Muddle: Difference between revisions

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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.

== 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.

The first step s1 of the muddle scale is the sum of the first t1 steps from P, the next step s2 the sum of the next t2 steps after that (after the previous t1 steps), the next step s3 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=

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=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|~~EDO]]s contain more potential MOS scales than smaller ones.

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=

MOS muddles seem to be as old as MOS, although the name "muddle" is new. Page six of [[Erv_Wilson|Erv Wilson]]'s [http://www.anaphoria.com/mos.PDF seminal article on MOS scales] shows a 17-tone MOS subset of [[41edo|41edo]] as "parent scale" and a 7-tone MOS pattern as "target scale shape". They are also present in the work of [[Kraig_Grady|Kraig Grady]], who prefers the term "bi-level" (see [~~http~~://anaphoriasouth.blogspot.com/2011/05/pentatonic-family-pt-1.html this blog entry] and [~~http~~://anaphoriasouth.blogspot.com/2012/01/pentatonic-family-pt2.html this one]). The word "muddle" comes from [[Gene_Ward_Smith|Gene Ward Smith]].

MOS muddles seem to be as old as MOS, although the name "muddle" is new. Page six of [[Erv_Wilson|Erv Wilson]]'s [http://www.anaphoria.com/mos.PDF seminal article on MOS scales] shows a 17-tone MOS subset of [[41edo|41edo]] as "parent scale" and a 7-tone MOS pattern as "target scale shape". They are also present in the work of [[Kraig_Grady|Kraig Grady]], who prefers the term "bi-level" (see [https://anaphoriasouth.blogspot.com/2011/05/pentatonic-family-pt-1.html this blog entry] and [https://anaphoriasouth.blogspot.com/2012/01/pentatonic-family-pt2.html this one]). The word "muddle" comes from [[Gene_Ward_Smith|Gene Ward Smith]].

=Non-MOS Muddles=

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 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.
+== 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.
+The first step s1 of the muddle scale is the sum of the first t1 steps from P, the next step s2 the sum of the next t2 steps after that (after the previous t1 steps), the next step s3 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=
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 =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|EDO]]s contain more potential MOS scales than smaller ones.
+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=
-MOS muddles seem to be as old as MOS, although the name "muddle" is new. Page six of [[Erv_Wilson|Erv Wilson]]'s [http://www.anaphoria.com/mos.PDF seminal article on MOS scales] shows a 17-tone MOS subset of [[41edo|41edo]] as "parent scale" and a 7-tone MOS pattern as "target scale shape". They are also present in the work of [[Kraig_Grady|Kraig Grady]], who prefers the term "bi-level" (see [http://anaphoriasouth.blogspot.com/2011/05/pentatonic-family-pt-1.html this blog entry] and [http://anaphoriasouth.blogspot.com/2012/01/pentatonic-family-pt2.html this one]). The word "muddle" comes from [[Gene_Ward_Smith|Gene Ward Smith]].
+MOS muddles seem to be as old as MOS, although the name "muddle" is new. Page six of [[Erv_Wilson|Erv Wilson]]'s [http://www.anaphoria.com/mos.PDF seminal article on MOS scales] shows a 17-tone MOS subset of [[41edo|41edo]] as "parent scale" and a 7-tone MOS pattern as "target scale shape". They are also present in the work of [[Kraig_Grady|Kraig Grady]], who prefers the term "bi-level" (see [https://anaphoriasouth.blogspot.com/2011/05/pentatonic-family-pt-1.html this blog entry] and [https://anaphoriasouth.blogspot.com/2012/01/pentatonic-family-pt2.html this one]). The word "muddle" comes from [[Gene_Ward_Smith|Gene Ward Smith]].
 =Non-MOS Muddles=