Music: A mathematical offering – David Benson

This book was a gift from a former student and, to my shame, I hadn’t got round to reading it until this summer.  It took a while (even after I had started!).

It’s a fairly academic book about the maths behind music. 

I was expecting to read about patterns made by sand on drumskins, tuning-systems, Pythagorean commas, etc, and while these did feature, there was so much more too. 

The Musical Offering

To start with, having some musical knowledge was useful.  The title, for example, is a play on the title of an extraordinary work by JS Bach (The Musical Offering).  Frederick the Great, King of Prussia asked Bach to improvise on a melody which the King provided (and perhaps composed).  Bach later wrote a collection of variations on the theme, including pieces where you can play the music forwards and backwards simultaneously, or with one part at half the speed of another, or in mirror image, and culminating in a 6-part fugue.

Pythagorean comma 

The Pythagoreans knew about music and vibrating strings.  They knew that an interval of an octave corresponds to a ratio of 2:1 (nowadays we would say the frequency of the note is doubled) and a perfect fifth is 3:2 (these values can be used to create ‘harmonics’ on a guitar or violin).  On a piano, if you go up in fifths, it takes 12 of them before you arrive back at the same note (seven octaves higher than where you started).  For example:

C – G – D – A – E – B – F# – C# – G# – Eb – Bb – F – C

This should mean that 1.5 (from the ratio 3:2) to the power of 12 is equal to 2 (from the ration 2:1) to the power of 7 (because that final C is 7 octaves higher than the initial one).  Unfortunately it doesn’t:

1.5^12 = 129.746…, while 2^7 = 128

The discrepancy is known as a ‘Pythagorean Comma’. 

This comes about because of the sleight-of hand where G# moved up a fifth to make Eb.  On a keyboard we have to treat G# and Ab as being the same note – whereas in this perfect Pythagorean world they are different notes.  If we continue to go up in fifths, remaining with sharps (rather than switching to flats) we get:

C – G – D – A – E – B – F# – C# – G# – D# – A# – E# – B#

Again, keyboard players would see B# as being identical to C natural, but in fact (as the Pythagorean comma tells us) it is a little higher than the C.

It doesn’t matter if we decide to go down in fifths: the D-double-flat we end up on is lower than the C natural it is deemed to be equivalent to.

And it doesn’t matter if we go up in perfect fourths (ratio 4:3).  12 of them gives 31.569, which is less than the 5 octaves value of 32.

Even a major third is unhelpful.  The ratio is 81:64, and three stacked major thirds ‘should’ make an octave (C – E, E – G#, Ab – C – but there’s the sleight of hand with G#=Ab again).  (81/64)^3 = 2.027…, which is a shade above 2.)

Back to the book!

Lots of cool stuff like this, which I could understand and which taught me lots of new things. 

Some almost incomprehensible (to me) stuff.  I did struggle through the chapter on Fourier Theory, but was relieved the author said this could be skipped!  Here’s a sample:

 

And some lovely stuff about transformations, wall-paper patterns, etc.  See some neat symmetry here:

In summary, if I didn’t have any prior knowledge at all I probably wouldn’t have finished it.  And clearly it is aimed at those in the intersection between maths and music (without one or the other you probably won’t get much out of this).  I did glaze over at the extent of the calculus that was involved.  But I learned some cool new things too. Worthwhile.