There is no mystery to the harmonic series. Let's be plain about it, it's just 1/1, 1/2, 1/3, 1/4... that's all it is. Let's say you start off with an A note at 440Hz (that means the sound wave runs at 440 cycles per second). If you go down the harmonic series from there the next note is 1/2 that frequency, that is 220Hz, another A, followed by 1/3 of 440Hz, which is 146.66Hz, a D, and so on. If you go up it's the same thing, but you have to flip the ratio over so for the second term you multiply 440Hz by 2, then by 3 for the third note and so on. Simple. This is how a natural trumpet produces different notes without pistons.
One problem that takes a bit more thought is that your brain perceives tone logarithmically. The size of a tone in hertz is different depending on where the tone is in the musical spectrum. If you want to look up the maths you can google it. I'm not great with logarithms so I won't attempt an explanation, but I've got the equations working which is sufficient for me at present.
I found myself swimming in a sea of information, so I made a spreadsheet to try and gain control of it. You can download it to your computer in the file menu in google docs and have a play with it if you like.
A few notes about the spreadsheet:
You can adjust the absolute frequencies and the note identification names by changing the note in the purple box labelled "Tonic:" Everything will follow through automatically from there.
"Partials" is the name of the harmonic elements that go into making a tone. It includes both overtones and the fundamental.
"12-TET" means 12-Tone Equal Temperament, our modern system of tuning.
The "Diff. (Cents)" column tells you how much a harmonic is out of tune in our modern system.
The purple box labelled "Identification Resolution:" specifies the maximum value allowed in the "Diff. (Cents)" column to lead to the following columns being filled in. I've set it at 14 (14 cents out of tune!) so that major thirds are identified, but 14 is a completely unacceptable level of compromise. If you set it to 50, all harmonics are identified as their closest diatonic equivalent, even if they're a quarter tone out. If you set it to 2 you get medieval organum.
So what can we do with this information? We can use the values produced to tune to Pythagorean tuning or other flavours of just temperament. We can use it to examine the relationship between harmony and equal temperament. But the two things that excite me are:
1. All of those unused harmonies which have no equivalents in modern tuning. The enharmonic possibilities and unfelt emotional affects that may be hidden in these gaps between consensus reality really inspire me. There are legitimate notes here that no one plays and few people have ever consciously heard.
2. I have noticed a pattern in the harmonic series. I'm sure other people have spotted it, but I need to know exactly how this works. The fifteenth partial, identified as a seventh, is actually a major third of a fifth. You can justify this by looking at their tuning. A fifth is 1.96 cents out of tune, a major third is -13.69 cents out. The fifteenth partial is 1.96-13.69=-11.73 cents out. This is because each of the overtones of the fundamental have their own overtones. The relationship of their position in the series follows the pattern 1, 2, 4, 8... in complex ways. This is what I will explore today.
(part 4 to follow)
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