Structuring ‘warped passages’ part 2

In my last post I spoke about the very elementary design of warped passages – the ‘isofingering’, the expansion durations, the tempi. In this post I want to pick up where I left off: the various parameters that I ‘pre-structure’ in each section and which form the basic identity-relations between sections of the work. I’ll go through them one-by-one.

Determinacy level

This parameter controls what is determined and what is indeterminate or resultant in the score – mostly with regard to pitch. I numbered the basic degrees of this parameter like this:

  1. Pitch is determined, all sound production parameters are subordinate to the production of the pitch (alongside other traditional parameters like dynamics, although these too are subordinate to the pitch);
  2. Pitch is determined, but a secondary parameter called ‘instability’ (including sound producing parameters: air pressure, direction of air stream, and rolling of instrument) acts directly on the notated pitch to push it towards jumping up or down to other strata of underblowing, to destabilise the pitch between layers, or to simply bend the pitch and change its colour;
  3. Pitch is not specified, only the fingering and duration. Instead, what is specified is air pressure, direction of air stream, and rolling of instrument towards or away from the upper lip. These sound production parameters are independent of each other at this degree of indeterminacy, unlike at the second degree. This means that the pitch structure, while partially determined by the fingerings given (each fingering can really only give 3 or 4 different pitches, and slight alterations of intonation), the actually sounding pitch at any point will be more or less indeterminate (it will change from flute to flute, player to player, performance to performance), and most certainly resultant: the effect of various parameters determined by the score, as well as the performer’s own interpretational choices.
  4. Only fingerings and overall duration of the section or sub-sections are determined. Everything else is left up to the performer, which means, of course that these sections are basically improvised.

Across the 14 sections I decided that there would be 6 sections at degree 1, 3 sections at degree 2, 3 sections at degree 3, and 2 sections at degree 4.

Presence of pitch layers

The next thing to structure is the particular underblowing layers that I want present in each section. Obviously, in sections where determinacy is at degree 3 or 4 (where pitch is not specified), I have to count them as having all layers present. For the other two though, I either have total control over pitch layers present, or the at least control over the ‘centre of gravity’ of the pitch domain. So you get the picture of what I’m talking about, here’s a rough fingering chart:

Screen Shot 2016-04-04 at 6.30.53 PM

You probably can’t read that particularly well, but the top pitch is the fingered one, and you’ll see that for about half the fingered pitches there are 2 other layers below, and for the other half there are 3 layers. What I did was to label the top (fingered) pitches ‘layer 1’, the next set of pitches below are much less stable and only available in half the fingerings, these were called ‘layer 2’, the next-to-bottom layer was ‘layer 3’, and then the very bottom is of course ‘layer 4’.

Seeing as the 5 sections that are either determinacy degree 3 or 4 have all the pitches in them, I organised the other sections to focus mostly on individual strata. So the 6 sections of determinacy degree 1 had the following pitch layers: 1; 3; 4; 3,4; 1,2,3; 1,2,3,4. The 3 sections of determinacy-2 had only the single layers (of course these contain other layers by way of the ‘instability’ parameter): 1; 3; 4.

The other major parameters that I pre-structured in this way concerned the temporal or rhythmic dimension of the work: number of fundamental rhythmic groups, temporal tendency (or denominator), degree of ‘refraction’ or ‘mediation’ of this tendency (or numerator).

With these parameters I did not try to determine them in advance, but, once I was clear in my mind on how they would be applied and the overall scope they offered, I could choose quite freely within them as I began each section. More on this later.

Number of rhythmic groups

This simply meant the number of groups into which I would divide the pulses of each section (determined by the duration of the section and the number of fingerings within it). Seeing as there are 30 fingerings (and rests), I decided to have the groups always as factors of 30 (so 5, 6, 10, 12, 15, and 30). This parameter has a strong impact on the next two parameters, insofar as it determines the degree of ‘resolution’ of a section, and thus is a factor in its rhythmic fluidity.

Temporal tendency

This parameter determined the number of pulses (at the given tempo) within each rhythmic group (which usually, but not always, coincided with the bar). I decided to have 5 degrees of this: 1. exponential acceleration; 2. slower acceleration; 3. totally flat, equal division; 4. slow deceleration; 5. exponential deceleration. In reality, this was not adhered to in any systematic way, particularly with regard to the distinction slow/exponential. My approach was to create a formula of the kind y=x^a, generate a sequence of the number of rhythmic groups in the section, and then fit that proportionally to the overall number of pulses in the section. I would vary the a to create the degree of curvature of the section:

Obviously 0 would give a completely even distribution, since any number to the power of 0 is 1.

Where a =>0<1 the sequence would have a slow curve that was quicker at the start and slower at the end.

Where a = 1 there was a sequence of a constant increase (of course this does not mean at all that it will be perceived as constant, as Stockhausen pointed out in …how time passes…).

Where a = >1 there was a properly exponential increase.

These (apart from a=0) generated decelerating structures (since the durations increased). To get accelerating structures I could either go into negative exponents, or simply invert the positive sequences, which I chose to do mostly, out of ease.

(I also on occasion added another constant integer to boost the bottom of a curve while keeping its exponential character).

Degree of ‘mediation’ of temporal tendency

For this parameter I would do basically the same as the previous one. I would generate a sequence of numbers according to a certain simple exponential formula (with varying degrees of exponentiality, if that’s a word). Rather than use this as a basic tendency or sequence, though, I use it as a set of proportions, an abstract pool of numbers that have a particular proportional relation. Which is to say I would permute the sequence generated by the formula, and not apply it in its original form. This permuted sequence would be applied over the top of the ‘temporal tendency’ as the numerators in what is now a series of polyrhythms. I chose to keep the total pulses for the numerator sequence to be the same as the denominator, which would mean the realised rhythm would oscillate around the pre-established tempo of the section – mediating or refracting the overall tendency of the section.

Here’s an example which hopefully will make it a bit clearer. A section of 30 semiquavers with 6 groups at tempo: quaver=164.

Screen Shot 2016-04-04 at 7.42.40 PM

As you can see in the final row of the spreadsheet, there’s still a tendency towards acceleration, but it has been mediated by the number sequence that is in the numerator.

Here’s what it looks like in the score (minus the other parametric structures, which are irrelevant here):

Screen Shot 2016-04-04 at 7.48.01 PM

Of course at such small sections like this, you could intuitively do these number sequences without having to go through the mathematics. But once you have sequences of 30 or 60 or more, the maths becomes very useful indeed, and it all allows you to have a much clearer control over the temporal logic of the work.


Structuring ‘warped passages’ part 1

My previous posts (here and here) should give a taste of the basic concepts that I’m working with in my flute solo warped passages. So here I wanna get technical.

Before I elaborated any structure, the first thing I did was to workshop the sonic and technical space of the piece with Hannah in a couple of sessions. I quickly realised that since the underblowing technique the work was based on provided many ‘passages’ through the same set of fingerings, the work could well be a sequence of different approaches to the one ‘isofingering’. I roughly thought would be made up of 20-odd fingerings.

Each instantiation of the isofingering I decided would be of a different temporal expansion (inspired by Stockhausen’s method of formula composition). But the question was how to get the right proportional relations between the durations of iterations. I wanted to have a slow exponential increase from the smallest to the largest expansion since this would, for perception, represent a more real differentiation than an increase by stable number. Here’s the sequence that I came up with:

Screen Shot 2016-03-28 at 8.24.06 PM

In which the degree of increase of the increase (starting with 0.5 secs) itself increases arithmetically by 0.25 secs each iteration. Such a formula could be mathematically represented, of course, but I was still doing things quite intuitively at this stage (something that has very much changed while writing this piece), so I just tried a couple different ways of having an increase from one iteration to the next until I reached this sequence. This meant that the total number of 14 iterations was a result, not predetermined (except that I wanted more than 10 iterations and less than 14). Likewise the total duration of the work, 9’07.5”, was a result of this experimentation, rather than planned in advance (except, also, that I wanted it to be less than 10 mins and greater than 8 mins).

Ok, with that done I decided that each iteration would have 25 fixed fingerings and 5 rests in variable positions in the sequence. This is the ‘isofingering’:

Screen Shot 2016-03-28 at 8.47.25 PM

(In which the top note is the fingered pitch and any lower notes are possible underblowings… Different flutes might have slightly different results, and the specified accidentals aren’t exact, only more or less accurate but dependent on various factors).

This was quite ‘intuitive’; I think it has a good balance of disjoint motion, stepwise ascent and descent (which are ‘warped’, as you can see, in the underblown layers).

I also then arranged the expansions in a sequence that I liked:

Screen Shot 2016-03-28 at 9.01.27 PM

Which is the interpolation of two divergent sequences: 5,6,4,7,3,8,2,9,1,10, and 13,12,14,11.

To determine tempi for each section it’s quite simple: you just divide the number of fingerings (30 in each case) by the duration of the iteration, and then multiply by 60secs to get the tempo (which I would give in quaver = x bpm). In the case of very fast passages, I would divide the final result by two and give only a semiquaver to each fingering; in the case of very slow passages, I would double or quadruple the bpm and then give either a crotchet or minim to each fingering.

For example. The 5sec iteration would be: 30/5=6; 6*60=360; 360/2=180. Therefore quaver=180 and each fingering takes 1 semiquaver.

Another example. The 79.25sec iteration would be: 30/79.25=0.3786; 0.3786*60=22.7129; 22.7129*4=90.8517. Therefore quaver=91 and each fingering takes 1 minim.

After this I began basic determinations for each section (and here’s where it get’s interesting I promise). These are the parameters that I structured for each iteration at this level of the work:

  • determinacy level (1=pitch determination; 2=pitch determination with deviations; 3=action notation determined, pitch result indeterminate; 4=fingerings determined, other parameters improvised)
  • pitch/underblowing layers present
  • number of rhythmic groupings (5,6,10,12,15,30)
  • temporal tendency, or denominator of rhythmic groups (expansion, stable, contraction)
  • degree of ‘mediation’ or ‘refraction’ of the tendency, or numerator of rhythmic groups (even, uneven, radically/exponentially uneven)

In my next post I’ll pick apart each of those parameters and show how they are structured.


The meaning of ‘warped passages’

Since returning from Melbourne I’ve been more or less immersed in my new solo flute piece for Hannah, warped passages. It’s coming along surprisingly quickly, and this may be because it has the clearest structural logic that I’ve yet come up with. A full draft will be done within 2 weeks and then there’ll be some time for some testing things out with Hannah and her feedback, before the ‘final’ version takes shape on paper.

I’ll try to outline the basic structure of the work and how I’m writing it in my next post, but first a thought on the title.

After deliberating for quite a while about the title for this piece, I have decided (at least for now) to call it warped passages, after the very enjoyable book by theoretical physicist Lisa Randall (who coincidentally wrote the libretto for Hector Parra’s Hypermusic Prologue). Naturally I’m not attempting any direct translation of the science (which I know only as a lay person), but instead taking a metaphor or two from it. Really there are two essential intended meanings behind the title:

  1. The warped nature of the passages through the entangled dimensions (the ‘characteristically interdependent’ parameters that the flute, in its particular use in this piece, installs). Warped because the space isn’t ‘flat’ but ‘curved’ or ‘curled’ in different ways, either temporally in the sense that each section of the work is shaped by a specific time-structure that is often tendentially accelerating or decelerating (which is kind of like the ‘curved’ space theorised by general relativity), or in the sense that certain parameters kind of ‘curl’ around other parameters like in many extradimensional theories. (It is also, more mundanely, but perhaps just as legitimately, a basic description of the piece: the solo will comprise a series of ‘warped’ (or ‘wonky-sounding’) musical ‘passages’).
  2. The other metaphor which I really like, and which is a little more abstract has to do with the warped passage that the imagination has to take when trying to reincorporate, into the four-dimensional world of our experience, higher dimensional geometry, which can be mathematically determined. For example the drawing of tesseracts (4-dimensional ‘hypercubes’), which Lisa Randall explains as though a 4-dimensional (5-dimensional including time) cube passed through a 3-dimensional space and we tracked the changing view we had of it across the time it took to pass through (much as, if a 3D sphere passed through a 2D page, you would see over time a growing and shrinking circle). This has two main resonances for me:
    1. The warped passage from an extra-dimensional instance of the work to a lower-dimensional instance: e.g. from the multi-dimensional object of the instrument, to a lower dimensional representation of a work in its score (which seizes upon and structures a number of parameters, but leaves a whole number of others as indeterminate); or from the multi-dimensional structure as represented in the score, to a kind of synthetic expressive imagination in the mind of the performer; or from the extra-dimensional performance itself to the limited cognitive filter applied by the listener in attempting to grasp the work.
    2. On an ‘extramusical’ level, which relates to the work less explicitly, but only through a kind of structural (and spiritual?) affinity, there’s the idea of complete equality and freedom (the ‘Idea of Communism’ as Badiou would call it), an idea easy to imagine in abstraction, but incredibly difficult to grapple with and integrate into the concrete world that we know. Thus the warped passages the human subject (collective or individual) must take to construct a model, in the real world, that approximates, as close as possible with the dimensions and ranges we have available, this ‘communist’ ideal.

Well, I hope some of that comes across in the work itself.