"My partner, Greg Back, was taking a course at California State University at Northridge called 'The Physics Of Music.' Included in the course was a description of the model Pythagoras used to figure out the rule of 18, the formula that determines fret placement. That model implies that a string is under constant tension. However, a string is not under constant tension on a guitar neck, and he neglected to take in account that it's easier to press down the string in the middle than right next to the nut. That increase in tension directly translates to pitch, so notes will always be sharp at the first two frets because the model he used did not include that fact."
There are two errors here. He repeats his assertion that Pythagoras devised the rule of 18 and implies that he was working on the physics of fretted instruments when, in fact, fretted instruments didn’t exist. Pythagoras’s monochord was a device for investigating the harmonic relationships in a vibrating string and he used it to develop the Pythagorean scale which formed the foundation for subsequent development of the idea of harmony but which was quite different from the equal temperament scale.
The second error (highlighted in red) is a piece of complete nonsense displaying a massive ignorance of basic physics.
He’s correctly observed that the strings feel tighter closer to the nut and quite wrongly inferred from this that the tension is greater closer to the nut!
Tension is a constant load; it can’t be greater closer to the nut. The strings feel stiffer close to the nut because a displacement here subtends a steeper angle than at the middle of the string so more of the vector component of the string’s tension is directed against the guitarist’s finger. This means that the resistance to displacement can be greater here although the actual stretching effect may be less.
The increase in pitch is actually linked not to these subjective observations but only to the increase in strain applied to the string.
Although a given displacement close to the nut does result in a more rapid increase in strain than the same displacement towards the middle of the string this is actually an irrelevance because in a properly set up guitar the string presents a shallow angle to the fingerboard plane; as a result of this it always experiences more displacement when fretted at the octave fret than at the first fret where clearance is very small.
When a string is fretted it's forced into a longer path between nut and saddle, stretching it, and the percentage by which it's stretched is an indicator of strain. On a properly adjusted guitar the string clearance is always greater at the high frets than the low ones the so the string is always stretched more the higher up the fingerboard you go.
An average guitar has a string clearance at the first fret about five times less than at the twelfth and a brief calculation (courtesy of Pythagoras) gives us an estimate of the strain around a quarter to a fifth of that at the twelfth; the traditional method of intonating a guitar means that any displacement of the saddle to correct the intonation at the twelfth has about half the effect at the first fret, so the first fret is actually being over-compensated.
The spreadsheet shown at left uses data from two guitars that I recently set up to arrive at this conclusion.
Ockham’s Razor urges us that when multiple solutions to a problem present themselves one should select the simplest one and it seems to me that when Buzz discovered that his first fret was sounding sharp the simplest answer was that his guitar hadn’t been set up and tuned properly!
