Introduction

Bach wasn't explicitly interested in mathematics but his music reveals a harmonic beauty which can be expressed in mathematical terms. Following article was posted by me on the newsgroup alt.music.j-s-bach in reply to someone's request to express some views on Bach's music. (More on accessing and posting articles on the Usenet here).

Usenet Article

Here's a scientific take on the issue:
Our Western musical language (and most of the ethnic music I know) is based on the circle of fifths. The circle of fifths can be closely approached using division by 3 and multiplication with 2 ... I actually wrote about this topic: see Constructing Scales.

Many don't see the significance of this, but I've come to see number as one of the most fundamental concepts in the universe (with the help of Plato, Pytagoras, etc...), virtually all else in the universe is 'fuzzy'.

I think it amazing that the interaction of these numbers (creating harmony and rhythm) creates such beautiful and intricate music. Not unlike the beautiful Mandelbrot images that can be gotten by iterating a very simple equation.

Bach's music is somewhat mathematical because in some ways, he thinks like a mathematician, that is, he can see patterns, see works in their totality. He also applied number to issues such like the number of bars a work had, etc ... A great book that explores this issue is Godel, Escher and Bach.

More significantly, he was able to apply rules to aesthetic issues, such as the well-known 'consecutive fifths and octaves' rules. I think this combination of aesthetics and his ability of seeing patterns is what makes Bach's music great.

I think that by listening or playing Bach, the mind is given a taste of the totality of the universe (I know, big words), but to my mind this accounts for the spiritual experience that Bach's music brings.

For example, so-called 'super-particular' ratios, namely proportions like 2:1, 3:2, 4:3, 5:4, 6:5, 7:6, 8:7, 9:8, where the top of the fraction is one bigger than the bottom. Many of these are musically significant:

• 2:1 octave
• 3:2 fifth
• 4:3 fourth
• 5:4 major third
• 6:5 minor third
• 8:5 minor sixth
• 9:8 major second
• 16:15 minor second

The German mathematician Andreas Sparschuh has suggested applying this 'super-particular' notion to the circle of fifths, by considering super-particular factorisations of the Pythagorean Comma as representing tempering operations. For example, he reinterpreted Werckmeister Nr.3 to represent the Pythagorean Comma 531441/524288 (i.e., 3^12/2^19) as the product of four super-particular ratios: (6561/6560)(205/204)(153/152)(513/512)

To bring this back to Bach, Sparschuh has also proposed that the squiggle on the cover sheet of Das Wohltemperirte Clavier represents a division of the Pythagorean Comma into the following 9 super-particular factors: 531441/524288 = (615/614) (921/920) (345/344) (1161/1160) (435/434) (651/650) (975/974) (1461/1460) (657/656).

Back in 1999 he proposed a different super-particular factorisation also based on the WTC squiggle. Recently, he has offered an interpretation of the Bach monogram (also used by Kellner) and this leads to a different factorisation. Consideration of Suppig's squiggles (similar to Bach and also from 1722) has led him to yet another super-particular factorisation.

Note, however, that many of these sources can be given an alternative acoustic interpretation as beats within an octave. See Bach Tuning.

After Aristotle, there were several Greek theorists who devoted themselves to mathematical computations, the favorite problem seeming to be to find as many ways as possible of dividing the perfect fourth, or the ratio 4 : 3, into what they called super-particular ratios - that is to say, a series of fractions in which each numerator differed from the denominator by unity. They had observed that all the ratios discovered by Pythagoras had this character, 1/2, 2/3, 3/4, 8/9, and they attributed magical properties to the fact, and sought to demonstrate the en-tire theory of music by the production of similar combinations.

It is in Ptolemy's record of the determinations of Didymus (born at Alexandria, 63 B. C.) that the true tuning of the first four tones of the scale occurs. This is it :