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Fret Placements

Discussion in 'Luthier's Corner' started by daveze, Dec 7, 2003.

  1. I've finally decided to quite whinging to myself about how I don't have enough money to buy the stuff I 'need' to start my bass. So, I'm do what I can, and thats planning everything out. Finally actually drawing scale pictures and stuff, rather than doodling in my notepad during lectures.

    I'm planning to use 900mm as the scale length (about 35.4"), cause its a roundish number, and its a long scale, which will mean a tighter B. I know that construction plays a bigger part in B tension, but this is my first bass, so construction will not be quite the luthier extraordinaire. Enough waffle, and to my question: I've looked at JP's thingy, build-your-own-guitar's jobby, used the 18 (17.817) rule-a-ma-jig but the problem is, should the 5th fret distance be exactly a quarter of the scale length, and the 7th fret a 3rd? Cause thats where the harmonics are, so shouldn't the fret be at exactly the same place? Or am I simply wrong? Or, because its only discrepancies of less than 1mm (which I can't measure, but for the 5th and the 7th, I round them 'away' from each other), does it just not matter enough to be heard in most situations?

    A lot of froth for little question, but I wanna know, cause I'm actually making progress towards getting the thing started, as opposed to talking big. I kinda have a headstock pic, placed the tuners and stuff, but haven't really decided how I want the 'feel' of the contours yet, I'll post them later this arvo. And I have a big A3 bit of paper on the floor next to me, waiting for the body to go down.

    Josh D
  2. RobbieK


    Jun 14, 2003
    Actually its an excellent question. The chromatic scale is divided into 12 equally sized intervals (semi-tones). It's been like this for around 300 years. In this tuning system, if you get a frequency, say 440Hz (A above middle C) and multiply it by the twelfth-root-of-two (or two to the power of one-twelfth), which is about 1.059, you get 446.16Hz - the frequency of the next semi-tone up (Bb). Multiply this product by the twelfth-root-of-two again and you get the next semi-tone up (B) and so on and so on. (Of course dividing by the twelfth-root-of-two heads downwards, giving Ab, G etc.) As a consequence, all intervals except for the octave have been streched or squashed slightly compared to their similar natural interval.

    Another way to look at it is if you tune, say low C and G on a piano so that they are "perfectly" in tune (ie. the fundamental frequency of the G is exactly half that of the G harmonic of the C), then tune a D to the G in the same way, then an A to the D and so on, when you get around to C again and compare it to the original C, it'll be almost a quarter of a semi-tone sharp. This is very similar to what we are doing as bass players when we tune using the old 5th and 7th fret harmonics method. (Remember, the harmonic on the 5th fret is a double octave of the open string, but the harmonic on the 7th fret is a 12th - ie an octave and a 5th). But with 4ths and 5ths, the difference from their natural intervals is only slight (less than a quarter of a semi-tone over 12 tunings), and in our case, the error is only compounded 2 or three times.

    3rds and 6ths, however are actually quite a distance from natural intervals. Grab a six string guitar and tune the G and B strings by ear so that they don't beat (that warbling sound you hear when two oboes attempt to play in unison :p). Now fret a D on the 3rd fret of the B string and compare it to the open G. The interval is way flat!

    This equally tempered tuning system is known as "Equal Temperament" :D (For more info, try a Google search on this expression, also try "Pythagorian Comma", "mean tone", "just intonation" and "sexy sweedish bikini girls". :meh: ) So in practice it means that the 4ths and 5ths are a tiny bit flat of natural over tones (that's why your fret calculations for the 5th and 7th frets seem a bit iffy) and the 3rds and 6ths are actually rather sharp (compare your measurement for the 4th fret to one-fifth of your scale). I guess over time our western ears have just got used to hearing these intervals slightly out of tune. Of course many cultures still base their tunings systems on natural overtones and in the 70's some composers even started devising their own tuning systems (check out the Music of Harry Partch).

    It seems the reason equal temperament was favoured over just intonation was that it allowed guys like Bach and Telemann to modulate to keys furthur and furthur from the tonic and to use more and more chromaticism, and also to play consecutive pieces in distant keys without retuning their harpsichords and organs etc. I've also read where many people blame the growth in popularity of the accordian through Europe for the demise of many scales etc. from Eastern European folk music. The theory is that the accordian is a loud and portable instrument that is tuned more or less permanently (to equal temperament) during its construction in a factory. The instrumentalists and vocalists playing along with it had little choice but tune up to it.

    Our fret positions, of course, are based on this system. Several years ago I did an Excel spreadsheet that calculates 36 fret positions for any given scale. If you are interested, I might be able to dig it up from back ups of my old computer.

    Kinda strange, isn't it, to find out that 11 of the 12 intervals of the chromatic scale, our fundamental system of musical tuning, are actually fudged a bit for convenience. :(
  3. pilotjones

    pilotjones Supporting Member

    Nov 8, 2001
    The short answer is: no, the fifth fret is not exactly at 1/4 the theoretical string length, and the seventh is not at exactly 1/3. The 12th is at exactly half, the 24th at exactly 3/4, the 36th at exactly 7/8, but all others are off.

    Use a fret calculator, such as JP's which is included as the last sheet of my string calc sheet (end of this thread). And don't use a sheet or program that uses an iterative method (each fret figured from the previous), unless you know that it uses enough digits to make the accumulated repetitive roundoff error insignificant.
  4. Okay sweet. I think I'll use the one that JP did up, I downloaded it in my searches for the answer to my question yesterday.


    Josh D
  5. pilotjones

    pilotjones Supporting Member

    Nov 8, 2001
    In case you're still interested, here's a chart of the frets and the harmonic positions (as laid out for a 35" scale). I could provide the spreadsheet if anyone's interested.