I was chatting with two of my girlfriends at Starbucks last night, one of which is finishing up her master’s in education. She is doing her student teaching now and was talking about the math lesson she prepared for a second grade class. Of course I was all ears. Math was my strongest subject in school. I learned how to add and subtract before learning to read and at age 9, I solved my first algebraic equation. None of my friends liked math. I was the only who could get excited over a good math problem. I had plans to major in mathematics in college but once I completed calculus I, my passion began to fade. I think the math department was slightly disappointed when they learned that I had not pursued a mathematics major, but Our Lord had other plans. Music, unexpectedly pulled me in and the interesting part is that I was probably a stronger mathematician than I was a musician.
But as I began my studies of music analysis, the glories of mathematics remained with me. When I was a sophomore in college, I completed a math project using Microsoft Excel in which I calculated the frequencies of all 88 notes played on the piano.
The lowest note on the piano is A, which has a frequency of 27.5 Hertz. That means the string vibrates 27.5 times per second. To find the frequency of the note A# (A-sharp), which is one half step above, you multiply 27.5 by the 12th root of 2. The 12th root of 2 refers to some number multiplied by itself 12 times that will give you something close to 2. Why are we talking about the 12th root of 2? Because the octave consists of 12 half steps.
The 12th root of 2 in computer lingo or on a graphic calculator is expressed as 27.5 * ^ 1/12. The 12th root of 2 expressed as a decimal is about 1.0594631 (rounded). That means if you take that decimal and multiply it by itself 12 times, you will get close to the number 2. The 12th root of 2 is an irrational number just like PI
Oh and here is a little side note, the asterisk (*) stands for multiplication because if you use the traditional multiplication sign, it might get confused with a variable X that you find in algebra. The caret sign (^) is used to indicate an exponent. So if you want to say 2 squared, you write 2 ^ 2. To express a square root of a number like the square root of 4 you write 4 ^ 1/2. Note that you express the exponent as a fraction for square roots, cube roots, fourth etc). So if you want to say the cube root of 8 you would say 8 ^ 1/3. The cube root of 8 is 2 because 2 * 2 * 2 = 8.
Now on excel you can use one formula to solve all the frequencies so you don’t have to do it 87 times. The formula that I came up with is:
Y = 27.5 * 2 ^ (x/12)
Y (the frequency of a note) = 27.5 (the given frequency of the lowest note on piano) * (multiplied by) 2 ^ (X/12). Okay, I know the factional exponent looks strange with the X and all. The best way is to show you.
The X stands for the number of half steps away from the given note, A. For A#, we substitute X with 1 because A# is one half step above A.
Substitute 1 for X and we get
Y = 27.5 * 2 ^ (1/12)
Y = 29.16 (roughly)
Now if I wanted to find the frequency of the next note B, substitute X with 2 (two half steps away from the given note A). How does this work? What you are really doing is 27.5 * 2^1/2 * 2^1/2. Since you are multiplying 2^1/2 by itself you are really doing 27.5 * 2^2/12. Meaning you are taking the 12th root of 2 and then squaring it. the Denominator equals the root so in this case, the 12th root of 2 and then squaring it. The numerator refers to the power (in this case the 2 on top means to square it).
Below are my findings for all 88 frequencies.
Here is a line graph of all the frequencies. Notice the shape of the graph. The higher you go, the larger the gap between each of the frequencies. Frequencies always double at the octave. Therefore, if you play A above middle C on the piano, the frequency is 440. The next A above that would have a frequency of 880.
Music and math go hand and hand. In math we have substitution where you substitute numbers or expressions in place of letters. In music we do have chord substitution. Don’t get me getting on that discussion. I love secondary functions in both math and music!
If you found this whole thing confusing don’t worry about it. I must confess that I posted this help preserve the memory. I was quite proud of myself after I completed this. I never considered myself a genius, but that was a very high moment in my life because it was my own individual project.
I believe that all things, both living and non living, are a reflection of the Holy Trinity, separate entities that are all connected as one. I always believed in a common oneness in everything since everything that is comes from God.