Tuesday, June 07, 2005

MM IV: Relativity of Pitch to Tempo & Interval to Rhythm

I'm going to make an asside here that I've been thinking about for a few days now. One of the intriguing areas of musical research over the past fifty years or so has been the probing into the temporal continuum between pitch to tempo and interval - or harmony - to rhythm. The basic premise is this: If you have a piece of music composed in the key of A minor or A major for instance, and your pitch relativity is based on an A at 440Hz, there is a continuum that you can get by transposing that note A at 440Hz down eleven octaves that will take you through the "blind spot" below human hearing, and then back again into human perception where the former pitch will emerge as a simple series of rhythmic pulses which will strike the ear as eighth notes at 110 beats per minute. Whether or not there is a subconscious link between keys and tempos has yet to be difinitively proven, in my estimation, but it remains curious that 110 BPM is a popular song tempo that is quite pervasive, just as the number of popular tunes comosed on the pitch level of A is. This is purely anacdotal evidence to be sure, but my gut tells me there is something to this idea. Of course, dance rhythms are probably also related to the mean tempo of the human hearbeat, so I wouldn't expect there to be any cut-and-dried conclusions emerging from these inquiries anytime soon, but I think it is a very valuable thing for a composer to consider.

If we take this concept and apply it to the superparticular ratios that appear in the Natural Harmonic Overtone Series, the former intervallic combinations do not reemerge into human perception as a series of even pulses, but as repeating palindromic rhythms (Rhythms that read the same forwards and backwards) with dividable axes of symmetry. These rhythms are the result of the interference between the two wave periodicities of the notes involved in the interval. Regardless of the pitch level of the speciffic interval, the rhythm will always be the same. Now, previous evaluations of my intellectual proclivities make me aware that I am a visually oriented thinker versus someone who can take numbers and extrapolate patterns from them - and this is true for a remarkable percentage of musicians, especially composers - so a useful way to look at this phenomenon is in standard musical notation, which can easily be done. The following example was put together using a technique from Joseph Schillinger's "Theory of Rhythm" book, which is the first chapter in his monumental "Theory of Musical Composition".



Starting with the octave and it's ratio of 2:1, the process is to first find the common denominator between the two terms in the ratio. Since 2*1= 2, our common denominator is two in this example. You can set the time signature up to be the common denominator - as I have done in the first three examples above - or you can use either the major generator (2 in this case), or the minor generator (1 here) as I did in the final two examples which are longer and more complex. Below that put the major generator, which is a half note here, and under that put the minor generator, which are a pair of quarter notes. The resultant rhythm is where there are attacks by either the major generator or the minor generator. The first attack is always accented because that is the only place where the two terms have simultaneous attacks, and that is also the point at which the pattern repeats.

As we can see from this experiment with the octave ratio of 2:1, the rhythmic pattern that emerges is an accented attack followed by an unaccented attack. This is your basic Rock & Roll backbeat (Tells you something about the simplicity of pop rhythms, regardless of the surface ornamentations, no?).

Repeating the same process with the perfect fifth's ratio of 3:2 we get a CD of 6, a major generator of 3, and a minor generator of 2. As I said, the resultant rhythm may be grouped into two measures of 3/4 time or three measures of 2/4 time. To use this in a rhythmicly structured phrase in 3/4 time, for example, you could use the first half of the rhythm as the antecedant, and the second half as the consequent penultimate to the cadence that terminates the phrase. It would look like this in an eight bar phrase if a 1= 1/4 note, a 2= 1/2 note, and a 3= a dotted 1/2 note:

||2, 1|2, 1|2, 1|2, 1|2, 1|2, 1|1, 2|3||

This is simple but highly effective and corresponds to the rhythm of many popular and traditional pieces that have existed for over a millennium. Those ancients just discovered this intuitively. The same thing can be done with the resultant rhythms generated by the perfect fourth's 4:3 ratio, the major third's 5:4 ratio, the minor third's 6:5 ratio, or any other interval you wish to calculate a resultant rhythm for. In either the time signature for the major generator or the minor generator at that. This is but one way these rhytmic patterns generated by the ratios in the Natural Harmonic Overtone Series can be applied; the limit to how many different musical levels you can apply these on is only governed by your imagination.

Beyond this, the rhythmic resultants for triads and seventh chords can also be calculated in this manner, though for really complex harmonic structures with several genarators I'd want a computer program I could plug the terms into because I personally hate working with numbers. Someone out there with a decent knowledge of C++ or even BASIC ought to be able to put something together for this. It sure would be a boon to blockheads like me with limited inherant numerical abilities (Hint, hint).

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