The Kite Guitar
- 1 The Kite Guitar
- 2 For Luthiers
- 3 About 41-EDO
- 4 Tunings
- 5 Fretboard Charts (downmajor tuning)
- 6 Relative and Absolute Tab
- 7 Tuning Instructions
- 8 Chord Shapes (downmajor tuning)
- 9 Translating 12-edo Songs to 41-edo
- 10 Original Compositions
- 11 Further materials
The Kite Guitar
The Kite guitar (or bass, mandolin, banjo, etc.) uses 41 divisions of the octave instead of 12. 41-tET or 41-edo approximates 7-limit just intonation to within 3-6¢, and chords sound gorgeous! But a guitar with 41 frets per octave is impractical. The Kite guitar cleverly omits every other fret. Thus while the frets are closer together than a standard guitar, they're not so close as to be unplayable. The interval between open strings is 13 steps of 41. 13 is an odd number, thus all 41 pitches are present on the guitar. Each string has only half of the pitches, but any adjacent pair of strings has all 41.
Omitting half the frets in effect moves certain pitches to remote areas of the fretboard, and makes certain intervals difficult to play. Miraculously, it works out that the remote intervals are the ones that don't work well in chords, and the ones that aren't remote are the ones that do work well. For example, the sweet 5-limit major 3rd, a 5/4 ratio, is easily accessible, but the dissonant 3-limit major 3rd 81/64 isn't. (3-limit & 5-limit refer to the largest prime number in the frequency ratio.)
In addition, important 7-limit intervals like 7/6, 7/5 and 7/4 are easy to play. This means the Kite guitar can do much more than just play sweet Renaissance music. It can put a whole new spin on jazz, blues and experimental music. The dom7 and dom9 chords are especially calm and relaxed, revealing just how poorly 12-tET tunes these chords. But dissonance is still possible, in fact 41-tET can be far more dissonant than 12-tET. And 41 notes means that the melodic and harmonic vocabulary is greatly expanded, allowing truly unique music that simply isn't possible with 12 notes.
The Kite guitar has almost twice as many frets as a standard guitar. Even with these additional frets, the Kite guitar is still quite playable. The interval between open strings is usually a major 3rd, not a 4th. Thus new chord shapes must be learned. However, the Kite guitar is isomorphic, meaning that chord shapes can be moved not only from fret to fret but also from string to string. Thus there are far fewer shapes to learn. (Open tunings, which are non-isomorphic, are also possible.) Tuning in 3rds not 4ths reduces the overall range of the guitar. Thus a 7-string or even an 8-string guitar is desirable.
Left: Caleb Ramsey’s 8-string Shawn guitar by Agile, refretted by Vivian Cecylia Tylińska
Right: Jim Snow's 7-string Ibanez guitar, refretted by himself.
Squire 6-string (left) and Yamaha 6-string (right), both refretted by Matthew Autry
Five early prototypes with cable-tie frets, all refretted by Kite Giedraitis:
A simple 12-bar blues by Aaron Wolf:
An arrangement of Auld Lang Syne by Aaron Wolf:
Triadic Etude by Kite Giedraitis (midi demo)
Open-tuning recordings by Jason Yerger:
- https://soundcloud.com/sacred-skeleton/modified-kite-guitar-take-1 (clean)
- https://soundcloud.com/sacred-skeleton/modified-kite-guitar-take-2 (fuzz)
Videos and Podcasts
Kite Giedraitis and Aaron Wolf were hanging out one afternoon playing guitar, when Spencer Hargraves (Terry Maple, Jock Tears, Redrick Sultan) texted that he had a gig that night in Portland. They invited him over, and excitedly filled him in on the Kite guitar.
Short cell phone videos with Jacob Collier:
http://tallkite.com/videos/JacobCollier3.mp4 (1 min 29 sec)
Kite, Aaron and Caleb Ramsey with Jacob after his June 2019 show in Portland, on the sidewalk outside the venue.
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. Or purchase a pre-slotted fingerboard from PrecisionPearl.com. It comes radiused, tapered and inlaid, so all you need to do is glue it on and put in the frets. Replacing 24 old frets makes 41 new frets, but the last few are very tightly spaced. One might instead replace 21 old frets to make 36 new frets. Every 4th fret has a dot, and every 12th fret has a double dot.
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 8ve apart, not a unison. To get a unison, when you fret the string, play the 2nd harmonic with your other hand. With your forefinger or middlefinger, 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¢.
A 6-string Kite guitar 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. Microtonalist and luthier Tom WInspear can provide custom string sets at his website www.winspearinstrumental.com. 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. For an 8-string electric guitar with a downmajor 3rds tuning running from a low D up to vF, 9.5 12 15 21 26 32 42 52 (thousands of an inch, the first 3 strings are plain) would be similar to the standard 10-46 for EADGBE.You can use an 18 or 19 plain instead of the 21 wound if you prefer, but I would prefer 21w. 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. For a 27" scale, 9 11 14 20 25 31 40 50 is best (the 2nd string could be 11.5, if you can find it)."
Tom's method involves extrapolating from familiar string sets. Another method involves calculating a 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 is 440 * (2 ^ (-7/12 + (21 - 13*N) / 41)). For the Nth string of 6, it's 440 * (2 ^ (-7/12 + (8 - 13*N) / 41)). 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. 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.
If the frequency doesn't change:
- A longer scale will require a higher tension and/or a smaller gauge
- A higher tension will require a longer scale and/or a bigger gauge
- A bigger gauge will require a higher tension and/or a shorter scale
41-edo approximates just intonation very closely. Prime 3 is extremely accurate, and primes 5 and 7 are both flat, which means their errors partially cancel out in ratios such as 7/5. Unfortunately prime 11 is sharp, so the errors add up, and 11/10 is nearly 11¢ sharp.
The 41 notes can be named with ups and downs. A sharp equals four ups, and a minor 2nd equals three ups.
|C||^C||^^C = vvC#||vC#||C#||^C#|
|vDb||Db||^Db||^^Db = vvD||vD||D|
|P1||^1||^^1 = vvA1||vA1||A1||^A1|
All 41 intervals:
- P1 ^1
- vm2 m2 ^m2 ~2 vM2 M2 ^M2
- vm3 m3 ^m3 ~3 vM3 M3 ^M3
- v4 P4 ^4 ~4 vA4/d5 A4/^d5 ~5 v5 P5 ^5
- vm6 m6 ^m6 ~6 vM6 M6 ^M6
- vm7 m7 ^m7 ~7 vM7 M7 ^M7
- v8 P8
The most dissonant intervals are the off-perfect ones ^1, v4, ^4, v5, ^5 and v8.
This chart shows 41-edo in terms of 12-edo. "-ish" means ±1 edostep. The 12 categories circled in red correspond to the notes of 12-edo. The two innermost and two outermost intervals are duplicates.
These charts show the 7-limit and 11-limit lattices with 41-edo names and edosteps in place of ratios. Each color represents a separate plane of the lattice. In the 2nd lattice, the ~3 note in the center next to the P1 is simultaneously 49/40, 60/49, 11/9 and 27/22.
Tuning the Kite guitar to EADGBE doesn't work, because the conventional chord shapes create wolves. For example, the usual E major chord shape 0 2 2 1 0 0 would translate to either 0 3 3 2 0 0 = E vB vE G# B E, or else 0 4 4 2 0 0 = E ^B ^E G# B E. Either way, the chord contains 3 wolf octaves and two wolf fifths. In addition, the major 3rd isn't 5/4 but 81/64.
The standard tuning is the downmajor tuning, in which adjacent open strings are tuned a downmajor 3rd apart. Alternative tunings use an upminor 3rd or an upmajor 3rd. All three tunings are isomorphic, thus there is only one shape to learn for any chord. A "semi-isomorphic" tuning alternates downmajor and upminor 3rds, and every chord has two shapes. In addition, there are open tunings such as DADGAD.
Open tunings become more playable with the use of a "half-fret capo". From Jason Yerger's liner notes (see the "Recordings" section):
"A couple of improvisations on a guitar loaned to me by Kite Giedratis. The guitar is fretted to 41 notes per double-octave, i.e. every other note of 41 notes per octave, using movable cable ties. On these tracks I modified the fretting slightly by moving the 2nd fret down one step of 41edo and then put a capo behind it, effectively moving all the frets above it UP by one step of 41edo, so that the frets all give odd-numbered pitches from 41edo instead of even-numbered ones. This gives frets for approximations to the ratios 21/20, 12/11, 9/8, 7/6, 6/5, 5/4, 9/7, 4/3, 11/8, and 10/7 relative to the open strings, which makes it possible to let the open strings ring out against pitches fretted low on the neck when the open strings are tuned to DADGAD or DGDGAD, my two favorite open tunings.
Without the offset I introduced, the normal fretting on Kite's guitar would have the lowest frets approximating 28/27, 16/15, 10/9, 8/7, 32/27, 11/9, 81/64, 15/11, 7/5, and 16/11, which doesn't work well for the open tunings I like but is rather designed to have the open strings tuned in parallel 3rds (5/4 or 6/5), for an isomorphic layout that facilitates chords built by stacking 3rds. I found that tuning somewhat challenging, being so unlike any open string tunings I've ever used before, and most of the intervals between non-adjacent open strings are rather discordant. Other players, whose styles don't lean as heavily on open strings and drones the way I do, may find Kite's original design preferable to my modification.
But anyway, the two designs can coexist on the same fretboard by simply inserting an extra fret between the 1st and 2nd instead of moving the 2nd fret lower as I have done, and by varying the tuning of the open strings as you please. It's a fantastic way to access the resources of 41edo on a guitar, without having an absurd number of very closely-spaced frets!"
How to implement the half-fret capo trick: An extra fret slot is cut to allow insertion of a temporary fret in between the 1st and 2nd (permanent) frets. (If the guitar has a zeroth fret, the temporary fret can go between the 0th and 1st frets.) The slot stops short of the treble side of the fretboard. So gravity holds it in place, plus of course the capo. The temporary fret has the barbs on the side of the tang filed off. The extra slot is a bit wider, so the fret can be pulled out easily. It goes in from the side, under the strings, so the strings don't need to be loosened. It can be inserted and removed on stage between songs. The fret is a bit longer, sticks out about 1/2 inch, so that you can pull it out easily.
Fretboard Charts (downmajor tuning)
This chart is in relative not absolute notation, meaning it shows intervals not notes. At the bottom is P1, a perfect unison. This is the tonic of the scale, or the root of the chord. This chart shows all the intervals within easy reach of this note, up to an octave. There are four "rainbows": one of 2nds, one of 3rds, one of 6ths, and one of 7ths. These plus the 4th, 5th, 8ve, and a few other notes add up to 25 of the 41 notes. Every single ratio of odd-limit 9 or less appears here.
This chart is the same, but extends much further. Some ratios change in the higher octaves, e.g. 16/15 becomes not 32/15 but 15/7.
This chart extends even further, showing the "rainbow zones" and the "off zones". When two guitarists play together, it's very natural for one to play chords in the lower rainbow zone, and another to solo in the higher rainbow zone. The open strings tend to be in an off zone, unless the tonic is fairly close to the nut, or else up around the 14th fret.
This chart shows the actual notes of an 8-string Kite guitar. The notes circled in red are the open strings of a 12-edo guitar. The ideal string gauges for this tuning are discussed in the "For Luthiers" section. Every 4th fret has a dot, and every 12th fret has a double dot. Three dots equals a 5th.
A 6-string guitar is usually tuned to the middle 6 strings of an 8-string:
This chart shows all the notes, not just the natural ones. But it's too much work to memorize all this. Just learn where the 7 natural notes are, and learn your intervals. Since the open strings don’t work as well, one tends to think more in terms of intervals than notes anyway.
Some keys are somewhat awkward to play in. For example, a vG scale is either too close to the nut to have a plain major 2nd, or else way up at the 16th fret where the fret spacing becomes too cramped to play comfortably. There's a "sweet spot" for the tonic on the lowest 3 strings, from about the 5th fret to about the 12th fret. This defines a 3x8 rectangle containing 24 keys, roughly every other one of the 41 possible keys. The lowest string of an 8-string is tuned to vD not D so that the common keys of C, G, D, A and E fall in this sweet spot. D is tuned to A-440 standard pitch, to bring these 5 keys as close to 12-edo as possible. D agrees exactly, A is 2.5¢ sharp of 12-edo, E is 5¢ sharp, and so forth along the spiral of 5ths.
In 12-edo, all 12 keys are needed so that a vocalist can get within 50¢ of their optimal range. In 41-edo, using only these 24 keys, one can get within 30¢ of the optimal range. 30¢ from optimal is sufficient, 15¢ from optimal is overkill, so the other 17 keys aren't really needed. The 24 most comfortable keys on a 6-string guitar are: A vBb ^Bb vB ^B C ^C vDb Db ^C# D vEb ^Eb vE E ^E vF ^F Gb ^F# G vAb ^Ab ^G#.
Printable charts, one of scale degrees, the other of the three main scales. In the latter, some scale degrees appear more than once. In general, use the one that agrees with the current chord.
Relative and Absolute Tab
Since the fretboard is isomorphic, any interval can be expressed in relative tab as a vector. This is particularly useful for in-person oral instruction of chord shapes. For example, in the downmajor tuning, going up 2 strings and down 1 fret always takes you up a perfect 5th. In relative tab, that move is spoken as "plus-two, minus-one", and written as (+2,-1). The downmajor 2nd is at "oh, plus-three", (0,+3). The downmajor 3rd is at "plus-one, oh", (+1,0).
Every interval appears in several places on the fretboard. Typically one is within a few frets and another one is many frets away. Mentally grouping four frets together into one dot facilitates large jumps up and down the fretboard. For example, the octave is (+3,+1) and also (+1,+14). A jump of 14 frets is a "3 and 2" jump, meaning 3 dots plus 2 frets. Thus the octave is at "plus-one, plus-three-and-two", or (+1,+3+2). The 5th at (0,+3+0) is spoken as "oh, plus-three-and-none", or alternatively "oh, plus-three-dots". The unison is plus-two minus-three-and-one, (+2,-3-1). An upward jump of 11 frets could be called either plus-two-and-three or plus-three-minus-one. Note that plus-three-oh means up three strings, but plus-three-and-none means up three dots.
Notes can be referred to similarly in absolute tab, which names each string/fret combination, i.e. each location on the fingerboard. For example, a low E on an 8-string is at "eighth and first", written (8th, 1st), meaning 8th string, 1st dot. This is particularly useful when one wants to tell another guitarist what key they are in, without having to use note names. For example, one might be in the key of "sixth, second and three", (6th, 2nd & 3) meaning 6th string, 3 frets above the 2nd dot. "Sixth and two" (6th, 2) means 6th string, 2nd fret. "Sixth and oh" (6th, 0) means the open 6th string.
Unlike relative tab, absolute tab doesn't require isomorphism, and can be applied to any guitar, as long as the dot locations are agreed on. For 12-edo, dots are at frets 3, 5, 7, 9, 12, etc., thus the 10th fret is fourth-and-one.
Note that in absolute tab, strings are numbered in descending order, but in relative tab, a positive move is an ascending move. Thus moving from the 3rd string to the 1st string is plus-two, not minus-two.
The Kite guitar in downmajor tuning can be tuned by ear using the octaves at (+1,+3+2). The open 6th string should be an octave bellow the 5th string's 14th fret. This can be written as (6th, 0) = (5th, 3rd & 2). We can double-check the tuning using the unisons at (+2,-3-1). Thus the 6th string at the 13th fret should match the open 4th string, and (6th, 3rd & 1) = (4th, 0). Finally, the 3rd harmonic of the 6th string should match the open 1st string (technically it should be half a cent sharp of it). Here are the full tuning instructions for a 6-string guitar:
|(6th, 0) = (5th, 3rd & 2)||(6th, 3rd & 1) = (4th, 0)|
|(5th, 0) = (4th, 3rd & 2)||(5th, 3rd & 1) = (3rd, 0)|
|(4th, 0) = (3rd, 3rd & 2)||(4th, 3rd & 1) = (2nd, 0)|
|(3rd, 0) = (2nd, 3rd & 2)||(3rd, 3rd & 1) = (1st, 0)|
|(2nd, 0) = (1st, 3rd & 2)||(6th, 0) harmonic at 3rd dot = (1st, 0)|
In the upminor tuning, the octave is at (+1,+3+3), and (6th, 0) = (5th, 3rd & 3). The unison is at (+2,-2-3), and (6th, 2nd & 3) = (4th, 0).
In the upmajor tuning, the octave is at (+1,+3+1), and (6th, 0) = (5th, 3rd & 1). The unison is at (+2,-3-3), and (6th, 3rd & 3) = (4th, 0).
The Kite guitar can be tuned to a specific pitch using the EDOtuner, a free strobe tuner for microtonal guitars (requires Reaper or ReaJS). Presets for the Kite guitar can be found here: http://tallkite.com/misc_files/js-MIDI_EDOtuner_txt.ini In Reaper, select Options/Show REAPER Resource Path and put the .ini file in the Presets folder.
Chord Shapes (downmajor tuning)
Chords are named using ups and down notation, see also the notation guide for edos 5-72. Briefly, an up or down in the chord name immediately after the root affects the 3rd, 6th and/or the 7th, but not the 5th or 9th. Chord progressions are written as Cv7 - vEb^m6 - Fv7 or Iv7 - vbIII^m6 - IVv7.
A fairly exhaustive survey of 41-edo chords is here: The_Kite_Guitar_Chord_Shapes_(downmajor_tuning)
Here's a printer-friendly chart to get you started, with and without fingerings:
Translating 12-edo Songs to 41-edo
Obviously, the Kite Guitar can do much more than simply play conventional music. But a good starting place is to take what you know and find it on the Kite Guitar. Translating 12-edo music is sometimes problematic but never impossible. Quite often the translated version sounds better, because it's so well tuned.
One way to translate a conventional song is to first translate it to 7-limit JI, perhaps visualizing it on a lattice, keeping in mind that 41-edo tempers out the Layo, Ruyoyo and Saruyo minicommas. Then translate the JI to 41edo. Another way is to use the spiral charts in the "About 41-edo" section.
Often there is only one obvious way to translate a song. I - V - VIm - IV becomes Iv - Vv - vVI^m - IVv. Sometimes there are multiple obvious translations. For example, the first 3 chords of "When I Was Your Man" are II7 - IIm7 - I. That could become vII^7 - vII^m7 - Iv, or it could become ^IIv7 - ^IIvm7 - Iv.
In general, chords shouldn't use offperfect intervals (^4, v4, ^5 or v5). Downmajor is preferred over upmajor. Upminor is preferred for most folk, but downminor is preferred for most blues. Avoid plain major and minor 3rds and 6ths.
Comma pumps, other than the aforementioned minicommas, cause pitch shifts, or occasionally, a tonic drift. The two most common commas that cause issues are the Gu and Ru commas. The choice of which two chords in the pump contain the pitch shift can be tricky. Generally, root movement by an offperfect interval is avoided. This usually necessitates a root movement by a plain major or minor interval.
For example, I - VIm - IIm - V7 - I is a Gu pump. Without the pump, I - VIm would be translated as Iv - vVI^m, to avoid shifts. The roots would move by a vM6. With the pump, this might translate to Iv - VI^m - II^m - Vv7 - Iv. The first root movement is by a M6. The tonic and the major 3rd both shift between the I chord and the VI chord.
Likewise, I7 - IV7 - V7 - I7 is a Ru pump. The usual translation is Iv7 - IVv7 - Vv7 - Iv7, with the 4th shifting between the IV and V chords. Another example is Im7 - bIIIm6 - bVII7 - IV7 - I7. The root movements are m3, P5, P5, P5. Without the pump, the m3 movement would be translated to vm3. With the pump, to avoid an ^5 movement, the translation is Iv7 - bIII^m6 - bVIIv7 - IVv7 - I.
For rapid comma pumps of only two measures, a shift halfway through the pump is often best. See Kite's translation of "I Will".
One way to hide pitch shifts is to voice the two occurrences of the pitch in different octaves. Another way is to omit the 5th in one of the chords. Thus in the Gu example, the 2nd chord might be VI^mno5.
In much music, especially pre-20th-century music, the dissonance of the dom7 chord is what drives the V7 - I cadence and gives the music momentum. But 41-edo's smooth v7 chord is like a guard dog that smiles and wags its tail at strangers instead of barking. It's too relaxed! And the 7-limit intervals can sound out of place in a pre-20th-century context. One might instead use Vv,7 (down add-7, with a plain minor 7th) or Vv,^7 (down up-7, with an upminor 7th). For example, Am - G - F - E7 can be translated as A^m - ^Gv - ^Fv - Ev,^7. (This also avoids a pitch shift.)
For 20th-century music, a Vv7 chord is often appropriate. But when a stronger V7 - I cadence is desired, a V^7 chord often works. For example, IIm7 - V7 - IM7 could be translated as either II^m7 - Vv7 - IvM7 or IIvm7 - Vv7 - IvM7. But the v7 chord is actually smoother than the vM7 chord, so the latter progression feels unfinished. Better is II^m7 - V^7 - IvM7. The II^m7 chord has two notes in common with V^7. It feels somewhat like a V11no1no3 chord. If a 9th is added to the V^7 chord, there are three common notes, and the progression feels even more connected.
However, if the I chord has no 7th, or a minor 7th, either II^m7 - Vv7 - Iv or IIvm7 - Vv7 - Iv works. The IIvm7 chord is more connected to the V chord than II^m7.
Actual song translations are on separate xenwiki pages, grouped by translator. if you have any translations, feel free to create your own page and link to it here!
- Triadic Songs
- Tetradic Songs
- Barbershop, World Music, Microtonal Music, etc.
- Underground Revolution
- I Hear Numbers
- Triadic Etude
- Blues in G
- II-V-I riff in A downmajor
- The Relative Song
- The Walking Song
- Latrizo Pump riff
The Kite Tuning (May 2019 paper announcing the discovery, 16 page pdf)
Scales on the Kite Guitar Pentatonic, diatonic, semitonal, chromatic and microtonal (rough draft on the bottom of this page)