Musical Idea 37: Intervallic Dance

06.21.2014 § Leave a comment

If you’re not yet convinced that aharmonics is absolutely bonkers, then consider that there is more out there than tonal movement. How about tonal stretching. This is where you get a new knob which scales the intervals. Let’s call it the interval knob. You turn this puppy to the right and all the intervals between notes get proportionally bigger. You turn it to the left and they all get proportionally smaller until everything is the same pitch. You could have another knob which controls the location of the pitch epicenter onto which all the pitches collapsed or from which they all repel, and that epicenter can be in motion during such stretches. The pitches repelled can simply move outside of human hearing range or be programmed to wrap back around using pitch circularity. More than one epicenter could compete, pushing pitches in both directions.

And certainly we can get this knob for rhythm, too. Unlike with pitch, it would be slightly boring to simply increase the size of all intervals proportionally, since with rhythm that would amount to nothing other than a tempo change. However, if you change intervals independently, or in more complex relationships with each other, you can get some unprecedented musical effects.

For example, if you have a repeating rhythm going X – – X – – – – X – – X – – – – – – , you could have the 3 interval slow to become 5, while the 5 slows to become 7. It’s a different slowing rate: the first interval is slowing by 3/5ths while the second is slowing to 5/7ths. The 7 could speed up to become a 3. Once that’s complete, go another step so that this new 5 becomes the next 7, the new 7 becomes the next 3, and the new 3 becomes the next 5. The intervals keep rotating through each other, and we feel this since their rotation process is slow enough with respect to their repetition in the music (we’ll explore a similar type of time vs. time in a later chapter, Yaw).

This type of transformation where intervals are not changing proportionally but according to arrayed instructions raises a question of implementation (see figure below). You have two options:

  1. Preserve the atomic intervallic unit. In this case, the length of a meter will change, since as the constituent intervals cycle through they will add up to different total durations.
  2. Preserve the meter duration. In this case, the length of the atomic intervallic unit of duration will have to change proportionally to the increased atomic units a meter contains.

Top: for the X–X—-X–X—— (3,5,3,7 -> 5,7,5,3 -> 7,3,7,5 -> loop) example, preservation of the atomic duration results in the meter gradually changing from a total 18 units, to 20 units, to 22, then back to 18. Bottom: If, on the other hand, the meter duration is preserved instead of the atomic, then the atomic durations will gradually change (as the counts of them the intervals take up are also gradually changing!) to keep pace so that metrical divisions keep steady.

One could further experiment with allowing intervals (not individually, categorically by length) to randomly reassign each transformation, rather than going through this simple cycle.

The restriction that every original interval length be made present could be lifted, too, so that by chance some times all intervals could come out as 3’s, for instance.

Or the intervals could follow cycles but different cycles, perhaps even of different lengths.

You could compromise by changing your intervals not proportionally, but still according to a clear pattern. You could, for example, increment each interval up or down by one each. For instance, turn your interval knob to make a 3-3-4-2-4- (16) rhythm turn into a 4-4-5-3-5- (21) one and further to 5-5-6-4-6- (26).

Or the other way into a 2-2-3-1-3-. Once you hit a 1 using this method, you would have to split the intervals (like splitting stock) to proceed. For simplicity and listenability I recommend splitting by 2 but you could do it by 3, 4, 5, or anything really. What I mean is: treat your 2-2-3-1-3- now like a 4-4-6-2-6-, from which you can reduce into a 3-3-5-1-5-. And then as a 6-6-10-2-10- to become a 5-5-9-1-9-. Each time you push against this wall you distort the rhythm in a different way than you were doing moving without 1’s, which can be interesting. But you can also reverse out and move away from 1’s again at any time. From 3-3-5-1-5- you can return to 4-4-6-2-6-, or perhaps sneakily you could feign like you were pushing against the 1 and treating yourself like a 6-6-10-2-10- however move the other direction into a 7-7-11-3-11-!

(top) Plenty of interesting sounds can be gleaned from simply incrementing the size of your intervals. (bottom) However, you can get even more complex effects by splitting your interval. In the second step, we split by 2 to allow reduction of the smallest interval. In the fourth step we split by 3 but instead of reducing we expand, at which point all intervals being a multiple of 2 we can actually collapse by that for our next move.

(top) Plenty of interesting sounds can be gleaned from simply incrementing the size of your intervals. (bottom) However, you can get even more complex effects by splitting your interval. In the second step, we split by 2 to allow reduction of the smallest interval. In the fourth step we split by 3 but instead of reducing we expand, at which point all intervals being a multiple of 2 we can actually collapse by that for our next move.

So far we’ve been gradually changing our intervals whether or not the grid they span is gradually changing or not. This would be the equivalent in pitch of constantly glissandoing, but just as with aharmonics when the notes changed discretely while the pitch knob turned smoothly, we can also adjust the intervals in a punctuated way while the underlying resolution changes gradually.

Suppose, for example, we have a repeating bar of the rhythm XX-X–X–, or 1233. Suddenly, in a jump from one bar to the next, it is replaced with the similarly patterned (isomorphic) rhythm X-X–X—X—, or 2344. Next it can move to 3455, then 4566. Here, the rhythm is simply switching; this alone is not an act of intervallic stretching. However you could add another effect to make it so.

Let’s say that by the time we should reach otherwise 6788 the intervals have gradually stretched back to the initial 2344. In other words, we have a four-step process of punctuated incrementation of the intervals being canceled out by a gradual process by which the first interval speeds by 3x, the second speeds by 7/3x, and the third and fourth speeds by only 2x.

In other words, at the moment the “2344” switches to a “3455”, the rhythmic ratios across the switch will indeed be 2:3, 3:4, and 4:5. However by that moment the 2 will already have sped up by one quarter of the 3x it must speed up by by the end of this four-step process, the 3 will have already sped up by one quarter of its ultimate 7/3x, and likewise for the 4’s. Thus we can feel both the continuous nature of the stretching as well as the incremental relation of the punctuated proportional changes of the individual intervals.

This is not so much intervallic dancing, but it involves non-live stretching of intervals applied in a way kind of more like the later Psychoshuffle chapter insofar as you set a window of effect.

Suppose you have a stretch of music which features a number of different rhythmic or pitch intervals. Find the average of all of those intervals, and change every interval to that. You’ve uniformed the music. It will have the same total duration or pitch span, but be mono-intervallic within that.

You can create a new version of your music now. Choose a percentile. If an original interval is smaller or larger than that, permit it; otherwise use the average.

Once you’ve created this new version, it may be quite a bit larger or smaller than the original, so it may be appropriate to then stretch its entirety back to the same total span as the original; if you do so, the recurring interval — the cap — will be different from the average.


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