Intervallic Runs, Introduction

The principle behind intervallic runs is to make up single-note lines based on the intervals between consecutive notes rather than the notes themselves. In this way you can construct patterns that don't have a direct association with any chord. It is a neat way to produce the kind of 'outside' sound that you often hear in the playing of Michael Brecker and Allan Holdsworth.

I always found 'outside' playing very difficult because it only works well when it is played at a decent speed. When you play it slowly, as you have to when you are trying to come up with a hip line, it sounds ridiculous. So it is a skill that is hard to acquire unless you have a very good ear for slow-motion lines or you are very quick to learn to play a new line fast (neither of which I am particularly good at). A few years ago I came across Schoenberg 12-tone rows, however, and by studying their interval sequences I finally found a systematic way to approach one specific type of outside sounds. If you are interested in the gory details, you can read the paper I wrote on the subject, Interval Sequences In 12-Tone Rows (submitted in a personal capacity to Journal of Integer Sequences on August 17th 2012, rejected on September 23rd 2012, so I now publish it here on the m3guitar website). The technical field is best described as 'applied combinatorics' and from the viewpoint of most musicians it is very abstract. The information is presented in condensed form below but if you are not keen on numbers and maths, you can jump straight to the section TTR Sound Examples at the bottom of this page. Right now, the only examples available with tablature are the 18 TTRs using two intervals and a selection of 48 All-Interval Sequences taken from the complete set of 1,928. More will be added.

Schoenberg 12-Tone Rows (TTRs)

A TTR is a permutation of the 12 notes within an octave. If we use clock notation, a TTR is a list of the integers from 1 to 12 in some randomly chosen order. So

is a TTR and so is

In total there are 12! (the exclamation mark denoting factorial) TTRs which is close to 480 million.

Interval Sequences (IS)

If we look at the intervals between consecutive notes in TTR2, we get

The intervals are given as numbers between 1 and 11. If a note is moved up by a whole tone, the interval is 2; if it is moved down by a whole tone, the interval is 10, not -2. The first interval is the difference between the second note and the first note, and the last interval is the difference between the last note and the first note. It is seen that IS(TTR2) contains 1 (three times), 2 (once), 5 (twice), 6 (twice), and 9 (three times). That means we can classify the interval sequence of TTR2 in a compact way, without taking the order of the intervals into account, as 1(3) 2(1) 5(2) 6(2) 9(3).

Transformations of Interval Sequences

As a way to add variation to a given TTR it is often modified by one or more of four transformations: transposition (shifting the pitches up or down by a constant amount), retrograde (reversing the sequence so that it plays backwards), inversion (flipping the direction of the pitch changes), and a cyclic shift (splitting the sequence up into two parts, and swapping them). Changing the starting note is equivalent to transposition and it is easy to see that transposition does not make any difference to an IS. That brings down the number of unique ISs in TTRs by a factor of 12 to around 40 million.

Let's have a look at some interesting subsets of that vast amount of combinations: 1-, 2-, 3-intervals, All Intervals, and R12 symmetry.

One Interval

There are only four different intervals that produce a TTR when repeated 12 times. It is the chromatic scale, ascending and descending, and the cycle of fifths, ascending and descending. Their ISs are written as all 1s, all 5s, all 7s, and all 11s. Musically, they are trivial and they are mentioned here only for completeness.

Two Intervals

With two intervals there are 18 ISs after removing duplicate sequences produced by one or more of the four transformations.

With only two intervals in a 12-tone row the sound is somewhat repetitive but even so it gives you an idea about the many characters of TTRs.

Three Intervals

With three intervals there are 327 ISs after removing sequences that can be produced by the four transformations. In terms of variation it is a very rich set and yet it is sufficiently small that it is feasible to go through in full. It is the most logical place to start when you are first approaching 12-tone rows. Here are some examples.

All-Interval Sequences (AIS)

An AIS is a TTR whose IS contains all the numbers from 1 to 11. Since an IS contains 12 elements, and it cannot contain 0 because all the notes in a TTR are distinct, one number in an AIS must occur twice. That number is always 6. Here is an example of an AIS,

and here is its IS,

The set of AIS was the first to receive attention from researchers. It happened at the dawn of the computer age, and in a classic paper from 1965 Bauer-Mengelberg and Ferentz calculated the 1,928 unique AIS. 12-tone rows constructed from AIS sound sort of random in a weird and wonderful way. They are hard to play and hard to memorise.

Interval Sequence Equals Retrograde (R12)

The intervals of an R12 sequence are symmetric around its center. That means if you play the intervals from beginning to end you get the same line as if you play the intervals from end to beginning (backwards). Since the number of intervals is the even number 12, the interval at position 6 equals the interval at position 7; the interval at position 5 equals the interval at position 8; and so on up to the last interval which must equal the first interval. Here is an example.

12-tone rows constructed from R12 intervals have a beautiful balance to them in the sense that you finish the line in a way that is similar to the way you start it. They almost always sound good.

Notation of 12-Tone Rows (TTRs)

The clock notation provides a very good way to visualise a TTR to the extent that for the examples I am not going to bother with any other type of notation except tablature. On the clock a TTR is shown as a set of lines that jumps between the hours until all 12 hours are covered, and it then returns to the starting point which is marked with a circle (this type of notation is not my idea. It has been used in the literature for decades).

1(4) 4(8) 1(3) 2(6) 3(3) AIS R12

TTR Sound Examples

Musically, inversion is a very useful transformation, and all the 12-tone rows on this site, with the exception of All-Interval Sequences, include the inverse after the original. Below are some sound samples that demonstrate the different qualities of 12-tone rows from the classes listed above (click sequences to play the mp3).

Class Sequence Inverse
1-interval All-5s All-7s
2-intervals [3 3 3 7 3 3 3 7 3 3 3 7] [9 9 9 5 9 9 9 5 9 9 9 5]
3-intervals (1-2-3) [ 1 2 1 2 1 2 2 3 3 3 2 2] [11 10 11 10 11 10 10 9 9 9 10 10]
3-intervals (2-3-4) [ 2 2 2 2 3 4 4 3 3 4 4 3] [10 10 10 10 9 8 8 9 9 8 8 9]
3-intervals (3-4-5) [3 3 3 4 4 5 4 5 4 5 4 4] [9 9 9 8 8 7 8 7 8 7 8 8]
3-intervals (4-5-6) [4 5 4 6 4 6 5 5 5 6 4 6] [8 7 8 6 8 6 7 7 7 6 8 6]
AIS [ 1 2 4 7 6 3 6 5 11 9 10 8] [11 10 8 5 6 9 6 7 1 3 2 4]
R12 [9 1 7 6 9 10 10 9 6 7 1 9] [3 11 5 6 3 2 2 3 6 5 11 3]