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How to make a Kite Guitar

Revision as of 23:02, 16 June 2021 by Wolftune (talk | contribs) (→Fret and Marker Placement: split Fret placement and Fret Markers)

General Design Considerations

Assuming one is using one of the isomorphic all-3rds tuning, a Kite Guitar with 6 strings is a little limiting. 7 strings or even 8 is better. A slightly longer scale, say 27", is nice because it makes the frets less cramped. 12-edo 7- and 8-string guitars often have longer scales anyway, good if you're converting one.

Best to replace the entire fretboard, rather than removing the frets and putting new frets in the old fretboard. The 41-edo 5th is 702.5¢, so two frets will be only 2.5¢ away from the old ones, two will be 5¢ away, etc. So the old and new fret slots overlap, making conversion difficult. The following table shows the distance from the old fret to the new fret for close pairs. One could just use the old slots (or even the old frets) and accept a few cents error. But in certain keys a 5¢ error will make the 5/4 that's already 6¢ flat a full 11¢ flat.

distance between fret slots (center to center, scale = 25.5")
old fret 7 10 14 17 20 21
new fret 12 17 24 29 34 36
cents 2.5¢ -5¢ 5¢ -2.5¢ -10¢ 7.5¢
distance 0.024" -0.040" 0.032" -0.013" -0.045" 0.032"

Removing the entire fretboard also has the advantage that you can get a pre-slotted computer-cut fretboard fairly cheaply (roughly $75-100) that has extremely accurate slot placement.

One way to get an 8-string acoustic is to convert a 12-string guitar. The neck will be sufficiently strong and there will be enough tuners. There's fewer strings but more courses, so the new fretboard may need to be wider than the old one. The fretboard overhang can be filled with bondo to create a nice-feeling neck. Another possibility is to convert a 6-string classical nylon-string to 7 or 8 strings. The fingerboard is wide enough that it may suffice as is. The tension is low enough that an extra string or two won't break the guitar. The 3 holes on each side of the headstock that the tuner pegs go through can be filled and 4 new holes drilled.

In any given key, the Kite guitar has multiple "rainbow zones" on the neck. Assuming the tonic falls in the "sweet spot" between the 4th and 11th fret, it takes about 28 frets to provide 2 zones in every key, but it takes the full 41 frets to provide 3 zones. This 3rd zone increases the range the lead guitarist has to solo in by a 5th or so. The highest frets are very tight, but still playable melodically. Chording is very difficult.

The fret spacing is 1.7 times tighter than a 12-edo guitar. This chart compares it to the standard fret spacing. The spacing between the nut and the first fret is about the same as the space between the 12-edo 9th and 10th frets. Increasing the overall scale length will widen the spacing.

 

Fret Placement

To place the frets on a Kite guitar, simply replace the 12th root of 2 = 1.059463 with the 41st root of 4 = 1.034390.

As an alternative to doing the work yourself, various suppliers can make pre-slotted fingerboards. One option is PrecisionPearl.com which can make fingerboards complete with radius, taper and inlays. With that option, all you need to do is glue it on and put in the frets.

Fret Markers

Dots (fretboard markers) are placed every 4 frets in a cycle of single-double-triple. So, the 4th fret has a single dot, the 8th fret has double dots, the 12th fret has triple dots, and then the 16th fret is back to single, and so on. Thus, a 36-fret guitar has 18 dots, and a 41-fret guitar has 19 dots.

String Gauges

A 6-string Kite guitar tuned in 3rds can be strung with a standard set of strings, but it's not ideal. The high strings will be somewhat slack, and the low strings will be somewhat tight. To find the appropriate gauges, use the D'Addario method: calculate each string's tension from its unit weight, length and pitch (frequency) by the formula T = (UW x (2 x L x F)2) / 386.4. For open strings, the length is the guitar's scale. The frequency in hertz of the Nth string of 8 strings tuned in the standard downmajor 3rds with a low string of vD is 440 * (2 ^ (-7/12 + (21 - 13*N) / 41)). For a 6-string guitar in mid-6 tuning, N ranges from 2 to 7. Or use the frequency table below. The unit weight is pounds per inch, and is a function of string gauge and string type (plain vs. wound, etc.). D'Addario has published their unit weights, thus the individual tensions can be calculated for a given set of strings. One can work backwards from this and select string gauges/types that give uniform tensions using this spreadsheet: http://tallkite.com/misc_files/StringTensionCalculator.ods The desired tension depends on the instrument, and of course personal taste. A steel-string acoustic guitar might have 25-30 lbs. tension for each string. A 12edo 25.5" electric guitar strung with a standard 10-46 set has 15-20 lbs. With a 9-42 set it has 13-16 lbs.

  • A longer scale means a higher tension or a smaller gauge or a lower pitch (frequency)
  • A higher tension means a longer scale or a bigger gauge or a higher pitch
  • A bigger gauge means a shorter scale or a higher tension or a lower pitch
  • A higher pitch means a shorter scale or a higher tension or a smaller gauge

Microtonalist and luthier Tom WInspear can provide custom string sets at his website www.winspearinstrumental.com. His approach is to extrapolate from familiar string sets. He says this about string gauges: "Gauges can be scaled at the same ratios as frequency. A 41-edo downmajor 3rd is 2^(13/41) = 1.2458, thus from string to string the gauge changes by 24.58%. But you can't do that across the plain to wound transition. To tune to different keys, increase the gauges by 5.95% for each 12-edo semitone of transposition, or 1.705% for each 41-edostep. All this assumes a 25.5" scale. For a scale of S inches, multiply each gauge by 25.5/S and round off. For scales longer than 25.5", err on the side of heavier and round up, as longer scales do feel more flexible loaded with the same tension. Likewise, for scales less than 25.5", err on the side of lighter and round down. However, the plain strings should always be rounded slightly down, and should utilize .0005" increment plain strings where available."

JustStrings.com sells custom gauges singly or in bulk. Recommended (somewhat light) gauges for a 27" acoustic guitar: 11.5 15 18 24 30 36 46 56 (3 plain, 5 wound). For a 25.5" or 26.5" electric: 10 13 16 22 26 32 42 52, the wound 4th string could instead be a 19 plain.

Saddle and Nut Compensation

Checking the intonation and compensation is similar to standard 12edo guitars but some of the fret comparisons are unavailable. The best comparisons to use for the Kite Guitar are described in the following chart.

  • fret = the Kite Guitar fret number
  • harmonic = the harmonic which shares the fret's pitch class
  • octave difference = how many octaves from the fretted pitch to the harmonic
  • cent offset = the precise difference from the harmonic to the fretted pitch (ignoring octaves)
  • location(s) = where to play the harmonic(s) separate from the fret in order to produce a same-octave comparison between the harmonic with the open string versus with the fret depressed
Best frets to check for Kite Guitar intonation setup
fret harmonic octave difference cent offset location(s) to play shared harmonic
12 3 1 +0.5¢ 2/3 of string, in-between frets 32 and 33
24 9 2 +1¢ 7/9 or 8/9 of string: in the picking area past the fretboard
37 7 1 -3¢ 6/7 of string: in the picking area past the fretboard
41 2 0 0 1/2 of string: at the same fret 41

Further discussion and description of intonation compensation steps:

Method #1: To find the saddle compensation on a standard guitar, one compares the harmonic at the 12th fret with the fretted note at the 12th fret. For the Kite guitar, by a weird coincidence, one does the same! But the 12th fret now makes the 3rd harmonic, not the 2nd. Thus the two notes should be an octave apart, not a unison. If using a tuner, this is not a problem. But if using your ear, a unison is easier to hear than an octave. To get a unison, when you fret the string, play the 2nd harmonic with your other hand. With your forefinger or middle finger, touch the string midway between the 32nd and 33rd frets. Then stretch your hand and pluck with your thumb as close as you can get to the midpoint between your finger and the bridge. If this isn't feasible (e.g. with a bass guitar), you can capo the string at the 12th fret and use both hands to play the harmonic. And to be extremely precise, the fretted note should be 0.48¢ sharper than the harmonic. The 3rd harmonic is 701.96¢ and the 41-edo interval is 702.44¢.

On a standard guitar, there's a formula for saddle compensation. Move the saddle point back by about 0.015" for every cent that the 12th fret note is sharp of the open string's 2nd harmonic. The 0.015" figure is more precisely the scale length times ln(2)/1200, which is scaleLength/1731. Saddle compensation flattens the 12th fret note twice as much as the open string note. So if the 12th fret note is 3¢ sharp, flattening the open string note by 3¢ (about 0.045") flattens the 12th fret note by 6¢, and the interval between them is flattened by 3¢ to an exact octave.

On a Kite guitar, the scaleLength/1731 formula still holds. But saddle compensation affects the 12th fret note only one and a half times as much as the open string note. (Because an octave is 2/1 and a fifth is 3/2.) Hence for each cent of sharpness, one must flatten by two cents. For example, suppose the 12th fret note is 2¢ sharp of the 3rd harmonic. It's supposed to be 0.48¢ sharp, so the actual sharpness is only 1.5¢. (In practice, if one's tuner isn't this accurate, one might simply round down a bit.) Move the saddle point back by twice this, 3¢ or 0.045". This will flatten the open string by 3¢ and the 12th fret note by 4.5¢, narrowing the interval by 1.5¢ to an exact 41-edo 5th. On the saddle, mark a point 0.045" behind the exit point, and file up to the mark.

Method #2: The first method serves as a rough check of the saddle points. But it's much safer to check multiple frets. The cents table below (printable pdf here) has the pitch of every single note on the fretboard. The 2nd page of the pdf omits some redundant information to make room to pencil in discrepancies in cents. But the open strings aren't reliable, because the nut is not yet compensated (nut compensation must be done after saddle compensation). Use a capo to remove the nut issue. Capo the string at the 1st fret (or 2nd or 3rd, if the capo doesn't fit your 8-string very well). Tune the capo'ed string to the table, then compare the other frets to the table. Important: do not remove the capo during this process, as that will change the tension, and thus the pitch. It's usually sufficient to check every 4th fret, i.e. every dot. Look for the general trend. If the saddle point is too far back, the higher frets will be increasingly flat. Too far forward, and they will trend sharp. If there's an outlier that breaks the pattern, check its neighboring frets. No guitar is perfect. If some frets are sharp and some equally flat, that's the best you can get. Once you find the trend, estimate how much cents error would be expected at the 5th dot, which is almost an octave. That's roughly how many cents to compensate by. (To be super-precise, you could increase the cents by about 3%, so that 6¢ becomes 6.2¢.) Compensate as in method #1 with the scaleLength/1731 formula.

Nut compensation can be done similarly to a standard guitar, by comparing the open string to the fretted notes. But since the Kite guitar is so much more in tune, extra care might be taken here. One can shorten the fingerboard by around 0.030" (more if the nut action is high) to slightly overcompensate, then de-compensate empirically by filing the front of the nut to move the exit points back. One can determine the exact amount to file by finding the sharpness in cents with a tuner, then using the scaleLength/1731 formula. The front of the nut can be filed lengthwise to move all the exit points at once, or up and down to move individual exit points.

Final notes:

  • String gauges affect compensation, so try to choose the correct gauges first.
  • One can avoid nut compensation by using a zero fret.
  • Electric guitars have easily adjustable saddles. Adjustable saddle and nut for acoustic guitars: https://www.portlandguitar.com/collections/bridges

For more on saddle and nut compensation, see

Tables

Cents

Every note on the Kite Guitar fretboard. The outer columns show the dots on the fretboard. The low note is vD and the tuning is in downmajor 3rds. The note names in the table are 12-edo, not 41-edo. The low vD is written "D -29.3", meaning 12-edo D minus 29.3¢. The 7 natural notes in 41-edo are bolded and underlined italic. The full set of 41-edo names are here: File:The Kite Tuning 5.png

8th string 7th string 6th string 5th string 4th string 3rd string 2nd string 1st string
0 D -29.3 F# -48.8 A +31.7 C# +12.2 F -7.3 A -26.8 C# -46.3 E +34.1
1 D +29.3 F# +9.8 Bb -9.8 D -29.3 F# -48.8 A +31.7 C# +12.2 F -7.3
2 Eb -12.2 G -31.7 Bb +48.8 D +29.3 F# +9.8 Bb -9.8 D -29.3 F# -48.8
3 Eb +46.3 G +26.8 B +7.3 Eb -12.2 G -31.7 Bb +48.8 D +29.3 F# +9.8
* 4 E +4.9 G# -14.6 C -34.1 Eb +46.3 G +26.8 B +7.3 Eb -12.2 G -31.7 *
5 F -36.6 G# +43.9 C +24.4 E +4.9 G# -14.6 C -34.1 Eb +46.3 G +26.8
6 F +22.0 A +2.4 C# -17.1 F -36.6 G# +43.9 C +24.4 E +4.9 G# -14.6
7 F# -19.5 Bb -39.0 C# +41.5 F +22.0 A +2.4 C# -17.1 F -36.6 G# +43.9
* * 8 F# +39.0 Bb +19.5 D +0.0 F# -19.5 Bb -39.0 C# +41.5 F +22.0 A +2.4 * *
9 G -2.4 B -22.0 Eb -41.5 F# +39.0 Bb +19.5 D +0.0 F# -19.5 Bb -39.0
10 G# -43.9 B +36.6 Eb +17.1 G -2.4 B -22.0 Eb -41.5 F# +39.0 Bb +19.5
11 G# +14.6 C -4.9 E -24.4 G# -43.9 B +36.6 Eb +17.1 G -2.4 B -22.0
* * * 12 A -26.8 C# -46.3 E +34.1 G# +14.6 C -4.9 E -24.4 G# -43.9 B +36.6 * * *
13 A +31.7 C# +12.2 F -7.3 A -26.8 C# -46.3 E +34.1 G# +14.6 C -4.9
14 Bb -9.8 D -29.3 F# -48.8 A +31.7 C# +12.2 F -7.3 A -26.8 C# -46.3
15 Bb +48.8 D +29.3 F# +9.8 Bb -9.8 D -29.3 F# -48.8 A +31.7 C# +12.2
* 16 B +7.3 Eb -12.2 G -31.7 Bb +48.8 D +29.3 F# +9.8 Bb -9.8 D -29.3 *
17 C -34.1 Eb +46.3 G +26.8 B +7.3 Eb -12.2 G -31.7 Bb +48.8 D +29.3
18 C +24.4 E +4.9 G# -14.6 C -34.1 Eb +46.3 G +26.8 B +7.3 Eb -12.2
19 C# -17.1 F -36.6 G# +43.9 C +24.4 E +4.9 G# -14.6 C -34.1 Eb +46.3
* * 20 C# +41.5 F +22.0 A +2.4 C# -17.1 F -36.6 G# +43.9 C +24.4 E +4.9 * *
21 D +0.0 F# -19.5 Bb -39.0 C# +41.5 F +22.0 A +2.4 C# -17.1 F -36.6
22 Eb -41.5 F# +39.0 Bb +19.5 D +0.0 F# -19.5 Bb -39.0 C# +41.5 F +22.0
23 Eb +17.1 G -2.4 B -22.0 Eb -41.5 F# +39.0 Bb +19.5 D +0.0 F# -19.5
* * * 24 E -24.4 G# -43.9 B +36.6 Eb +17.1 G -2.4 B -22.0 Eb -41.5 F# +39.0 * * *
25 E +34.1 G# +14.6 C -4.9 E -24.4 G# -43.9 B +36.6 Eb +17.1 G -2.4
26 F -7.3 A -26.8 C# -46.3 E +34.1 G# +14.6 C -4.9 E -24.4 G# -43.9
27 F# -48.8 A +31.7 C# +12.2 F -7.3 A -26.8 C# -46.3 E +34.1 G# +14.6
* 28 F# +9.8 Bb -9.8 D -29.3 F# -48.8 A +31.7 C# +12.2 F -7.3 A -26.8 *
29 G -31.7 Bb +48.8 D +29.3 F# +9.8 Bb -9.8 D -29.3 F# -48.8 A +31.7
30 G +26.8 B +7.3 Eb -12.2 G -31.7 Bb +48.8 D +29.3 F# +9.8 Bb -9.8
31 G# -14.6 C -34.1 Eb +46.3 G +26.8 B +7.3 Eb -12.2 G -31.7 Bb +48.8
* * 32 G# +43.9 C +24.4 E +4.9 G# -14.6 C -34.1 Eb +46.3 G +26.8 B +7.3 * *
33 A +2.4 C# -17.1 F -36.6 G# +43.9 C +24.4 E +4.9 G# -14.6 C -34.1
34 Bb -39.0 C# +41.5 F +22.0 A +2.4 C# -17.1 F -36.6 G# +43.9 C +24.4
35 Bb +19.5 D +0.0 F# -19.5 Bb -39.0 C# +41.5 F +22.0 A +2.4 C# -17.1
* * * 36 B -22.0 Eb -41.5 F# +39.0 Bb +19.5 D +0.0 F# -19.5 Bb -39.0 C# +41.5 * * *
37 B +36.6 Eb +17.1 G -2.4 B -22.0 Eb -41.5 F# +39.0 Bb +19.5 D +0.0
38 C -4.9 E -24.4 G# -43.9 B +36.6 Eb +17.1 G -2.4 B -22.0 Eb -41.5
39 C# -46.3 E +34.1 G# +14.6 C -4.9 E -24.4 G# -43.9 B +36.6 Eb +17.1
* 40 C# +12.2 F -7.3 A -26.8 C# -46.3 E +34.1 G# +14.6 C -4.9 E -24.4 *
41 D -29.3 F# -48.8 A +31.7 C# +12.2 F -7.3 A -26.8 C# -46.3 E +34.1

Frequencies

41-edo frequencies in Hertz. D is tuned to standard A-440 pitch. vA is roughly 432hz, and ^G/vvB are roughly the ubiquitous 50hz/60hz mains hum.

0th octave 1st octave 2nd octave 3rd octave middle-C 5th octave 6th octave 7th octave 8th octave
C 16.305589 32.611178 65.222357 130.44471 260.88943 521.77886 1043.5577 2087.1154 4174.2308
^C 16.583595 33.167191 66.334382 132.66876 265.33753 530.67505 1061.3501 2122.7002 4245.4004
^^C / vDb 16.866341 33.732683 67.465366 134.93073 269.86146 539.72293 1079.4459 2158.8917 4317.7834
vC# / Db 17.153908 34.307817 68.615633 137.23127 274.46253 548.92506 1097.8501 2195.7003 4391.4005
C# / ^Db 17.446378 34.892756 69.785512 139.57102 279.14205 558.28410 1116.5682 2233.1364 4466.2728
^C# / vvD 17.743834 35.487669 70.975337 141.95067 283.90135 567.80270 1135.6054 2271.2108 4542.4216
vD 18.046362 36.092724 72.185449 144.37090 288.74179 577.48359 1154.9672 2309.9344 4619.8687
D 18.354048 36.708096 73.416192 146.83238 293.66477 587.32954 1174.6591 2349.3181 4698.6363
^D 18.666980 37.333960 74.667919 149.33584 298.67168 597.34335 1194.6867 2389.3734 4778.7468
^^D / vEb 18.985247 37.970494 75.940988 151.88198 303.76395 607.52791 1215.0558 2430.1116 4860.2232
vD# / Eb 19.308941 38.617881 77.235763 154.47153 308.94305 617.88610 1235.7722 2471.5444 4943.0888
D# / ^Eb 19.638153 39.276306 78.552613 157.10523 314.21045 628.42090 1256.8418 2513.6836 5027.3672
^D# / vvE 19.972979 39.945957 79.891915 159.78383 319.56766 639.13532 1278.2706 2556.5413 5113.0825
vE 20.313513 40.627026 81.254051 162.50810 325.01621 650.03241 1300.0648 2600.1296 5200.2593
E 20.659853 41.319706 82.639412 165.27882 330.55765 661.11530 1322.2306 2644.4612 5288.9224
^E 21.012098 42.024197 84.048393 168.09679 336.19357 672.38714 1344.7743 2689.5486 5379.0972
vF 21.370349 42.740698 85.481397 170.96279 341.92559 683.85117 1367.7023 2735.4047 5470.8094
F 21.734708 43.469416 86.938833 173.87767 347.75533 695.51066 1391.0213 2782.0426 5564.0853
^F 22.105279 44.210559 88.421118 176.84224 353.68447 707.36894 1414.7379 2829.4758 5658.9515
^^F / vGb 22.482169 44.964337 89.928675 179.85735 359.71470 719.42940 1438.8588 2877.7176 5755.4352
vF# / Gb 22.865484 45.730968 91.461936 182.92387 365.84774 731.69549 1463.3910 2926.7819 5853.5639
F# / ^Gb 23.255335 46.510669 93.021339 186.04268 372.08535 744.17071 1488.3414 2976.6828 5953.3657
^F# / vvG 23.651832 47.303664 94.607329 189.21466 378.42931 756.85863 1513.7173 3027.4345 6054.8690
vG 24.055090 48.110180 96.220359 192.44072 384.88144 769.76287 1539.5257 3079.0515 6158.1030
G 24.465223 48.930446 97.860892 195.72178 391.44357 782.88714 1565.7743 3131.5485 6263.0971
^G 24.882349 49.764698 99.529395 199.05879 398.11758 796.23516 1592.4703 3184.9406 6369.8813
^^G / vAb 25.306586 50.613173 101.226346 202.45269 404.90538 809.81077 1619.6215 3239.2431 6478.4861
vG# / Ab 25.738057 51.476115 102.952229 205.90446 411.80892 823.61783 1647.2357 3294.4713 6588.9427
G# / ^Ab 26.176885 52.353769 104.707538 209.41508 418.83015 837.66031 1675.3206 3350.6412 6701.2825
^G# / vvA 26.623194 53.246388 106.492775 212.98555 425.97110 851.94220 1703.8844 3407.7688 6815.5376
vA 27.077112 54.154225 108.308450 216.61690 433.23380 866.46760 1732.9352 3465.8704 6931.7408
A 27.538770 55.077541 110.155081 220.31016 440.62032 881.24065 1762.4813 3524.9626 7049.9252
^A 28.008299 56.016599 112.033197 224.06639 448.13279 896.26558 1792.5312 3585.0623 7170.1246
^^A / vBb 28.485834 56.971667 113.943335 227.88667 455.77334 911.54668 1823.0934 3646.1867 7292.3734
vA# / Bb 28.971510 57.943020 115.886039 231.77208 463.54416 927.08831 1854.1766 3708.3533 7416.7065
A# / ^Bb 29.465467 58.930933 117.861867 235.72373 471.44747 942.89493 1885.7899 3771.5797 7543.1595
^A# / vvB 29.967845 59.935691 119.871381 239.74276 479.48553 958.97105 1917.9421 3835.8842 7671.7684
vB 30.478789 60.957579 121.915158 243.83032 487.66063 975.32126 1950.6425 3901.2850 7802.5701
B 30.998445 61.996890 123.993780 247.98756 495.97512 991.95024 1983.9005 3967.8010 7935.6019
^B 31.526961 63.053921 126.107842 252.21568 504.43137 1008.86274 2017.7255 4035.4510 8070.9019
vC 32.064487 64.128974 128.257949 256.51590 513.03179 1026.06359 2052.1272 4104.2544 8208.5087
C 32.611178 65.222357 130.444714 260.88943 521.77886 1043.55771 2087.1154 4174.2308 8348.4617
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