In the paradox of Achilles and the tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles arrives somewhere the tortoise has been, he still has some distance to go before he can even reach the tortoise.See the linked Wikipedia article for some of the solutions to the problem. This paradox is designed to show that motion is an illusion. We know the argument because it was preserved in Aristotle's Physics where he offers solutions:
There are two ways of looking at time: either as a series of discrete points, or as Aristotle says, "indivisible nows," or as a smooth continuum without discrete points. The basic assumption lying behind most music and musical notation is that time is a series of discrete points which are represented as pulses or beats. The whole notion of a time signature is based on the idea of discrete beats. All music software that I know of is based on the idea of recurring pulses or discrete beats.Aristotle (384 BC−322 BC) remarked that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small. Aristotle also distinguished "things infinite in respect of divisibility" (such as a unit of space that can be mentally divided into ever smaller units while remaining spatially the same) from things (or distances) that are infinite in extension ("with respect to their extremities"). Aristotle's objection to the arrow paradox was that "Time is not composed of indivisible nows any more than any other magnitude is composed of indivisibles." (Footnotes from Wikipedia)
But alongside this concept of time, which seems to provide the interior skeleton, as it were, of music, is the idea of smooth continua. We see this other concept of time in the use of rubato, accelerando, ritardando and even in the idea of "grace" notes that fall outside of the discrete pulses. All these elements go against the Procrustean bed of the discrete beat.
My new piece for violin and guitar, titled "Dark Dream" makes use of the contrast between the two notions of time: time as a series of pulses and time as a smooth continuum. One basic motif and the most important structural element is a two second figure that moves smoothly between repeated notes slower than eighths to ones faster than sixteenths and back again and at the same time a dynamic range between ppp and mf is also traversed. Both of these are smooth continua. They take up a regular amount of time, which they have to do because the two instruments are expressing the motif against one another most of the time. That is, as one instrument speeds up and gets louder, the other is slowing down and getting softer, so they are constantly circling one another. Only at the end do they articulate the motif together. But within this regular amount of time, there are no discrete pulses.
The rest of the piece uses regular pulses. So the basic contrast built into the structure of the music is between music of pulses and music of smooth transitions. Some other music I know of that does something similar is the early "phase" music of Steve Reich, though there the realization is quite different. In pieces like "Piano Phase" Reich has one instrument play a repeating pattern in a fixed pulse while the other instrument, playing the same pattern, slightly accelerates so as to move one unit ahead. This is repeated several times until the two instruments are in unison again. What I am doing is having certain sections of the piece using only smooth continua while others use a set pulse. Yes, there are other pieces that do this, of course! But they do it in various ways that are, to my knowledge, a bit different than I do and the contrasting sections are perhaps not contrasting in the same conscious way.
Let me illustrate what I am talking about with the first page of the score of "Dark Dream:"
The "feathered" beams indicate a smooth accelerando and ritardando and this is combined with a smooth crescendo and decrescendo. In the third and fourth systems we see another kind of smooth transition with the glissandi between two fixed pitches.
There are certainly other composers that have explored these kinds of possibilities! So I don't want to claim that kind of unique innovation. But I hope that I am at least exploring some interesting possibilities in this piece. I am planning a recording of it, so I hope to be able to post that at some point so you can hear for yourselves.
Comments welcome, as always.