Munit: Difference between revisions

The purpose of munits is that they form fragments of musical scales which can be combined to form many different scales, generalizing what people sometimes do with tetrachords. That is, while an entire scale can technically be thought of as a munit with the 2/1 (or whatever octave of equivalence) as the framing interval, the intended use is that the framing interval is smaller than the equivalence interval, so that several munits may fit into one scale. For instance, the "pentachords" that Paul Erlich studies in his 22-EDO pentachordal scales are also munits, so that "munit" is something of a catch-all term generalizing both tetrachords, pentachords, etc. But also, there is no requirement that munits have 4/3 as the framing interval, so that we can also look at munits with other intervals on the outside, such as 5/4 (which may be useful when looking at the [[MOS]] scales of [[magic]] temperament.

Not only do munits serve as a basic set of symbols from which scales can be formed, they also can serve as something of a [[Wikipedia:minimal pair]] for xenharmonic tunings. A minimal pair, in linguistics, is a set of words that differ in only one phoneme, but have very different meanings. They are useful for learners of a new language to demonstrate that two different phonemes really are ''different'' in the language in question, rather than (for instance) intended to represent the same phoneme spoken with two different "accents." When learning a new tuning, one tends to project munits from the old tuning onto the new tuning without realizing it, so it is important to deliberately focus on individual munits that differ as a way to learn about the "logic" of the new tuning, and how key intervals are split into certain step-size patterns which differ from the old tuning.

One may note that, althought we gave "5/4: LsL" as an example above, it is not the only way one could subdivide 5/4 into two large and one small step. In general, given some munit, we can derive additional munits, called '''modes''' or '''rotations''' of the munit, which are obtained by simply starting the munit at a different place. Thus, we also have the additional munits "5/4: sLL" and "5/4: LLs".

The original definition of "munit," introduced munits as having a framing interval which has some rational interpretation, and in which the pattern of steps subdividing it is given as a pattern of relative step sizes, such as "5/4: LsL". This basically parallels one common usage of scale - for instance, when we say "the diatonic scale" (in a generic sense, without regard to tuning), we typically mean something like 2/1: LLsLLLs, for which the outer interval is a (possibly tempered) 2/1, and for which the exact tuning may vary (for instance, from 31-EDO to 12-EDO or even Werckmeister III and so on). However, because there are several slight variants of what a "scale" is to begin with, so when we look at fragments of scales, we similarly get variants of what a "munit" can be. It doesn't take much effort to quickly run into other slight variants of this definition which are also useful, and which also may be called "munits" in the same sense.

The interval of equivalence of a scale, corresponding directly to the framing interval of a munit, has some similar variation in use. For instance, the LLsLLLs scale is typically assumed to have an "octave" as the interval of equivalence, but it is quite common for this "octave" not to be a perfect 1200 cents, but rather stretched or compressed slightly (as is common with pianos, or also the [[TOP tuning]]). Similarly, when we talk about a munit like 4/3: LLs, we typically do not assume the 4/3 needs to be perfectly just; rather it could be tempered with the exact size of the tempered 4/3 varying somewhat.

In this article we will give some examples of munits which are common, starting with the familiar and then veering into the xenharmonic. It can be very useful to ''learn'' these munits, as they tend to appear in many different tuning systems, for which they are important to internalize, rather than projecting 12-EDO based munits in situations that they don't apply.

Some important munits from (septimal) [[meantone]]:

It is important to note, however, that in this situation these aren't just scale-building-blocks, but also expectations. For instance, if you hear two whole-step sized intervals, you expect to get a 5/4, two whole-step and one half step is a 4/3, and three whole steps is a 7/5. Thus, at the very least, someone who has internalized these munits would expect the outer interval of the LLL munit to be, if not 7/5 specifically, much less strongly consonant than 4/3.

Here are some altered versions of the above munits, taken from [[Superpyth]] temperament:

As a result, when moving from meantone to superpyth, we get different harmonic properties than we'd typically expect when the same pattern of step sizes. In meantone, two whole steps yields a 5/4, so we may expect that to sit into consonant chords a certain way; thus we may have an expectation that an LL munit up from the tonic is a chord tone of a very simple, consonant, otonal 4:5:6 chord. But in superpyth, we instead get 9/7, which may sit in the harmonic setting in an entirely different way; it forms a 14:18:21 instead, which needs to be treated somewhat differently than a 4:5:6 chord, so that our expectations in this regard may not be entirely correct.  

In addition to having munits with the same step pattern and different framing intervals, we have munits with the same framing interval but different step size patterns. Here are some examples from porcupine temperament, and in particular the porcupine-8 MOS:

Possibly most critically of all, 9/8 is now subdivided into an Ls pattern, where the L is an approximate 10/9 and the s is an approximate 25/24! This is ''very'' different from anything in Western music, where we are very used to treating 9/8 as one step, unless we are working in a purely chromatic setting. But now we have to get used to there being a passing tone in between our 1/1 and 9/8.

We get some different things if we look at porcupine-7 instead and draw our munits from that. Now we will let the "L" be a 9/8 whole step, and the "s" a 10/9~11/10~12/11-ish ~164 cent size step:

We have the same thing for 6/5 and 4/3 (although we have relabeled the former "L" as "s"), but the rest have changed. 5/4 now has only one passing tone between 1/1 and 5/4, similarly to before, but it is no longer exactly in the middle of the interval as it is in meantone - and in porcupine, the difference between 9/8 and 10/9 is exaggerated, so that it is even further from the middle than 8:9:10. 9/8 is now just one large step again, and 3/2 is four steps.

Lastly, as an informal note, we tend to get some very interesting things when we look at decatonic scales. For instance, one may look at the  [[7L_3s]] LLsLLsLLsL MOS of suhajira temperament, in particular the 13-limit extension tempering out 64/63, 78/77, and 169/168. Then, we get some very interesting munits:

* 7/6: LL

* 11/9: LLs

* 4/3: LLsL

* 3/2: LLsLLs

We get another, possibly more familiar set when looking at [[neutral]] temperament. In this situation, we will use "A" to refer to an augmented second-ish-sized step, "L" to refer to a large step, "n" to refer to a neutral step, and "s" to refer to a half step, so that we have A > L > n > s in the relative size ordering:

The last munit listed above, 32/27: nn, does ''not'' correspond to any traditional jins in maqam music, but listed here regardless as being a very useful step pattern to learn in its own right. The basic idea of subdividing a minor third into two equal parts is very exotic and unfamiliar within Western music (although it wasn't always!), but it appears so frequently that it is a useful thing to learn!

Lastly, without regard to any particular tuning or temperament, this is an informal roundup of some munits that are very common in different xenharmonic tunings. One may want to learn these, particularly if one is leaving the realm of meantone temperament.

Note that we are kind of stretching the notation a bit at the end, where we have two different entries for 2/1: LLLs, depending on if we are thinking about the L's as "major thirds" or "minor thirds." Strictly speaking, these both correspond to the same 2/1: LLLs munit, but there are different suggested size ranges for the L's and s's. We could extend our notation for "munit" to deal with this situation, but for now we will leave it as is as the meaning is clear enough.

It is noteworthy that certain pairs of these munits can seriously challenge our preconceptions about what kinds of "interval category" we are dealing with. For instance, it is noteworthy to look at the following pair of munits:

Note that none of these terms are in any way mathematically precise, but it does show you just how much of these informal perceptions can change amid a totally different backdrop of whole steps and leading tones. And as a last, personal anecdote from the author, I (Mike Battaglia) have found munits to be very useful to study this phenomenon; I used to basically project all kinds of munits onto JI intervals without realizing it. For instance, I kind of assumed 11/7 "was" an augmented fifth, 9/7 "was" a type of major third, etc, or at least one of a few potential JI intonations thereof. I didn't realize that the ''feeling'' I associated with 11/7 would totally change with that LLsL munit rather than the LLLL one, and likewise with 9/7 as LLs rather than LL, and how much of these feelings derive from some kind of internalized logic regarding how these things are subdivided into patterns of steps with all kinds of subliminal expectations. My experience has been that learning to hear intervals drawn amid a different backdrop of step size patterns can lead to, all kinds of feelings and associations with the interval in question changing as well. It does seem to me that the perception of where the "leading tones" are located seems to be an important part of this perception.

[[Category:Tetrachords]]