Math as Art: Part 3

For the past 3 days, I have been obsessing over the letters: ABCD.










Every axiom I used combined with every “clever” set of rules, produced the most monotonous dribble I have ever heard. I was ready to admit defeat and go back to writing cute little melodies when I realized I had forgotten the most important factor!!!  When I was first conceiving of this project, one of the first thoughts I had was this:

How many unique pitch classes at a minimum will allow a piece of music to be interesting?

My answer? It depends.

One pitch is all that is necessary if the piece is minimalist and the goal is to explore the timbral and sonorous  shades to a single pitch.

Two pitches are all that is necessary if your goal is to exploit the relationship between the two.

However, if your goal is to write a melodic, tonal piece of music inheriting the functions of the classical western tradition….

No less than Five pitches are necessary.

I had assumed this intuitively, but had not performed any experiments to prove it for myself. Well, I can assure you that the last 3 days proved to me without a doubt that no less than 5 pitches are absolutely necessary to create a piece of music that is melodic, tonal, and following in the traditions of western classical music.  With my new assumption, I set out creating a new set of 5 pitch axioms and a set of rules which both imitated formal structures from classical music.  Remember, my goal is to combine classical mathematics and classical music theory in their pure forms to reach my hypothesis that it can be done without threatening either.

So, I chose to use a musical form that is self-referential for the axioms: Rondo




For the rules, I used the idea of building harmonic structures:

ruleA = “AE”;
ruleB = “ACE”;
ruleC = “BD”;
ruleD = “CBDAE”;
ruleE = “EA”;
What resulted was a two-voice counterpoint that has a clear tonal center, with clear melodic themes that repeat and interact without ever becoming monotonous.

I’ll let you hear a short excerpt for yourself:

Now, to “wash” it with my human hand, adding intuitive musicality such as dynamics, phrasing, rhythmic variation, and articulation.


Math as Art: Part 2

Immediately following part 1 of this blog series going live, I received an assortment of cool links to and recommendations for other related projects/books/pieces. I have to say I was very happy to see the interest in this topic! What I wasn’t prepared for was the reaction I would get from my Professor{Discrete Structures, Network Theory, Cryptography, Complexity Theory}.  While excited by the project and the idea of expressing abstract mathematics through sound/music, he had issue with my choice of algorithm:

The Mandelbrot set M is defined by a family of complex quadratic polynomials

P_c:\mathbb C\to\mathbb C

given by

P_c: z\mapsto z^2 + c,

where c is a complex parameter. For each c, one considers the behavior of the sequence

(0, P_c(0), P_c(P_c(0)), P_c(P_c(P_c(0))), \ldots)

obtained by iterating P_c(z) starting at critical point z = 0, which either escapes to infinity or stays within a disk of some finite radius. The Mandelbrot set is defined as the set of all points c such that the above sequence does not escape to infinity.

A mathematician’s depiction of the Mandelbrot set M. A point c is coloured black if it belongs to the set, and white if not. Re[c] and Im[c] denote the real and imaginary parts of c, respectively.

More formally, if P_c^n(z) denotes the nth iterate of P_c(z) (i.e. P_c(z) composed with itself n times), the Mandelbrot set is the subset of the complex plane given by

M = \left\{c\in \mathbb C : \exists s\in \mathbb R, \forall n\in \mathbb N, |P_c^n(0)| \le s \right\}.

You see, the Mandelbrot Set and fractals in general are considered “pop” math.  My complexity (and primes) loving Professor brought to light an important topic: Does pop have any place in high art, and if so what is it?

It seems like an old hat that has been worn many times, by Andy Warhol, by Roy Lichtenstein, by Michael Daugherty, and many other prominent artists and composers.  I think, for me the better question is: Where does COMPLEXITY fit into my practice?

If my goal with this whole thing is to try to find a way to express mathematics in sound, does it matter what function/algorithm I chose?  Now, I understand the desire for an elegant solution to this problem. I appreciate the beauty in abstract mathematics but I try to follow a simple programmers crede: Keep It Simple Stupid (KISS).

This doesn’t mean to limit yourself to remedial tasks but to only use the level of complexity that is necessary for the task.  I chose this algorithm and topic in mathematics as an entry point to the greater problem of expression of mathematic properties in music.  Fractals are familiar and even easy to understand by the average person. My goal is not to alienate but to share this abstract theory that math and music can co-exist in their pure forms without the degradation of either.  Perhaps I’ll gain enough trust to take my audience on the long road to primes…but not today.

Math as Art: Part 1

I often bounce ideas off my husband, sometimes to get his initial gut reaction to something musical, and other times I use him to verbalize concepts I have difficulty realizing in practice.  Often with the later, I see his eyes glaze over as he just nods and smiles while I talk AT him…usually once I start referring to frequencies and algorithms and nested loops.  You see, my husband has an incredible innate gift of musicality.  He understands music theory and form intuitively.  And he is a man of patience to humor my intellectual ramblings:

“Look at these two axioms and how they interrelate when they are used recursively!”

“Oh, yeah…cool. That’s really interesting.”

“I hypothesize that the resulting melody will actually be tonal and aesthetically pleasing because we, as human beings tend to like things built recursively. Just look at the Golden Mean, and fractals, and flower petals…”

…and I’ve lost him.

Now, if I were to by-pass all the jargon and formula and just play the resulting music for him, he would just GET it.  He would understand exactly what is happening and even point out weaknesses and how I might make it stronger.  He is magic and I am a mere mortal who has to explore and research and use trial and error to figure out how to make these ideas a musical reality. So, I decided to try sparing him the process so he can savor the result.  What better sounding board to use freely with little to no guilt than a blog?  Over the course of the next few weeks, I plan to use this blog to help me through my latest obsession: Fractals.

And so it begins…

Fractal: A definition

A fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension and may fall between the integers. Fractals are typically self-similar patterns, where self-similar means they are “the same from near as from far”. Fractals may be exactly the same at every scale,  or they may be nearly the same at different scales. The definition of fractal goes beyond self-similarity per se to exclude trivial self-similarity and include the idea of a detailed pattern repeating itself. (Wikipedia:

Inspired by the study of weather patterns and Strange Attractors, I decided to use 2 axioms which at first glance are similar, but have very slight differences. Once repeated recursively, given a set of rules, these differences will explode into 2 very unique sets.
As an example:

Axiom 1

{A, B, A, B, C, A, B}


Axiom 2

{A, B, A, B, D, A, B}


A = AB


C = BC

D = AC

Step 1

Axiom 1 {A, B, A, B, C, A, B} =

{A, B, A, B, C, A, B, A, B, C, B, C, A, B, A, B, C}

Axiom 2 {A, B, A, B, D, A, B} =

{A, B, A, B, C, A, B, A, B, C, A, C, A, B, A, B, C}

At this point, the differences have been abstracted further by the use of recursion. It is difficult to see, and even more difficult to hear the differences between the two.

Step 2

Axiom 1 {A, B, A, B, C, A, B, A, B, C, B, C, A, B, A, B, C} =

{A, B, A, B, C, A, B, A, B, C, B, C, A, B, A, B, C, A, B, A, B, C, B, C, A, B, C, B, C, A, B, A, B, C, A, B, A, B, C, B, C}

Axiom 2 {A, B, A, B, C, A, B, A, B, C, A, C, A, B, A, B, C} =

{A, B, A, B, C, A, B, A, B, C, B, C, A, B, A, B, C, A, B, A, B, C, B, C, A, B, B, C, A, B, A, B, C, A, B, A, B, C, B, C}

Now, just at the 2nd iteration, we can see the similarity grow farther apart:

{A, B, A, B, C, A, B, A, B, C, B, C, A, B, A, B, C, A, B, A, B, C, B, C, A, B, C, B, C, A, B, A, B, C, A, B, A, B, C, B, C}

{A, B, A, B, C, A, B, A, B, C, B, C, A, B, A, B, C, A, B, A, B, C, B, C, A, B, B, C, A, B, A, B, C, A, B, A, B, C, B, C}

This is a very simple example, using only 4 rules on 3 variables, but it gives the general idea behind this new piece I’m working on. I’m thinking the working title “Strange Attractors” is appropriate.


sarah j ritch, pyr interview!


sarah in the red room

When did you first become interested in music?

Hmm, you’re asking for a bit of family history here. I can’t remember becoming interested in music. It has just always been there, probably because of my family. My Mom was an amateur cellist until her early 20′s, my Dad is a brass player and conductor (notably of the U.S. Army band in San Francisco during the final days of the Presidio), my Grandma sang opera, one brother plays violin, another played sax and flute, plus various other relatives who played various other instruments (including auto-harp!). Music is just part of life, like air and sunshine and thunderstorms. Please don’t judge me for the cheesiness of that line, but it’s true! I’ve always loved moving to music and making sounds.

Man, that’s a lot of music in your family, it seems like it was pretty much inevitable that you would start playing. Although, you could’ve also rebelled by completely rejecting it too I suppose. With all those musicians on hand, did your family ever play music together?

My brothers and I joked around about starting a grungy Hanson type band, but no. The closest we ever came to playing music together was solfegging the violin and cello parts to various symphonies on many long drives between Vegas and Reno (nerd alert).

What was it that drew you in to music?

Growing up, my Mom always encouraged me to pursue all my interests (probably because of my attention span issues). I’m what you would call a “high stress functioner,” or someone who needs a multitude of things going on at once in order to stay focused. If you give me one thing to focus on, I can’t. So, (through generous community support because we were dirt poor) my Mom had me in ballet, gymnastics, piano lessons, girl scouts, and various after-school academic clubs. I’m really lucky that so many people were able to make this happen for me. When I say we were dirt poor, I mean dirt. Section 8 housing, homeless shelters, WIC, food bank, seven people in a two bedroom apartment, moving every six months kind of poor. Our Christmases were provided by the churches and public donations and I remember a few occasions where I was told to go get a clean rock from the yard for stone soup. . . .

Continued at:

sarah j ritch, pyr interview!

ESPLORAZIONI: Featuring work of Aaron Einbond









I have always told my students that music is maths realized as art.  I think this always comes up because I have a tendency to explain music theory fundamentals as if I were teaching arithmetic.  Most of my students, being in the 4th Grade through Jr. High academic range always pick this up and say “Wait, is this music or math?!”  I love seeing the sparkle in their eyes when I explain my view of maths and music…it’s like I just made something boring (presumably maths) into something mystical.

So, naturally, I am drawn to the increasingly expanding field of computer music.  I’m talking low dirty dark magic computer science and skilled music composition.  There aren’t many people working in this new genre who are a genuine mix of coder/composer, but those who are give me hope for the future.  Ensemble Dal Niente is presenting a concert this Saturday which features a new work by one of these special sorts of person, Aaron Einbond.  Aaron is a composer for both electronics and acoustic instruments as well as is a researcher in computer science!  His new work Without Words combines soprano, nine players, and live electronics.

Made with CataRT

Aaron explains the above here :

“In the plot, each dot represents a short sample as performed by Amanda DeBoer, and the axes correspond to the timbral qualities of the sounds.  I explored this plot to assemble the samples into micro-montages, which I use in the electronic component of the work, or ask Amanda to recreate live.  I also map them to other audio recordings, so that the performers are asked to recreate a recording of frogs, the sun, or of Wallace Stevens reading.  Because samples of Amanda and the other players were my starting point, their unique sonic identities are woven into the fabric of the work as well.”

And how does one notate this idea? Aaron was kind enough to give us a little peek into the score:

You can check out the realization of this piece live this Saturday. Details given below.


Saturday, June 9, 2012 7:30pm

Nichols Concert Hall
1490 Chicago Avenue
Evanston, IL
Tickets: $20 general/$10 students, available at the door