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I was wondering why the white dots in the fretboard are usually placed in fret numbers 3, 5, 7, 9, 12, 15, 17, 19, and 21.
I know that their general purpose is for counting, but what I wonder is: Why are they placed on precisely those frets, and not different ones?
For example: Why are there two frets between the dots on 7 and 9, and three frets between the dots on 9 and 12?
The spacing is designed to offer useful milestones on the fretboard. Take the notes on the 6th string (in standard tuning), for example:
Maybe lost to history, the dots are references for the Fibonacci series, which when harmonics are considered, give a pure major chord with octave and perfect fifth redundancies. The dot at the 9th fret marks 2/5ths of the string length. The harmonic there is of the 5th partial (the major third) -- this is the one that throws you off.
This is my observation, and it is so easy once seen, that I have little doubt that many people have stumbled across it. Nonetheless, Gibson and Martin guitars had nothing to say about it, and so far as I could see, nothing to see in google searches. THE DOTS MARK THE HARMONICS COMPRISING A PURE MAJOR CHORD.
Are the dots an atavism from pre-equal tempered tuning? In historical paintings of fretted instruments, or in museums, when are the earliest dots seen in their modern location??
string
open = 1 = do
dots
12 = 2nd partial = do (octave over root)
7 = 3rd partial = sol (perfect 5th + octave over root)
5 = 4th partial = do (double octave over root)
9 = 5th partial = mi (double octave + major third over root)
3 = 6th partial = sol (double octave + perfect 5th over root)
The frets that are highlighted are the equal-tempered equivalents of the basic fractions, as seen in the table below. (The first column is the fraction, the second column is the corresponding number of half-tones):
1 0 = 0 6/5 3.156 ~ 3 4/3 4.980 ~ 5 3/2 7.020 ~ 7 5/3 8.844 ~ 9 2 12 = 12 (from now on, it's simply an octave up, i.e., 12/5 15.156 ~ 15 times 2 for the fractions and +12 for the frets) 8/3 16.980 ~ 17 3 19.020 ~ 19 10/3 20.844 ~ 21
One can ask why 6/5 and not 7/5, 8/5 or 9/5. I don't know exactly, but the fractions used have the lowest possible sum of numer+denom in the interval [1,2]. (I'm not 100% precise here, we omit 5/4 which gives 3.863 ~ 4, but you wouldn't place two dots adjacent since it would only confuse people, would you?)
This as well explains why 9 got there and not 10: It corresponds to a very low-entry fraction. One can as well notice that you can try flajolets on bars 12, 7 and 5 easily, and with some adjustments on 3, 4 and 9 as well, but these don't sound quite strong. Flajolets on other positions are even more difficult.
They are at important intervals: minor third, fourth, fifth, major sixth, octave.
Then they repeat for the second octave: minor third, fourth, fifth, major sixth.
I always thought the ninth fret marker was simply aesthetics. Being that the marker after the twelth is three frets away, the marker before the the twelth is the same distance away. It centers the twelve marker. Practically speaking, the twelve marker is also more noticeable.
The white dots are typically placed at harmonic points on the string. The first harmonic of the open string is the octave, halfway along the string at the 12th fret. The second harmonic is one-third the string length, and that falls at the seventh fret. On some models, you'll see that fret marked with a double-dot as well, as it's a very prominent harmonic:
The woman holding the guitar here is pointing at the octave marker, but you can see the seventh fret is also double-dotted.
You'll notice the above guitar also has fewer dots; there isn't one on the third fret, and the second octave doesn't match the first. The above layout is a more harmonic-oriented one; the average player, with a little practice, can coax out a harmonic at each of the marked frets. Now, because harmonics, by their nature, are Pythagorean "just temperament", while the guitar's fretboard is designed for "equal temperament", the actual location of the harmonic usually won't be over the fret marked by the dots, so this layout isn't an exact guide but it's useful at a glance.
Most fretboards have the frets marked as you describe, and this is a "compromise" layout of sorts that can be repeated on both octaves of the fretboard, so the guitarist can play in both octaves without losing his place. This layout is more a straight-up positioning guide, but it is still useful for identifying harmonic points.
Other harmonics of the open string besides the ones in the above picture are less prominent, though a skilled player with a well-made guitar can get them to speak. Jaco Pastorius was famous for coaxing third and even second-fret harmonics from his Jazz Bass: Jaco Pastorius - Loop Jam (incorporates part of his "Portrait of Tracy" solo).
I think it's to signify intervals as much as anything else. The 12th fret is the most obvious - it's the octave. The 3rd fret is pretty obvious because it gives you some general orientation for most open chords. The 5th and 7th frets are valuable because they signify the 4th and 5th interval of the open strings.
After 45 minutes or so reading people's answers to the ninth fret question, then 15 minutes of pondering, I came up with a crappy answer:
The NINTH FRET is a position of all flatted notes:
Db Ab Fb Cb Gb Db --or-- Db Gb Cb Fb Ab Db
(The order depending upon whether You look at the notes from first string to sixth or vice-versa.) But then, the fourth fret is also a position of all flatted notes. I know we aren't supposed to call B a Cb or E an Fb, though there are exceptions. Yeah, somebody, please come up with the answer to this ninth fret conundrum! In any case I won't stop thinking about this.
