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

I also use the Oscillator in Apple Logic which confirms the exact frequency being produced for each offset on the piano note.

Here’s the same data, 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 some time 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). The guitar seemed to match the piano pretty well!

Using the True Temperament offsets for the curvy frets, I’ve explored how to achieve the Earth frequencies on the TT guitar, as follows

  • Column-AG are the notes
  • Column-AH are the Earth frequencies we’re trying to achieve on the guitar for each of those notes
  • Column-AK is the calculated Equal Temperament frequencies based on A = 435.7 Hz (an offset I’ve chosen as it seemed to provide the most “hits”)
  • Column-AL is the True-Temperament offsets for each notes, as explained on the TT website.
  • Column-AM is the resulting calculated frequency in Hz once the TT offset (column-AL) has been applied.
  • Column-AN shows the difference in Hertz between the frequency we’re striving for (column-AH) and the frequency we get on the True Temperament guitar (column-AM). I’ve coloured these according to “white” as a perfect hit, green as a near miss, yellow as acceptable, and red as a poor match.

In this way, Bb, D, Eb, and F are “exact”; A, G and Ab are a near miss; C-sharp is OK; and B, E and F-sharp are off. The average difference between the frequencies we’re trying to get and the frequencies we do get is 0.3 Hz (it would take on average more than 3 seconds for you to hear a dissonant “beat” between harmonics).

Regarding the “red” ones: The good news about B is that the “Geo” B is 245.76, and the “magic” B is 242.73 Hz (see the three “Geo”, “Magic”, and “432” columns of this chart) – so with the TT guitar giving 244.39 Hz shown in column-AM above, we’re getting a tempered blend between the two extremes.

Similarly with E: while the Geo E is 327.68 Hz, the “Magic” E is 323.63 Hz, so our True Temperament frequency of 326.26 Hz is a blend of these.

To tune a TT guitar, you need a tuner with 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 chromatic tuning configuration called “ERT” which you can use if you have this tuner and have registered it on the Peterson site. This tuning does not take into account the offsets of the True-Temperament guitar. It is basically the offsets to tune any note exactly to the Earth frequency on any instrument. On the TT guitar, it gives the satisfaction of tuning the guitar’s open strings exactly to the Earth frequencies, but you’ll have to bend the fretted notes so they line up across strings.
  • I’ve also created a hybrid Earth/TT tuning on the Peterson site called “ETT” which gives you the offsets from column-M above – that is, the Earth frequencies as offset from standard 440 Equal Temperament, taking into account the recommended offsets for the True-Temperament guitar to match the table above.
  • I’ve also created another tuning on the Peterson site called “JTT“, which is simply each of the 6 to 8 strings on a guitar with the recommended True-Temperament offsets applied. To make this match the chart above you can set your reference frequency for “A” on your StroboClip itself to 436.9 Hz or 435.7 Hz.
  • Other tuners I use include the Sonic Research ST-200 Turbo-Tuner, which comes with the True Temperament offsets pre-programmed. It also supports custom tunings, and I’ve programmed mine with the offsets above.
  • Also, on Android, the StroboPro app supports True Temperament tuning and is both accurate, and a good way to validate your tuning from the other tuners.

Conclusion on the True-Temperament guitar:

It sounds more in-tune than other guitars I’ve tried, and at least some of our Earth frequencies are exact or close to exact. As shown in the video above, if you play a note or a harmonic in place on the neck and then play along to it, those other notes are mostly in tune. It’s also easy to play – with no extra frets to confuse you. To me, it’s worth it to have the satisfaction of being mostly in-tune with the Earth, and it solves the problems of the “gap” intervals which we get on the piano keyboard tunings where everything sounds perfect except for a big gap between two of the third-harmonic notes. The tempered aspect of the TT frets keeps us something which is more harmonic than Equal Temperament

The other beautiful thing about the Manton Matriarch 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 regardless of changes of temperature or humidity.

This is actually my 4th TT guitar neck, but now I’ve now figured out how best to tune it, and the EverTune and the impeccable intonation of the Manton guitar means that I can really experience the precision of these tunings on a consistent basis.

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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