Musical Idea 17: Quantization Styles

01.25.2014 § Leave a comment

This entry expands on the idea in the previous entry: quantization gravity.

Recall the idea of “underlying ideal.” Let’s use pitch as our example. More often than not, the ideal frequencies don’t actually get heard when a quantization lattice, in this case a tuning system, is exerting gravity on them; its pitches have moved toward or perhaps even all the way in perfect alignment with the closest available frequencies in the tuning. For example, if our music calls for an ideal pitch of frequency 1760 Hz but the tuning system only allows frequencies that are multiples of 100 Hz, then this pitch will be quantized toward or to 1800 Hz (it should go without saying that a tuning system can have as many or as few member frequencies as we can imagine, with as little or as much rhyme or reason to them as we desire).

Let’s also recall from the previous entry that more than one lattice can operate on an ideal at a time. Choosing the frequency to quantize to gets a little more complicated in this case. If every lattice is exerting the same strength of gravity, then an ideal note will still simply be drawn to the closest lattice point. But if the strength of gravity varies from one lattice to another, then that may not be the case. If closeness of approximation of pitch and gravitational pull are weighted equally, then if one lattice point is twice as far away but also its gravity is more than twice as strong (than the nearest lattice point on the other side of the ideal), then it will win out.

Here now we will consider four different styles of quantization gravity:

  1. Discrete

    Quantization is winner take all. There is no such thing as tug-of-war on the underlying ideal — strength and distance are considered, and the best approximant is snapped to exactly.When you dial up the strength of the underdog lattice, note that it is unlikely that the pitches switching from the incumbent will happen simultaneously. Suppose that one pitch that recurs over and over in the underlying ideal music is much closer to its closest approximation in the underdog lattice, but has nonetheless been snapping all the way to its not nearly as close but still closest approximation in the incumbent lattice simply because its gravity is so strong. Well, this pitch will be one of the first to switch over to the underdog when it begins to gain strength, while pitches which truly are closer to the incumbent will be the last to switch.

    In the discrete style, the non-simultaneity of the switching of pitches from tuning to tuning suffices to produce the desired fluidity of transitioning.

    The discrete method may be preferable when you desire to rule out all pitches between (not members of) any of your chosen quantization — it will sound much more harmonically coherent.

  2. Blending

    Instead of winner-take-all, each pitch in the original music is quantized to each tuning system’s closest approximant and all of them are sounded together at loudnesses according to their closeness and the gravitational strength. At the point in the discrete style where an ideal would make a switch from one lattice to another, in the blending method the two lattices will be equiloud, 50/50.This style, in contrast to the discrete one, will wreak havoc on harmony, having a propensity for extreme discordance, but has potential to be pleasant (such as in the acoustic beatings of a gamelan).

  3. Aleatory

    The aleatory style, like the discrete, is winner-take-all: only one lattice point successfully asserts itself on a note at a time. However, it boasts an element of chance: the percentage of presence that would be the case in the blending style now becomes the probability that this lattice will win. This one will also sound plenty harmonically coherent.

  4. Sliding

    The sliding style is the only one to permit frequencies which are in none of the underlying ideal nor any of the quantization lattices. This is because the sliding style is the one that works maybe the most like actual gravitational pull: if two lattice points are pulling an ideal in equal and opposite directions, their effects will cancel out and nothing will happen, but if one lattice point is slightly stronger, it will pull the ideal slightly toward it, but not all the way.The sliding style, if you adjust the gravitational strength of your lattices a lot, will result in an almost psychotic amount of glissando. 

Other things to consider when designing your quantization gravity world:

  • The closeness of the frequency approximation and the percentage in the tuning do not have to be weighted equally, though. We could decide that at a particular time or place, strength matters much more than distance.
  • We can also alter the nature of gravity from exponential to linear, or to an s-curve, etc; in the latter case, that is, there may be a certain distance beyond which gravity of a lattice point can have little effect on ideal pitches, but within which it has a very strong effect.
  • We haven’t been specific yet about whether it is only the closest approximation in a given lattice which exerts its gravity on an ideal pitch, or whether we consider that all lattice points exert their gravities at all times. In the latter case, a cluster of pitches in a lattice could all contribute to pulling pitches in, like extra mass; in some cases this could be desirable, in others not.

The final note today is that quantization style is a musical aspect, and you can certainly blend between quantization styles. You can even get meta on it by quantizing the use of quantization styles to certain entities and using different quantization styles to move between those quantizations. Unnh.

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