TRANSCRIPT of the captions for the video: Harpsichord tuning: building a scale by ear - Bradley Lehman(Some of them might be difficult to read as superimposed on the bright background...and there's a LOT of text going by in three minutes, because the tuning process is so quick!)
Tetrasecting a major 3rd: fitting four evenly-tempered 5ths/4ths into a given size of major 3rd
In this demonstration we are starting from an "A" tuning fork. The technique works in the same manner, regardless of that A's frequency (440, 415, 410, or whatever). There are no calculations involved here.
We will set up some size of major 3rd as our boundary, on the notes F-A. Then, we will fit all the 5ths of the sequence F-C-G-D-A evenly into that boundary, with consistent tempering.
We will work in the region of tenor A, where it is easy to hear the quality of 5ths and 4ths. Here is our A, to start. All octaves are always pure.
Now, we choose some arbitrary but tasteful size of major 3rd, setting F below A. Today I choose to use a pure major 3rd, to do "1/4 comma" tempering. (Standard tuning literature explains the "comma"....)
If we would choose some slightly wide interval for F-A, we could do some other size of meantone (or "regular") tempering such as 1/5, 1/6, 1/7 comma, etc. It's a continuum. Analog, not digital.
Now, the fun little trick and the main point of this demonstration. We want to put G exactly halfway (a "mean" or average tone) within the major 3rd F-A, without having to guess.
We build a TEMPORARY reference C as pure 4th below our F, and a temporary reference D as pure 5th below our A. No tempering yet. There's the C, and here comes the D.
We have some bunch of error to burn off among the 5ths, and we haven't used any of it yet. So, let's put the "mean" G halfway, i.e. with 1/2 the error on each side of it, geometrically.
We play around with our G until the QUALITY of the C-G 5th is similar to that of the D-G 4th. And the checkpoint for that is: the D-G 4th will beat as triplets against the speed of the C-G 5th's duplets.
3 against 2, a normal musical listening skill deployed here. Triplets in the D-G 4th against duplets in the C-G 5th. When we find that spot, we know our G is exactly mean (average) within F-A.
OK, now we have our G with 1/2 of the tempering error on each side of it. Split that 1/2 into two 1/4s, each, and we're done! It's like folding a piece of paper twice to get four equal pieces.
From our G and A, put middle D with the same quality of error from both of them. And working similarly from our F and G, put middle C with the same quality of error from both of those.
The same 3-to-2 technique holds true here. The A-D 4th beats as triplets against the duplets of the 5th from G-D. And similarly, the G-C 4th gives triplets against the duplets of the F-C 5th.
F-C, C-G, G-D, D-A all the same size as one another: tempered 5ths each having 1/4 the total amount of error we're burning off. (One whole comma, here, since we started with a pure major 3rd.)
Let's not forget to correct our earlier temporary C and D, in that lower octave. Bring them in as pure octaves from the C and D we just built.
And that's the whole thing. Whatever size of major 3rd we started with at F-A, we have now tetrasected it: we have split it into four equal parts, giving geometrically equal error to each 5th in F-C-G-D-A.
Elapsed time to set up those five core notes, by ear: less than two minutes.
To do all the remaining notes of a meantone (regular) temperament, or a modified-meantone temperament, we now have the reference of quality for all the core 5ths, as to their slight sourness/impurity.
See also: Tuning a harpsichord in Bach's temperament - Bradley Lehman