Can we tune to the microtones of Taylor’s ‘stretched tuning’ without a cent-display tuner to help us? What would be the minimum level of technology needed to do this? And what is ‘inharmonicity’ all about? (tl;dr: I may have invented the world’s least efficient tuning method)
Section by section:
- What are ‘stretched’ tunings? t
- Inharmonicity: bending vibrations: t
- What is our puzzle? t
- Solution 1: ‘Basic beating’: t
- Solution 2: ‘Beat pairing’: t
- Does this answer the question? t
- And would it actually work? t
- Further questions & avenues: t
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• What are ‘stretched’ tunings? •
Last year I posted some tuning-based queries on a few online guitar forums, seeking criticism and community input before I published my Ultimate Tuning Guide articles in Guitar World (…which eventually became this project). Among many excellent avenues of peg-winding discussion, there was particular interest in the idea of ‘stretched tunings’.
Specifically, the ‘stretched’ layout used by songwriting legend James Taylor. For decades, he’s been subtly but precisely flattening each of his strings by different amounts – to counteract intonation issues, chiefly related to uneven capo pressure and the ‘inharmonic’ nature of vibrating strings (more on this below).
- Watch Taylor explain (and yeah, that’s a TV not a giant tuner):
His strings are lowered by these cent values (1 cent = 1% of a semitone):
E6: -12 | A5: -10 | D4: -8 | G3: -4 | B2: -6 | E1: -3
‘Inharmonicity’: bending vibrations
In broader terms, a ‘stretched’ tuning is any arrangement which sets the strings slightly flatter than the desired pitches (/range clause). (range is stretched, strings are slackened) Pianos (ac
All are designed with ‘inharmonicity’ in mind:
- What is ‘inharmonicity’? Your strings will probably sound buzzy and slack. To some degree this is part of the dulcimer tone we want, although shorter-necked guitars and thinner string sets may struggle when tuned very low.
- We also have to contend with ‘inharmonicity’ – essentially, the fact that the physical vibration of a string is essentially a small, ever-shifting set of ‘bends’. This means that – in literal terms – all strings always ring a little sharp of their fundamental (as they do with any type of bend).
- In standard tuning we don’t really need to worry about this very much, as the effect is miniscule, but it comes into play with low, loose tunings, especially if we pluck forcefully. Hear inharmonicity in action on my downtuned 6th string (read more about it in my multimedia essay Guitar tuning in fine detail):
- add cumulative step-by-steps for method
…what is ‘inharmonicity’? If you think about it, the oscillation of a string is itself a form of ‘bend’…meaning that all strings actually ring a little sharp of their fundamental frequency (like any bend, the vibration raises the string’s pitch
Not the only reason…but all linked
Other ‘stretches’: James Taylor is far from the only guitarist to ‘stretch’ their tuning: in fact, virtually everyone who tunes low ends up doing it in some form (or else the guitar would sound out of tune) – and you’ve probably tuned your thickest strings slightly low to accommodate the pressure of a high capo before…)
• What is our puzzle? •
A couple of Reddit users wondered whether you’d need a digital ‘cent tuner’ to reach James Taylor’s tuning – a question which, for some reason, fascinated me. Thanks to HectorTigo and Shmeehay on r/jazzguitar:
To formalise our puzzle:
- Can you precisely tune to James Taylor’s microtonal ‘stretch’ with nothing but a reference note as your starting point? What is the minimum level of ‘technology’ required to do this to genuine cent-level accuracy?
In short, there’s no practical solution (tl;dr: ‘god help anyone who needed to do this in real life’). But I think I’ve found a curious set of steps which, strictly speaking, would allow you to do it with strong ears and no electronic tech at all. It’s hilariously convoluted, but the puzzle is intriguing…
So here’s my best solution for how to reach the microtones of James Taylor’s ‘stretched tuning’ without the direct help of electronic tuners, or if you have a tuner which doesn’t work to the cent level. (Also: any actual scientists out there, please check my workings…)
• Solution 1: ‘Basic beating’ •
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 pairs of natural harmonics. This method doesn’t solve our central puzzle – but does lay the conceptual foundations for how we can solve it with ‘harmonic beating’.
‘Harmonic beating’ is very much a legitimate ear method. It’s used regularly by piano tuners (e.g. Peter Oberg, a piano tuner turned guitarist-luthier, tunes the <5fr> and <7fr> harmonic pairs to beat against each other at exactly at the right rate, putting the strings in with each other very, very precisely).
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.
Table 1 – ‘Beating rates’ vs. 12tet EADGBE
|String||12tet (Hz)||JT’s nudge (cents)||JT’s freq. (Hz)||12tet 12fr NH (Hz)||JT’s 12fr NH (Hz)||Diff. (Hz)||Beat rate (secs)|
Binaural beats check: Now the fun part – we can check our results 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: summary
- Have another acoustic guitar tuned to 12tet EADGBE
- 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
• Solution 2: Mapping ‘beat pairs’ •
Method 1 requires another 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 (I think). So, for demonstration’s sake, here’s a way you could get a guitar to J.T.’s ‘stretched tuning’ from just an A440 reference tone.
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 basically 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:
Table 2a – Calculating ‘reference’ pitches
|Step||Reference position||Semito. from A440||JT’s nudge||NH’s ‘pure 5ths’||Total from A440||Ref. pitch (Hz)|
- (italics = we’ve already tuned this string to its ‘stretched’ frequency)
Table 2b – Calculating ‘matching’ pitches
|Step||Tuning position||Semitones from A440||JT’s nudge||NH’s ‘pure 5ths’||Total from A440||Match pitch (Hz)|
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.
- Table 2c – Determining our desired ‘harmonic beat rates’
|Step||Reference position||Ref. pitch (Hz)||Tuning position||Tuning pitch (Hz)||Diff. (Hz)||Beat length (secs)|
Method 2: summary
- 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 near-unison NH pairs which will cover all 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? •
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 log table notebook, which for some reason is kind of pleasing. Although solving the equations by hand would take a while…
- Where n is semitones, n = 12*(log(fn/440)/log(2))
- Where fn is frequency, fn = 2^(n/12)*440
• And would it actually work? •
Well…not in any kind of practically useful way. 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 notable 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. I think…but am not a real vibratory scientist: feedback, corrections, & suggestions very much welcome!
- The best answer still comes from r/jazzguitar user Shmeehay, the asker of the original question, in response to an early article draft:
• Further Learning •
- More on stretched tunings: James Taylor’s own overview from above – and watch Paul Davids compare the sounds of ‘stretched’ and ‘natural’ tunings (below). Peterson strobe?
- More on James Taylor: in a
- Benedetti: https://www.youtube.com/watch?v=TYhPAbsIqA8
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