Modal UDP Notation

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Modal UDP Notation

Modal UDP notation is a way to uniquely specify a particular 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.

Any MOS has an ambiguous choice of generator: for example, the generator for meantone[7] can be viewed as either the perfect fourth or the perfect fifth. If the generator is chosen by picking the one which is the larger specific interval in its generic interval class, also called the chroma-positive generator, then the UDP notation for any mode is U|D(P) - where P is the number of periods per equivalence interval, U is the number of chroma-positive generators going up from the tonic times P, and D is the number of chroma-positive generators going down from the tonic times P. If P = 1, then it can be omitted, so that the UDP notation is simply U|D.

This choice of generator means that the UDP notation simultaneously describes the following properties of the mode in question:

  1. How many scale degrees are of the "larger" or "major" variant, vs the "smaller" or "minor" variant.
  2. How many generators need to be stacked up vs down from the tonic it requires to generate the mode.

Examples can be found below.

Mathematical Definition

A 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).

Given a monotone periodic scale S, suppose it is also a MOS or DE scale. Let the generator S(m) = g be such that g ≥ S(i+m)-S(i) for all i. If Q is the period of S, let u be the largest integer such that 0<=u<Q and S(m*u) = g*u, and d the largest integer such that 0≤=d<Q and S(-m*d)=-g*d. If S(P*Q) = octave, so that P is the number of periods to an octave, let U = P*u and D = P*d. Then the UDP notation for the given mode is is U|D(P). If P=1 we may omit it and just write U|D.

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 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.

Rationale

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.

Within the first interpretation, the diatonic scale's Ionian and Dorian modes would be adjacent, since they begin on adjacent notes within the scale. But within the second interpretation, Ionian and Lydian modes would be adjacent, since they occupy adjacent positions along the brightness-darkness continuum. Ionian and Mixolydian would similarly be adjacent.

Movement to an adjacent mode "up" in this paradigm means a single interval will become sharpened, and moving "down" means that one will become flattened. For example, the movement "up" from Ionian to Lydian sharpens the 4th scale degree, and the movement "down" from Ionian to Mixolydian flattens the 7th.

This interpretation is what UDP notation generalizes.

Examples

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.

The chroma-positive generator for porcupine[7] is the larger 7th, which is about ~11/6; as a consequence, porcupine[7]'s Lssssss mode is 6|0, and sssLsss is 3|3. Likewise, mavila[7]'s ssLsssL anti-Ionian is 1|5, and Mavila[9]'s LLsLLLsLL "Olympian" mode is 4|4.

It should be noted that the chroma-positive generator will vary from MOS to MOS even within the same temperament. For example, the chroma-positive generator for meantone[7] is the ~3/2, but is the ~4/3 for meantone[5].

MOS's

  • Meantone[7] Ionian, LLsLLLs: 5|1
  • Meantone[7] Aeolian, LsLLsLL: 2|4
  • Mavila[7] Anti-Ionian, ssLsssL: 1|5
  • Mavila[7] Anti-Aeolian, Herman Miller's sLssLss mode: 4|2
  • Porcupine[7] Lssssss: 6|0
  • Porcupine[7] Lssssss mode, but altered with 7/4 instead of 11/6: 6|0 b7
  • Porcupine[7] sssLsss: 3|3
  • Diminished[8] sLsLsLsL 0|4(4)
  • Diminished[8] LsLsLsLs 4|0(4)
  • Triforce[9] LLsLLsLLs: 6|0(3)
  • Meantone[5] minor pentatonic, LssLs: 3|1
  • Meantone[5] major pentatonic, ssLsL: 0|4
  • Sensi[11] LLsLLLsLLLs: 8|2
  • Pajara[10] Static Symmetrical Major, ssLssssLss: 4|4(2)
  • Mavila/Mabila[9] Olympian, LLsLLLsLL 4|4

MODMOS's

  • Melodic minor is 5|1(1) b3, abbreviated 5|1 b3 for short, but could also be 3|3(1) #7, abbreviated 3|3 #7 for short.
  • Paul Erlich's standard pentachordal major for Pajara[10] is 4|4(2) #8, or alternatively 6|2(2) b3.

Diagram

UDPTable.gif