Just Intonation Guitar Design

Well kids, this is all very improving but how about we build a guitar with the frets in all the right places to hit all the right notes?

It won’t be able to play in every key – it will sound (slightly) out of tune with equal-temperament instruments like piano and normal guitars – even when they are tuned to A = 432 Hz.  But it will be acutely in-tune with itself and the frequencies we have identified as being “the still points in the turning universe”.  It will be truly musical, in the deepest possible sense.

It will work well with fretless instruments such as fretless bass and stringed instruments such as violin and cello.  And with high-end digital keyboards – where it is even possible to tune every key to the magic frequency – (it can be done in Apple’s Logic software as well – I’ll add this to the blog at some point).

The ideal instrument will probably have bent frets, like the True-Temperament instruments here.  But unlike us, who would bend the frets to ensure that each note is exactly harmonically correct, True-Temperament has bent their frets to ensure Equal Temperament at each fret.  Sometimes the potential of an invention is not fully realized by its inventor!

So, until I get my act together and produce my own neck with bent frets, then I am stuck with three options:

  1. Fretless – way too difficult to play unless you’re Guthrie Govan, and basically, he doesn’t attempt to play chords
  2. Split  frets – like Tolgahan Çoğulu – but then you can’t bend the strings when you play – and what fun is that?!
  3. Straight frets which are placed in harmonically correct places – but which will not always yield the in-tune note on all strings at every fret

The best compromise for me is option 3 – and the good news is that a man called Jon Catler makes FreeNote guitar necks with the frets at all the right places – (and quite a few more!)  I took the plunge and purchased the FreeNote 24-Fret Just Intonation Neck and put it on on a Stratocaster body I had bought from Stew-Mac.  It fit perfectly without intercession from a luthier.

FreeNote 24 fret Just - all frets

The only trouble was, it’s 24 frets per octave.  That’s a lot of frets, and even at the octave and fifth, the frets were so close together that it would require a change to my guitar technique which wasn’t compatible with the big chords and quick fills technique I’ve developed over the years.  I can get confused with 12 frets per octave – but 24 is ridiculous!

So, I did an analysis in the table below, where I substituted as the string tuning each of the 11 possible tones from the harmonic scale we worked out in the first section, and calculated mathematically what frequencies would occur at each fret.

Catler frequencies

In the table, the second column is the note that the string is tuned to.  The fourth row indicates the fret of the FreeNote neck, and the second row indicates the harmonic fraction being generated at that fret.  If the frequency generated at the fret matches one of the frequencies we’ve determined to be “correct” then I’ve indicated the resulting frequency and note in black text; if it’s close, then I’ve indicated that in gray text.  The column on the far right tallies the number of “hits” I get for that string overall.

As you can see, some string tunings did better than others: F, C and B-flat all achieved 7 hits per octave (100%).  Also, some frets had a good cluster of hits on them, while certain frets yielded zero to just one “hit” – so, it seemed to me, those were frets that could be removed.  And in fact, the red in the 4th row in the table above indicates those frets which I did actually end up removing; so now the guitar looks like this:

FreeNote 24 Fret Just - less frets.jpeg

With only the frets in place that yield notes within my simple harmonic range, I then created “tuning tables” to try and combine the string tunings to yield the maximum hit rates (i.e. B-flat, F, C, A, E-flat.) into workable 6-string guitar tunings – as you can see, below.

No surprises – tunings where B-flat and F predominate have the best hit rates.  I explored open-B-flat and an open-9th tuning.  These yielded the best “match counts” of 44 matches across 6-strings within the first 12-frets of the guitar.  But B-flat is a very low tuning for a standard length neck  (although it would work well if FreeNote also made baritone necks).  So for this neck, F would seem to be the best option.

In the tables below, the left-hand column indicates which string we’re talking about – and the column next to that is the note it’s tuned to.  The harmonic fractions of the fret position are shown in the second row, the 4th row is the original FreeNote fret-position; and the 5th row is what most people would call this fret now that we’ve removed all the “red” frets above; you’ll note that we kept two “third frets” on the neck!

B-flat & F

The next set of tunings.  open-E-flat, Concert-F and open-D – yielding 38, 37 and 35 hits respectively:Other tunings

With the next tier of tunings, we’ve now dropped from 44 hits to just 27 for open-D, 26 for open-C, and 25 for open-E-flat:Other tunings 2

The final kicker is the tuning everyone is used to: whether you play concert-E or open-E, the number of hits is pretty dismal: concert-E yields just 22 hits compared to 44 for concert-F or open-F:E tunings

So, it does make sense that if you take my path and procure a FreeNote 24 Fret Just Intonation neck, that you tune your guitar either to concert-F or concert-D – even if you don’t remove any frets.  With either of those two tunings, you are getting a respectable hit rate of 37 or 35 hits – and you’ll get the satisfaction of playing your instrument in harmony with the cosmos (at least for most frets) – and you’ll learn to avoid the frets that are not in this harmonic series – because they just won’t sound as sweet.  You can also mark them with nail-polish,  which is what I did!

If you want to get more out of the guitar, then I would recommend open-F, or even open-F 9th.  Or, look at the big chart and create your own tuning – perhaps with just the notes Bb, F, C, Eb and A – to get the maximum number of pure notes out of your guitar neck.

Stay tuned (pun intended!) and I’ll add some guitar riffs recorded with a regular guitar, and as recorded with this guitar, soon – so you can compare!

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