![]() Multiplying (instead of dividing) 110 by the same 1.05946 will give you the equal tempered pitch one half-step up, which is 116.54Hz or A#. say the open string length (650mm) pitch is A110Hz. ![]() Since pitch is inversely proportional to string length, you would simply perform the opposite operation. Of course, as has been mentioned, the problem is controlling all the variables where the only thing changing is the string length.Īs an example of the relationship, to calculate the first fret position, you could divide the total string length, say 650mm, by 1.05946 (12th root of two) to get 613.52mm for the first fret-to saddle distance. You could look at each successive fret as a graphical representation of an increase in pitch of 100 cents (100 cents per semitone) from the pitch produced by the previous fret-to-saddle string length. ![]() Look at how fret spacing is calculated for equal temperament. There is a simple relationship between string length and pitch. Robert England wrote: It seems to me that there should be a mathematical relationship between the vibrating string length and the amount of sharp or flat pitch at the 12th fret. Usually these shifts don't effect more than one or two notes near the resonant pitch, which means you can't correct for them at the saddle or the nut: you'd have to shift sections of individual frets.Īll of this, of course, can be rendered moot by a player who doesn't have consistent left-hand technique. Notes lower in pitch than a top resonance are displaced downward, and higher ones are shifted upward. The moving bridge effects the intonation of notes that fall close to the pitch of a strong top resonance, or an air resonance that can drive the top. It also makes the straightforward calculation of string stretch more complicated, by making it non-linear. Relief is the lengthwise concavity in the neck profile that allows for better string clearance, and thus a broader dynamic range. Some real thorny issues come in with 'relief' in the neck, and the fact that the top and bridge are moving. Again, this might possibly be amenable to a mathematical treatment. The progressively sharper upper partials cause the percieved pitch to be higher than the pitch of the string fundamental would indicate, and this gets worse with shorter strings. Some of Byer's work involves correcting for the inharmonicity caused by string stiffness. you might have problems finding all the information you want, such as the Young's modulus of the nylon core material, but a few experiments could get that. Fletcher and Rossing give equations for this in 'The Physics of Music'. ![]() In theory, if you know the physical characteristics of the string, it's length, and the amount it's displaced when you fret it, you can calculate the tension change and thus the pitch rise. Doing the math is no substitute for doing it on the guitar.īyers compensation is great in theory, but as soon as you change string brands, or change the action you're sometimes back at square one, re-doing the intonation to get it better. You will probably have to make 2 or 3 saddles before it's perfect, but it's the only real, "on the street" way to get it right for any particular guitar. IMO, a better way to determine intonation is simply by taking the guitar, setting it up with the action and strings you like, and checking the intonation with a strobe tuner. Many old school spanish makers don't even compensate the scale length, but use a different fret placement scale that in part makes up for this. What about the system used for fret placement? They're not all the same. Is it going to take into account the string brand, tension, action height, string stretch? Plus you've got other variables like fretting pressure, playing and tuning style used by the player. I find it highly unlikely that you're going to be able to find an equation that is capable of capturing all the relevant variables to determine the exact compensation for each string. Sorry, I cannot help you on the math part. ![]()
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