Modal UDP notation

Modal UDP notation is a way to uniquely specify a particular rotation, i.e. mode, of a MOS scale. Its name is derived from "up|down(period)", abbreviated as U|D(P). If accidentals are specified, it can also be used with MODMOS scales.

This mode notation system was designed by Mike Battaglia and Gene Ward Smith, although the basic idea is really due to George Russell and his Lydian Chromatic Concept, of which the UDP notation is basically a generalization to different MOS's.

Introduction

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 scales.

MOS scales are typically generated by repeatedly stacking some generator on top of itself from the tonic. For example, the diatonic scale is generated by repeatedly stacking the fifth on top of itself, until you have 7 notes. Other MOS scales are generated similarly.

For any such generator, we can obtain different modes of the same scale by changing how many generators we stack "up" vs "down" from the tonic. For instance, in the diatonic scale, if we have all fifths going up, that's Lydian mode. If we have all fifths going down, that's Locrian mode. Dorian has three fifths up and three fifths down, and so on.

Assuming we go with the diatonic scale for now, and choosing the "fifth" as our generator, the above principle leads to a certain "ranking" or "ordering" of the modes, from "most fifths up" to "most fifths down" (assuming our tonic is "C"). Behold, the picture worth a thousand words:

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.

Relative vs parallel modes

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.

Often, the modes are explained first in terms of a static set of unchanging base pitches, such as the white notes on a keyboard. Then, the individual modes are defined by picking different choices of tonic within that pitch set. As a result, this leads to the sequence of relative mode positions, based on where the tonic appears: C Ionian, D Dorian, E Phrygian, F Lydian, G Mixolydian, A Aeolian, and B Locrian. This is the perspective taught in many beginner treatments of modal harmony.

In comparison, the sequence we have given is the sequence of parallel modes with the same tonic (C), and with the pitch set changing. When we change the number of generators going "up" or "down," this equates to sharpening or flattening a note in the mode. As a result, taking one of the generators away from the stack going "down", and re-stacking it on top of the stack going "up", leads to the sharpening of a note, and vice versa if a generator is placed downward. This perspective is one of the first things usually taught in jazz music school, and other musical programs that focus on modal harmony.

It is somewhat unfortunate that the "relative" perspective is usually the only one given in beginner treatments of modal harmony, given that the "parallel" perspective of "brightest to darkest" modes is far more important to how modal harmony really works. To understand why, note that the "brightest to darkest" sequence that we gave has the following properties:

The number of steps between two modes in the above sequence is exactly equal to the number of notes in which they differ.
Modes that are adjacent in this sequence differ by only one note, and so are "as close as possible."
A modulation from one mode to another will sound much more "related" if the modes have more common tones than if they don't.
As a result, this gives you a sort of "modulation space" for the modes of any scale.
This sequence also simultaneously shows how the usual major (Ionian) and natural minor (Aeolian) modes are placed on a continuum of "brightness" to "darkness."
The sequence makes it easy to see things such as, for instance, that Phrygian is one step "darker" than Aeolian, and that Locrian is two steps "darker." Likewise, Lydian is one step "brighter" than Ionian, and Mixolydian and Dorian are in between.

The first four properties basically say that our sequence has turned the set of modes into what mathematicians would call a "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.

Generalizing to arbitrary MOS scales: bright and dark generators (chroma-positive and chroma-negative)

It is fairly straightforward to see that we can extend the above way of thinking to any MOS scale. 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.

Each MOS scale is also formed by a stack of generators. So, we can likewise vary how many are going up and down from the tonic, plot the resulting modes according to this sequence, and in general develop a "brightness"/"darkness" spectrum for any MOS scale. The resulting ranking of modes will have similar properties as our ranking for the diatonic scale.

The only wrinkle: in general, each MOS scale has two possible choices of generator. For example, in the above diatonic example, we chose the fifth as our generator, but we could have also chosen the fourth. That is, the diatonic scale is not just generated by stacking fifths from the tonic, but could also be thought of as stacking fourths. If we had chosen that, then Locrian would have all the fourths going "up" from the tonic, and Lydian would have all the fourths going "down", and we would have gotten an inverted table as a result.

So why did we pick the fifth for the diatonic scale? And for an arbitrary MOS scale, how do we choose the generator?

It so happens that there is a useful way to answer this question. Each MOS scale can be thought of as having a bright generator (or chroma-positive generator) and a dark generator (or chroma-negative generator):

The bright generator increases the number of "sharps", or "large-sized" intervals as you stack more of them going "up" from the tonic;
The dark generator increases the number of "flats", "small-sized" intervals as you stack more of them going "up" from the tonic;

where the term "large-sized" and "small-sized" refer to that each interval class in an MOS scale has two different sizes: a large and a small one.

In our case, it is easy to see that for the diatonic scale, the fifth is the bright generator. The more fifths you have going up from the tonic, the more sharps you have in your mode, or the more "major" or "augmented" intervals you have. The fourth is likewise the dark generator, as increasing the number of fourths going up means you are increasing the number of "flats," or of "minor" or "diminished" intervals, etc.

The standard convention, then, is to choose the *bright generator* for any MOS scale as our standard generator. This means that the "up" direction doesn't just mean more generators going up, but also means note pitches going up in general, as more of the intervals in your mode or "sharpened" or made into the "large" variant.

A useful method to quickly find the bright generator is that it is always the generator that is the "large" variant of its generic interval class. As an example, in the diatonic scale we know that the generators are the fourth and the fifth. Of the two, the perfect fifth is the "large" type of fifth in the diatonic scale, whereas the "small" variant of fifth is the diminished fifth, so it is the bright generator. Likewise, the fourth is the "small" fourth, whereas the "large" fourth is the augmented fourth, and it is the dark generator.

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.

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

The UDP notation

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

Get the bright generator for the MOS scale.
For the mode in question, determine how to generate that mode by stacking bright generators "up" and "down" from the tonic.
Get U, the number of bright generators going "up."
Get D, the number of bright generators going "down."
The UDP notation for the mode is simply U|D.

That's it. So for the Ionian mode of the diatonic scale, for instance, we have that Ionian is 5|1, and Aeolian is 2|4.

For mavila[7], the formerly "bright" generator becomes the dark one and vice versa, so that we have that the sLssLss (anti-Aeolian) mode is 4|2, and the ssLsssL (anti-Ionian) mode is 1|5. Note that anti-Aeolian is indeed "brighter" than anti-Ionian in mavila[7], due to how sharps and flats are inverted in the anti-diatonic scale relative to where they'd be in the diatonic scale.

Note that for mavila[9], the fifth again becomes the bright generator. This demonstrates an interesting property: the choice of bright or dark generator can be different for the different MOS scales of the same temperament. So in mavila[9], the LLLsLLLLs "superionian" mode is now 7|1, and the sLLLLsLLL "superphrygian" mode is now 1|7.

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.

Scales with multiple periods per octave

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

Let P be the number of periods per octave.
Calculate U as in the last section, but also multiply it by P.
Calculate D as in the last section, but also multiply it by P.

Then the UDP notation for an arbitrary mode of a multi-period MOS scale is simply notated U|D(P).

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

The reason that we multiply by P is that we get the following properties:

U describes how many scale degrees are of the "larger" or "major" variant, per octave
D describes how many scale degrees are of the "smaller" or "minor" variant, per octave
When we start chromatically altering the notes of the mode, we often do that differently for each individual sub-period of the octave, so it is a more useful perspective to look at the total number of "sharps" and "flats" in the octave, rather than each sub-period

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

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

Mathematical definition

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 \gt 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\lt j[/math] implies that [math]S(i) \lt 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 \lt Q[/math] and [math]S(m·u) = g·u[/math], and [math]d[/math] the largest integer such that [math]0 ≤ d \lt 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.

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 scales

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] 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 scales

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.

Things to watch out for

Requiring the generator to always be bright causes certain issues:

The bright and dark generators may "flip" based on the size of the generator and how many notes are in the corresponding MOS scale.
- 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.