Hi,
I am reading the book for the second time and I thought I should clarify as I go along.
I have taken a lot of physics classes and my difinition of harmonics is whole number mulitiples of a fundamental frequency. Which is stated in the book.
The natural Harmonic series confuses me a little. I would expect a freq of 440 to have 880, 1320, 1760 etc harmonics. Does 1320 Hz relate to an "octave and a fifth" I know somehow it must equate to 1320 Hz, especially since the next harmonic is the second octave but I don't know how.
Thanks
Cheers,
Max
Yes, 1320 would be an octave and a fifth in Pythagorean tuning.
I didn't cover Pythagorean tuning in the book, but I might as well here. The second overtone is a fifth above the first overtone, which gives a fixed ration for perfect fifths. In Pythagorean tuning, all notes are tuned to this ratio... 1320/880 = 1.5, so the E note at 1320 is a perfect fifth above the A880.
There's a few problems with Pythagorean tuning, though. The first is the 'Pythagorean comma'. If you continue to go up by 3:2 multiples, you should be cycling around the circle of fifths. Eventually, you'll get back to A... except you won't. Take a nice low A like 27.5, and apply the 3:2 rule 12 times to go around the circle - you get A3568. If you multiply A27.5 by 2 seven times (12 fifths is seven octaves), you get A3520.
The next problem is that Pythagorean tuning makes every note different. F# and Gb will not be the same note. Early instruments were built for specific keys along Pythagorean tuning, but the introduction of the keyboard made that awkward - keyboards should be able to play in any key. They tried various split key arrangments, none of which were very workable, and abandoned it altogether. Our modern tuning is called 'even temperament', because we split the octave into 12 equal steps, with each step being the 12th root of 2 above the last - so 12 steps is a perfect octave.
Doing that makes a perfect fifth 7 x the 12th root of 2 over the fundamental - just a touch flat from the natural harmonic. However, it makes all octaves perfect, and eliminates the Pythagorean comma.
So the Pythagorean ratios, which create the overtones we hear in harmonics (and add to the overall timbre of the guitar in general) don't always add up to the modern notes you'd expect - and the higher the overtone, the more it's off. That's why I phrased things like I did in the book - the 12th overtone is "3 octaves and a bit less than a sixth" etc.
(edited to fix a typo)
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Great Explanation Noteboat,
Ill re-read your post a few times to cement it in, but it makes sense already.
Thanks for clearing that up.
Cheers,
Max
