Modal UDP notation: Difference between revisions

The above diagram, hopefully, makes fairly plain what is going on. As you can see, the "Lydian" mode has the most fifth going "up" - 6 up, 0 down - and also has the most "sharps." Locrian has the most generators going "down" - 0 fifths up from the tonic, 6 fifths down - and also has the most "flats." This is sometimes phrased as saying that Lydian is the ''brightest'' diatonic mode, whereas Locrian is the ''darkest''.

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 scales shortly.

The bright generator can also easily be found using modular arithmetic and the modular inverse. If your scale has <math>L</math> large steps, <math>s</math> small steps, and <math>T</math> total steps, the bright generator will always be <math>s^{-1} \mod T</math>, where the result denotes the number of steps ascending from the tonic. As an example, for the diatonic scale we have <math>L=5</math>, <math>s=2</math> and <math>T=7</math>, so <math>2^{-1} \mod 7 = 4</math>, and indeed 4 steps ascending  from the tonic is the bright generator: the perfect fifth. (Note that using this convention, the tonic itself maps to <math>0</math> steps rather than to <math>1</math>, so the result is one less than the conventional name: the "fifth" is <math>4</math>, the "fourth" is <math>3</math>, etc.)

The dark generator can be found similarly as <math>L^{-1} \mod T</math>. So for the diatonic scale, we have <math>s=2</math> and <math>T=7</math>, and <math>5^{-1} \mod 7 = 3</math>, where 3 steps ascending from the tonic is the perfect fourth.

Note that if '''P'''=1, it can be omitted, so that the UDP notation is simply '''U'''|'''D'''.

* The quantity '''U''' + '''D''' + '''P''' is always the number of notes in the scale

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

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

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

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 ascending melodic minor scale 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.

* Porcupine[7] Lssssss mode, but altered with 7/4 instead of 11/6, is 6|0 b7.

** The major pentatonic scale is a subset of the major scale. The genchain is simply shortened on either end. Thus one would expect the major pentatonic scale contained within Meantone[7] 5|1 to be Meantone[5] 4|0. But instead it's Meantone[5] 0|4. This is because changing the size of the MOS scale often changes the generator to its octave inverse. Meantone[5]'s bright generator is the 4th not the 5th.

** Because the MOS type that a temperament's MOS represents is tuning-dependent, the choice of bright generator can be tuning-dependent. For example, assume you are using approximately but not exactly 700¢ for dominant meantone. Then Dominant[12] is either [[7L 5s]] (if generator < 700¢), in which case the bright generator is the fourth, or [[5L 7s]] (if generator > 700¢), meaning that the bright generator is the fifth. As a result, Dominant[12] 7|4 is ambiguous.

* It's impossible to determine the bright generator in a non-MOS scale like Meantone[8], thus Meantone[8] 5|2 is ambiguous.