Groups and music

I have admired jazz musicians for many years, and envy their ability to improvise tunes over a chord sequence.   I recently signed up to a jazz workshop at the music school, and one of the excercises that we have been assigned got me thinking about a link between the excercise and mathematical group theory.  This kind of group is what I refer to in the title, so if you were expecting something about rock or jazz bands, I apologise for the disappointment.  But bear with me; this only needs junior-school maths, and it doesn't hurt a bit.

First, a gentle introduction to group theory.  Mathematicians study what they are pleased to call groups, because groups are the smallest entities, systems, things, whatever you want to call them, in which you can solve equations like 3+X=7 or 12.5 x X=73.4.   (Lower case x signifies multiply)   More generally, you can solve a*X=b, where a and b are known members of a set (such as the whole numbers, or any number that exists) and * is a way of combining two elements of the set to produce a third one (e.g. addition, multiplication)

A group is a combination of a set of things, and an operation that can be performed on them, such that a*X=b can always be solved to a single value for X.  If we call the set of all the whole numbers N, and the set of all possible numbers R, then in the above examples, the group exemplified by 3+X=7 can be written as (N,+) and the group exemplified by 12.5 x X=73.4 as (R,x)   Note that (N,x) is not a group because (for example) 3 x X=11 does not have a solution that is a whole number.   (Excercise for the reader:  Is (R,+) a group?)

The sets N and R both have an infinite number of members, but we can make smaller groups that are easier to get one's head around.  Perhaps the most trivial is the one with just one member, 0.  We can draw the set by listing its members inside brackets like so: {0} and the group is drawn as ({0},+).  It has only one equation, 0+X=0 and it has only one solution, X=0 which is not surprising.

The next size up has two elements, and we can draw it ({0,1},+)    The + this time has to represent addition modulo2, that is, if we get an answer of 2 or more, keep subtracting 2 until we get to 0 or 1.  There are four equations of the form a+X=b, one for each of the two possible values of a and b, and all the equations are soluble.

I need to show an example of a group that has a subgroup, so let's look at the group ({0,1,2,3,4,5},+)  The + represents addition modulo6, so if we get a result of 6 or more, keep subtracting 6 until we get a number from 0 to5.   The group has 6 elements, and mathematicians would say it has an order of 6.  The set {0,2,4} is a subset of {0,1,2,3,4,5} and is a group under the same operation of addition modulo6. That is ({0,2,4}, +) is a subgroup of ({0,1,2,3,4,5},+)   (So if we start only with the elements {0,2,4} then no matter how many times we use the + operation, we never get a result that is outside of this set.)  The order of the subgroup is 3, and 3 exactly divides the order of the original group that is 6.

It is a general truth that the order of a subgroup divides the order of the group exactly.  (So, for example, a group with a prime order has no subgroups)

On to music, specifically western music that has an 8-note scale that comprises 12 semitones.  We'll make the approximation that all the semitones are equal (not strictly true for perfect scales, but widely adopted in western music).  We'll also equate all the notes with the same name, so we don't distinguish between F# and the same note one or more octaves higher or lower.

We have a set of notes {C, C#, D, D#, E,F,F#,G,G#,A, A#, B} each of them different by a semitone.  This is rather like our example group above, but of order 12 instead of 6.  In going from C to C# we add a semitone, and we can keep adding until we get back to C, just like our group.   There are subgroups.  If we start at C and go up in a minor third, (3 semitones) we get to D#, up again by the same amount to F#, then to A and back to C.  The order of this subgroup, 4, exactly divides the order of the main group (12).  4 is the smallest multiple of the number of semitones that is also a multiuple of 12.

If we look at the interval that has 9 semitones, we go from C to A, to F#, to D# and back to C.  This is our minor third group (backwards)  and has order 4 because 4 is the smallest multiple of 9 that is also a multiple of 12.   The interval of 8 semitones gives us a subgroup of order 6; the "whole tone" scale.

An important and common interval in music is the fifth (e.g. from C to G); it has 7 semitones.   In our jazz workshop we are assigned to learn the minor 7th arpeggio in all keys, starting with C and moving to the next key by going up a 5th (to G, then 7 semitones more on to D and so on).  

Musicians call this the cycle of fifths, and it cycles through the whole octave because the smallest multiple of 7 that is also a multiple of 12 is 12x7.

Circle of fifths from education.davidspencer.ca