Muddle: Difference between revisions

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''Muddling'' refers to a process of mapping one [[Periodic_scale|periodic scale]] onto another one; the scale thus produced is a ''muddle''. By periodic scale is meant a scale that repeats "at the something" (octave, tritave, something else). There are two necessary components: a '''parent scale''' and a '''target shape''', or simply "parent" and "target". The parent scale can be any periodic scale at all; the target scale is not exactly a scale -- it's the outline or ''shape'' of a scale -- and it must be defined in terms of units or degrees which comprise the steps. If the period is an octave, this means the target scale will be a subset of an [[EDO|EDO]].

The simplest sort of muddle is a MOS Muddle, which is a sort of second-order [[MOSScales|MOS Scale]] and is useful for generating usable subsets of larger MOS scales and for navigating [[Regular_Temperaments|Regular Temperaments]]. This article will mostly deal with MOS muddles, but it should be remembered that this process can be generalized to ~~other structures~~.

The simplest sort of muddle is a MOS Muddle, which is a sort of second-order [[MOSScales|MOS Scale]] and is useful for generating usable subsets of larger MOS scales and for navigating [[Regular_Temperaments|Regular Temperaments]]. This article will mostly deal with MOS muddles, but it should be remembered that this process can be generalized to arbitrary sequences of steps.

== Construction ==

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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~~==

===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~~, the parent~~ scale ~~is any MOS~~ scale ~~large enough~~ that ~~taking subsets~~ of ~~it would be musically useful. The~~ target scale is ~~something like 12122. This could be~~ the ~~form~~ of an ~~actual scale~~ (~~an MOS subset of [[8edo|8edo]],~~ in ~~this~~ example), ~~but as~~ a ~~target~~ scale~~, we are interested in its general ''shape'', which suggests~~ a ~~way of bunching intervals~~ of the parent scale. If ~~we apply~~ the target ~~to an equal-step scale~~, ~~we arrive at a standard MOS. Eg~~. If ~~our~~ parent scale is ~~[[8edo|8edo]] -- with steps 11111111 -- and our target scale is 12122~~, then the ~~resulting~~ scale is ~~(1)(11)(1)(11)(11) = 12122 --~~ the ~~same as our~~ target scale. ~~But if~~ our parent scale ~~is some other MOS scale, say 22222223 (a subset~~ of ~~[[17edo|17edo]]),~~ applying ~~the 12122~~ target scale ~~generates~~ (2)(22)~~(2)(22)(23) = 24245 -- not an MOS scale~~. ~~The latter scale, which we can call~~ a ~~''muddle''~~, ~~has some melodic similarity to~~ the target ~~scale of 12122, but belongs to a different temperament family entirely. Choosing a different mode (rotation) of either the parent scale or~~ the ~~target scale may produce a different muddle~~.

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.

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.

{| class="wikitable"

|+

!Scale

! colspan="17" |Step pattern

|-

!17edo

|1

|-

!Parent scale (22222223)

| colspan="2" |2

| colspan="3" |3

|-

!Muddle scale (24245)

| colspan="2" |2

| colspan="4" |4

| colspan="2" |2

| colspan="4" |4

| colspan="5" |5

|}

Starting with a different mode (or rotation) of either the parent or target scale results in a different muddle altogether; see the examples below.

=== 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.

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

==Examples==

===1.===

===Running example===

To continue with our example of a parent scale of 22222223 and a target shape of 12122, here are all the muddles that can result from different rotations of the parent scale:

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<ul><li>22222223 parent with 12122 target gives (2)(22)(2)(22)(23) = 24245</li><li>22222232 parent with 12122 target gives (2)(22)(2)(22)(32) = 24245</li><li>22222322 parent with 12122 target gives (2)(22)(2)(23)(22) = 24254</li><li>22223222 parent with 12122 target gives (2)(22)(2)(32)(22) = 24254</li><li>22232222 parent with 12122 target gives (2)(22)(3)(22)(22) = 24344</li><li>22322222 parent with 12122 target gives (2)(23)(2)(22)(22) = 25244</li><li>23222222 parent with 12122 target gives (2)(32)(2)(22)(22) = 25244</li><li>32222222 parent with 12122 target gives (3)(22)(2)(22)(22) = 34244</li></ul>Notice that not all of these rotations produce unique muddles. The unique muddles are 24245, 24254, 24344, 25244, and 34244.

===2.===

===Another example===

Here is a diagram showing the muddles available with a 55755757 parent scale ([[Sensi|Sensi]][8] in [[46edo|46edo]]) and a 12122 target scale. Note that this combination produces MOS scales as well as muddles.

[[File:sensi_pentatonics.png|alt=sensi_pentatonics.png|sensi_pentatonics.png]]

=== 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).

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.

{| class="wikitable"

!Scale

! colspan="12" |Step pattern

!Comments

|-

!12edo

|1

|Original edo from which the parent scale comes from

|-

!Parent scale (2212221)

| colspan="2" |2

|1

| colspan="2" |2

|1

|Parent scale

|-

!Muddle scale (22323)

| colspan="2" |2

| colspan="3" |3

| colspan="2" |2

| colspan="3" |3

|Outcome of using 11212 as a target (result is another mos)

|-

!Muddle scale (23241)

| colspan="2" |2

| colspan="3" |3

| colspan="2" |2

| colspan="4" |4

|1

|Outcome of using 12121 as a target (result is not a mos)

|}

==Comments==

~~MOS Muddles~~ always have more than two sizes of step -- either three or four sizes. Whereas MOS scales have two varieties of interval for each interval class (eg. a "large step" and a "small step"), muddles have potentially two varieties within each variety (eg. two sizes of "small step" and two sizes of "large step"). Parent scales that are close to equal (eg. [[Maximal_evenness|maximally even]] scales) will produce muddles that are closer in sound to the target scale. Larger parent scales contain more potential muddles than smaller ones, just as larger [[EDO]]s contain more potential MOS scales than smaller ones.

With the exception of trivial muddles, mos muddles always have more than two sizes of step -- either three or four sizes. Whereas MOS scales have two varieties of interval for each interval class (eg. a "large step" and a "small step"), muddles have potentially two varieties within each variety (eg. two sizes of "small step" and two sizes of "large step"). Parent scales that are close to equal (eg. [[Maximal_evenness|maximally even]] scales) will produce muddles that are closer in sound to the target scale. Larger parent scales contain more potential muddles than smaller ones, just as larger [[EDO]]s contain more potential MOS scales than smaller ones.

==History==

@@ Line 1: / Line 1: @@
 ''Muddling'' refers to a process of mapping one [[Periodic_scale|periodic scale]] onto another one; the scale thus produced is a ''muddle''. By periodic scale is meant a scale that repeats "at the something" (octave, tritave, something else). There are two necessary components: a '''parent scale''' and a '''target shape''', or simply "parent" and "target". The parent scale can be any periodic scale at all; the target scale is not exactly a scale -- it's the outline or ''shape'' of a scale -- and it must be defined in terms of units or degrees which comprise the steps. If the period is an octave, this means the target scale will be a subset of an [[EDO|EDO]].
-The simplest sort of muddle is a MOS Muddle, which is a sort of second-order [[MOSScales|MOS Scale]] and is useful for generating usable subsets of larger MOS scales and for navigating [[Regular_Temperaments|Regular Temperaments]]. This article will mostly deal with MOS muddles, but it should be remembered that this process can be generalized to other structures.
+The simplest sort of muddle is a MOS Muddle, which is a sort of second-order [[MOSScales|MOS Scale]] and is useful for generating usable subsets of larger MOS scales and for navigating [[Regular_Temperaments|Regular Temperaments]]. This article will mostly deal with MOS muddles, but it should be remembered that this process can be generalized to arbitrary sequences of steps.
 == Construction ==
@@ Line 8: / Line 8: @@
 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==
+===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, the parent scale is any MOS scale large enough that taking subsets of it would be musically useful. The target scale is something like 12122. This could be the form of an actual scale (an MOS subset of [[8edo|8edo]], in this example), but as a target scale, we are interested in its general ''shape'', which suggests a way of bunching intervals of the parent scale. If we apply the target to an equal-step scale, we arrive at a standard MOS. Eg. If our parent scale is [[8edo|8edo]] -- with steps 11111111 -- and our target scale is 12122, then the resulting scale is (1)(11)(1)(11)(11) = 12122 -- the same as our target scale. But if our parent scale is some other MOS scale, say 22222223 (a subset of [[17edo|17edo]]), applying the 12122 target scale generates (2)(22)(2)(22)(23) = 24245 -- not an MOS scale. The latter scale, which we can call a ''muddle'', has some melodic similarity to the target scale of 12122, but belongs to a different temperament family entirely. Choosing a different mode (rotation) of either the parent scale or the target scale may produce a different muddle.
+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.
+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.
+{| class="wikitable"
+|+
+!Scale
+! colspan="17" |Step pattern
+|-
+!17edo
+|1
+|1
+|1
+|1
+|1
+|1
+|1
+|1
+|1
+|1
+|1
+|1
+|1
+|1
+|1
+|1
+|1
+|-
+!Parent scale (22222223)
+| colspan="2" |2
+| colspan="2" |2
+| colspan="2" |2
+| colspan="2" |2
+| colspan="2" |2
+| colspan="2" |2
+| colspan="2" |2
+| colspan="3" |3
+|-
+!Muddle scale (24245)
+| colspan="2" |2
+| colspan="4" |4
+| colspan="2" |2
+| colspan="4" |4
+| colspan="5" |5
+|}
+Starting with a different mode (or rotation) of either the parent or target scale results in a different muddle altogether; see the examples below.
+=== 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.
+It is also possible muddling to result in a mos, even if both the parent and target scales are mosses; see the examples below.
 ==Examples==
-===1.===
+===Running example===
 To continue with our example of a parent scale of 22222223 and a target shape of 12122, here are all the muddles that can result from different rotations of the parent scale:
@@ Line 20: / Line 70: @@
 <ul><li>22222223 parent with 12122 target gives (2)(22)(2)(22)(23) = 24245</li><li>22222232 parent with 12122 target gives (2)(22)(2)(22)(32) = 24245</li><li>22222322 parent with 12122 target gives (2)(22)(2)(23)(22) = 24254</li><li>22223222 parent with 12122 target gives (2)(22)(2)(32)(22) = 24254</li><li>22232222 parent with 12122 target gives (2)(22)(3)(22)(22) = 24344</li><li>22322222 parent with 12122 target gives (2)(23)(2)(22)(22) = 25244</li><li>23222222 parent with 12122 target gives (2)(32)(2)(22)(22) = 25244</li><li>32222222 parent with 12122 target gives (3)(22)(2)(22)(22) = 34244</li></ul>Notice that not all of these rotations produce unique muddles. The unique muddles are 24245, 24254, 24344, 25244, and 34244.
-===2.===
+===Another example===
 Here is a diagram showing the muddles available with a 55755757 parent scale ([[Sensi|Sensi]][8] in [[46edo|46edo]]) and a 12122 target scale. Note that this combination produces MOS scales as well as muddles.
 [[File:sensi_pentatonics.png|alt=sensi_pentatonics.png|sensi_pentatonics.png]]
+=== 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).
+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.
+{| class="wikitable"
+!Scale
+! colspan="12" |Step pattern
+!Comments
+|-
+!12edo
+|1
+|1
+|1
+|1
+|1
+|1
+|1
+|1
+|1
+|1
+|1
+|1
+|Original edo from which the parent scale comes from
+|-
+!Parent scale (2212221)
+| colspan="2" |2
+| colspan="2" |2
+|1
+| colspan="2" |2
+| colspan="2" |2
+| colspan="2" |2
+|1
+|Parent scale
+|-
+!Muddle scale (22323)
+| colspan="2" |2
+| colspan="2" |2
+| colspan="3" |3
+| colspan="2" |2
+| colspan="3" |3
+|Outcome of using 11212 as a target (result is another mos)
+|-
+!Muddle scale (23241)
+| colspan="2" |2
+| colspan="3" |3
+| colspan="2" |2
+| colspan="4" |4
+|1
+|Outcome of using 12121 as a target (result is not a mos)
+|}
 ==Comments==
-MOS Muddles always have more than two sizes of step -- either three or four sizes. Whereas MOS scales have two varieties of interval for each interval class (eg. a "large step" and a "small step"), muddles have potentially two varieties within each variety (eg. two sizes of "small step" and two sizes of "large step"). Parent scales that are close to equal (eg. [[Maximal_evenness|maximally even]] scales) will produce muddles that are closer in sound to the target scale. Larger parent scales contain more potential muddles than smaller ones, just as larger [[EDO]]s contain more potential MOS scales than smaller ones.
+With the exception of trivial muddles, mos muddles always have more than two sizes of step -- either three or four sizes. Whereas MOS scales have two varieties of interval for each interval class (eg. a "large step" and a "small step"), muddles have potentially two varieties within each variety (eg. two sizes of "small step" and two sizes of "large step"). Parent scales that are close to equal (eg. [[Maximal_evenness|maximally even]] scales) will produce muddles that are closer in sound to the target scale. Larger parent scales contain more potential muddles than smaller ones, just as larger [[EDO]]s contain more potential MOS scales than smaller ones.
 ==History==