On Nontransposeable Series

This is a poem that I wrote for my friend.  It’s partly stolen from a poem he wrote for me. 

I've got a theory - no! really - that I'm 
Chewing on like a bored dog at a chair leg.
And I'm going to rehearse it here
Then I'll send it to you - whether you like it or not - 

This is no fulsome account of the "creative life"
Nor of falling down, per se, but of another way,
Of looking at the same thing; it is just the same
As a person may be a parent or a lover, depending. 

Or the way a mountain is called one thing in this village,
And another thing in that village. When all the while
They are the same thing from the outside, just like
Cathedrals and Observatories, Constellations and Heroes. 


Part I.  Observatory.  I've become diverted by a thing that I am calling the non-transposable series. It is fair enough that Messiaen was so very worked up about limited transposition, and he made a lot of it. But what I attempt, then, (keeping in mind just how much of a priest of modulo 12 I have become, just how much equal temperament enables such technical naval gazing, but as Berio put it, "without even a small mountain on the horizon...") starts with a question of mode. 

It is helpful here for me to loosen the screws on the currently accepted half-step-whole-step conception of the liturgical mode - keep in mind Latin modus, measure, extent, quantity - as Messiaen did, of course, in including "rhythmic modes", but was curtailed in his thinking of intervallic modes.

His modes derived from what is, apropos of the two-faced mountain, both technical (modern, post-enlightenment, "scientific", positivist, logical) and mythical (religious, embodied, spiritual, fanatical) in that he sought to highlight the mechanical features, e.g. limited transposability, all the while being predicated on the step model that finds its roots in European medieval plainsong. 

For my part, and for my own edification, my own "folk science" (I don't care if someone's already figured this out – as I think Stockhausen may have in his “nets”), I wish to leave aside the liturgical/modal roots and go for broke with interval class.  In doing so I have latched on to one thing so far:  if I repeat ordered intervals, either the one alone, or in patterned pairs of two, eg +1,+1,+1, etc, or -1,-2,-1,-2, etc, or +1,+4,+1,+4, etc, occasionally I find a non-transposable series.  

This may seem initially unremarkable because the first one of these series is the chromatic scale.  Then, as you already know:

+1 etc – this is the chromatic scale, and may not be transposed (1 only)

+2 etc - this is the whole tone scale (Messiaen's first mode) and it may be transposed 1 time (2 total)

+3 etc - this is a fully diminished 7th chord and it may be transposed 2 times (3 total)

+4 etc - this is an augmented triad and it may be transposed 3 times (4 total)

+5 etc - this is the circle of 4ths, and may not be transposed (1 only)

+6 etc - this is the tritone and it may be transposed 5 times (6 total)

Given one interval class we come across two non-transposable Series (in bold above).  Note that one may be considered a diminution of the other, in species counterpoint terms, that is, IC5 may be ‘filled in’ by IC1.

All well and good, and something I figured out on a long road trip when I was 19 and trying to keep myself from falling asleep at the wheel.  (It’s a fine adolescent realization, no?)  Messiaen of course provides the step of presenting IC patterns, but does not go beyond IC4 (major 3rd) in his 6 modes, and sticks like dogma with the “step”, IC1 and 2.

Messiaen’s Modes:

1. 222222
2. 12121212
3. 211211211
4. 11131113
5. 114114
6. 22112211

My project now is to take my cue from Messiaen and his world, but leave aside the necessity of the “step” and move forward into the light.  I’ll now move in patterns, starting with a simple oscillation between two interval classes (maintaining order, not mixing + and -).  It gets dull very quickly, so I’ll cut to the chase.  Of the 15 possibilities:

1-2,  1-3,  1-4,  1-5,  1-6,  2-3,  2-4,  2-5,  2-6,  3-4,  3-5,  3-6,  4-5,  4-6,  5-6

Five of these repeated series are non-transposable: 1-4, 1-6, 2-3, 2-5, and 3-4.  It is here where I first draw back the curtain on that silly old wizard, admonish myself at my own foolishness, and notice that, of course these are all non-transposable, because they are all simply fractured versions of the same thing; they all add up to 5 or 7 (the inversion of 5), and therefore are all only elaborated versions of the circle of 4ths. 

It would be timely here to point out that I’m looking at a total series, that is to say, all the possible combinations of interval classes that may create a non-transposable series, are being looked at from either end, as it were.  These simple patterns, this-that-this, are ways of breaking up IC5.  Well, so is the chromatic scale.  1+1+1+1+1, 1+1+1+1+1, etc.

With this in mind, all I need to do is to discover all the possible ways that a 5 or a 7 may be broken into patterns of smaller numbers.   (I recognize that as far as absolute interval is concerned, this may also be multiples thereof.


One thing has become clear.  When it comes to transposition – the way that we navigate around our equal tempered system, the currency–the dollar, the meter, the atom, the axiom, the alpha and omega–is not the chromatic scale.  That’s a trick of the light.  It is IC5, the circle of fourths. 


Part II. Cathedral.  I have been digging in the dirt because my body needs it, and what my body needs my mind needs.  At the old Highfield House in Dennistoun, Glasgow, later a council day nursery, before it was purchased in its entirety by friends, I have been spending my time digging out the old foundations of some old structure.  It was covered in brambles and about 3 inches of root-networked sod.  It’s hard, sweaty, grimy work.  I often pull hypodermic needles out of the earth (in my leather gauntlet-like gloves!) from the abandoned days when it served as an addict’s shooting gallery.  Underneath all this is some past person’s labor of love.  A solid foundation of brick and stone.  We’ll build something on it again. 

I’ve also been teaching myself to play the various non-transposable series, as one would play scales. The combination of activity means that my keyboard gets dirtier and dirtier.  These series are so much unlike a scale; each of the series in the group with two interval classes (below) is circular, arrives where it started after touching on each note twice.  I have done this for some time with less awareness in what have long called “the circle of thirds”.  See video 1.  circle of thirds.  This is simply one of my non-transposable series, 343434 etc., although I arrived at it via harmonic extension in jazz.  A major or minor triad sets up a pattern that may be extended to the 7th, then the 9th, then the sharp 11th and the 13th, whilst maintaining that 343434 (minor) or 434343 (major) pattern.  The assumption in jazz is that a 15th is simply two octaves up, and that’s that for your extended harmony.  However, if one remains true to the pattern, we get the #15, which continues a kind of looping eternal triad.  See video 2. Infinite triad.

VIDEO 1: Circle of Thirds

 VIDEO 2: Infinte Triad

The other patterns I’m practicing, with their inversions, are 1-4 and 2-5 (which I find harmonically dull). 





Example 5: 3-4


The next video, Video 3. +4+3 inside the octave, demonstrates what happens if you keep the ordered +4+3 series, played like an arpeggio, inside an octave.  And Video 4. -4-3 inside the octave is the same thing in inversion.  These are written in example 5 above.  Of the other examples I have started to learn 1-4, shown in Video 5 and Video 6.

VIDEO 3. +4+3 inside the octave

 VIDEO 4. -4-3 inside the octave Playing with the infinite third - in 10ths (compound 3rds), and with chords

VIDEO 5. +1+4 etc inside the octave

 VIDEO 6. -1-4 etc inside the octave


What use is this to being a composer? or an improviser? or educator? researcher? lover? maker? doer?  What use is the telescope to seeing?  The labratory to understanding?  What use is the church to worship?  What is the use of the technical to the poetic?