Just Intonation Instrument Design

Well kids, this is all very improving but how about we build a just-intonation instrument? First – tuning an electronic keyboard, and then we’ll get into guitars. The offsets I provide could be used for tuning any instrument, really.

electronic Keyboard instrument

The easiest one is a keyboard instrument – especially if you have the Apple “Logic” software where you can set an offset for each note in the octave. The table below is the collection of frequencies from the the harmonic series I explored on the home-page, which I think sound sonorous together, with their off-sets in cents. I haven’t averaged or adjusted the frequency for any of them, but I have chosen specific harmonic intervals. e.g. I used the 9th of the 5th of Eb to give me a value of 388.8 Hz for a G, rather than 384, because I think 388.8 Hz sounds sweeter with the other frequencies. You can use this table below to set your offsets from either A=432 Hz Equal Temperament, or from A=440 Hz Equal Temperament.

Note that depending on whether I use B-flat as the fundamental tone from which all the other harmonics emanate, or E-flat, the A-flat note comes out to a different frequency (403.2 Hz when generated from Bb, vs. 409.6 Hz when generated from Eb). Because Eb at 307.2 Hz also corresponds to one beat per Helek (the ancient Hebrew “second” of 3.333 modern seconds), I’ve chosen Eb as the true fundamental tone, a fifth below Bb, even though I couldn’t detect any funny vibrational wobble for this frequency on my tone generator as I did for Bb and F as documented on my home-page). I’ve written about the Helek and how my “magic frequencies” directly coincide with the speed of rotation of the Earth, here.

I recommend you use the pink “Eb columns” on the right side of the table as your off-sets – either from 432 Hz or 440 Hz Equal Temperament.
( http://www.sengpielaudio.com/calculator-centsratio.htm ).

Guitars

OK, let’s get into guitars. Unless we’re going to play slide all the time, we need a guitar with the frets in all the right places to hit all the right frequencies – harmonically. (It won’t be able to play in every key because of the lemma – which I’ve discovered is a fractal reflection of your starting frequency in the cycle-of-fifths, and not a random gap – here.) Guitars with the kind of fret placement we’re talking about 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. Ideally, it will hit the frequencies in the table above.

It will work well with fretless instruments such as fretless bass and stringed instruments such as violin and cello – and horns!  And with high-end digital keyboards which have been set with the offset in cents for each note – as described above.

The ideal guitar will probably have bent frets, like the True-Temperament instruments here.  But unlike me, who would bend the frets to ensure that each note is exactly harmonically correct according to Just Intonation, True-Temperament instead has bent their frets into a compromise temperament they call the Thidell Formula One Temperament – which has offsets from Equal Temperament published here.  

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
  4. Go with the True Temperament approach and be in better harmonic intonation, but not harmonically exact.

True Temperament Necks and Fret-boards

My most recent adventure has been with the 4th option, True Temperament – and I’ve now had these frets and necks applied to three guitars. Look closely – the frets are “bent”!:

G&L Legacy, Reverend Club King 290, Tanglewood parlour guitar – with True Temperament frets

It is a compromise, but once you learn to tune them relatively, it doesn’t seem to matter too much whether you tune to an open chord, or with regular concert tuning – the result is more sonorous than an equal temperament instrument. In this way, I was able to tune in my favored tuning, which is Open-Eb 7th – which is, I reckon, how Robert Johnson tuned his guitar – see my blog about this here.

I use the Sonic Research ST-300 tuner, which allows me to create a custom tuning using these exact offsets – which saves a lot of messing about for tuning an electric guitar.

FreeNote Guitar

My earlier attempt was option 3.   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 wouldn’t be compatible with the “big-chords-and-quick-fills” technique I’ve developed over the years.  I get confused with 12 frets per octave – so 24 is ridiculous!  If you do a bar chord on this guitar, you sort of have to choose notes from the micro-frets next-door for it to sound good.  And I don’t have that kind of precise mind!  So, I tried to simplify by identifying the key frets that I really need.

This required a fair bit of analysis to figure out which frets would yield the frequencies I’m after depending on what note the string was tuned to.  I created the table below, where I put a different starting frequency in the second column for each string, and calculated what frequencies would be produced by the Jon Catler JI neck at each fret, and bolded those that matched the target frequencies we worked out in the first section.

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”, 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 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 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 (presumably – it got stolen!):

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, Open-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 I’ve removed all the “red” frets above; you’ll note that I 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, or Open-Eb – even if you don’t remove any frets.  With either of those three tunings, you are getting a respectable hit rate of 38 to 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!  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.

I have to say though – it’s one more thing to worry about – not hitting the out-of-tune frets, and this is why, when my FreeNote guitar was stolen, it was in a way a blessing because it gave me permission to explore the True Temperament guitars.  Not that they’re perfect either – but they are easy to play.  Now, if I can just persuade Anders Thidell to make me a neck specifically for Open-Eb 7 tuning, we’ll be golden!

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