Yesterday in A rose by any other name… we saw that we could analyze things mathematically, even when we aren’t dealing with numbers. We had taken the alphabet as our set of elements and defined an operation. In this way we arrived at an algebra (a set together with any operations) using letters instead of numbers. Today, we will work along the same lines, defining an algebra using musical notes. With this we can then look at things like, what is musical symmetry? Can we relate this idea to something that is actually discussed in music theory? Are there more conclusions to made about music from our construction?
There are 12 chromatic notes in music theory which are referred to as a, a#(bb), b, c, c#(bb),…,g,g#(ab). If we were then to continue, we would start again at a and repeat. So, what can we do with these notes? Well we can play them.
In music, a chord is any two or more notes played at the same time, for example, ace is an a minor chord and ac#e would be the a major chord. For our purposes, we will expand this set to include the possibility of playing 0 or 1 note at time. We would denote playing no notes with a rest sign, -. Therefore, if we play any number of notes at the same time, we end up with a chord. Then the set of all chords will be all possibilities for what can be played at one point in time.
Making chords isn’t an entirely different idea than making words. That is, we have a starting set and we combine those starting elements together. However, with chords, we have some additional properties that hold.
One such property is that if we play the chord ‘aa,’ this is the same as playing the chord ‘a.’ That is, repeating a note, even if at a different octave, won’t change the resulting chord. Another property that holds is that if we play the chord ‘ab,’ we hear the same thing as if we had played the chord ‘ba.’ Since we are playing both simultaneously, there is no distinction in order.
Therefore, in the case of chords, we would have abbcbabcdg would be the same chord as abcdg or gdcba. In order to determine if two chords are the same, we can rearrange them and delete any repeats to see if we end up with the same thing.
One more property of importance about chords is that we can tell the difference between different intervals. That is ‘bc’ would sound different than ‘ab,’ beyond just the fact the they are different notes, because the b and c are only a half step apart and the a and b are a whole step apart. If we want to maintain this property of notes we will have to describe this. Therefore, we define the # operation on the set of chords as (x)# means to move all the notes up a half step. In order to move up n half steps, we will write (x)n#.
If we now take what we decided above, we have that we can construct a model of musical harmony. If we let C be the set of all possible chords, then we have that (C,•,#,-) is the set of chords together with the operation of play at the same time, •, and the operation move up a half step, #, and – is the chord with no notes.
With this model, we can note some nice properties of chords as we explained above.
- x•y=y•x, since the order the notes are played does not matter. That is, •, is commutative.
- x•x=x, so every chord is idempotent with respect to •.
- (-•x)=(x•-)=x for all chords x, that is, the rest or empty chord acts as an identity on the set.
- (x)12# =x, so have that notes and chords will repeat every 12 times we move them up half a step.
- If x is a single note, every other single note can be written as (x)n# for some n with 0 ≤ n <12. Since the rest can be written as, -, and all other chords can be written using the singles notes combined with the use of •, we get that C is cyclic. That is, it is generated by a single element, namely any single note.
Now that we know that C is cyclic and that a is one of its generators, we know that we can completely determine any symmetry (automorphism) of C by determining what it does to a. Since the image of a will also be a generator of C, we have that a must be and can be mapped to any other single note through the mappings defined as x is mapped to (x)n#, that is, the mapping that raises everything by n half notes. Hence, these are all the symmetries of C.
If we look at this mapping in terms of music, it would be representative of a key change. That is, if we map a to c and the rest of the notes are mapped accordingly as moving up 3 half notes, then the key of A major would be mapped to the C major, or the key of a minor would be mapped to the key of c minor.
In order to further justify the accuracy of this model, I would suggest that if we listen to a song in one key, then listen to it in another key, the song will sound the same to us. Note, however, that these mapping do not include the mappings from major to minor keys since this would cause some of the notes to move by differing intervals. Using the same reasoning, if we did listen to a song in A major, then in a minor, we could distinctly tell the difference in sound and tone. We therefore have the interpretation that musical harmony and chords would be completely determined up to a key change.
We have already arrived at meaningful results for the set of all chords; however, we can go further with this model. Things like key changes between major and minor can be modeled, but this would require something other than automorphisms. Furthermore, if we had decided that the highest or lowest note in the chord was important to distinguish, we could make some changes to our definition to get out a meaningful model. In order to look at entire songs, instead of just the chords being played at a single point in time, we can again make modifications to cover this situation. That is, the given model is useful in more than just the modest applications used so far.
While we often feel like mathematics is the domain of numbers, we have now seen different cases where mathematics shows up in the places we thought we be least likely. If we use any objects, not just numbers, and define meaningful relations on them, we can use mathematical modeling to find meaningful results in the situation we are interested in, even if the given situation normally falls within a different field of study. Symmetry, in particular, seems to show up as a question of interest quite frequently in such cases.
I would like to thank the students that I worked on this topic with while at Carthage College.