Musical Puzzle: Is it possible to tune to James Taylor’s microtonal ‘stretched tuning’ without a cent-display tuner? What would be the minimum level of technology needed to do this precisely? What can we learn from the process?
Part of RJ’s ‘World of Tuning‘
I recently posted some ideas around guitar tuning on Reddit, to get some feedback before I published my Ultimate Tuning Guide article for Guitar World (Dec 2019). Along with many other excellent questions, there were a few that brought up fingerpicking legend James Taylor’s ‘stretched tuning’.
For years, he’s been deliberately tuning his guitar a little flat to counteract various subtle intonation issues, relating to the pressure of a capo and inharmonicity (essentially the fact that all vibrating strings are bending slightly as part of their oscillation, meaning they actually ring a little sharp). He lowers all his strings precisely as follows (sourced from the video below), with a cent being 1/100th of a semitone:
E6: -12 | A5: -10 | D4: -8 | G3: -4 | B2: -6 | E1: -3
A couple of the Reddit questions touched on whether you’d need a digital cent tuner to reach his tuning, which I thought was an absolutely fascinating idea. In practice – yes, you definitely would. But (I think) there’s a curious workaround, meaning that strictly speaking you could do it with strong ears and no electronic technology at all. It’s pretty much entirely impractical, but the puzzle is intriguing! Thanks to HectorTigo and Shmeehay on r/jazzguitar:
So this is what I came up with, although the tl;dr is ‘god help anyone who needed to do this in real life’. Guessing this is never likely to happen, meaning all this will never be of any direct practical use to anyone. But if anyone wants to listen to some nice meditative overtone beating while learning some of the science…
This is my best guess at how to get to James Taylor’s ‘stretched tuning’ without the direct help of electronic tuners, or if you have a tuner which doesn’t display (or accurately calculate) to the cent level. (also: any proper scientists out there, please check my workings…)
If you happen to have another pre-tuned guitar around (or could tune one, but only have a non-cent display tuner to work with), then you can go about things by achieving the right ‘beating rate’ between different natural harmonic pairs.
This a legitimate ear method used by some experts (e.g. Peter Oberg, a piano tuner turned guitarist-luthier, can tune the <5fr> and <7fr> harmonic pairs to beat precisely at the right rates, putting the strings exactly in).
Beating rates (or rather, our perception of them) are equal to the frequency difference between the two waves in question. e.g. one at 420Hz and one at 423Hz (equivalent to ~12 cents apart) will seem to beat together with a frequency of 3Hz, or 3 times per second (this is why the beating will slow and ‘disappear’ as the strings come closer together).
So, we can calculate the open string frequencies of James Taylor’s ‘stretched’ tuning using a semitone-to-frequency calculator, subtracting, e.g., 0.12 semitones from the standard 6str frequency. All the strings vary by less than 2Hz. We then use these values to work out the desired beating rate between the <12fr> natural harmonics of each guitar, doubling the frequencies to account for the octave jump.
Now the fun part – we can check this with a binaural beat generator. Put in any frequency pair, and the beat rate values from the final column will come to life.
Method 1 steps:
- Get another acoustic guitar into precise 12tet tuning
- Note down the frequencies of all <12fr> N.H. for 12tet and JT tunings
- Calculate the ‘desired beat rates’ between them
- Tune the guitar by counting the beat rates…as best you can
The issue with the approach above is that you’d have to have another precisely pre-tuned guitar around, and an accurate, non-cent tuner. But the original question asks if it can be done “with only a tuning note as reference”. My second method would, in theory, allow you to do this. So, for demonstration’s sake (and because I want to know what it sounds like), here’s a way you could get a guitar to J.T.’s ‘stretched tuning’ from just an A440 reference tone (I think).
To solve the puzzle, we have to rework the ‘classic’ natural harmonic tuning method, tuning the <5fr> and <7fr> on adjacent strings to the desired beat rates. As covered in the main article, <7fr> harmonics are actually ~0.02 of a semitone sharp (due to the equal temperament vs. just intonation issue), which is why I recommend choosing unison pairs for regular tuning purposes, even if you need frets to do it (e.g. 6str <12fr> with 5str 7fr).
But here, we can’t do that, as the guitar doesn’t offer the right non-fretted unison pairs to jump across the strings (In fact, we’re just stuck on our A reference pitch, as no other string has a ‘pure’ A harmonic). And we can’t use the frets, because in J.T.’s tuning all fretted notes will come in slightly less flat than their open string.
(This is actually why the tuning exists at all – J.T. only came up with it to even out the intonation of his guitar when it is fretted/capoed. It is ‘stretched’ on the open strings to minimise intonation error in the higher positions where much of his music lies, i.e. if open 6st is tuned 12 cents flat, 12fr 6st will ring out less than 12 cents flat, but its harmonic will retain the original 12 cent drop.)
In any case, we aren’t aiming for ‘beatless’ unison. We only need two things from our pairs – they must be unfretted (i.e. only natural harmonics or open strings), and clear enough that we can pick up on their ‘beating rate’ (i.e. they must be in unison rather than octaves, as beating doesn’t really work the same way if the frequencies aren’t close – by some estimates they must be within a 7:6 ratio of each other). Also, we must place them in a sequence that avoids ‘overwriting’ any strings already tuned in a previous step.
Tweaking the <5fr> vs. <7fr> method is the best way to do this. First, we can tune our A string’s <5fr> natural harmonic against the A440 reference pitch. To flatten the A by the required 10 cents, get the beating rate to 2.5Hz (i.e. each beat takes 0.4 seconds). We can count this as 5 beats every 2 seconds, which, despite effectively being a 5:2 polyrhythm, is a surprisingly manageable task if you practice a bit, especially with a clock/wristwatch.
(As before, 2.5Hz is calculated as the difference between the two input frequencies. Cents are a relative measure – there is no overall ‘frequency to cent’ conversion scale, but frequency differences can always be described in cents. So to account for the <5fr> natural harmonic, we have to multiply J.T.’s lowered open A string frequency by 4 (‘doubling it twice’ for the two-octave jump). This gives us (440-437.4658)=2.5342)
Then, we separately calculate the desired beat rates for both sides of our harmonic pairs. I’ve done this by working out how many semitones they are from A440, as that’s the input the semitone-to-frequency calculator takes.
It can be a little intricate, as we have to take into account J.T.’s lowered open strings as well as adding 2 cents if the harmonic in question is at the deviant <7fr> position (as it will be justly-intoned rather than equally-tempered). So, for each side – the ‘reference’ and the ‘matched’ frequencies for each eventual pair:
Now, we have the info we need to calculate our desired beating rates, meaning we can (in theory) tune the guitar very precisely. You can match the first two strings (A and D) directly to the A440 reference, but have to go in an exact right order after that. Go through in the ‘right to left, next line down’ fashion of a book. Be careful to get the pairs the right way round – some matched strings are higher than their references this time.
Method 2 steps:
- Calculate the ‘desired beating rates’ between the A440 reference pitch and each of the <5fr> Astr and <7fr> Dstr in J.T.’s desired intonation, and tune those two harmonics to match the calculated rates
- Map out a sequence of N.H. (near-unison) pairs that will cover all the strings without using any fretted positions (only N.H. and open strings), and which doesn’t ‘overwrite’ any strings already tuned
- Calculate the desired frequencies of all 12 pair values, working step-by-step through the sequence above
- Calculate the ‘desired beating rates’ from these frequencies, and – again staying in exact sequence – tune the N.H. pairs accordingly…
Does this answer the question? Would it actually work in practice?
Yeah, ok, using online frequency converters would violate a blanket ban on digital help. But there’s still not a tuner in sight with the second method, and all this could be done with just a known reference tone and the log functions on a scientific calculator.
Or even, in theory, with nothing electronic at all – just an A440 pitch pipe and a notebook of log tables, which for some reason is kind of pleasing. Although solving the two interlocking equations by hand would take a while…
- Where n is semitone difference, n = 12*(log(fn/440)/log(2))
- Where fn is frequency, fn = 2^(n/12)*440
And even if you followed the whole method perfectly, it would probably come out a bit wrong when you play your guitar due to the way intonation and inharmonicity varies between different instruments.
You could tune precisely to Taylor’s system (using this approach or a digital tuner), but still be out on the fretboard, as the low notes are pushed out in different ways if your guitar has a different scale length or string thicknesses (…after all, his tuning was formulated by ear from the quirks of his own guitar).
And this method can’t be generalised either – i.e. we can’t adapt it to any possible tuning, as some tunings don’t offer a full set of unison pairs. Although I think most of the common ones could be done with close-enough pairs and maybe some 5ths.
Ultimately, there’s no direct use for any of this really – it’s just an interesting puzzle. (Also interesting how a Reddit thread about efficient tuning techniques somehow led me to come up with what might be the most inefficient tuning method of all time).
But it is conceptually possible to reach J.T.’s exacting, microtonal tuning with nothing electronic at all. Any mathematically-inclined piano-tuners out there? (I’m firmly guessing yes tbh).
n.b. I’m not an actual scientist…corrections and suggestions very welcome at email@example.com. And the best summary of my answer comes from r/jazzguitar user Shmeehay, the asker of the original question, in response to the first draft:
George Howlett is a South London-based musician and writer. I play guitar, tabla drums, and santoor (Himalayan dulcimer), and write about topics loosely related to jazz, rhythm, and global improvised music. Currently I’m a musicologist for Darbar, write ‘Beyond the Repertoire‘ lessons for Guitar World, and release music as Rāga Junglism. See the site for more – or email firstname.lastname@example.org