Music Mathematics Part IV. Timeless Music Theory: composing with fractals on purpose

Read Parts 1 through 3 first  

I think pure noise, as wonderful as it is, provides too easy an answer to our challenge to develop timeless music. While it is infinitely variable, it always sounds the same.

Let’s come up with timeless music that has discernable patterns that can’t be fully described in the two dimensions of amplitude and time, even if we use them to reproduce it.

We’ll use a means by which music was stored and reproduced prior to audio recording. Classical musical scores also have two dimensions, with time again running from left to right, at least, but rather than showing the amplitude of air pressure in the vertical dimension, we indicate frequencies to be played or sung: for example a dot on the second space in a treble clef means a note with that changes the air pressure 440 times a second.

Classical composed music works if these notes, which are packets of frequencies, follow one another fast enough so that there is a perception of a flowing pattern, the way a cartoon flipbook looks continuous if your thumb is fast enough.

Since classical compositions use an up / down dimension with notes, and a left / right dimension of time, let’s produce music with the same pattern at multiple scales, similar to using different size rulers to measure the coast or duration of the fractal Sinatra in Part I.

I’ve selected a five note pattern that Arnold Schoenberg uses as a basis of his second string quartet. The notes are A Bb A B D, or on the piano, intervals of up one note, back down one note, up two notes, and finally up three notes.

We can transpose these intervals around the piano. It’s not only easy to hear the relationship, but moving up three notes from a low note is the same sonic distance as moving up three notes from a high note (this is based on math associated with Pythagorus, and may be a future blog topic). For now, adding or subtracting notes to a pattern is like climbing or descending steps near the middle of a long staircase, and moving up one step, back down one, up two and up three changes your relative position similarly.

Here’s an example with Schoenberg’s pattern played on low notes on the piano one time, while the high notes play the pattern five times as fast, so that there are five repeats, and we hear the same melody at two time scales. The fast phrase adds the frequencies to the slow notes, so the melody changes to higher or lower notes, in the way someone appears taller or shorter depending on where they stand on the staircase.

Let’s expand on this and listen to the PubliQuartet play this five note phrase at four different time scales: this is the Fractal Variation from my String Quartet #3, The Essential, which is based on math manipulations (Download the score). The quartet plays the phrase in four voices, one in which the five note phrase repeats 125 times, another slower scale where it is played 25 times, another 5 times, and finally one only time. Despite the fact that there are only four time scales, and every note uses the same steps in that five note phrase, the notes add to each other so that the music constantly changes and nothing precisely repeats.