next t3 steps after that (after the previous t1+t2 steps), and so on, where the last step sm is the sum of the last tm steps from P. For example, if s1 is made from the first 3 steps of P (p1, p2, and p3), then the next step p2 is the sum of the next t2 steps after p3, meaning the sum starts at (and includes) p4.

===MOS muddles===

In the case of a mos muddle, both the target and parent ~~scale~~ describe some sort of mos. ~~Ideally~~, the parent scale is large enough from which a subset of its notes is musically useful. As a running example, let's use 22222223 (1L 7s) as a parent scale and 12122 (3L 2s) as a target ~~scale~~.

In the case of a mos muddle, both the target and parent describe some sort of mos. Typically, the parent scale is large enough from which a subset of its notes is musically useful, whereas the target describes a grouping of steps from the parent rather than an edo. As a running example, let's use 22222223 (1L 7s, step ratio 3:2) as a parent scale and 12122 (3L 2s) as a target shape.

Applying the construction rules to our example results in the parent's steps being grouped as such: (2)(22)(2)(22)(23). This results in a muddle scale with a step pattern of 24245 -- not a mos since it has more than two step sizes. This new scale has melodic properties similar to that of the target ~~scale~~, but has other properties that fall outside the target ~~scale~~; when viewed under regular temperament theory, for example, both the target and muddle fall under different temperament families.

Applying the construction rules to our example results in the parent's steps being grouped as such: (2)(22)(2)(22)(23). This results in a muddle scale with a step pattern of 24245 -- not a mos since it has more than two step sizes. This new scale has melodic properties similar to that of the target, but has other properties that fall outside the target; when viewed under regular temperament theory, for example, both the target and muddle fall under different temperament families.

Another way to conceptualize a muddle is to consider that the parent scale already describes a subset of an edo (17edo in our example), and the target describes finding a subset of that subset.

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=== Trivial muddles ===

If every step size in the parent scale is the same size k, then the muddle scale is similar to the target, only with k times as many divisions. If every step size in the parent scale is 1, then the muddle scale is identical to the target ~~scale~~. Using our running example, this is like having a parent scale of 22222222 or 11111111; applying our target ~~scale~~ on either results in 24244 (the same as the target but with twice as many divisions) and 12122 (the same as the target) respectively.

If every step size in the parent scale is the same size k, then the muddle scale is similar to the target, only with k times as many divisions. If every step size in the parent scale is 1, then the muddle scale is identical to the target. Using our running example, this is like having a parent scale of 22222222 or 11111111; applying our target on either results in 24244 (the same as the target but with twice as many divisions) and 12122 (the same as the target) respectively.

It is also possible for muddling to result in a mos, even if both the parent and target ~~scales~~ are mosses; see the examples below.

It is also possible for muddling to result in a mos, even if both the parent and target are mosses; see the examples below.

==Examples==

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=== A muddle that's itself a mos ===

Consider a parent ~~scale~~ of 2212221 (or 12edo diatonic) and a target ~~scale~~ of 11212 (or 2L 3s). Given the construction rules previously described, the resulting mos muddle has a step pattern of (2)(2)(12)(2)(2)(21), or 22323, which itself another mos (~~specifically~~ the same as the target but with different step sizes).

Consider a parent of 2212221 (or 12edo diatonic) and a target of 11212 (or 2L 3s). Given the construction rules previously described, the resulting mos muddle has a step pattern of (2)(2)(12)(2)(2)(21), or 22323, which itself another mos (the same as the target but with different step sizes).

Rotating either scale can result in a muddle scale that is not a mos, in that it has more than two step sizes. For example, rotating the target to 12121 instead results in a mos muddle with a step pattern of 23241.

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 == Construction ==
-Let the target scale T be a sequence of steps [ t1, t2, t3, ... , tm ], the parent scale P be a sequence of steps [ p1, p2, p3, ... , pn ], and the resulting muddle scale S be a sequence of steps [ s1, s2, s3, ... , sm ]. Note that the number of steps in P must be equal to the sum of all ti from T. Also note that both ti and pi are both numeric values, as with si.
+Let the target shape T be a sequence of steps [ t1, t2, t3, ... , tm ], the parent scale P be a sequence of steps [ p1, p2, p3, ... , pn ], and the resulting muddle scale S be a sequence of steps [ s1, s2, s3, ... , sm ]. Note that the number of steps in P must be equal to the sum of all ti from T. Also note that both ti and pi are both numeric values, as with si.
 The first step s1 of the muddle scale is the sum of the first t1 steps from P, the next step s2 is the sum of the next t2 steps after that (after the previous t1 steps), the next step s3 is the sum of the next t3 steps after that (after the previous t1+t2 steps), and so on, where the last step sm is the sum of the last tm steps from P. For example, if s1 is made from the first 3 steps of P (p1, p2, and p3), then the next step p2 is the sum of the next t2 steps after p3, meaning the sum starts at (and includes) p4.
 ===MOS muddles===
-In the case of a mos muddle, both the target and parent scale describe some sort of mos. Ideally, the parent scale is large enough from which a subset of its notes is musically useful. As a running example, let's use 22222223 (1L 7s) as a parent scale and 12122 (3L 2s) as a target scale.
+In the case of a mos muddle, both the target and parent describe some sort of mos. Typically, the parent scale is large enough from which a subset of its notes is musically useful, whereas the target describes a grouping of steps from the parent rather than an edo. As a running example, let's use 22222223 (1L 7s, step ratio 3:2) as a parent scale and 12122 (3L 2s) as a target shape.
-Applying the construction rules to our example results in the parent's steps being grouped as such: (2)(22)(2)(22)(23). This results in a muddle scale with a step pattern of 24245 -- not a mos since it has more than two step sizes. This new scale has melodic properties similar to that of the target scale, but has other properties that fall outside the target scale; when viewed under regular temperament theory, for example, both the target and muddle fall under different temperament families.
+Applying the construction rules to our example results in the parent's steps being grouped as such: (2)(22)(2)(22)(23). This results in a muddle scale with a step pattern of 24245 -- not a mos since it has more than two step sizes. This new scale has melodic properties similar to that of the target, but has other properties that fall outside the target; when viewed under regular temperament theory, for example, both the target and muddle fall under different temperament families.
 Another way to conceptualize a muddle is to consider that the parent scale already describes a subset of an edo (17edo in our example), and the target describes finding a subset of that subset.
@@ Line 58: / Line 58: @@
 === Trivial muddles ===
-If every step size in the parent scale is the same size k, then the muddle scale is similar to the target, only with k times as many divisions. If every step size in the parent scale is 1, then the muddle scale is identical to the target scale. Using our running example, this is  like having a parent scale of 22222222 or 11111111; applying our target scale on either results in 24244 (the same as the target but with twice as many divisions) and 12122 (the same as the target) respectively.
+If every step size in the parent scale is the same size k, then the muddle scale is similar to the target, only with k times as many divisions. If every step size in the parent scale is 1, then the muddle scale is identical to the target. Using our running example, this is like having a parent scale of 22222222 or 11111111; applying our target on either results in 24244 (the same as the target but with twice as many divisions) and 12122 (the same as the target) respectively.
-It is also possible for muddling to result in a mos, even if both the parent and target scales are mosses; see the examples below.
+It is also possible for muddling to result in a mos, even if both the parent and target are mosses; see the examples below.
 ==Examples==
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 === A muddle that's itself a mos ===
-Consider a parent scale of 2212221 (or 12edo diatonic) and a target scale of 11212 (or 2L 3s). Given the construction rules previously described, the resulting mos muddle has a step pattern of (2)(2)(12)(2)(2)(21), or 22323, which itself another mos (specifically the same as the target but with different step sizes).
+Consider a parent of 2212221 (or 12edo diatonic) and a target of 11212 (or 2L 3s). Given the construction rules previously described, the resulting mos muddle has a step pattern of (2)(2)(12)(2)(2)(21), or 22323, which itself another mos (the same as the target but with different step sizes).
 Rotating either scale can result in a muddle scale that is not a mos, in that it has more than two step sizes. For example, rotating the target to 12121 instead results in a mos muddle with a step pattern of 23241.