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.
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:
- 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);
- 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;
- 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.
- 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:
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.
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 a = 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.
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):
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.