Talk:Tuning A Kite Guitar To 31edo or 62edo

Derivation of the formulas used in the spreadsheet

Let L = the guitar's scale length and len = string length when fretted

For the nth fret, usually len(n) = L * 2 ^ (-2n/41)

Move the saddle points back by some fraction of the scale length x, with x << 1

len(n) = xL + L * 2 ^ (-2n/41)

The frequency is f(n) = k / len(n) where k = some constant

The interval from the open string to the nth fret's note as a ratio is f(n) / f(0)

f(n) / f(0) = len(0) / len(n) = [xL + L] / [xL + L * 2 ^ (-2n/41)] = [x + 1] / [x + 2 ^ (-2n/41)]

The interval in cents is 1200 * log (f(n) / f(0)) / log (2)

Thus cents (n) = 1200 * ln ([x + 1] / [x + 2 ^ (-2n/41)]) / ln (2)

We can choose x so that any one fret (but only one) is in tune with the open string. Suppose we want the nth fret to align exactly with 31edo. The nth fret is 2n steps of 41edo and 3n steps of 61.5edo. Thus it approximates 3n steps of 62edo, and 1.5n steps of 31edo. The cents must be 1200 * 1.5n / 31, hence the ratio must be 2 ^ (1.5n/31).

f(n) / f(0) = [x + 1] / [x + 2 ^ (-2n/41)] = 2 ^ (1.5n/31)

x + 1 = [x + 2 ^ (-2n/41)] * 2 ^ (1.5n/31)

x - x * 2 ^ (1.5n/31) = 2 ^ (-2n/41) * 2 ^ (1.5n/31) - 1 = 2 ^ (1.5n/31 - 2n/41) - 1

x = [2 ^ (1.5n/31 - 2n/41) - 1] / [1 - 2 ^ (1.5n/31)]

Then we use this value of x in our cents formula to calculate the cents of each fret.

--TallKite (talk) 10:42, 13 August 2023 (UTC)

72edo

It is possible to interpret 1 fret of a Kite guitar as 7\144 skip fretting, to get 72edo on every other fret. The error per fret is only 0.203¢. --TallKite (talk) 21:31, 13 August 2023 (UTC)