The Fredösphere

Follow these links for Nesting Instinct, Parts the Ith and IIth.

I kept my promise; don't keep your distance. I have completed the 2-part, quadruply nested counterpoint example. You can listen to a sound file in mp3, and you can see the score in pdf.

Remember that, in the score, there are 4 treble staffs, but they are all for the one soprano part. That's why the measures are all identical vertically. Likewise for the bass. The redundant parts simply represent new layers of canonic relationships. I drew diagonal lines to help you better see each of the four canons in their various entrances.

The result of all these nested canons is that, by measure 34, the soprano part is echoing measures spread throughout the piece: ms. 33 in the bass, which in turn echoes ms. 31 in the soprano, etc. etc. As I guessed (and proved empiracally in one frustrating evening), I had to begin at the end and work backwards. I had to start with the measure with the most canonic freight; otherwise it would need to agree contrapuntally with several contradictory measures.

But that's not quite true. I wanted the first four measures of the piece to be, well, let's just say I wanted it to have a certain distinctive sound. (Listen for yourself to find out what I mean.) It turned out I could write those ahead of time without creating an impossible puzzle.

After measure 34, a kind of shearing occurs and the various canonic relationships become contradictory. Yet that measure doesn't work as an ending. I had made sure measure 34 contained candential material, so I simply added mearsures 35 and 36 and put a tonic chord there as an abrupt close to the piece.

Okay, I solved the puzzle, but does it work musically? I see failure and success here. The canonic relationships don't really grab the ear like I hoped. My original, doubly nested example was much more successful in that regard. The problem is, since measures are reused so much, they don't sound distinctive enough at the canons' entrances. Nevertheless, the mathmatical system gives the whole peice a rigor and form which (to my ear) is detectable on a subjective level. If I may dare make this judgment, I think it's the most tasteful 36 beats of music I've ever written.

Oh, but that brings us to the First Law Of The Fredösphere: Good taste is not enough!

Labels: NestedCounterpoint

My research into the exciting new field of nested counterpoint has made progress and today I will share my discoveries. (If you are going to need a glossary for the counterpoint terminology that's coming up, here's one.) I have decided to attempt a two-part canon at the octave. (I toyed with a tritone interval just to show off but decided not to push my luck.) I intuited that a nesting scheme based on the fibonacci sequence would be fun and might be aesthetically pleasing. It turns out, that is the minimum amount of time between entrances of the nested canons, for reasons which I will explain later if I can find a way without this post colasping under its own weight.

This crude chart shows what I mean. I will call the two voices Soprano and Bass, since Merrill and Lynch are probably under some kind of trademark protection. The brackets above the dashed line belong all to the soprano and represent the various entrances of its nested canons. Below the dashed line are those of the bass. At the bottom are beat numbers. I decided to number the beats backwards for reasons which seem compelling to me and may become obvious to you but I just don't feel like trying to articulate.

The soprano is the leader and begins at beat 34. The bass line follows at a time interval of 13 beats. This means that, when everything comes to a halt at beat "0" the base will have been in for 21 beats. These parts are the conventional, outermost canon and are represented by the A brackets.

The first nested canon begins at the B brackets. The leader-follower relationship switches back and forth; I really haven't worked out if that is switching is required, but at least I know it works. The bass' B theme is 13 beats long and the soprano will sing just the first eight beats of it.

Please realize that these inner brackets do not represent new voices entering. They simply represent new canonical relationships coming into play. Therefore, in the bass for example, the first beat of B is exactly the same music as the 9th beat of A (and must be since they are sung by only one voice.)

In a similar way, the soprano then leads with canon C which is five beats long (and the bass sings the first three beats of it) followed by the two beats of canon D, lead by the bass, with the soprano singing only one beat of it.

By the time we reach the end, we have a beat that is heavily burdened with canonical relationships. Extrapolating from the standard rules of how to write a canon systematically, I can see that I will have to start with the last beat and work backwards. This will guarantee that I will end up with a canon that follows all these rules. This working backwards will not unfortunately guarantee that the results will be beatiful. But then, counterpoint was never about beauty, was it? It's all about showing off.

My next installment should be much easier to follow because it will use a musical score instead of some goofy chart. So stay with me.

Sound of a hand briskly slapping your face several times. Odor of smelling salts.

Stay with me! Stay with me!

Labels: NestedCounterpoint

I started writing some 4-part counterpoint as an exercise. (See it here in a pdf file.) I stumbled across a contrapuntal technique I'm geeked about. The example shows the first two voices entering, with the tenor imitating the bass at a 30-beat interval. Now look carefully at the two sections outlined in the fourth measure: the bass is imitating the tenor at a 3-beat interval. This happens while the tenor is imitating the bass -- in other words, the counterpoint is nested.

Now, I am under no illusion that I have invented this. But I don't recall anything like this being taught in any counterpoint text I've seen, and I don't recall noticing it in any music. OTOH it wouldn't always jump out at you where it did occur. I would love to know if other examples exist.

Naturally there is no theoretical limit to the number of times the nesting occurs. In this case it was easy to do because the temporal intervals involved are different in scale, by a factor of 10 (30 versus 3). If the intervals were very close (say 6 versus 3), it might work but not be recognizable as such; it would probably start sounding like the same few notes repeated several times.

I'd like to take a crack at several nested levels. No promises how soon I do it. I'd be pretty happy if I could just learn to write counterpoint as good as Dmitri Shostakovich.

Labels: NestedCounterpoint