Tuning A Kite Guitar To 31edo or 62edo
One fret of the Kite guitar is 2 steps of 41edo or 1 step of 20.5edo. But it can also be thought of as 3 steps of 61.5edo. And 61.5 is very close to 62, which is twice 31. Thus every other fret of the Kite guitar is very close to Skip fretting system 31 3 7. This allows one to approximate 62edo on a 41edo guitar. And by ignoring every other fret, and playing only on the dots and the frets midway between the dots, one can play in 31edo!
In the table below, the 62edo column is the desired tuning. The 61.5edo column is the actual 41edo tuning of the guitar. There's an error of 0.47¢ per fret. The errors add up quickly. For example the 12th fret note is not a 31edo 5th from the open string but a 41edo 5th. At the top of the neck, these errors are nearly 20¢. Nevertheless, assuming open strings are not used, within any one region of the fretboard, the errors are only a few cents. An interval that spans 6 frets is only 2.83¢ sharp of 31edo. Thus while the whole chord may be quite sharp, the harmonies within the chord are reasonably in tune.
These errors can be greatly reduced by adjusting the intonation at the guitar's saddle. This is especially easy on an electric guitar. This makes a quasi-edo, an irregular tuning in which each step from fret to fret is a different size. One can choose which fret to align at, meaning which fret is in tune with 31edo. For example, aligning at the 12th fret makes the interval from the open string to the 12th fret be an exact 31edo 5th. Frets 1-11 will be less than 0.3¢ sharp. But fret 22 is 2¢ flat, and fret 29 is 5¢ flat. Better to align at a higher fret!
One can align at the 20.5th fret. This allows one to capo at the 0.5 fret and intonate so that the 21st fret's harmonic matches its fretted note, very convenient. (But best to remove the capo after intonating, unless one likes 124edo!) The 20.5th fret alignment is shown in the table below. The maximum sharpness is only 0.83¢ (11th fret) and the flatness is only perceptible above around the 28th fret. Furthermore, the error per fret improves for frets 1-32, and is nearly imperceptible around the 12th fret. Even at the very top of the neck, a 4-fret interval is less than 3¢ flat, and a 6-fret interval is less than 4.2¢ flat.
One can get more accuracy (but a less convenient intonation process) by aligning at a higher fret. The table also shows a 26th fret alignment. The error of a 6-fret interval at the top of the neck is under 2.5¢. This spreadsheet displays the various errors for various alignments. The formulas are explained in the discussion tab of this page. The spreadsheet also calculates the required saddle adjustment as a percentage of the scale length. For the 20.5th fret, it's 0.564%. For the 26th fret, it's 0.508%. For the usual 25.5" scale length, both figures are slightly over 1/8". The direction of adjustment is away from the nut.
| fret | 62edo
target |
61.5edo
(41edo) |
error
from 62 |
difference
of errors |
quasi-62edo aligned
at the 20.5th fret |
quasi-62edo aligned
at the 26th fret | |||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0.0¢ | 0.0¢ | 0.0¢ | 0.0¢ | 0.0¢ | 0.0¢ | 0.0¢ | ||||
| 1 | 58.1 | 58.5 | 0.5 | 0.47 | 58.2 | 0.1 | 0.14 | 58.2 | 0.2 | 0.17 | |
| 2 | 116.1 | 117.1 | 0.9 | 0.47 | 116.4 | 0.3 | 0.13 | 116.5 | 0.3 | 0.16 | |
| 3 | 174.2 | 175.6 | 1.4 | 0.47 | 174.6 | 0.4 | 0.12 | 174.7 | 0.5 | 0.15 | |
| * | 4 | 232.3 | 234.1 | 1.9 | 0.47 | 232.7 | 0.5 | 0.10 | 232.9 | 0.6 | 0.14 |
| 5 | 290.3 | 292.7 | 2.4 | 0.47 | 290.9 | 0.6 | 0.09 | 291.1 | 0.8 | 0.13 | |
| 6 | 348.4 | 351.2 | 2.8 | 0.47 | 349.0 | 0.7 | 0.08 | 349.3 | 0.9 | 0.12 | |
| 7 | 406.5 | 409.8 | 3.3 | 0.47 | 407.2 | 0.7 | 0.06 | 407.4 | 1.0 | 0.10 | |
| ** | 8 | 464.5 | 468.3 | 3.8 | 0.47 | 465.3 | 0.8 | 0.05 | 465.6 | 1.1 | 0.09 |
| 9 | 522.6 | 526.8 | 4.2 | 0.47 | 523.4 | 0.8 | 0.04 | 523.7 | 1.1 | 0.08 | |
| 10 | 580.6 | 585.4 | 4.7 | 0.47 | 581.5 | 0.8 | 0.02 | 581.9 | 1.2 | 0.07 | |
| 11 | 638.7 | 643.9 | 5.2 | 0.47 | 639.5 | 0.8 | 0.01 | 640.0 | 1.3 | 0.05 | |
| *** | 12 | 696.8 | 702.4 | 5.7 | 0.47 | 697.6 | 0.8 | -0.01 | 698.1 | 1.3 | 0.04 |
| 13 | 754.8 | 761.0 | 6.1 | 0.47 | 755.6 | 0.8 | -0.03 | 756.2 | 1.3 | 0.02 | |
| 14 | 812.9 | 819.5 | 6.6 | 0.47 | 813.6 | 0.7 | -0.04 | 814.2 | 1.3 | 0.01 | |
| 15 | 871.0 | 878.0 | 7.1 | 0.47 | 871.6 | 0.7 | -0.06 | 872.3 | 1.3 | -0.01 | |
| * | 16 | 929.0 | 936.6 | 7.6 | 0.47 | 929.6 | 0.6 | -0.08 | 930.3 | 1.3 | -0.03 |
| 17 | 987.1 | 995.1 | 8.0 | 0.47 | 987.6 | 0.5 | -0.10 | 988.3 | 1.2 | -0.04 | |
| 18 | 1045.2 | 1053.7 | 8.5 | 0.47 | 1045.5 | 0.4 | -0.12 | 1046.3 | 1.2 | -0.06 | |
| 19 | 1103.2 | 1112.2 | 9.0 | 0.47 | 1103.5 | 0.2 | -0.14 | 1104.3 | 1.1 | -0.08 | |
| ** | 20 | 1161.3 | 1170.7 | 9.4 | 0.47 | 1161.4 | 0.1 | -0.16 | 1162.3 | 1.0 | -0.10 |
| 21 | 1219.4 | 1229.3 | 9.9 | 0.47 | 1219.3 | -0.1 | -0.18 | 1220.3 | 0.9 | -0.12 | |
| 22 | 1277.4 | 1287.8 | 10.4 | 0.47 | 1277.1 | -0.3 | -0.20 | 1278.2 | 0.8 | -0.14 | |
| 23 | 1335.5 | 1346.3 | 10.9 | 0.47 | 1335.0 | -0.5 | -0.23 | 1336.1 | 0.6 | -0.16 | |
| *** | 24 | 1393.5 | 1404.9 | 11.3 | 0.47 | 1392.8 | -0.8 | -0.25 | 1394.0 | 0.4 | -0.18 |
| 25 | 1451.6 | 1463.4 | 11.8 | 0.47 | 1450.6 | -1.0 | -0.27 | 1451.8 | 0.2 | -0.20 | |
| 26 | 1509.7 | 1522.0 | 12.3 | 0.47 | 1508.3 | -1.3 | -0.30 | 1509.7 | 0.0 | -0.22 | |
| 27 | 1567.7 | 1580.5 | 12.7 | 0.47 | 1566.1 | -1.7 | -0.33 | 1567.5 | -0.2 | -0.25 | |
| * | 28 | 1625.8 | 1639.0 | 13.2 | 0.47 | 1623.8 | -2.0 | -0.35 | 1625.3 | -0.5 | -0.27 |
| 29 | 1683.9 | 1697.6 | 13.7 | 0.47 | 1681.5 | -2.4 | -0.38 | 1683.1 | -0.8 | -0.30 | |
| 30 | 1741.9 | 1756.1 | 14.2 | 0.47 | 1739.1 | -2.8 | -0.41 | 1740.8 | -1.1 | -0.32 | |
| 31 | 1800.0 | 1814.6 | 14.6 | 0.47 | 1796.7 | -3.3 | -0.44 | 1798.5 | -1.5 | -0.35 | |
| ** | 32 | 1858.1 | 1873.2 | 15.1 | 0.47 | 1854.3 | -3.7 | -0.47 | 1856.2 | -1.9 | -0.38 |
| 33 | 1916.1 | 1931.7 | 15.6 | 0.47 | 1911.9 | -4.2 | -0.50 | 1913.9 | -2.3 | -0.41 | |
| 34 | 1974.2 | 1990.2 | 16.1 | 0.47 | 1969.4 | -4.8 | -0.53 | 1971.5 | -2.7 | -0.44 | |
| 35 | 2032.3 | 2048.8 | 16.5 | 0.47 | 2026.9 | -5.3 | -0.57 | 2029.1 | -3.2 | -0.47 | |
| *** | 36 | 2090.3 | 2107.3 | 17.0 | 0.47 | 2084.4 | -5.9 | -0.60 | 2086.7 | -3.7 | -0.50 |
| 37 | 2148.4 | 2165.9 | 17.5 | 0.47 | 2141.8 | -6.6 | -0.64 | 2144.2 | -4.2 | -0.53 | |
| 38 | 2206.5 | 2224.4 | 17.9 | 0.47 | 2199.2 | -7.2 | -0.68 | 2201.7 | -4.8 | -0.57 | |
| 39 | 2264.5 | 2282.9 | 18.4 | 0.47 | 2256.6 | -8.0 | -0.72 | 2259.1 | -5.4 | -0.60 | |
| * | 40 | 2322.6 | 2341.5 | 18.9 | 0.47 | 2313.9 | -8.7 | -0.76 | 2316.6 | -6.0 | -0.64 |
| 41 | 2380.6 | 2400.0 | 19.4 | 0.47 | 2371.1 | -9.5 | -0.80 | 2374.0 | -6.7 | -0.67 | |
The intra-string gaps must of course be tuned to 31edo. One could use major 3rds (10\31), but this makes the major 2nd harder to reach. The best choice is downminor 3rds (7\31). Seven of these add up to a perfect 12th, implying the Sepru/Orwell temperament.
This fretboard chart is for a 9-string guitar. The cents are in exact 62edo, since the chart is meant to be moveable, and the errors vary along the fretboard. Harmonics 8-16 are bolded. All bolded notes are on even-numbered frets, thus all 13-limit intervals are contained in 31edo. However slightly better approximations of 9/8 and 13/8 exist in 62edo. These are shown in parentheses. The better 13/8 (832¢) is less accessible, but the better 9/8 (213¢) is much more accessible.
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
|---|---|---|---|---|---|---|---|---|---|
| -6 | 852 | 1123 | 194 | 465 | 735 | 1006 | 77 | 348 | 619 |
| -5 | 910 | 1181 | 252 | 523 | 794 | 1065 | 135 | 406 | 677 |
| -4 | 968 | 39 | 310 | 581 | 852 | 1123 | 194 | 465 | 735 |
| -3 | 1026 | 97 | 368 | 639 | 910 | 1181 | 252 | 523 | 794 |
| -2 | 1084 | 155 | 426 | 697 | 968 | 39 | 310 | 581 | 852 |
| -1 | 1142 | (213) | 484 | 755 | 1026 | 97 | 368 | 639 | 910 |
| 0 | 0 | 271 | 542 | 813 | 1084 | 155 | 426 | 697 | 968 |
| 1 | 58 | 329 | 600 | 871 | 1142 | (213) | 484 | 755 | 1026 |
| 2 | 116 | 387 | 658 | 929 | 0 | 271 | 542 | 813 | 1084 |
| 3 | 174 | 445 | 716 | 987 | 58 | 329 | 600 | 871 | 1142 |
| 4 | 232 | 503 | 774 | 1045 | 116 | 387 | 658 | 929 | 0 |
| 5 | 290 | 561 | (832) | 1103 | 174 | 445 | 716 | 987 | 58 |
| 6 | 348 | 619 | 890 | 1161 | 232 | 503 | 774 | 1045 | 116 |
| 7 | 406 | 677 | 948 | 19 | 290 | 561 | (832) | 1103 | 174 |
| 8 | 465 | 735 | 1006 | 77 | 348 | 619 | 890 | 1161 | 232 |
| 9 | 523 | 794 | 1065 | 135 | 406 | 677 | 948 | 19 | 290 |
| 10 | 581 | 852 | 1123 | 194 | 465 | 735 | 1006 | 77 | 348 |
| 11 | 639 | 910 | 1181 | 252 | 523 | 794 | 1065 | 135 | 406 |
| 12 | 697 | 968 | 39 | 310 | 581 | 852 | 1123 | 194 | 465 |
| 13 | 755 | 1026 | 97 | 368 | 639 | 910 | 1181 | 252 | 523 |
| 14 | 813 | 1084 | 155 | 426 | 697 | 968 | 39 | 310 | 581 |
| 15 | 871 | 1142 | (213) | 484 | 755 | 1026 | 97 | 368 | 639 |
| 16 | 929 | 0 | 271 | 542 | 813 | 1084 | 155 | 426 | 697 |
| 17 | 987 | 58 | 329 | 600 | 871 | 1142 | (213) | 484 | 755 |
| 18 | 1045 | 116 | 387 | 658 | 929 | 0 | 271 | 542 | 813 |
| 19 | 1103 | 174 | 445 | 716 | 987 | 58 | 329 | 600 | 871 |
| 20 | 1161 | 232 | 503 | 774 | 1045 | 116 | 387 | 658 | 929 |
A 41edo Kite guitar has a distinct rainbow zone and complex zone. But here every even-numbered fret has a low-odd-limit ratio. Both 8/7 and 4/3 are 4 frets up. Both 10/7 and 5/3 are 6 frets up. And both 9/8 and 9/5 are 8 frets up. Thus the rainbow zone is all even-numbered frets, and the complex zone is all odd-numbered frets. Thus the two zones are interspersed. This makes the layout somewhat more awkward than a 41edo Kite guitar. Furthermore, the 9/8 major 2nd is hard to reach. It's either 6 frets back or 8 frets forward. The major 3rd is 2 frets forward, so playing do-re-mi means an awkward 8 fret jump. Or two 6 fret jumps, which is worse. Thus one might want to use 62edo's 213¢ major 2nd. But beware, it makes a 716¢ 5th with 3/2.
Once properly intonated, the 62edo Kite guitar can be tuned by ear. Play an open string (or its octave harmonic) and match it to the note one string higher, on the 16th fret.
Example riff that pumps 81/80:
/ I VIm
|| . . 6 . | . . 6 . | . . . . | . . . . |
|| . . . . | . . . . | . . . . | 14 . . . |
|| . 10 . 10 | . 10 . . | . . 10 . | . 10 . . |
|| 8 . . . | 8 . . 8 | . 8 . 8 | . . 8 . |
|| . . . . | . . . . | . . . . | . . . . |
|| . . . . | . . . . | 12 . . . | . . . 12 |
\
IIm V,v7 \
| . . . . | . . . . | . . . 2 | 6 2 . . ||
| . . 14 . | . . 14 . | . . 2 . | . . 2 . ||
| . 12 . 12 | . 12 . . | . . . . | . . . . ||
| . . . . | . . . . | . 6 . . | . . . 6 ||
| 16 . . . | 16 . . 16 | 4 . . . | . . . . ||
| . . . . | . . . . | . . . . | . . . . ||
/