UDP: Difference between revisions

Modal UDP notation is a way to uniquely specify a particular [[Periodic_scale#Definition-Rotations|rotation]], ie mode, of any MOS. Its name is derived from up|down(period), or U|D(P). If accidentals are specified, it can also refer to the MODMOS's of those MOS's as well.

The basic premise is fairly simple: we will begin by using the diatonic scale as an example, and then show how this extends to arbitrary MOS's.

You will note that next to each mode is a little signature - Lydian has 6|0, Ionian has 5|1, etc. This is the UDP notation for the mode! Fairly simple, and we will generalize to arbitrary non-diatonic MOS's shortly.

If you aren't that familiar with modal harmony, you may note that the above is a slightly different perspective than the one you may have seen in "beginner" introductions to the subject, as well as a different sequence of modes.

The first four properties basically say that our sequence has turned the set of modes into what mathematicians would call a [https://en.wikipedia.org/wiki/Metric_space "metric space"]. This means that we don't just have a set of objects, anymore, but we can now see how "close" or "far" they are from one another (or how "similar" or "different" they are, if you like). The last two properties show us that there is a real perceptual parameter involved, and that our sequence shows how the modes rank according to it.

It is fairly straightforward to see that we can extend the above way of thinking to any MOS. In general, we can even extend this to MOS/DE scales that have multiple periods per octave, such as the octatonic diminished scale. We will start by focusing on the single-period case, so that the period is just an octave.

The "bright generator" has sometimes also been called the '''chroma-positive generator''', and likewise the "dark generator" has sometimes been called the '''chroma-negative generator''', because of which direction they shift intervals in an MOS by its chroma (the chroma for any MOS is the difference between the large and small step, which is also the difference between the large and small third, fourth, etc).

Given the above, the UDP notation for an arbitrary mode of an arbitrary MOS is simple:

For more exotic scales, such as porcupine[7], we have that the Lssssss mode is 6|0, and that the sssLsss mode is 3|3.

We can generalize this to arbitrary MOS/DE scales with multiple periods per octave as follows:

As an example, the Pajara[10] Static Symmetrical Major of ssLssssLss would have UDP notation 4|4(2).

A [[Periodic_scale|periodic scale]] S associates an interval S(i) to every integer i, such that there is a period (strictly, a quasiperiod) Q>0 and an interval of repetition R such that S(i+Q) = S(i)+R. Q is chosen so as to be minimal; there is no smaller period. S is monotone if i<j implies that S(i)<S(j).

For example, consider the quasiperiodic function Ionian(i) = V[(i+3 mod 7)+1] + 31ceil((n+4)/7)-49, where V = [5, 10, 15, 18, 23, 28, 31]. This has period 7, and Ionian(7) = 31, where the tuning is [[31edo|31edo]] so that 31 represents an octave. Going up from 0, it has values 0, 5, 10, 13, 18, 23, 28, 31, 36, 41 ... corresponding to 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ..., and going down from 0, it gives 0, -3, -8, -13 ... corresponding to 0, -1, -2, -3 .... This gives the Ionian, or major, mode of the diatonic scale. Then Ionian(4)=18, the fifth, and 18 >= Ionian(i+4)-Ionian(i) for all i. We have Ionian(4)=18, Ionian(8)=36, Ionian(12)=54, Ionian(16)=72 and Ionian(20)=90. However, Ionian(4*6) = Ionian(24) is 106, which is less than 6*18 = 108. Hence the largest value for which Ionian(4*u) and 18*u are equal is u=5. Similarly, Ionian(-4)=-18, but Ionian(-8) is -34, not -36, and so d=1. Since Ionian(7)=31, which is the octave, P=1, so U=u=5, D=d=1, and the UDP notation for Ionian is 5|1(1), or simply 5|1.

While the simplest interpretation of the modes is that they're only cyclic permutations of one another, a more advanced interpretation often utilized in musical styles where modal theory is prominent is to understand them as varying on a continuum from "brightest" to "darkest," meaning "most sharps" to "most flats" or "most major" to "most minor." This is the same as arranging the modes in descending order by the number of chroma-positive generators which go up from the tonic.

This interpretation is what UDP notation generalizes.

For example, the proper generator for meantone[7] is the perfect fifth, because it's larger than the other specific interval it shares a class with, the diminished fifth. Consequentially, meantone[7]'s Ionian mode is 5|1(1), which is 5|1 for short, because it contains five chroma-positive generators up from the root and one down, as in the diagram F-[C]-G-D-A-E-B for C ionian. This also means it has five "sharper" scale degrees - the second, third, fifth, sixth, and seventh - and one "flatter" scale degree - the fourth. If we want to sharpen the fourth to turn it into an augmented fourth, we arrive at 6|0 or [C]-G-D-A-E-B-F#. Conversely, Aeolian mode, with only two sharp scale degrees - the second and fifth - is 2|4. We can add accidentals as well, so that meantone's harmonic minor is 2|4 #7.

* Paul Erlich's standard pentachordal major for Pajara[10] is 4|4(2) #8, or alternatively 6|2(2) b3.