Muddling refers to a process of mapping one 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.
The simplest sort of muddle is a MOS Muddle, which is a sort of second-order MOS Scale and is useful for generating usable subsets of larger MOS scales and for navigating Regular Temperaments. This article will mostly deal with MOS muddles, but it should be remembered that this process can be generalized to other structures.
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, 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 -- 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), 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.
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:
- 22222223 parent with 12122 target gives (2)(22)(2)(22)(23) = 24245
- 22222232 parent with 12122 target gives (2)(22)(2)(22)(32) = 24245
- 22222322 parent with 12122 target gives (2)(22)(2)(23)(22) = 24254
- 22223222 parent with 12122 target gives (2)(22)(2)(32)(22) = 24254
- 22232222 parent with 12122 target gives (2)(22)(3)(22)(22) = 24344
- 22322222 parent with 12122 target gives (2)(23)(2)(22)(22) = 25244
- 23222222 parent with 12122 target gives (2)(32)(2)(22)(22) = 25244
- 32222222 parent with 12122 target gives (3)(22)(2)(22)(22) = 34244
Notice that not all of these rotations produce unique muddles. The unique muddles are 24245, 24254, 24344, 25244, and 34244.
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. 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 EDOs 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's seminal article on MOS scales shows a 17-tone MOS subset of 41edo as "parent scale" and a 7-tone MOS pattern as "target scale shape". They are also present in the work of Kraig Grady, who prefers the term "bi-level" (see this blog entry and this one). The word "muddle" comes from 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, MOS Cradle scales, other muddles, scales of temperaments with rank higher than 2, 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).
As one example of a muddle with a non-MOS parent, 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.