"My father was a math teacher and, growing up, he taught me that music and math have so much in common! When I founded the Exponential Ensemble it was important for me to develop educational programs connecting music with math, science and other subjects taught in school as a way to help students make new and meaningful connections with those subjects."
Pascal Archer
Founder, Exponential Ensemble
J. S Bach Canons
"Exponential Ensemble did a wonderful dive into the 14 Canons based on the bass line from Goldberg Variations by Bach with my orchestra classes. Exponential Ensemble struck a magnificent balance of going deep enough to be impactful and memorable, while also moving fast enough to engage the class and keep them wondering what was coming next. Never has the presentation of art and music alongside mathematical relationships been so engaging. Students spent the day finger-painting, composing, and listening to a beautiful demonstration of this lesser-known set of wonderful canons. By the end of class, every student took part in an in-class performance of canonic creations they made. It was an amazing day!!"
David West, Music teacher
The Celia Cruz Bronx High School
In 2019, American composer Robert Paterson wrote us a piece titled "Relative Theory" inspired by four important mathematicians and scientists: Blaise Pascal, Emmy Noether, Albert Einstein and Pythagoras. Paterson guided students through his compositional process, how he incorporated specific theorems in his music.
For example, according to legend, Pythagoras discovered the foundations of musical tuning by listening to the sounds of four blacksmith's hammers, which produced consonance and dissonance when they were struck simultaneously.
"In the beginning The Hammers of Pythagoras and in similar spots throughout the movement, I imagined blacksmiths in a blacksmith shop striking different anvils at varying rates of speed. The harmonic and melodic content in this movement is primarily intervallic, meaning, the melodic and harmonic material was created around specific intervals such as the perfect fifth, which is one of the most important intervals in the harmonic series for most standard instruments, and is also critical to Pythagorean tuning, a tuning system invented by Pythagoras. In this tuning system, the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, was chosen because it is one of the most consonant and easiest to tune by ear. The ascending run of notes in the bass clarinet part in the beginning is indeed inspired by a harmonic series, although it is not a true representation from bottom to top, but merely an ascending pattern of notes that sounds somewhat like a harmonic series. In this case, the sound of the notes in the ascending patterns and how they blend with the chords in the other instruments was more important to me than being theoretically accurate."
Robert Paterson, composer
The Music & Math educational program is based on a fascinating set of 14 canons (or rounds) by J.S. Bach. We use a math graph to learn J.S. Bach's melody. That method is especially efficient for students that can't read music yet.
The X axis represents the pitches (low to high) and the Y axis the time it takes to sing the melody.
J.S. Bach transforms his melody in many ways by changing its unit rate from 2:1 (half notes) to 4:1 (quarter notes), 8:1 (eighth notes) and 16:1 (sixtheenths notes). Additionally, he uses geometric transformations to reflect the melody.
Our Music & Physics Interactive Program focuses on the physics of sound and the inverse relationship between frequency and wavelenght (i.e. as you increase the frequency, the wavelenght decreases.)