Tune the open fourths directly to tempered fourths
I came up with my "ultimate method" after reading that cheap Dover book on Piano Tuning. I believe it is the most direct and accurate way possible (so please school me if I err!).
- Start with a reliable A 440. I use a tuning fork. [To use a tuning fork with a guitar: hold the ball end and strike the fork against a knee-bone with a swinging motion; choke up (like a baseball bat) on the handle so end of the ball is exposed (quickly, while the fork is still freshly singing); place the ball against the bridge of the guitar (holding it in contact with your fingers); a loud clear note will emerge from the guitar. I use my right hand for the striking and placing, then I switch hands and continue holding with my left, freeing the right hand to do a pinched harmonic and adjust the tuning peg.]
- Tune the 1/4 (5th fret) harmonic of the A string in unison with the A 440. (External A440 =unison= A-string 5th-fret Harmonic (double-octave))
- Tune the D to a perfect fourth above the A. (You can use the 2/3 (7th fret) harmonic if you like.)
- Nudge the D ever-so-slightly sharp, until you hear a very slow beat (about 3 beats over 5 seconds). (D-string =tempered-fourth-above= A-string)
- Tune the G from the D in the same way (just a smidgen sharp). (G-string =tempered-fourth-above= D-string)
- Tune the low E from the A in a contrariwise manner: scoop in from the D# side until you find the slow beat just before a perfect fourth. (E-string =tempered-fourth-below= A-string)
- Tune the High E from the low E. (You can use the 1/4 (fifth fret) harmonic if you like.) (High-E-string =unison= E-string 5th fret Harmonic (double-octave))
- Tune the B from the E the same way you took the low E from the A: scoop in from the A# side until you hear the slow beat. (B-string =tempered-fourth-below= High E-string)
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And finally, check your work with the open triads: D-G-B and G-B-E. They should sound balanced and distinct. You can hear all three notes, but none of them sting. If the triads don't sound good, nothing else will either: Do it again in exactly the same sequence. You can also check all the tempered fourths against each other. They should all sound like identical intervals.
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Note: It is very difficult to hear the beats. One trick I've found is to hold my right hand a few inches over the guitar as I sound the notes. I seem to be able to "feel" the beats more easily than I can hear them.
Tempered fourths gives you equal temperament. But for music that stays in a diatonic key, you can make a few slight adjustments. Starting from equal temperament, fret the root chord of the key. Then nudge the fifth up a smidgen (to shake off the temper), and scoop the third in a little flatter (a Helmholtz third). Remember which strings you altered so you can scoop them back into tempered from the "stationary" strings.
The importance of re-tuning from scratch every time is that you don't know where you made a mistake. Had you known you would have fixed it before moving on, right? If you start fixing the bad note, you'll end up chasing your tail. This goes for any tuning method (except the "use a tuner" method, I guess).
The words nudge and scoop deserve some elucidation. If you're tuning the upper note of a tempered fourth, it needs to be just a little sharper than the center of the perfect fourth. So the action you perform on the tuning head is to tune up to the perfect fourth and then a very small amount further up to "temper" the fourth. Tune perfect, nudge the temper.
But if you're tuning the lower note of a tempered fourth, it needs to be just a little flatter than the center of the perfect fourth. So you approach the perfect fourth from below (always tune up) but stop short by a very small amount. Scoop into the temper, merely approaching perfect.
You can use a different sequence if you like, if you remember what these 2 actions are for (tuning the upper/lower notes of a fourth). If you start with G, then it's scoops all the way. If you start with B (bizarre though it sounds) then it's all nudges. If you start with the low E, then you only have to scoop the B, the rest is all nudges.
I've made a video of tuning the guitar this way 5m35s. It's much easier to hear the beats thanks to the mpeg filters than it is in real life. This time I was pretty sure about E A D and G, so I took the E again from E, and the B the same way from E. I could also have adjusted G again from D, as it's a sort of independent branch.
Why/How this works. or More luser philosophy.
When you tune one note to another, the result is a ratio of frequencies. The musical apparatus of the mind is keen upon "harmonic" ratios: where several of the overtones of each tone coincide. This is perceived as "beats". If the two notes were runners on a track, the beat happens every time the faster runner laps and blows-a-raspberry-at the slower one. If they don't meet as often, the fast guy kinda zones out and is less offensive.
When you tune a perfect interval, you find two "zones" of beating around a central "can't really tell unless you focus on it" area. At the dead center between the beating zones is the perfect interval, essentially "infinite beating". But hovering around the dead center mark is the "temperate zone". The beats are slow enough that they aren't offensive. And this enables us to approximate Pythagorean intervals in any key using a base-12 logarithm. Thus, equal temperament.
The guitar has frets installed according to this same base-12 logarithm (they slowly, evenly, get closer together).
Everybody knows, when you start with C and go around the Circle of Fifths you end up not at C again but some straunge beastie called B#. What equal temperament does is fix that problem by making B# == C. But it has to drag all the other notes with it; just a little bit.
So the interval of the Fifth becomes a little flat (not too much). The interval of a Fourth becomes a little sharp (Since a Fifth plus a Fourth makes an Octave, they temper in opposite directions to keep the Octave in the same place.) The squeeze decends from the Fifth into the Triad where the interval of the Major Third is flattened (from a Pythagorean Major Third, that is). To counterbalance and keep the Fifth stable, the Minor Third has to go sharp by the same amount. And the smaller intervals get smaller adjustments.
The Helmholtz Major Third is actually a different animal entirely, it deviates even further from the Pythagorean than the Equal-Tempered Third does. But it catches a different circuit of the musical mind which is keen on overlapping frequency spectra. But to do full justice to these kinds of relations, you need many more than 12 distinct notes in an Octave. One of his keyboards had 30-something keys to the Octave; another one had over 100!