Won’t you lead me there?

Some thoughts on shapes:

A great master Ornette Coleman died today. When I was about 14, I heard a saxophone and trumpet playing a melody on the car radio. I didn’t stay around for the DJ’s announcement, and the melody stuck in my head. It kept coming back randomly in my mind for years – I could sing the whole thing, having only heard it one time. I remembered the shape of it. It was driving my a little crazy – I didn’t know anybody who could solve the riddle. When I was about 20, I picked up a copy of Ornette’s “Something Else!!!” and there it was, “When Will the Blues Leave.” The shape was what stuck with me, the contour of the melody, the break for the drums, the leaping at the beginning contrasted with the snakiness of the ending. This was an early composition, but those shapes! So many of the compositions could be identified by the shape of the melody alone – “Peace,” “Round Trip,” “Law Years,” “WRU.”

Recently I heard a rehearsal tape where Ornette is working with Ed Blackwell, describing what he wants from the drums. The focus is on something like, trying to get somewhere without sounding like you’re trying to get somewhere. A couple of quotes:

What I’m saying is that, when you were doing it then it was just like saying, you were playing a certain rhythm that you know you have to play because the next thing that’s coming up is going to justify what you just done. But cut that all loose.

The only thing I’m trying to get clear is that isn’t there a certain method that, the way you play ideas on the rhythm, is you can set it up to where it gives you a feeling that you’re going – like anybody listening would get a feeling like there’s something you’re going to get to? . . . What I’m saying, that is what’s the difference between you having to adjust your idea to sounding like we’re going somewhere than playing actually what’s right there. What you actually play, without making it sound like it’s going somewhere.

Rest in peace, Mr. Coleman, and thank you for leading us to so many places.

In my own practice, I’m working on some kind of “shape theory,” and recently taking another look at Allen Forte’s The Structure of Atonal Music, which has been on my shelf for about 15 years, cracked open and put away many times for lack of patience. Too much nomenclature that I don’t understand. And this book is from 1973, so I’m way behind. I wish that I could read what modern music theorists are up to because there are so many useful ideas, but I don’t have training in this area, and have always been turned off heavy jargon. So I end up going through things very slowly and translating them to more plain language. I admire the rigor of Forte’s book, but stuff like this is impenetrable to your average practicing musician (p. 7-8):

If we let A be [0,1,2] and t = 1, the process by which B is produced can be displayed as follows:


t= 1

A     B

0 -> 1

1 -> 2

2 -> 3

Observe that every element of A corresponds to an element of B and that the correspondence is unique in each case – that is, some element in B does not correspond to two elements in A. Thus, transposition can be regarded as a rule of correspondence-namely, addition modulo 12 – that assigns to every element of B exactly one element derived from A. We will borrow a conventional mathematical term to describe such a process and say that A is “mapped” onto B by the rule T. The notion of a mapping is more than a convenience in describing relations between pc sets. It permits the development of economical and precise descriptions which could not be obtained using conventional musical terms.

So, unless I’m mistaken, I read this as: “transpose the material up a semitone.” For example [C,C#,D] to [C#,D,D#]. Most things can be transposed to 12 locations. Seems to me that out of 351 pitch shapes, all but 16 fit this description (more on this later).

The next operation Forte describes is “inversion” (p.8):

Like transposition, the inversion process can also be described in terms of a rule of correspondence “I” which maps each element of a set A onto an element of a set B. The inversion mapping “I” depends upon the fixed correspondence of pc integers displayed in the following table:


0 <–> 0

1 <–> 11

2 <–> 10

3 <–> 9

4 <–> 8

5 <–> 7

6 <–> 6

Observe that in each case the sum of the integers connected by the double arrow is 12, and recall that 12 = 0 (mod 12). We say that 0 is the “inverse” of 0, 1 is the inverse of 11, and 11 is the inverse of 1, and so on. In general, if we let a’ represent the inverse of a, then:

a’ = 12 – a(mod 12)

For any set A, therefore, the mapping I sends every element of A onto its inverse, producing a new and equivalent set B.

Translation: “Take the intervals of your material and play them “down” instead of “up” (or vice versa). For example the intervals in the melody [C,E,G] (up major third, up minor third) played down instead of up from C makes [C,Ab,F] (down major third, down minor third). The word that I don’t really get in the quote is “equivalent” – the system here makes no distinction between sounds that are intervallic mirror images. For example, there is no way to distinguish between a major and minor triad, which are structural inverses (both called “3-11″ in the list of sets, Appendix 1, p.179). But musicians know that these are two different sonic realities. Personally, I can’t hear a minor triad as a “downward” major triad. I hear a sound that has a root, a 5th, and a pitch a minor third above the root. This doesn’t mean it’s not interesting to think about. In fact, I’m pretty obsessed with ideas about mirrors and symmetry. Steve Coleman has developed whole worlds of music based on this idea. But my ear tells me that the root of [F,Ab,C] is F. Structurally, it’s related to [C,E,G], but the sonic impression is distinct.

So Forte lists 12 trichords in his Appendix I. Trichords are structures with three notes. Usually, they are listed in some kind of “prime form,” or the inversion that has the smallest range. Kind of like “root position” for a common triad. Other people that I’ve seen also list 12 trichord types (John Rahn, Elliott Carter). I wonder what Schillinger’s version is – I’ve never been able to justify the cost of those giant books. I’m sure there are many, many versions of this, because it’s pretty easy to figure out. John O’Gallagher hipped me Peter Schat’s “Tone Clock,” which has to do with layering trichord shapes to partition the 12 tones.

O’Gallagher’s Twelve-Tone Improvisation is the book that is the most convincing study for me in this area. Rather than a list of numbers, he describes the 12 trichords intervallically, which is much more intuitive for improvising musicians. For example, Forte’s 3-3, with Pitch Class Set [0,1,4] could be thought of as [C,C#,E], or any of its 12 transpositions. But this set can also go “downward” (inversion), which would be [0,11,8], (each “mapped” pair has to add up to 12), or [C,B,Ab] and its 12 transpositions. Lots of calculating. Lots of focus on fixed pitches as opposed to structural content. O’Gallagher describes this structure with the more user-friendly “1 + 3″ or “3 + 1.” This would give you (starting on C): [C,C#,E] or [C,D#,E]. The intervals are shuffled within the same frame. You see right away what the content is. It may seem a trivial distinction, but this intervallic description relates much better to the way things move and connect. Of course, maybe it’s just a better tool for the job – Forte’s book is, I guess, more about analysis than composition or improvisation. I believe that John (having hung out and played with him) has effectively trained himself to hear these asymmetrical trichords (of which there are 7) as two sides of the same coin. Major and Minor triads are like two children from the same family. Or maybe twins, male and female.

Anyway, I’ve meditated on these 12 trichords for several years now, and decided that I hear 19 sounds. This is because 5 of the trichords are symmetrical (containing at least two of the same intervals) and their structure doesn’t change upon reflection. They may sound transposed, but the quality is the same. For example, reflect the diminished triad [C,Eb,Gb] around C to get [Gb,A,C] – same structure, transposed by a tritone. It has two identical intervals (a minor 3rd). The other 7 trichords in Forte’s list are asymmetrical (containing 3 different intervals), and sound structurally different when reflected. This makes 14 asymmetrical + 5 symmetrical, or 19.

I have a little trouble mentally with lists of numbers – for example in integer notation, [0,4,7] [0,3,8] and [0,5,9] are all major triads (Cma, Abma, and Fma). But this isn’t very apparent to me by just looking at the numbers. I know that these type of things are very far from what’s in my mind while improvising. Much easier for me is to draw the shape on a pitch circle and see what it looks like. The three sets above then look like this (imagine “C” at the 12:00 position, and read clockwise).

major rotations

It’s pretty easy to see that these are all rotations of the same shape. It took me a little while to get used to this notation, but after a some time it allowed me to see relationships and patterns very quickly, closer to the fluid way of thinking (or not thinking) while improvising. I’ve always liked the idea of visualization as a tool, but the idea of shapes conveying data efficiently was really kicked into high gear after taking a look at Edward Tufte’s “The Visual Display of Quantitative Information.”

Thinking about intuitive interfaces, consider the idea of the “Interval Vector” of a pitch set (another concept in Forte’s book). All this means is “how many of each type of interval there are” in a given set of pitches. The text goes through some pretty tortuous calculations to get all of the numbers (looking at the differences between each pair of numbers, etc, p. 14-15). Basically, it’s the problem of when you walk into a room and everybody has to shake everybody else’s hand. During the greeting, you figure out the difference in your ages (subtract 12 if this number larger than 11). Then count how many of each number you get for all possible handshakes. Long process, likely to make a mistake. On the circle, it’s pretty easy to look at the shape and see the intervals quickly and accurately. For the major triads above, just go around and look at each pitch: How many semitones follow any of the three pitches? None. How many whole tones? None. How many minor thirds? One. How many major thirds? One. How many Perfect fourths? One. How many tritones? None. You don’t need to go farther than this (the number of perfect 5ths is the same as perfect 4ths, and so on). This gives the interval vector (001110).  The interval vector thing reminded me right away of some things that been explained to me about devices used by Henry Threadgill where the interval content of a set of pitches is treated as a set of improvisational possibilities.

So anyway, I decided after some study that to my ear I’m on the 19 trichords side of the fence, or triads, or whatever you want to call them. Three notes that you think of as being a musical “word.”  Let’s try to crack the code – How are they all related to one another? Well, a useful way to meditate on information is on a mandala. Here’s one, with only one rule: All trichords that are connected by a line have to be only slightly different – you can move from one to another by moving one of the pitches by one semitone:


Any three pitches you name will be one of these 19 shapes. This particular array (or some tangled version of it with the same connections) is the only possible way to organize the shapes under this rule of close relationships. The numbers here give the old fashioned integer notation, starting from the white circle and reading clockwise. The center pitch of each triad is at 12:00, to see the shapes in a similar orientation. The dotted lines connect triads that are almost the same. They only differ by one pitch moving one semitone. For example, you can see that the minor triad (0,3,7) and the major triad (0,4,7) are connected by a dotted line. The difference between these two is the semitone movement of the 3rd, minor or major. The pathways along the dotted lines are possible “mutations” within the larger triad organism. The five symmetrical triads are the spine, and the assymetrical triads form the wings. At the head is triad (0,2,7), which has a special significance for those who study the music of John Coltrane – for example, the opening of “A Love Supreme,” (B,E,F#,B) which is (0,2,7) where E = 0.

I’ve made some other mandalas of these 19 shapes. It’s a way of getting the mind around a certain amount of information. See how much is actually there without getting overwhelmed. Find an analogy somewhere in nature maybe, to attach some symbolism to the data. For example, there is a progression of the moon phases called the “Metonic Cycle” that takes 19 years to complete. After 19 years, the moon will be at (very nearly) the same phase at the same time of year (this was one of the cycles measured by the ancient Greek “Antikythera Mechanism”). You could take any date and see what the moon phase is over the course of this cycle, associate each phase with a triad, and come up with a complete progression of the 19 triads. An idea of using one shape to get at an understanding of another.

Moon Phase Triad analogy

So after some study here, the next question is, let’s get past 3 notes – what do all possible pitch shapes look like? Is there a way to organize them in a way that is maximally useful for improvisors?

Well, first of all, how many are there?  Forte lists 220 pitch sets in appendix 1 (for some reason not including sets of 1,2,10,11, or 12 pitches, which I also don’t understand, since these are things that we use quite a bit). So, if you are on the 19 triad side of things (rejecting  the notion of sonic equivalency in inversion), you come up with 351 pitch sets. That is, all possible pitch shapes between 1-12 notes in 12 tone equal temperament. This is a lot of information, but doing some “humanizing” of the data can be a way in. Take the number 351. This divides up certain ways. For example, you could take it in smaller pieces, 13 groups of 27. This sounds almost like a calendar. Imagine a world where each pitch shape is a day, there are 9 days in a week, and 3 weeks in a 27 day month. There are 13 months, and if you put a vacation day in between each one, plus an extra on January 1 (for the hangover), you get (13 x 27) + 14, or a 365 day year. Map your regular calendar onto this one. Take a year and go through all possible shapes, and you must learn something.

So I figured I should do that. It’s a way of using one familiar, ingrained system (the calendar) to get into a less familiar bunch of information (“learn a word a day” type of thing). I mean, it’s just these 12 pitches, shouldn’t we know the materials of our trade inside and out? The main question is, how to order the shapes? Forte’s ordering is based on the “Cardinal Number” of each set (how many notes in the shape), and an ordering within each cardinality based on an permutative algorithm that finds smallest intervals. On the surface, this ordering looked tedious to me for practice purposes. It would be good to change it up a bit more. What if the order could somehow imitate the seasons, one big cycle? This would logically start with a single pitch and end with the chromatic scale. Those define a minimum and maximum range of pitches. So then you could just go through visually and arrange the shapes by the range that they cover, which is very useful musically. Then the whole cycle is like a slow filling in of the chromatic scale. The pitch sets get appropriately full as the cold winter months march on. The 13 months of the tonal calendar with 14 empty days could look like this: 00 cover

Recently I started doing a daily study of each of these pitch shapes, following this order, one per day. I’m about two weeks in. June 18th is the next day off. Right now is the time of year to deal with shapes that span 8 semitones. There’s a bit of a Thelonious Monk theme going on. June 10th was (0,3,5,6,8) or [C,Eb,F,Gb,Ab], which you hear at the beginning of the melody of “Little Rootie Tootie” (Ab,Eb,C,Gb,F) and also at the beginning of “Bolivar Blues” (transposed to Bb). June 11th was (0,3,5,7,8) or [C,Eb,F,G,Ab], which you hear (transposed up a major 3rd) at the beginning of “Monk’s Dream” (C,E,G,A,B). But maybe this is changing, because tomorrow (June 12th) is (0,3,6,7,8), or [C,Eb,Gb,G,Ab], which goes along with the final lyric of Hoagy Carmichael’s “Skylark”:  “Won’t you lead me there?”

– Brooklyn, 11 June 2015