Kite Guitar: Difference between revisions

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[[File:Jacob Collier with a Kite Guitar 6-26-19.jpg|none|thumb]]

== For Luthiers ==

~~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 [https://precisionpearl.com~~/ 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.

'''Fret and Marker Placement'''

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¢~~.

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 [https://precisionpearl.com/ 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 (fretboard marker), and every 12th fret has a double dot. Thus a 36-fret guitar is a 9-dot guitar.

~~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 [https://www.winspearinstrumental.com/ 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)."~~

'''String Gauges'''

~~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 [https://www.daddario.com/globalassets/pdfs/accessories/tension_chart_13934.pdf 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.

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. 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 is 440 * (2 ^ (-7/12 + (21 - 13*N) / 41)). For a 6-string guitar, N ranges from 2 to 7. The unit weight is pounds per inch, and is a function of string gauge and string type (plain vs. wound, etc.). D'Addario has [https://www.daddario.com/globalassets/pdfs/accessories/tension_chart_13934.pdf 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 will require a higher tension and/or a smaller gauge and/or a lower frequency

* A higher tension will require a longer scale and/or a bigger gauge and/or a higher frequency

* A bigger gauge will require a shorter scale and/or a higher tension and/or a lower frequency

* A higher frequency will require a shorter scale and/or a higher tension and/or a smaller gauge

If the frequency ~~doesn~~'t ~~change:~~

Microtonalist and luthier Tom WInspear can provide custom string sets at his website [https://www.winspearinstrumental.com/ 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. 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)."

* 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~~

'''Saddle and Nut Compensation'''

* ~~A bigger gauge will require a higher tension~~ and/~~or a shorter scale~~

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. 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 flattens the 12th fret note by 4.5¢, narrowing the interval by 1.5¢ to an exact 41-edo 5th.

Nut compensation can be done similarly to a standard guitar. 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 can 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. Here's an adjustable saddle for acoustic guitars: https://www.portlandguitar.com/collections/bridges

For more on saddle and nut compensation, see

* https://www.doolinguitars.com/intonation/intonation4.html (Mike Doolin)

* http://schrammguitars.com/intonation.html (John and William Gilbert)

* https://www.proguitar.com/academy/guitar/intonation/byers-classical (Greg Byers)

== About 41-EDO ==

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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.

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 7-limit (49/40 and 60/49), 11-limit (11/9 and 27/22) and 13-limit (16/13 and 39/32).

[[File:The Kite Tuning lattices-2.png|left|thumb|400x400px]]

[[File:The Kite Tuning lattices-3.png|center|thumb|400x400px]]

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== Tunings ==

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.

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 three 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.

@@ Line 57: / Line 57: @@
 [[File:Jacob Collier with a Kite Guitar 6-26-19.jpg|none|thumb]]
 == For Luthiers ==
-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 [https://precisionpearl.com/ 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.
+'''<u>Fret and Marker Placement</u>'''
-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¢.
+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 [https://precisionpearl.com/ 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 (fretboard marker), and every 12th fret has a double dot. Thus a 36-fret guitar is a 9-dot guitar.
-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 [https://www.winspearinstrumental.com/ 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)."
+'''<u>String Gauges</u>'''
-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)<sup>2</sup>) / 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 [https://www.daddario.com/globalassets/pdfs/accessories/tension_chart_13934.pdf 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.
+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. 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)<sup>2</sup>) / 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 a 6-string guitar, N ranges from 2 to 7. The unit weight is pounds per inch, and is a function of string gauge and string type (plain vs. wound, etc.). D'Addario has [https://www.daddario.com/globalassets/pdfs/accessories/tension_chart_13934.pdf 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 will require a higher tension and/or a smaller gauge and/or a lower frequency
+* A higher tension will require a longer scale and/or a bigger gauge and/or a higher frequency
+* A bigger gauge will require a shorter scale and/or a higher tension and/or a lower frequency
+* A higher frequency will require a shorter scale and/or a higher tension and/or a smaller gauge
-If the frequency doesn't change:
+Microtonalist and luthier Tom WInspear can provide custom string sets at his website [https://www.winspearinstrumental.com/ 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. 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)."
-* 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
+'''<u>Saddle and Nut Compensation</u>'''
-* A bigger gauge will require a higher tension and/or a shorter scale
+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 <u>interval</u> 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. Hence for each cent of sharpness, one must flatten by <u>two</u> 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 flattens the 12th fret note by 4.5¢, narrowing the interval by 1.5¢ to an exact 41-edo 5th.
+Nut compensation can be done similarly to a standard guitar. 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 <u>over</u>compensate, then <u>de</u>-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 can 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. Here's an adjustable saddle for acoustic guitars: https://www.portlandguitar.com/collections/bridges
+For more on saddle and nut compensation, see
+* https://www.doolinguitars.com/intonation/intonation4.html (Mike Doolin)
+* http://schrammguitars.com/intonation.html (John and William Gilbert)
+* https://www.proguitar.com/academy/guitar/intonation/byers-classical (Greg Byers)
 == About 41-EDO ==
@@ Line 149: / Line 167: @@
-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.
+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 7-limit (49/40 and 60/49), 11-limit (11/9 and 27/22) and 13-limit (16/13 and 39/32).
 [[File:The Kite Tuning lattices-2.png|left|thumb|400x400px]]
 [[File:The Kite Tuning lattices-3.png|center|thumb|400x400px]]
@@ Line 155: / Line 173: @@
 == Tunings ==
-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.
+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 three 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.