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How do you measure that a note is 14% out of tune from a "true" G#6? Is a difference that small even audible? It's not that I'm doubting your work - I'm just curious because I have never seen notes portrayed/measured like that - but then I am not an expert in music theory.
You probably have, just in a different way. When you see a chromatic tuner tell you how many "cents" out of tune you are, it's telling you what percentage of the way you are to the next note.
Here's the quick explanation of how I figured it out. Every octave doubles the vibrating frequency of the string. In western music, we divide that octave into 12 steps, so we need a ratio to increase the frequency by so that, when we do it 12 times, we will have ended up doubling the frequency.
For this, we need the 12th root of 2, which is 1.0595. If you take a number, and then multiply it by 1.0595, and then multiply the result by 1.0595, and keep doing that 12 times, the number will get doubled.
So that ratio, 1.0595, is the ratio between any two notes in western music (for those of you who know about even-tempering, stay quiet. I'm trying not to make things too confusing). If you measure from the bridge of your guitar to any fret, and then measure from your bridge to either of the adjacent frets, the ratio of those distances will be about 1.0595 (or 1/1.0595, depending upon which length you put on top of the division).
So, suppose I take a note and go up 7 frets. What's the ratio between those two notes? Well, it's going to be 1.0595 * 1.0595 * 1.0595 * 1.0595 * 1.0595 * 1.0595 * 1.0595. And that comes to about 1.5... or a nice tidy ratio of 3:2. That is why power chords sound so solid. The fifth harmonizes with the root really well because their vibration frequencies are related 3:2.
So, with harmonics, we sort of work the problem from the other direction. We start with a ratio and then see how many of those 1.0595 steps we need to go up to get that ratio.
Now, when you touch the string 1/2 the way along its length (over the 12th fret) to get the 2nd harmonic, you're forcing the string to vibrate at
2x its normal frequency. When you touch the string at 1/3 of the way along its length (over the 7th fret) to get the 3rd harmonic, you're forcing the string to vibrate at
3x its normal frequency. 1/4 of the way, is
4x the normal frequency, and so on.
So, when you touch the string at 1/7 of the way along its length (at fret "2.7"), the string will be vibrating at
7x its normal frequency.
But what
note is that? Well, to find out, we start with 1 and start multiplying by 1.0595 until we get to 7. Well, to save us some time (go read about logarithms if you want to see how it's done), I'll just tell you that, to get to 7, you'd need to multiply by 1.0595 a total of
33.69 times.
Uhhh... okay. But we can't go up 33.69
frets. We can either go up 33 or 34. So, because 33.69 is closer to 34, we're going to say that the 7th harmonic is
closer to going up 34 frets (2 frets short of 3 full octaves). But the harmonic is going to be a little flat of that. It's actually going to be 31% flat (because .31 is the difference between 33.69 and 34).
You can test this by playing the 7th harmonic (over fret "2.7") on the A string. That should give you a note just flat of a G three octaves up (15th fret on the high-E). Play them together. Now,
hear how it's a little flat? That's what 31 "cents" of pitch sounds like.