Category Archives: research

A Visual Representation of 3 against 4 against 5

Visual polyrythm 3/4/5

If you’ve ever been curious how 3 against 4 against 5 lines up…well here it is.


Searching for the Sound of π – Graphic investigations

It is amazing to me that the number π, something that we rely on for figuring out the universe we live in, is an is an irrational number, which means that it cannot be expressed as a ratio of two integers AND a transcendental number – a number that cannot be produced with a finite sequence of algebraic operations (powers, roots, sums, etc.)

Infact, to carry on the calculations of the universe, we need to carry this number out 39 decimal points!

Despite its complexity, we see it when we look at two circles of different sizes. There is a constant between the to circles, a ratio between their diameter and circumference (this ratio is called pi), we can see it, but we cannot mathematically express it. We can intuit it, but we cannot write it down. Crazy!!!

These illustrations below, are part of my search to find relations between this number and its possibilities in music.

Here is just 2 bars of a rhythm based on the number Pi. Can you find where it repeats???

[haiku url=”″ title=”PiRhythm1 by Ben Stapp”]


Graphic representations of Micro-Functional Tonality and Introduction

I came up with Micro-Functional Tonality so that I could use a language that would address more advanced chord progressions, found primarily in late-romanticism, French impressionism and 20th century American folk music in a way that treated these idioms as part of a larger whole, and less as exceptions/extensions of common practice tonality.  These advanced progressions are also found in Renaissance music and Baroque music but they are often put in a category of their own, associated very loosely and unconvincingly to common place tonality.  The term chromaticism is used often to lump these exceptions in a category that sets them apart from tonality.  MFT (micro-functional tonality) treats non-traditional harmonies in the same way it treats ‘tonality.’  The same system is used to evaluate both seemingly disparate systems.

MFT (micro-functional tonality) supposes that tonal relations are really only established on a chord to chord basis and that a major or minor key is a bi-product of a more complex system of inter-relations.

Besides my own private research into the subject, I also write music that demonstrates this system.  In the end, however, I am humbly reminded that this theory is really only a musical, theoretical explanation into the working of my own compositions.  I do however believe that this theory, which I use to compose, helps explain to me, in a straight forward way, harmonic techniques of other composers.