Kite's thoughts on the mathematical basis For the kite guitar
Consider this sequence:
9/8 -- 7/6 -- 6/5 -- 5/4 -- 9/7 -- 4/3
These are sequential notes on one string, and this is the essence of what makes the Kite guitar so playable. 1/1 must of course be on the next string down, to make 5/4 etc. easy to reach. Since 6/5 and 5/4 add up to 3/2, we know that 3/2 is on the next string up. And 3/2 plus 4/3 is 2/1, so we know that 2/1 is on the next string up after 3/2. From this we get the whole layout. Note that this holds not only for the downmajor tuning but also the upminor and upmajor tunings.
What if this sequence from 9/8 to 4/3 was all we knew of the guitar? Could we deduce that it is in 41-edo? No, but we can work backwards from the notes to deduce what commas are being tempered out. From 9/8 to 6/5 = 16/15 = 2 frets. From 6/5 to 5/4 = 25/24 = 1 fret. Thus (25/24)^2 = 16/15, which gives us the Laquinyo or Magic comma. This comma is the difference between (5/4)^5 and 3/1, thus its pergen is (P8, P12/5).
Let's extend this to the 7-limit. 9/8 to 7/6 = 28/27 = 1 fret. 6/5 to 5/4 = 25/24 = 1 fret. Thus 28/27 = 25/24, which gives us the Ruyoyo comma 225/224. Laquinyo and Ruyoyo gives us 7-limit Laquinyo/Magic. You can likewise extend to 11- or 13-limit.
19, 22 and 41 all temper out both commas, so 19edo and 22edo can mimic Kite guitars if you tune the open strings in 3rds. But you can also simply place the frets according to a rank-2 tuning. Say you tune Magic/Laquinyo so that your 5th is 700+c cents. 3/1 is 1900+c, and 5/4 is one fifth of that, 380+c/5. 32/27 is 300-3c, and 32/27 is 5 frets. Thus 1 fret = 60 - 3c/5.
Now c can be anything, and many values won't give you any edo at all. But certain values of c result in 19, 22, or 41 edo. 41 arises from doubling 19 and 22 and looking at every 3rd edo: 38-41-44. Or you can triple 19 & 22 and take every 3rd edo. 57-60-63-66. So 60 and 63 are also possibilities. Instead of every other fret of 41, it would be every 3rd fret of 57 or 60.
So for 19 and 22, we have every fret of the edo, for 41 we have every other fret, and for 60 and 63 we have every 3rd fret.
For 19, 22, 41, 60, etc., the ratios within the immediate area of 1/1 are all the same, except for how well tuned they are. But the ratios up the neck are quite different. With 41, 12 frets up 1 string make 3/2. But in 19, 3/2 is 11 frets. In 22, it's 13 frets. In 41, the octave is up 1 string and up 14 frets, which I write as (+1,+14). In 19, its (+1,+13), and in 22, it's (+1, +15). But in a non-edo tuning of Laquinyo/Magic, the 8ve doesn't appear anywhere up the neck.
This is a huge disadvantage to a non-edo Kite guitar, because when two guitarists play together, with a 41-edo tuning, one guitarist can solo 14 frets above the other.