Interactive Educational Programs

"Exponential Ensemble educational programs seamlessly integrate arts into science and math common core standards.  They provided an extremely engaging way for our students to make a powerful unknown connection."


Dana Taylor, Supervisor of Special Programs 

Monticello School District (NY)

On November 26, 2019 we visited the Morton School in Manhattan and performed Relative Theory by Robert Paterson, a brand new piece inspired by four important matehmaticians and scientists: Blaise Pascal, Emmy Noether, Albert Einstein and Pythagoras.  Paterson guided the students through his compositional process, how he incorporated specific theorems in his music. 


Relative Theory is made possible by the New York State Council on the Arts with the support of Governor Andrew Cuomo and the New York Statet Legislature.  


On December 9, 2017 we performed an intriguing canon (or round) by J.S.Bach at the Princeton Public Library.



The original melody can be viewed and sung using a math graph.


The X axis represents the pitches from low to high and the Y axis the time it takes to sing the melody. 



Bach used the concept of unit rate to modify the melody.  


In this picture we are experiencing the melody by walking (or running) at a rate of 2:1, 4:1, 8:1 and 16:1.  



Bach transformed the original melody by reflecting it onto the Y axis.



We invited students to create a reflected melody with a finger painting activity.  Once the students painted the original melody at the top we were able to create a reflection of the melody by folding the lower part of the paper onto the painted melody. 



Bach transformed each melodies using different unit rates and geometric reflections. 



Once we figured out how J.S. Bach transformed all the melodies we were able to build our own canon together assigning each instrument of the ensemble a specific melody to play. 


Math teacher testimony

Students from MS245 (The computer school) solving a fraction equation using rhythm along a timeline... (May 2016)

"It teaches us a new strategy on how to add fractions
and it is fun to do.  It is also a good review on fractions."


Mariana, 6th grader

The Computer School

Students extracting fraction equations from a word problem...

“I was extremely excited that my students who struggle with math were able to make a connection with the music and 
apply it to solve more complex math fractions problems.”


Heather Conklin

Math Teacher, RJK Middle School








Bethel Woods E3 Residency

Engage - Experience - Explore 


October 19 - 24, 2015.


During this special residency we worked with hundreds of middle school students from the RJK Middle School in Monticello (NY).  They learned how rhythm and fractions are closely related.  Also, how J.S. Bach wrote a canon using different unit rates and geometrical transformations.  


Our Music & Physics Interactive Program (below) focused on the physics of sound and the inverse relationship between frequency and wavelenght (i.e. as you increase the frequency, the wavelenght decreases.)

For bookings, contact us at

2014-15 Season

Thank you the NY Public Library (Bronx branch) for hosting us on January 8, 2015!  We had a wonderful time!


The students learned how J.S. Bach transformed a melody by changing its rate and reflecting the melody onto its axis...  They also created their own canon using the various options J.S. Bach composed.

In september 2014, we launched our "Music & Math Interactive Program" for middle school students. So far, we have performed at the Pelham Academy in the Bronx and Village Community School in Manhattan.  

During this concert, the students learn how JS Bach created a canon using math concepts such as Rate and Geometric Transformations; both concepts that are part of the Common Core Math Standards for middle school students. 

This concert features hands-on activities supporting the different math concepts that Bach used to create his canon.  In this example, the students are asked to walk on stage following the melodies played at different rates.  




Pianist and mathematician Tyler Wottrich helping students trace Bach's melody on an imaginary math graph.  





This concert lasts about 45 minutes and is tailored for large assemblies of students, grades 6-7-8. 




Coming up in January 2015, we will perform for students at The Cathedral School in Manhattan and members of the NY Public Library in the Bronx.




For bookings, contact us via email:


If you wish to make a fully tax-deductible donation towards our Music and Math Educational Concerts please visit our website: and click on "Donate"


Thank you so much for your support.

Exponential Ensemble


2013-14 Season

Instrumental Tuning Lectures with mathematician


Kevin Wilson and violist David Creswell.


In order to understand the difference between different tuning system, we need to be able to hear slight differences within a half step.  


A half step is made of 100 cents and you should be able to hear a 2 to 10 cents difference between two pitches.


By taking the Adaptive Pitch Test, you will ear those differences.


Adaptive Pitch Test:

Overtone series


A sound is made of overtones. For example, the overtones of an open string can be heard by dividing it in 1/2, 1/3, 1/4. 1/5, 1/6, 1/7, etc...


\sum_{n=1}^\infty\,\frac{1}{n} \;\;=\;\; 1 \,+\, \frac{1}{2} \,+\, \frac{1}{3} \,+\, \frac{1}{4} \,+\, \frac{1}{5} \,+\, \cdots.\!


See below a demonstration by violist David Creswell.

Physics of sound

Mathematician Kevin Wilson explained what sound is made of. Sound waves are created when air molecules are moving fast. The same applies to a string,  the air column in a wind instrument or the skin on a drum. 

An air column vibrating.

Pythagorean Tuning

We measure the differences in pitch with Cents.  


1 cent is 1/100 of a half step.  So, there are 1200 cents in an octave.


In Pythagorean Temperament, the scale is made by stacking pure fifths on top of each other.  But, the Pure Fifth, the best sounding fifth (when you divide the string into 3 parts, also called the "Third Partial" ir second overtone), is at a ratio of 3:2 and actually 702 cents up. As we stack Pure Fifths, making our scale, we need to add 2 cents each time, 12 times in a row (see circle of fifths).


Problem:  As a result, one we return to the starting pitch, we end up 24 cents higher, almost 1/4 of a half step!

Equal Temperament

In Equal Temperament, all half steps are the same: each half step is 1/12 of an octave.  That way, we can go around the circle of fifths as many times as we want and end up at the same pitches.  For example, to reach the fifth we go 7 half steps (or 700 cents) and the scale is made by stacking these fifths on top of each other.


In practice, all fifths sound just a little bit off but at least it is mathematically equal.  

Well Temperament

Nowadays, ensemble without a piano or fixed-pitch instruments (string ensembles, choruses, etc...) use Well Temperament made of Pure Fifths (3rd partial) and Pure thirds (5th partial).  



Using this graph, we make all 12 notes of the western scale by moving in fifths and thirds up or down. 

That way, we get the most combined resonance from the interaction of overtones while still being able to play in all keys.  You can see (above) that any triangle will make a major or minor triad with a perfect 5th or 3rd.

















J. S. Bach's "Well-Tempered Clavier" DOES NOT refer to this system. We don't know what he was using for his temperament, but there are various theories including one involving the "doodle" (below) on the front page of his manuscript.








Kevin Wilson Mathematician

Kevin will help us develop a brand new Music & Math Program based on instrumental tuning theories, such as Pythagoras', along with violist David Creswell. We are excited to explore this unique and complex topic connecting music and math!  


Kevin Wilson grew up on a dairy farm in Shelbyville, Kentucky. He began studying the piano at the age of four under the tutelage of his grandmother. But his other tutor was his father, who began teaching him about probability and baseball before he even remembers, and his Uncle Brian who gave him a copy of Visual Basic for his eighth birthday. To the betterment of the musical world, Kevin came slightly to favor mathematics (and, thanks to Brian, its computational aspects), earning a B.S. in the subject from the University of Michigan and a Ph.D. from Princeton. He is currently employed as a Data Scientist at Knewton, an education technology company in New York City, and concerns himself daily with modeling how students learn. In his spare time he co-directs a corps of volunteer software engineers and data scientists help non-profits in the City improve their technology infrastructure, remotely co-teaches Computer Science at Lee County High School in Kentucky, plays the piano occasionally, and works with Exponential Ensemble to develop interesting coursework at the nexus of mathematics and music.

David Creswell Violist 

Violist David Creswell will be collaborating with mathematician Kevin Wilson in develping a lecture series on instrumental tuning.  David has been performing in the New York area for more than 15 years as an orchestral, chamber, and studio musician, while also keeping a busy schedule as a soloist, teacher and music coach.  Mr. Creswell is Principal Violist of the Greenwich Symphony, the Long Island Philharmonic, and the NEPA Philharmonic.  Mr. Creswell also frequently performs with the New York Philharmonic, the New Jersey Symphony Orchestra, and the American Ballet Theater Orchestra, and the Mostly Mozart Festival.  He is the former Principal Violist of the Sarasota Opera Orchestra in Florida. As a chamber musician, Mr. Creswell has concertized with such renowned artists as Kathleen Battle, Sidney Harth, and Anthony Newman, and is a founding member of the Larkspur Trio. His recordings include numerous film scores and popular releases as well as chamber music and projects with music icons David Byrne, Rufus Wainwright, Erasure, Linda Thompson, and Rod Stewart. He was the violist for the original Broadway productions of the Tony Award nominated “Caroline, or Change,” “The Color Purple,”  “South Pacific” at Lincoln Center, Stephen Sondheim’s “Follies” from the Kennedy Center, and has played several seasons with the Radio Music Hall Christmas Spectacular.  He continues to collaborate on collegiate teaching projects for students of the Juilliard School with the renowned teacher Heidi Castleman.  In 2010, Mr. Creswell soloed in Mozart’s Sinfonia Concertante with the Long Island Philharmonic and concertmaster Erica Kiesewetter, and 2012 performed the Bartok Viola Concerto with the Greenwich Symphony.  

A word from Pascal Archer,

Founder of the Exponential Ensemble


"My father was a math teacher and, growing up, he taught me that music and math have so much in common!"  


Music and mathematics have a lot in common but most of the time these two subjects are taught separately in school. For the Exponential Ensemble, it is our goal to collaborate with math teachers in making that connection possible for their students and, as a result, deepening their understanding of specific math concepts. 

(below) Pascal Archer, founder and clarinetist, playing rhtyhmic combination against a steady beat on the drum.  The students have to find which rhythmic combination from the list is being performed.