Muddle: Difference between revisions

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.

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.

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

==Examples==

===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:

===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.

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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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.

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]].

As mentioned at the beginning, in a muddle, the parent scale can be any kind of periodic scale at all, including but not limited to [[MODMOS_Scales|MODMOS scales]], [[MOS_Cradle|MOS Cradle scales]], other muddles, scales of [[Regular_Temperaments|temperaments]] with rank higher than 2, [[Just_intonation|Just Intonation]] scales, etc. The target scale is a little less flexible, but it could be at least a MODMOS scale, a MOS Cradle scale or any other MOS subset or subset of a periodic equal-step scale. It's just important that the total number of "units" in the target scale (eg. the MOS Cradle 23132 has 2+3+1+3+2=11 units) be the same as the number of tones in the parent scale (thus the 23132 target scale must be applied to a scale of 11 tones).

<span style="">As one example of a muddle with a non-MOS parent</span><span style="">, if you take overtones 16-32 as the parent scale, you can apply a 2322232 target scale and get 1/1, 9/8, 21/16, 23/16, 25/16, 27/16, 15/8, 2/1.</span>