What is Fractal Geometry?
About Fractal Music Workshops
What Educators are Saying
About Harlan J. Brothers
What is Fractal Geometry?

Fractal geometry is a new, fascinating, and surprisingly accessible field of mathematics. It became recognized, and instantly popular, with the 1982 publication of Benoit Mandelbrot's seminal work
on the subject, The Fractal Geometry of Nature.
The book gave mathematicians new and powerful tools for describing the
rough, wiggly, and hitherto untamable shapes that permeate the natural
world. It also described purely mathematical constructions
that capture the eye, and defy the "common sense" of the more familiar
Euclidean geometry. 
The fundamental characteristic of fractal geometry is scaling symmetry;
the structure of a fractal object looks the same under
magnification. This characteristic is often referred to as
“scale invariance” or
“selfsimilarity.” Selfsimilar objects
are, in some specific sense, composed of smaller copies of themselves. 


The structure of objects ranging from bacterial colonies to galactic
clusters bears witness to the ubiquity of fractal scaling.
Not only does fractal geometry describe the shapes of trees, clouds,
mountains, lightning, and river tributary systems, but it also gives
rise to our very existence as human beings. Our vascular,
respiratory, and nervous systems are all characterized by selfsimilar
branching patterns. Scaling symmetry even turns up in the
behavior of flood cycles, social networks, and financial markets. 
Perhaps it is a reflection of our fundamental
nature that research has uncovered numerous examples of fractal
structure in many areas of artistic endeavor. Scaling symmetry is known to arise in art, architecture, poetry, literature, and music. It is therefore not surprising that, increasingly, fractal geometry is the tool that mathematicians, physicists, biologists, and scientists of many other persuasions turn to for gaining a deeper understanding of the universe.


About Fractal Music Workshops
Using the tools of fractal geometry, Fractal Music Workshops offer teachers and students alike the opportunity to explore many of the intimate connections between mathematics and music. Focused presentations and handson exercises promote understanding in an
intuitive and entertaining manner.
Each workshop is tailored to meet the needs of the participants;
mathematicians can learn about symmetry and powerlaws in music, musicians can learn about the
math inherent in their discipline, and everyone can gain a clearer understanding of the
fascinating field of fractal geometry.
Topics include:
Fundamentals of Fractal Geometry
 Classical symmetry
 Scaling symmetry
 Natural fractals
 Power laws
Music Theory
 Physics of musical pitch
 Harmonic series
 Scales and intervals
 Music notation
Fractal Analysis of Music
 Prerequisites
 Common mistakes
 Duration scaling
 Pitch scaling
 Motivic scaling
 Structural scaling
After
learning the necessary background, all participants have the
opportunity to compose their very own fractal music using the Fractal
Music Composer software designed for the Fractal Geometry
Workshop at Yale.
Full workshops run approximately 6 hours including a lunch break. Halfday labs and custom curricula can be designed to fit the needs of your department.
What Educators are Saying

“Harlan gets the
details right, gives a fine presentation of the big picture, and is
endlessly energetic and goodhumored about the teaching and learning
process. He has the rare balance of skill in mathematics and
in music necessary to teach fractal music. Benoit, Nial, and
I all are very grateful that Harlan shared his talent with
us.”
 Michael Frame, Professor of Mathematics, Yale University


“Innovative, knowledgeable, creative, engaging, and witty  Harlan is sure to inform, challenge, and delight anyone with an interest in math, music, or technology. His passion for fractal music is contagious!”
 Elaine Dinto, Professor of Mathematics, Naugatuck Valley Community College


“Harlan was well
prepared and very flexible in meeting the needs of our
teachers. In a few hours time, he was able to teach music
theory, fractal basics, and how to use the Fractal Music Composer
software, and our teachers were creating their own fractal music
compositions. It was incredible!”
 Betsey Davis, Instructor & Program Coordinator, Fractal
Institute 2007, MathScience Innovation Center


“Harlan’s teaching style is open and engaging. With his understanding of fractals and music, he presented an outstanding workshop. Even with a limited knowledge of music, I found the workshop absolutely fascinating.”
 Dennis Riordan, Teacher, Masuk High School, Monroe, CT


“The workshop on
fractals and music created by Harlan Brothers succeeds on several
levels. First, Harlan introduces the concept of fractals in a
clear and fascinating manner. Then, as the master teacher
that he is, he gives a succinct and engaging description of how
fractals can be discovered in musical compositions. The
software that he has developed allows even a musical novice to create a
fractal musical composition. I was excited and impressed to see how it became possible to appreciate and create fractal music.”
 Virginia Jones, Professor of Mathematics, Central Connecticut State
University


“Harlan’s passion for mathematical challenges is evident in his teaching.”
 Andrew Derer, Mathematics Educator, MathScience Innovation Center


“I have participated in Fractal workshops at Yale where I interacted with Harlan and saw his delightful presentation on Fractal Music. Because of my own interest in music (as an amateur musician and through my symmetry course for nonscience students) I can well appreciate Harlan's excellent understanding of music fundamentals, as well as his innovative work in developing the ideas of fractal music illustrated through his engaging audiovisual examples. I hope more people get a chance to attend his lectures.”
 Laurence I. Gould, Professor of Physics, University of Hartford


“I loved it. Using
the Fractal Music Composer really allowed me to see the fractal nature
of the music. It was cool that I could create my own music
and understand the fractal mathematics behind it.”
 Arletta Aleshire, Teacher, Teaching Geometry Using Calculators and
Computers, MathScience Innovation Center

About Harlan J. Brothers
Harlan Brothers is an inventor, musician, mathematician and the founder of Brothers Technology, LLC. Harlan has received six U.S. patents and has published numerous journal articles related to number theory and the fractal geometry of music. He is also an accomplished jazz guitarist and composer, having studied
composition at the Berklee College of Music in Boston.
Over the course of six years, Harlan worked with Michael Frame and
Benoit Mandelbrot at the NSF funded Yale University Fractal Geometry
Workshops. The workshops served to train educators in the
subject of fractal geometry with the goal of developing fractalbased
mathematics curricula for students in middle school through college.
During an informal discussion in 2003 regarding the general lack of
understanding associated with the concept of fractal music, Benoit suggested to
Harlan that he undertake a rigorous treatment of the subject. The
first result was a lecture and lab Harlan presented at the 2004 Fractal
Geometry Workshop at Yale. His most recent work on fractals and music, entitled "The Nature of Fractal Music," will appear as a chapter in Benoit Mandelbrot  A Life in Many Dimensions, published by World Scientific Publishing (May 2015).
Contact Information:
Harlan J. Brothers
Brothers Technology, LLC
PO Box 1016
Branford, CT 064058016
harlan@brotherstechnology.com
Cell: (203) 5896769
© 20042015 Harlan J. Brothers, All Rights Reserved.
