Have you ever had an idea that you thought was totally new and undiscovered in the history of humanity? Were you also disappointed when you found out that it was not new at all? It happened to me a while ago when I announced this article. It is my nature to keep my word and therefore I want to write this article about the “Pythagorean comma”. Over the past weeks/months, I talked with piano tuners, musicians, composers, even with my old cello teacher about this problem. I asked them two questions:
1. How can you solve this?
2. As a musician, how do you deal with this?
The answer was “You can’t solve this” and “Don't think about it”. Well, I do not know what about you, people, but for me, these answers are not acceptable. Are they?
Before you read any further, watch this instructional video that demonstrates the Pythagorean comma precisely.
(I uploaded the video to my google drive. This is the sharinglink)
Imagine for a moment that we will find a solution. Realize that this will open a completely new world of sound because with solving the Pythagorean comma, you get a slightly different, but certainly a new tone scale that implies a total new harmonic spectrum. You can create new chords that nobody has ever heard before. If you are a composer or an arranger, it is quite impossible that that idea does not excite you.
I am 100% sure that this problem can be solved because the problem is not the instrument. The problem is in the theory of music. I do have a few brain-spins on how to solve this problem, but I cannot do this alone. I do not know much about absolute hearing, specifically from how many cents someone with an absolute hearing notices that a certain note is to sharp or to flat. What is that range, for example?
The musical approach:
Let us say that you start on the lowest note on the piano and you stack one octave on top of that. This means that you double the frequency, which means that the ratio of an octave is "2". There are seven octaves on the piano, so "2" to the power of "7" gives you a multiplication ratio of 128.
Let us do the same with fifths. We have 12 fifths on the piano. Therefore, we start again on the lowest note and if we stack 12 fifths on top of the lowest piano note, we also reach the same note that when we stack seven octaves. From a musical point of view, the picture is complete and it looks like there is no problem.
However, if we count in frequencies (the fundamentals of sound and therefore also music), a small error occurs. In order to make a fifth, you have to multiply the frequency of the fundamental tone with a ratio factor of 1,5. So if you do that 12 times, that means that you multiply it by 129,75 and this is “The Pythagorean Comma!" This is mathematical evidence that seven octaves are not the same as twelve fifths!
Because the musical approach does not allow us to "hack the system", we have to go further with mathematics. It is somewhat difficult to write maths with a computer keyboard, I did it with pencil and paper. Yes, "The Old-School Way", but because I am "Old-School myself", I assume that this is allowed. I scanned my calculations to a .pdf file that is attached to this post (MATH - PC.pdf).
I understand that we aren't all mathematicians, so if you do not understand a certain reckoning or you cannot read my handwriting, do not hesitate to ask. At the end of the day, if you read this topic until here, I think that you also want to get this problem solved once and for all. Correct?