The World According to Beethoven and Euclid

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. Little do we realize however that in that music are some of the most beautiful and symmetric numerical systems. From simple arithmetical processes to things as complicated as Group Transformations, music is full mathematics.

I have grown to love and appreciate most everything about both of these worlds and would like to just talk about anything having to do with either of them (not necessarily how they complement each other). Any questions you might have, some observations, fun facts, homework help, anything to do with math and music is fair game and I would love to hear your insight and provide my own so everyone grows to appreciate, just a little, the beauty of math and music.

Also, as a piano player who is always looking for a new song to play, I know how frustrating and expensive it is to find pieces of sheet music out there and I myself have become a very avid electronic sheet music collector. I have tens of thousands of sheets of music that I have acquired and would love to share with everyone. I will begin to post the sheets I do have and if there is something you would like that you don't see just let me know and I either have it or will do my best to find it. I have accumulated many resources for that.

For my first journey into the World of Math and Music, I would like to show you the beauty of the mathematical structure which underlies the basics of Western music theory.

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). These notes are just multiples of frequencies. The lowest note on the piano is an A and it's frequency is 27.5. To get A# you simply multiply my 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. 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 lay out a little chart of the notes and do a little mathematical modeling we will begin to see some very interesting things:

C C#/Db D D#/Eb E F F#/Gb G G#/Ab A A#/Bb B C

And if we now associate numbers to all of these we get this:

C C#/Db D D#/Eb E F F#/Gb G G#/Ab A A#/Bb B C
0 1 2 3 4 5 6 7 8 9 10 11 0

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.

There are so many interesting applications to this and if you notice any or if you see any other structures you think are interesting comment about it and we can see if we could make a whole post out of it.

I will also be working on compiling a list of sheet music, but if there's anything you want now, just let me know and I will tell you if I have it or not, and if I do I will get it to you ASAP