Munit

A munit^{[idiosyncratic term]} (pronounced myoo-nit) is the combination of a musical interval, called a framing interval, and a pattern of musical intervals (typically represented in relative step sizes) subdividing that interval. An example munit would be "4/3: LLs," sometimes called the "major tetrachord."

In short, munits are fragments of musical scales, intended in some sense to generalize tetrachords, jins in maqam, etc. They are useful both as a method of building scales from smaller chunks, and also as a way to analyze our expectations, harmonic or otherwise, regarding how intervals are subdivided differently into step-size patterns in different tuning systems.

The concept of munit was proposed by Mike Battaglia in 2011.

Interpretation and usefulness

As fragments of scales

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.

As "minimal pairs"

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.

For instance, the munit "5/4: LL" is the "do re mi" munit which subdivides 5/4 into two equal parts, which is characteristic of meantone temperament. However, unless one is playing in meantone temperament, the munit "5/4: LL" typically does not exist! Instead, one gets munits such as "5/4: Ls" - with one larger and one smaller step, such as 9/8 and then 10/9 - or munits such as "5/4: LsL", which is particularly important in porcupine temperament (and thus 15-EDO or 22-EDO), where an important interpretation is "5/4: 10/9 25/24 10/9" (note that we have 25/24 = 81/80 in porcupine). There are also many others - but, typically, none which subdivide the 5/4 into two equal parts as they do in meantone. Note that the exact meaning of "L" differs in each of these munits, only referring to the relative sizes within each munit, and that we could have chosen any letter for the "5/4: LL" munit, since there is only one step size.

Thus, we can view, as a basic exercise, learning to differentiate the munits "5/4: LL", "5/4: Ls" and "5/4: LsL". Particularly, it is important to avoid the novice mistake of projecting the expectation of 5/4 into two equal parts onto non-meantone temperaments. (Of course, there is no requirement that the two "L's" in 5/4: LL be tuned perfectly equally, such as in a circulating temperament.)

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".

Specification

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.

For instance, given the "major tetrachord" munit "4/3: LLs", we can tune it to 19-EDO, so that we have an exact tuning in cents of "4/3: 189.473 189.473 126.316". Much we can talk about the "19-EDO major scale" as an individual scale, we can talk about "4/3: 189.473 189.473 126.316" as an individual munit. We can also refer to the 19-EDO major scale as a tuning of the generic LLsLLLs scale word, and thus, we can similarly refer to "4/3: 189.473 189.473 126.316" as a tuning of the 4/3: LLs munit. We could similarly refer to these in step sizes of 19-EDO so that we have the major scale as 3323332, and the major munit as 4/3: 332. All of these various usages of "scale" are common, and which usage is chosen is a matter of convention (for both scale and munit).

Sometimes, it is useful to talk about scales as a pattern of step sizes without suggesting any particular size hierarchy, so that we have aabaaab as a generic scale word representing both LLsLLLs and ssLsssL. Similarly, we can talk about the munit 4/3: aab, which could be either the major tetrachord, or something like an "anti-major" tetrachord.

Similarly, sometimes we talk about scales as though we have chosen a particular choice of rotation, so that when we say LLsLLLs we really are referring in particular to the Ionian mode thereof. But, sometimes we talk about scales as not having any particular choice of rotation at all, so that when we say the "diatonic scale" we are talking about the general set of all possible rotations thereof. If our scale is an MOS, we sometimes refer to the rotation-less interpretation using the number of large and small steps, so that 5L2s can be thought of as the diatonic scale without choosing any mode. Similarly, we can refer to munits such as 4/3: 2L1s, if the pattern of step sizes is an MOS, to refer to the entire collection of munits that one can form this way. It can be very informative to look at the different modes of exotic munits such as 5/4: 2L1s!

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.

Lastly, while this article is primarily focused on having a framing interval which has some rational interpretation, there is nothing preventing us from using other framing intervals based on anything, such as irrational tuning systems built on phi or e, etc. We may even take some liberties with the definition such that framing intervals are given as a range of sizes such as those based on[[Interval category|interval categories], so that we can say things like "M3: LsL" (where M3 is a generic "major third"-sized interval, perhaps somewhere in the 370-410 cent size range).

Examples

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.

Meantone

Some important munits from (septimal) meantone:

5/4: LL, the "do re mi" munit, subdividing 5/4 into two equal parts
6/5: Ls, or "do re me", subdividing 6/5 into a large and small step
4/3: LLs, the "major tetrachord," subdividing 4/3 into two large and one small step
7/5: LLL, which splits the 7/5 into three equal parts

In all of these examples, the "L" is typically a whole-tone-ish sized interval, and the "s" a semitone-ish sized interval.

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.

Superpyth

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

9/7: LL, the superpyth version of the "do re mi" munit, which now subdivides 9/7 into two equal parts rather than 5/4
7/6: Ls, similarly the superpyth version of the "do re me" munit
4/3: LLs, the "major tetrachord," subdividing 4/3 into two large and one small step - note this doesn't change
10/7: LLL, which splits the 7/5 into three equal parts

Similarly, we have that "L" is a whole-tone-ish sized interval, and the "s" a semitone-ish sized interval, although L is slightly larger than before and s slightly smaller. And as we can see, these patterns of step sizes now sometimes have different framing intervals. Now it is 9/7 that is being subdivided into two equal parts, and 7/6 being subdivided into Ls, and so on.

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.

Porcupine[8]

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:

5/4: LsL, the step size pattern you should expect to get you to 5/4! (and its rotations)
6/5: LL, the porcupine trichord splitting 6/5 into two equal parts
9/8: Ls, the munit subdividing 9/8 into a large and small step (10/9 and 25/24!)
4/3: LLL, the porcupine tetrachord dividing 4/3 into three equal parts
3/2: LsLLL, the porcupine-tempered version of the scale 1/1 10/9 9/8 5/4 11/8 3/2 (as intervals from the tonic)

In this situation, we have that the L steps are all the same quasi-major/neutral-second-ish sized interval which is the porcupine generator, representing both 10/9, 11/10, and 12/11, and the s steps are all the quasi-semi/quarter-tone-ish sized interval which in porcupine is simultaneously 25/24, 33/32, and 81/80 (among other things). Of course, all of the rotations of these munits are also relevant munits.

We note that we have some radically different things right away. For instance, we now have that 5/4 is subdivided into LsL, rather than LL, so we need to learn to expect two passing tones between the 1/1 and 5/4. Similarly, 6/5 is similarly subdivided into two equal-sized LL steps.

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.

Porcupine[7]

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:

5/4: Ls, the step size pattern you should expect to get you to 5/4! (and its rotations)
6/5: ss, the porcupine trichord splitting 6/5 into two equal parts
9/8: L, the munit subdividing 9/8 into a large and small step (10/9 and 25/24!)
4/3: sss, the porcupine tetrachord dividing 4/3 into three equal parts
3/2: Lsss, the porcupine-tempered version of the scale 1/1 9/8 5/4 11/8 3/2 (as intervals from the tonic)

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.

Suhajira[10]

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:

13/12: L
9/8: Ls
7/6: LL
11/9: LLs
9/7: LLL
4/3: LLsL
3/2: LLsLLs

So right off the bat we have some very interesting stuff! In the POTE tuning, our large step is 138.674 cents and our small step is 76.427 cents. So we have to learn that, again, 9/8 is not one step, but that there is a passing tone in between. Similarly, we must learn that 7/6 is subdivided into two large steps, and that 11/9 is subdivided into two large and one small step. Perhaps strangely, if we are used to superpyth, 9/7 is now subdivided into three equal parts rather than two, and 4/3 into three large and one small step. 3/2 is now a type of "seventh" rather than a "fifth", and so on. Learning to internalize these munits is an important part of forming the correct expectations regarding what harmonic properties to expect upon hearing a certain pattern of scale steps.

Neutral[7]

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:

11/9: nL
8192/6561: nns
4/3: nnL
4/3: sAs
3/2: LnnL
3/2: LsAs
3/2: LsLL

Most of these will likely be somewhat familiar to practitioners of maqam music as highly simplified representations of common "ajnas." Of course, the actual tunings and perception of maqam ajnas are very complicated and variable from region to region, which we will not deal with here, but we do this just to show that it is a very common and useful practice for many musical cultures to view their scales as being formed from simpler chunks, and that munits are simply intended to be another instance of this basic idea.

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!

Some interesting xenharmonic munits

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 below, the "L" and "s" may differ, between different munits, simply referring only to a relative pattern of step sizes within that munit. Thus we may write both "5/4: LL" and "6/5: LL", referring to a different "L" each time. I will write some rotation I think is particularly useful to learn, for whatever reason, but of course all of the rotations are useful!

9/8: Ls, with one passing tone between 1/1 and 9/8!
5/4: Ls, with a different L and s from the above. We subdivide 5/4 into two (audibly) different sizes.
5/4: LL, the meantone do-re-mi munit. If you aren't in meantone, this (probably) doesn't exist, so don't project it everywhere!
5/4: LsL, LLs, sLL, a very common 5/4 tetrachord. The LLs version may sound familiar to practitioners of Arabic music.
5/4: LLL, characteristic of negri temperament.
6/5: LL, totally unfamiliar to western music but almost ubiquitous everywhere else.
9/7: LL, characteristic of superpyth and machine temperaments.
9/7: LLs, characteristic of godzilla (with LL=~5/4) and hedgehog (with LL=~6/5) temperaments. Note the difference from the above!
4/3: LL, characteristic of semaphore temperament.
4/3: LLL, characteristic of porcupine temperament.
4/3: LLs, the major tetrachord.
4/3: Lms, among other things, "detempered" major tetrachord without two equal whole step sizes.
4/3: LLLL, characteristic of negri temperament.
4/3: LLLLL, characteristic of ripple temperament, as well as 12-EDO.
4/3: ssLs, the pentachord used in Paul Erlich's "standard pentachordal major" scale in 22-EDO.
11/8: LLs, characteristic of machine temperament, as in 11-EDO.
11/8: LLL, characteristic of glacial temperament, as in 13-EDO. Note how different this sounds from the above!
3/2: Lsss, characteristic of porcupine temperament.
3/2: LssL, a Rast-ish munit.
3/2: LLsL, a major pentachord.
3/2: LLL, characteristic of slendric temperament.
3/2: LLLL, characteristic of tetracot temperament.
3/2: LLLLL, characteristic of bleu temperament.
3/2: LLLLLL, characteristic of miracle temperament.
11/7: LLsL, a common pentachord from "machine" temperament.
11/7: LLLL, a pentachord from 11-limit meantone. Note how different this sounds from the above!
2/1: LLL, splitting 2/1 into three equal major thirds. Again, for many xenharmonic tunings, this doesn't exist!
2/1: ssL, splitting 2/1 into two major thirds and a residual diminished fourth, which is larger (as in meantone)
2/1: LLs, splitting 2/1 into two major thirds and a residual diminished fourth, which is smaller (as in superpyth)
2/1: LLLs, spitting 2/1 into two three major thirds and a residual diesis
2/1: LLLL, splitting 2/1 into four equal minor thirds. Again, for many xenharmonic tunings, this doesn't exist!
2/1: LLLs, splitting 2/1 into three equal minor thirds and a residual augmented second, which is smaller (as in meantone)
2/1: sssL, splitting 2/1 into three equal minor thirds and a residual diminished second, which is larger (as in superpyth)

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.

Relevance to interval categories

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:

11/7: LLsL, a common pentachord from "machine" temperament.
11/7: LLLL, a pentachord from 11-limit meantone. Note how different this sounds from the above!

Let's start with the latter 11/7: LLLL munit. It's pretty familiar sounding, and basically frames the 11/7 as a meantone "augmented fifth." We now have a kind of "whole tone scale"-ish vibe going on with it, in that there are no half steps, or leading tones, or anything like that. The "root third fifth" triad is now a meantone augmented triad, and it's entirely different from the above. It is basically what you get if you flesh a meantone augmented triad out with two additional passing tones.

That first munit, on the other hand, is really wild. We note that it has the same LLsL pattern from the very familiar "major pentachord" 3/2: LLsL munit, found in 12, but with 11/7 as the framing interval instead. As a result, this munit has a very particular sound, and one which is very exotic for listeners who are used to meantone. Note that this munit exists in 11-EDO, where that "s" step is a very effective leading tone of about 110 cents. A "root third fifth" triad drawn from this munit, in 11-EDO, is 7:9:11, so that the entire thing can be viewed as a stretched version of the usual meantone major pentachord, but with 4:5:6 replaced by 7:9:11, and with a "perfect fourth" of 11/8, and so on, with the leading tone between the 9/7 "third" and the 11/8 "fourth" in the same place.

In general, the first munit basically treats the 7/11 as an "augmented fifth," whereas the second one, on the other hand, does not. Instead, it's almost as if the entire 7:9:11 chord has become a "stretched major chord," with all of the whole steps and half steps landing in the same places as before, so that the 11/7 is kind of "stretched perfect fifth." Again, these are not mathematically precise terms, but just an informal attempt to explain the perception of 11/7 being four whole steps, vs being three whole steps plus a half step (whatever a "whole step" is).

We get a similar situation with these munits:

9/7: LL, characteristic of superpyth and machine temperaments.
9/7: LLs, characteristic of godzilla (with LL=~5/4) and hedgehog (with LL=~6/5) temperaments. Note the difference from the above!

Again, there is quite a difference between that 9/7 being two whole steps, vs being two whole steps plus a half step. And note that it isn't really the tuning that we're talking about - for instance, the ~9/7's of 19-EDO and 27-EDO are about the same size (only about 2 cents off), but the former has it split into an LLs tetrachord whereas the latter has it split into an LL trichord. Thus, there is a totally different sound going on.

As one subjective attempt to interpret what is happening, at least, the entire 9/7: LL munit seems to lend itself to the perception that it is perhaps an arpeggiated 7:8:9 chord, for instance, whereas 9/7: LLs doesn't have quite as simple of an interpretation, with that "leading tone"-sized "s" interval jamming up the works in that regard, leading to a very different "layout" of how notes may be grouped regarding their membership into potential chords.

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.

References

Munits were originally introduced in http://groups.yahoo.com/group/tuning/message/102052