What I Learned Writing an Album in Just Intonation

Most guitars are not properly in equal temperament intonation, because that requires compensation at the nut.

In particular, an unwound G string can be off. I got sick of three decades of listening to that out of my main guitar, and earlier this year finally did something out of it with satisfactory results:

https://mstdn.ca/@Kazinator/112332540341043265

Without this string, I cannot get these G and D power chords to be in tune simultaneously:

G:

  E   |---|---|-*-|
  B   |---|---|-*-|
  G  o|---|---|---|
  D  o|---|---|---|

D:

  E  x|---|---|---|
  B   |---|---|-*-|
  G   |---|-*-|---|
  D  o|---|---|---|

If you tune the latter D, then the open G is too flat in the former G. If you sharpen the open G, the latter D is out of tune.

The problem is that when we fret the A note on the G string near the nut, it stretches; the string goes sharp. So the interval between the nut and that fret is wider than a tone, without the compensation that shrinks it.

The other strings have the problem, but the unwound G has it worse due to lower tension relative to mass. I'm so happy with the G compensation on that guitar, that it needs nothing else.

Anyways, there are videos out there which do A/B comparisons between ordinary guitars and just intonation. They feature uncompensated guitars, so they perpetrate a fallacy.

The guitarist must experience a properly intonated guitar with nut compensation first, then decide whether going to exotic intonations is worth it.

The fight increases the faster/further you jump around the circle of 5ths. The 12TET fret spacing aligns octaves and 5ths with JI, so if you tune for a JI `I` chord, your other chords will be internally JI but only your `V` chord will be spaced perfectly from the `I` chord. The other chords will be closer to perfect JI the closer they are to `I` on the circle of fifths.

This microtuning isn't perfect in theory, but it's a favorable compromise for a lot of western music, especially for legato strumming where the inter-string intervals are heard more often than the inter-fret intervals.

Obviously C4 should be 256Hz. This has the important property that it makes C-3 2Hz, which is another name for 120bpm.

In equal temperament, this makes A4 430.5Hz.

When I started with violin, since my academic major was math, looked into the 12th root of 2 and ratios of small whole numbers, dissonance from beats, etc. stuff.

Then I learned (partly from a really good violinist):

(1) On violin, can use the ratios of small whole numbers to check intonation. So, get to check the intervals of octave, fifth, fourth, major third. That is, play two notes, each on its own string, together, listen for the beats, and adjust pitch (move a finger) until the beats go away.

That approach starts with the tuning of the violin, notes G, D, A, E, each on its own string, starting with the first G below middle C and going up by fifths (frequency ratio 3/2). The beats are easy to hear, and making the beats go away is what essentially all violin players do, whatever slightly different frequencies might come from other approaches to tuning.

(2) For the rest, there is a single, simple overriding principle: Make it sound good. Uh, there is no practical way to play a violin and get all the frequencies anywhere except close, varying with vibrato, and sounding good!

(3) The math of the 12th root of 2, etc. is too precise for violin -- can't expect to play frequencies to such precision and with vibrato, on that wooden box, with flesh fingers, real violin strings, bow rosin, etc. and really don't need to.

E.g., for the math, there is no good guarantee that a note played on violin is really accurately what the math and physics (a solution of a certain differential equation) define as a periodic function -- thus, lots of considerations of overtones start to become imprecise and, thus, moot.

Again, make it sound good! So, sure, for a major third, ratios of small whole numbers can be relevant, but for a minor third, mostly just guess at the semi-tone and make the interval sound good!

(4) For semi-tone guessing: The famous Bach E-major Partita starts with E (one octave above the open E string, half way along the string), D# (down one semi-tone), and then the E again. So, it's the three notes, played quickly, with the D# inserted and left quickly, with the musical effect an introductory bright heralding -- the relevance of the semi-tone as the 12th root of 2 is lost and instead just play the E with the little finger and for the D# squeeze in the next finger until it passes for a semi-tone and gets the heralding musical effect.

(5) For much more, can play the Bach Chaconne with its many chords, sometimes on all 4 strings -- no way can a violinist try to honor some precise tuning arithmetic for all the intervals played. E.g., the piece starts in D minor and with the "D minor" chord, that is, D, F, A. On a piano that would be the first D above middle C, the first F, that is, 3 semi-tones, above the D, and the A, a fifth above the D. But on violin that would have both the D and the F on the same string, so Bach put the F an octave higher, on the E string, a semi-tone above the E string. Sooooo, to play that opening chord, play the D and the A together, and, continuing right away all with one down bow, with the A and the F together. Then the F is from the index finger at a guess of a semi-tone above the E string. So, the bow starts on the D and A strings, both open (no fingers on the strings) and, thus, brilliant, and then both the open A string and the fingered F. So, the A and F 'would' be just an okay major third but the F is up an octave so that the A and F are an interval of 8 semi-tones and, actually, dissonant. Since the F is on the E string which is still brilliant even though fingered and not open, all three strings are brilliant, and the opening is dramatic and in a minor key. Some of the piano and orchestra arrangements make this first chord dramatic!

In all that for that opening chord, on violin, the 12th root of 2 and ratios of small whole numbers don't play a central role, and the same for the rest of the piece, nearly all chords.

Sure Bach knew very well what he was doing, indeed, in that music, SHOWING that could make music in all the keys, and had all those pages and pages, for violin, in another book, for cello, in another book, piano with the intervals and chords "sounding good".

Better reading IMO: Temperament by Stuart Isacoff. It is a small-ish music history book that covers the whole subject pretty thoroughly, with various tuning options that have been tried through the centuries, while treating the tradeoffs between them in a fair manner.