Harmonic Instrument Design

Well kids, this is all very improving but how about we build a harmonically tuned 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 MIDI 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 extrapolated from our “magic” Earth tuning. Note: that A is 436.9 Hz (neither 440 Hz, nor 432 Hz), for reasons explained, here.

I use http://www.sengpielaudio.com/calculator-centsratio.htm to calculate the cent offsets from the frequencies

Or, here’s the same thing, set up like a keyboard, showing just the offsets compared to 440 Hz Equal Temperament tuning:

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, 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.  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.

Well, I’ll cut to the chase. Last week, I received this stunning, new, custom guitar from Manton Customs, in Shropshire, UK:

True Temperament guitars and necks

One thing you’ll notice are the “curvy” frets on this guitar. This is the True Temperament (TT) fretting system (and their instructions on how to tune it). Now, some people say TT is designed to make your guitar more “Equal Temperament” like a piano. But, having seen this video, it looked like it was pretty harmonically aligned, so I went ahead and included it in the design for the Manton guitar.

Amazingly enough, I have found that I am able to almost exactly achieve the “Earth tuning” on this guitar! – just by following True Temperament’s standard tuning offsets, with the addition of setting the default A on my tuner to 436.9 Hz, instead of 440 Hz. So, “TT” is actually a harmonic temperament.

I spent yesterday tuning it up, and at the end, I played along (badly) to this rendition of Beethoven’s Moonlight Sonata (which I have manipulated so every note corresponds to the Earth tuning.

To my astonishment, the notes are almost identical, so, not believing it, I did this analysis, this morning. The yellow column on the right is the kicker: this is the deviation in cents on the Manton TT guitar compared to my Earth tuning. On average, 0.2 Hz deviation!

Earth tuning frequencies in column-C. Frequencies I can achieve on TT guitar, in column-M

I had thought eventually I would have to design my own fretting system – but now I don’t!

The other beautiful thing about the Manton Matriach True Temperament guitar is the inclusion of the EverTune bridge: once you figure out how to use this bridge, and you’ve tuned your strings – they will basically stay in-tune for ever!

Note, to tune a TT guitar, you need a tuner which you can off-set the strings or notes by a specific number of cents. There are several tuners which support this:

  • I use the Peterson StroboClip HD, and I’ve created an on-line tuning configuration called “ERT” which you can use if you have this tuner.
  • And the Sonic Research ST-200 Turbo-Tuner, which comes with the True Temperament offsets pre-programmed.
  • Also, on Android, the StroboPro app supports TT

This is actually my 4th TT guitar neck! But now I’ve really figured out how to tune it.

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

FreeNote Guitar

My earlier attempt was the FreeNote guitar neck – which adds extra frets – so you can play 11th harmonics etc.  I had taken 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.  It’s enough to keep track of 12 frets per octave – so 24 is a bit much for me!  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 – unfortunately, it got stolen!  But, in fact, the True Temperament neck is a better for the way I play guitar):

FreeNote 24 Fret Just - less frets.jpeg

4 thoughts on “Harmonic Instrument Design

    1. Hi Kite – that’s great. I used to live in Portland. Anyone trying to do this sort of thing is alright with me.

      Like

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