Findings #2 - Intervals (part one)

“The saxophone is an interval machine” - Steve Lacy

I thought I would make this blog post to follow up and go a little deeper on a connection I made in last week’s post concerning my suspicions about the connection between Steve Lacy’s compositions and the influence of Nicolas Slonimsky’s ‘Thesaurus of Scales and Melodic Patterns’. This isn’t going to be an examination of Lacy’s music or the historical evidence for his possible exposure to Slonimsky’s work, rather I’m going to apply techniques from the Thesaurus to an exercise from Lacy’s book ‘Findings’ that I’ve been experimenting with that I think unlock the sound of Lacy’s compositions and improvisations.

First, a disclaimer; the Thesaurus is a huge book. I have been working through it off and on for years (I got my first copy nearly 10 years ago, long before I could hope to understand it) and I feel like I am still scratching the surface. It is neither a collection of technical exercises nor a completely unified theory of making music. Instead the book is a thorough reference book of hundreds of possible ways of generating musical information with a mathematical, self contained logic to express all of these permutations. It’s a tough nut to crack but I’m only going to be grappling with one element, which happens to be the fundamental mechanism of the book but I digress, which I will leave to Slonimsky to explain for us:

“Scales and melodic patterns are formed by the processes of Interpolation, Infrapolation, and Ultrapolation. The word Interpolation is in common usage; here it signifies the insertion of one or several notes between the principal tones. Infrapolation indicates the addition of a note below a principal tone, Ultrapolation is the addition of a note above the next principal tone. Infrapolation and Ultrapolation result in the shift of direction, with the melodic line progressing in zigzags. Infrapolation, Interpolation and Ultrapolation may be freely combined, resulting in hyphenated forms: Infra-Interpolation, Infra-Ultrapolation, and Infra-Inter-Ultrapolation.” (Slonimsky, 1947)

A large portion of the Thesaurus deals with creating scales and melodic patterns by applying these embellishments to various equal divisions of octaves. For example the equal division of one octave into two equal parts would be C-F# (this can be observed in an example I shared in my post from last week found here). Much of what I’m going to share in a minute here applies to unequal divisions of the octave and for that we have to return to Lacy’s book ‘Findings’.

In Chapter 7 of ‘Findings’ Lacy lays out an exercise with which to practice every interval available on a given instrument. He constructed this exercise by writing out every single interval on every distinct pitch on the soprano sax from a minor second starting on low Bb (of which there are 32 if one presupposes the average natural range of the horn from low Bb to high F#) all the way to minor tenths (which there are 18 of) for a total of 373 intervals. He then took all these intervals, cut up the piece of paper and shuffled and reassembled them into a random order.

Lacy did this because he believed that practicing all of a single type of interval chromatically became monotonous and too homogenous so he did this to randomize the process and to treat each individual interval as a distinct unit. I think this is crucial when practicing a wind instrument where every individual pitch on each individual instrument has slightly different acoustic properties so this exercise allows you to really hone in on the specifics of those sounds.

Now I intend to revisit this chapter of ‘Findings’ in more detail as I think there is a tremendous amount of musical information and inspiration one can glean from working through the exercises he lays out and I have spent a lot of time experimenting with ways to get the most out of them. For now however we’re going to combine Lacy’s interval practice with Slonimsky’s embellishments to create some unusual shapes that you can use for practice, composition, or improvisation.

I’m going to demonstrate this method using an interval of a perfect fifth between the notes D and A, though it can and should be applied to any interval:

Diagram A

First we’ll start with the interpolation of one note:

Diagram B

These first five examples fit comfortably into the examples set in the Thesaurus but begin to diverge with the next example:

Diagram C

With unequal divisions of the octave (such as a perfect 5th in this case) you reach a point with interpolation where it crosses over into ultrapolation (see the first bar of diagram C ^), this is because the relationship between the two pitches is not equidistant as in the case with equal divisions of the octave which causes the interpolated notes to have a different relationship to the other primary note in the original interval. Once we reach the third bar of diagram C we are no longer interpolating but instead we are ultrapolating but with a twist.

If we were to ultrapolate this interval it would look like this:

Instead we get this:

This sounds more organic to my ears and opens up a lot of unpredictable interval relationships that you might not otherwise stumble upon. Once you have interpolated to your heart’s content you can then apply ultrapolation, infrapolation, Infra-Interpolation, Infra-Ultrapolation, and Infra-Inter-Ultrapolation. I've given you the first one here:

The purpose of this exercise isn’t to work through every single iteration on every single interval, it might take a decade to get through them all, instead this is just a process to break out of your usual shapes and scales and to oriented your ear on a more intervallic approach rather than a diatonic or chordal one. As you go through this process on some different intervals you will find some sounds that you like more than others, often I find diatonic patterns I wouldn’t have otherwise identified that I really enjoy. At the end of the day this little activity is more of a spiritual successor to Lacy and Sloninsky’s work than a literal attempt to emulate their process. I think there is a lot to be gained by asking questions and trying to uncover how our heroes might have come to their sounds and approaches.

Let me know in the comments if you try this out and find other ways to use these concepts in other ways. Happy practicing!