Engineering Math Software

How Math Is Helping One Man Push the Boundaries of Acoustic Music

** Guest post by Mike Gayette**

If you’ve ever played music, you know that an octave in Western music is generally divided into 12 equal half-steps or pitches: C, C#, D, D#, E, F, F#, G, G#, A, A#, B. What you might not know is that the intervals between those pitches have not always existed as an exactly perfect 1/12th the width of the whole octave.

Because of a computational difficulty (caused by dividing strings with prime numbers) that mathematicians have recognized since ancient times, we “temper” the interval between half steps. An approach to tuning that goes back to ancient Greece, just intonation relates the pitches to each other using rational ratios, and ignores the irrational ratios introduced by perfect half-steps.

Breaking from western tradition

But not everybody uses tempered pitches. Some avant-garde artists find themselves drawn to more mathematically pure intervals—what is called “just intonation.” An approach to tuning that goes back to ancient Greece, just intonation relates the pitches to each other using ratios and ignores the problems perfect half steps introduce.

It’s an interesting intellectual exercise to be sure. But why would any trained musician leave behind conventional pianos and guitars to experiment with these often foreign sounding pitches?

“Someone needs to push acoustic music forward,” says Cris Forster, a musician/builder and a long-time PTC Mathcad user. “And by experimenting with alternative, sometimes forgotten tunings, I’m hoping to make a contribution to the future of music.”

Indeed, Forster has dedicated the past 40 years to exploring and composing music featuring unique intonations, and inventing acoustic instruments to perform them.

One of his first creations is the Chrysalis—a large wheel with 82 strings on each side, played in a manner similar to a harp. But unlike a harp, the wheel can be freely turned so the musician can access all the strings easily. Here’s Forster playing it while reciting a Whitman poem.

Musical Mathematics

As you might guess, calculation software can play an important role when crafting original instruments. “A creative investigation into musical sound inevitably leads to the subject of musical mathematics,” says Forster. “It even leads to a reexamination of the meaning of variables.”

Forster has written a fascinating and comprehensive book on the subject. Musical Mathematics: On the Art and Science of Acoustic Instruments, which is packed with equations created in PTC Mathcad, seeks to describe alternative intervals and instruments, and the calculations necessary to make them.

In the book’s first chapter, Forster introduces a new unit of mass he calls the “Mica.” The Mica is like a slug (a mass that accelerates by 1 ft/s2 when a force of one pound (lbF) is exerted on it), but scaled to inches. With the Mica, he produces units that are more convenient to instrument builders, who work in inches rather than feet.

Unfortunately, while the new unit solves one problem, it introduces another. How do you represent a unit nobody’s ever heard of before in your calculations –without adding tedious conversion work?

For Forster, that was easy. He simply taught PTC Mathcad the unit he needed.

“I incorporated the Mica math unit into the PTC Mathcad user-defined files,” says Forster. “Now I can use Mica as a written word and it’s interpreted by PTC Mathcad in the same way as I write the lb for pound or lbf for pound force. The worksheet even gives me answers in Mica, too.”

Image: The Mica is now available as a worksheet option when Forster uses PTC Mathcad.

The accuracy of those answers impresses Forster, too. While writing the book, he vetted his calculations in PTC Mathcad and didn’t find a single conceptual or computational error.

To read more about Cris Forster’s work, including a detailed journal of the building of Chrysalis II, see his website. And when you’re done, head over to our PTC Mathcad page and download your free-for-life version copy today.

Learn more about unit conversion in PTC Mathcad.


Image: Forster works on the Chrysalis II in his San Francisco workshop.