The Relationship Between Music and Math

We are surrounded by two things everyday... Math and Music. Most of the time we don't even notice the math or we just choose to ignore it. But we notice music everywhere... Sometimes as soon as our radio alarm clock goes off in the morning we are surrounded by it. When you take music lessons, we realize that in that music, there are beautiful and symmetric numerical systems. From simple arithmetical processes to things as complicated as Group Transformations, music is full of mathematics.

I would like to show you the beauty so some of the mathematical structure which underlies the basics of Western music theory.

Upper stave: Claude Debussy's Première ArabesqueImage via Wikipedia

The Western musical system consists of 12 tones, or notes. We give these notes names using letters A, B, C, etc. and we go all the way up to G and then we start at A again. Now this only accounts for 7 notes and the other 5 come from things called accidentals which are noted by the #(sharp) sign or the b(flat sign). There is already math involved here. These notes are just multiples of frequencies. The lowest note on the piano is an A and its frequency is 27.5 Hz. To get A# you simply multiply by the 12th root of 2 and you get 29.135 and you keep doing this and after doing it 12 times you will get 55 which is 27.5x2. When you get a multiple of a frequency then it is the same note, up or down some number of octaves. So 27.5 (A) is the same note as 55 (A), just one octave apart. So every note you hear in music is just some frequency and is derived from this.

Now if we imagine adding one number to another as moving that number up the piano that many notes then we can see that if you take

0+1=1 This means that 0(or C) moved up one note is C#.
You can do this with any numbers as many times as you want as long as you mod out by 12, which means if you get a number higher than 12 when you add, simply divide that number by 12 and the remainder is your new number.

9+11=20 20/12= 1 with a remainder of 8, so 8 is our new number. So when you move A up 11 notes you will land on G#. And this works for any number of additions.

3+7+4+9=23=11mod12 This means that if you take D# and move it up 7 notes, then 4 more, then 9 more, you will land on a B.

You can even do this with whole chords.

C major = {C,E,G} = {1,5,8} I will say that when you add a number to a chord, you are adding that number to each note in the chord.

So C major, plus 7 = {1+7,5+7,8+7} = {8,0,3} which is a G# major chord.

An interesting note about this is that if you add a number to a major chord you will get a major chord and if you add a number to a minor chord you will get a minor chord. This is basically what you are doing when you transpose to a new key. You could do it to the entire scale and it is essentially the same thing as going from one key to another.

There are many things you can do with this idea, such as inverting chords and doing whole Group Transformations which will give very interesting musically related results, but the math is very tedious and deep.  This is just a glance at the very surface of the relations music has with math and is one of the reasons why I believe musicians are generally better in areas such as math or science.