20 August 2011

Review: Cris Forster: Musical Mathematics

On the Art and Science of Acoustic Instruments.

When writing an article about the transformation of "light" to sounds I looked around to find other sources of sound production than the usual music channels. The first one was a blog of  Steven Speciale "Mostly Noise" where I found a lot of modern music seldom heard. But I was looking for some microtonal music and found a lot of thoughts and music on the pages of Chris Vaisvil "Music & Techniques". He is composing and performing microtonal music on a variety of instruments. And here I found also the book of Cris Forster which was released in  2010. After a look at his homepage I thought, that it could not be a fault to order this book.






Musical Mathematics does not tell stories about hidden mathematical symbol patters in the music of J.S. Bach. It translates our musical ear training into a down to earth language: Mathematics. In this book you find no guessing or flowering words for the incomprehensible. You will find formulas and numbers.

It starts with the basics of sound described in physical / mathematical formulars: The principles of force, mass and acceleration. But this is not a walk-through like in normal mathematical books on one page. He takes his time to describe in words and drawings what is going on. And this is one advantage of the book. Although it deals with mathematics, it uses also its elegant language, but never lets you alone guessing what that formula means. Forster describes step by step how to achieve a solution. He never takes the attitude of a professor who throws the equation on your head and says: "Here is the solution, see for yourself how it works."



He introduces the MICA mass definitions (Mass (unit based on the) Inch Constant (for) Acceleration; but a formula to conversion to kg is given.

Before I continue: This is my only complaint about the book. I thought that the USA changed their measurement system to metrics. But Forster stays in the old system. And for non-US readers that is really a pain in the back: You have to use your calculator anyway but you constantly have to convert e.g. 13 15/16 inches to mm. And I can assure you: On these over 900 pages there are inches, inches, inches, ...

Next you will find the correlation for plain strings and wound strings between frequency,  length, tension and mass. It is the basis for the actual sound you get by a string. Of course the string's break strength for different material is also discussed. Another point is the geometric structure of wound strings with various material. If you ever tried to understand the catalogs of string manufacturers for cembalos: Here is a starter.



After these primarily remarks Forster continues with the behavior of strings. He shows transverse travelling and standing waves and simple harmonic motion in strings with pulses from different directions. The explanations are clear and facilitated by drawings.

from Wikipedia


You will find tables with movement of strings fixed on both end;, period and frequency equations;
length, frequency and interval rations of the harmonic series; all you ever wanted to know.

The next block described in the book is string harmonics. Harmonics are normally described in musical terms: A "fifth"  is the fifth tone in a scale. This does not mean a mathematical length ratio 1/5 . Foster will throughout the book talk about mathematical ratios in order to describe other scales. So he describes in detail the relation between harmonics, note names, frequencies, cents, modern length rations, frequency ratios, interval rations, interval names and cents.

This is much theory, so in Chapter 13, Forster proposes to build a Canon (Monochord). This is a simple instrument which you can build by yourself (plans, pictures and measurements are included). With this Canon it is easy to learn other scales. The instrument has a nut and a stationary bridge, 6 strings of identical length and form .and you can subdivide the strings with movable bridges. So it  has a theoretically infinite number of notes and can produce any number of new frequencies without changing the overall design of the instrument.

Harmonic/Melodic Canon© Photo by Will Gullette
(This photo above shows a more complicated instrument as the one described in Chapter 13)

You will not only learn about mathematics. Forster digs deep into history and researches musical, mathematical, and linguistic origins of length ratios from the old Greeks to modern systems. You will find all the old systems used in former times translated to measures you are used to.

Until now the world was in harmony: All harmonics were exact integer multiples of the fundamental frequency. But that does not fit into the actual sound you get. In reality stiffness of the material increases frequency. You get inharmonically induced dissonance. And you have to consider this. In the workshop the luthier knows that you have to give or take a little to produce the right sound. But Forster shows how to calculate the differences, the relation between soundboard strings, bending waves, and also geometric versus acoustic lengths.

To ease the calculations he introduces step by step conversion to the cent system (1200 cents = 1 octave)

The next big chapter after the mathematics of the strings introduces bars, rods and tubes. It is a parallel description of the world of resonant bodies.  You will e.g. learn how to tune marimba bars, acoustic resonators and tube resonators. You will find the harmonic motion of longitudinal traveling waves in air and
standing waves in tubes resonators; and tuning techniques for cavity resonators using dowels to stiffen the walls. Forster explains the placement of the tone holes simple flutes, gives equations for analyzing the tuning of existing flutes; then gives logarithmic equations for guitar frets, 12-tone equal temperament to just intonation.



You will notice that I start to summarize. The material is extensive and each chapter would be worth an in-depth description. When you study this book up to page 281, you should have the following knowledge:

  • the mathematical structure of the harmonic series
  • the distinction between different ratios (length, frequency, interval)
  • mathematical methods used in the division of canon strings
  • distinction between arithmetic and geometric progession
  • how to add and subtract musical intervals
  • how to convert length and frequency rations into cents
  • ...
Chapters 10 and 11 are in my eyes the most important parts of this book and I must confess that most of it is "terra incognita" for me. It is an encyclopedic overview of all know tuning systems and scales ever invented by musicians all over the world. It covers scales from times gone by to actual modern tunings. You will not only get a description and an historical evolution but also a mathematical declaration based on the explanations of the former chapters. So you are able to reproduce these scales.

Chapter 10 is titled: "Western Tuning Theory and Practice". In the first part, Greek classifications of ratios, tetrachords, scales and modes are mathematically and practically explained by tuning canon strings.These differentiations continues with the systems of Philolaus, Euclid, Aristoxenus, and Ptolemy. Of course, these tunings are also broken down to length ratios on a canon. Tables with Greek Enharmonic, Chromatic and Diatonic scales are provided. Meantone temperaments, well-temperaments and equal temperaments are treated in the same way. You will find for all these scale mathematically exact formulas and tables.This is, of course, no easy reading material although the math required could be done with a simple calculator. In the following sections different 12-tone Western scales are calculated and translated into cents. It is really astonishing just looking at the tables to see how different these scales are.

I had my nightmare when I tried to tune my clavichord with a tuning aid. The tuning was not the problem, but to hear the difference. I must admit, that my memory was too short to remember the differences. So the proposal of Forster to build a canon, where you can quickly change the tuning and hear the difference, might be a better approach than mine ;-)

I could continue to drop words about the next 100 pages were Forster goes through more scales than I ever heard of. For me it is more a chapter for reference. I can try to understand the differences of tunings, but I have to hear them in comparison to the scale we use to "understand" what is going on. And please do not think that this is a book with a popular background. Only for this chapter there are 364 notes.


And the encyclopedy goes on with "World Tunings" with "only"  300 pages. You may have noticed it already. This is not really a review. I scanned this book, got lost by reading interesting chapters, turned pages, took my calculator, read on, found another interesting point,...

But if you are interested in non-Western music, this book is definitely for you. Of course Forster is not talking about music. His subject is still "only" tuning and I would bet, that you will find nearly every tuning which is use in this world.
  • Chinese music
  • Indonesian Music (Java, Bali)
  • Indian Music (Ancient, South India, North India)
  • Arabian, Persian and Turkish Music


So if you hear some ragas and on the CD-booklet you find the name of the instrument you will find the tuning of this instrument in this book. I cannot even write the names of the instruments: My character set does not allow it. I do not even want to know how long it took Forster to gather that wealth of information and transform it into today's mathematics. I am sorry: I am not the one to make a critical review of these two last chapters. But you can be assured: I have never seen a compilation of musical tunings and scales and if I make a conclusion from the first chapters: The scales are right to the point. Otherwise he would not be able to turn these scales into playable tunings.




You can take your Canon and tune it accordingly and you will hear the magic of this tuning. I sometimes played on a Turkish Ud. Now I know how it worked. My former experiences with a Sitar are now more transparent. 

So this book is a culmination of what I have seen about mathematics and music. You will understand what system is behind tone, harmonics, scales, ...

It is a summary of tuning systems  through the centuries in the Western world.

It is a compilation of musical scales of the world never seen before.

Who must own this book?

  • Musical libraries
  • Musicians, who have a brain for mathematics
  • Composers, who dive into new scales, sounds, systems
  • Freaks, who like to listen to instruments from other worlds
  • me, but I still have a lot to cover in this book
PS: Of course there is an extensive bibliography, index, sixteen color plates, ...
PPS: Another critical point: You can't read it lying on a sofa. It is far to big and heavy ;-)