Mathematical Basis For The Kite Guitar
Consider the sequence 9/8 -- 7/6 -- 6/5 -- 5/4 -- 9/7 -- 4/3. These are sequential intervals on one string of the Kite Guitar, and this is the essence of what makes it so playable. 1/1 must of course be on the next string down, to make each of these ratios available to play as an interval. (If it were two or more strings down, the number of strings would be unwieldy.) Since 6/5 and 5/4 add up to 3/2, we know that 3/2 is on the next string up. And 3/2 plus 4/3 is 2/1, thus 2/1 is on the next string up after 3/2. From this we can derive the entire layout. The next chart shows the downmajor tuning, but this analysis also applies to the upminor and upmajor tunings. Alternate tunings merely shift the ratios on each string sideways.
What if this sequence from 9/8 to 4/3 was all we knew of the guitar? Could we deduce that it is in 41-edo? No, but we can work backwards from the sequence to deduce what commas are being tempered out. From 9/8 to 6/5 = 16/15 = 2 frets. From 6/5 to 5/4 = 25/24 = 1 fret. Thus (25/24)2 = 16/15, which implies the Laquinyo or Magic comma 3125/3072 = [-10 -1 5]. This comma is the difference between (5/4)5 and 3/1, thus its pergen is #37, (P8, P12/5).
This temperament can be extended to prime limit 7 and 11. From 9/8 to 7/6 = 28/27 = 1 fret. From 6/5 to 5/4 = 25/24 = 1 fret. Thus 28/27 = 25/24, which implies the Ruyoyo comma 225/224. Extending the sequence of ratios, we get 9/8 -- 7/6 -- 6/5 -- 5/4 -- 9/7 -- 4/3 -- 11/8. From 4/3 to 11/8 = 33/32 = 1 fret. Since 25/24 is also 1 fret, 33/32 = 25/24, which implies the Luyoyo comma 100/99. Tempering out Laquinyo and Ruyoyo and Luyoyo gives us 11-limit Laquinyo/Magic. Both Ruyoyo and Luyoyo are strong extensions, i.e. the pergen doesn't change.
This pergen is easily notated with a single additional pair of accidentals. From the pergen notation guide, which assumes that the octave is untempered, and the fifth is tuned to 700 + c cents, where c is a free variable:
Period = P8 = 1200¢, Generator = vM3 = ^4dd4 = 380¢ + c/5, Enharmonic = ^5dd2 = 0¢, ^5C = B##, and ^ = 20¢ + 3.8c
This table, also from the pergen notation guide, shows a 20-step section of the pergen's genchain. The first row shows the cents of each interval from the tonic, the second shows the interval name, and the third shows the note names in the key of D, using ups and downs notation.
Extrapolating fourthwards, 32/27 is 300¢ - 3c. This interval occurs from 9/8 to 4/3, which is 5 frets. Thus one fret is 1/5 of 32/27, which comes to 60¢ - 0.6c. Assuming a downmajor tuning, the adjacent-open-string-interval is the generator, 380¢ + 0.2c. This is the fretboard:
|520¢ + 2.8c + P8||580¢ + 2.2c + P8||640¢ + 1.6c + P8||700¢ + c + P8||760¢ + 0.4c + P8||820¢ - 0.2c + P8||880¢ - 0.8c + P8||940¢ - 1.4c + P8|
|140¢ + 2.6c + P8||200¢ + 2c + P8||260¢ + 1.4c + P8||320¢ + 0.8c + P8||380¢ + 0.2c + P8||440¢ - 0.4c + P8||500¢ - c + P8||560¢ - 1.6c + P8|
|960¢ + 2.4c||1020¢ + 1.8c||1080¢ + 1.2c||1140¢ + 0.6c||1200¢ (P8)||60¢ - 0.6c + P8||120¢ - 1.2c + P8||180¢ - 1.8c + P8|
|580¢ + 2.2c||640¢ + 1.6c||700¢ + c||760¢ + 0.4c||820¢ - 0.2c||880¢ - 0.8c||940¢ - 1.4c||1000¢ - 2c|
|200¢ + 2c||260¢ + 1.4c||320¢ + 0.8c||380¢ + 0.2c||440¢ - 0.4c||500¢ - c||560¢ - 1.6c||620¢ - 2.2c|
|1020¢ + 1.8c - P8||1080¢ + 1.2c - P8||1140¢ + 0.6c - P8||0¢||60¢ - 0.6c||120¢ - 1.2c||180¢ - 1.8c||240¢ - 2.4c|
Certain values of c result in edos. For example, c = 2.44¢ (1/41 of a 12-edo semitone) gives us 41-edo. Most values of c do not result in edos. Can the accuracy of the tuning be improved over 41-edo by freely varying c? Increasing c brings 5/4 more in tune, but quickly throws 3/2 out of tune. The optimal tuning of this temperament is very close to 41-edo, and as we'll see below, non-edo tunings are very impractical.
Which edos support this fretting? The edo must support the pergen (P8, P12/5). The simplest edo in which the 12th can be divided into 5 equal parts is the one in which the 12th actually equals 5 edosteps. This is 3-edo, which is much too inaccurate to be musically useful. However the set of all edos that support this pergen appear in the scale tree on the spine of the 2\3 kite, and on every 5th ripple line from the kite. (See Chapter 5.7 of Kite's book for an explanation of kites, spines and ripple lines.) These edos support the pergen, but may or may not temper out the required commas Laquinzo, Ruyoyo and Luyoyo. Only a few edos temper out these commas, even if we exclude Luyoyo from the comma list. They are:
- 5th ripple line: 19-edo, 22-edo
- 10th ripple line: 41-edo, (44-edo)
- 15th ripple line: 60-edo, 63-edo
- 20th ripple line: (82-edo), 85-edo
- 25th ripple line: 104-edo
The edos on the 5th line needn't omit any frets, those on the 10th line must omit every other fret, those on the 15th line must omit 2 out of every 3 frets, etc. Note that every other fret of 44-edo is identical to 22-edo, and every 4th fret of 82-edo is identical to every other fret of 41-edo. Thus there are only 7 possible edos:
|edo||frets omitted||frets per octave||prime 3 error||prime 5 error||prime 7 error||prime 11 error|
Let's look at each edo in turn. 19-edo conflates 10/9 with 9/8, and 8/7 with 7/6. The tuning errors are quite audible, especially for primes 7 and 11.
|505¢ + P8||568¢ + P8||632¢ + P8||695¢ + P8||758¢ + P8||821¢ + P8||884¢ + P8||947¢ + P8|
|126¢ + P8||189¢ + P8||253¢ + P8||316¢ + P8||379¢ + P8||442¢ + P8||505¢ + P8||568¢ + P8|
|947¢||1012¢||1074¢||1137¢||1200¢ (P8)||63¢ + P8||126¢ + P8||189¢ + P8|
|1012¢ - P8||1074¢ - P8||1137¢ - P8||0¢||63¢||126¢||189¢||253¢|
22-edo conflates 9/8 with 8/7, and 7/5 with 10/7. Again, many ratios are audibly out of tune.
|545¢ + P8||600¢ + P8||655¢ + P8||709¢ + P8||764¢ + P8||818¢ + P8||873¢ + P8||927¢ + P8|
|164¢ + P8||218¢ + P8||273¢ + P8||327¢ + P8||381¢ + P8||436¢ + P8||491¢ + P8||545¢ + P8|
|981¢||1036¢||1091¢||1145¢||1200¢ (P8)||55¢ + P8||109¢ + P8||164¢ + P8|
|1036¢ - P8||1091¢ - P8||1145¢ - P8||0¢||55¢||109¢||164¢||218¢|
41-edo tunes the ratios far more accurately than 19 or 22. All 9-odd-limit ratios are < 7¢ off, and most are < 4¢ off. However, unisons appear only on every other string, significantly increasing the complexity.
|527¢ + P8||585¢ + P8||644¢ + P8||702¢ + P8||761¢ + P8||820¢ + P8||878¢ + P8||937¢ + P8|
|146¢ + P8||205¢ + P8||263¢ + P8||322¢ + P8||380¢ + P8||439¢ + P8||498¢ + P8||556¢ + P8|
|966¢||1024¢||1083¢||1141¢||1200¢ (P8)||59¢ + P8||117¢ + P8||176¢ + P8|
|1024¢ - P8||1083¢ - P8||1141¢ - P8||0¢||59¢||117¢||176¢||234¢|
On a 60-edo or 63-edo Kite guitar, unisons appear only on every 3rd string, increasing the complexity even more. The 5ths are not as well tuned as in 41-edo. Neither are the majority of 9-odd-limit ratios.
|520¢ + P8||580¢ + P8||640¢ + P8||700¢ + P8||760¢ + P8||820¢ + P8||880¢ + P8||940¢ + P8|
|140¢ + P8||200¢ + P8||260¢ + P8||320¢ + P8||380¢ + P8||440¢ + P8||500¢ + P8||560¢ + P8|
|960¢||1020¢||1080¢||1140¢||1200¢ (P8)||60¢ + P8||120¢ + P8||180¢ + P8|
|1020¢ - P8||1080¢ - P8||1140¢ - P8||0¢||60¢||120¢||180¢||240¢|
|533¢ + P8||590¢ + P8||648¢ + P8||705¢ + P8||762¢ + P8||819¢ + P8||876¢ + P8||933¢ + P8|
|152¢ + P8||210¢ + P8||267¢ + P8||324¢ + P8||381¢ + P8||438¢ + P8||495¢ + P8||552¢ + P8|
|971¢||1029¢||1086¢||1143¢||1200¢ (P8)||57¢ + P8||114¢ + P8||171¢ + P8|
|1029¢ - P8||1086¢ - P8||1143¢ - P8||0¢||57¢||114¢||171¢||229¢|
An 85-edo Kite guitar would be tuned to 27\85 3rds. Unisons would appear only on every 4th string. 85-edo has larger errors than 41-edo for prime 3 and prime 7. Again, the majority of 9-limit ratios are not as well-tuned as 41-edo.
104-edo is the only edo which improves tuning over 41-edo, despite its larger prime 3 error. However, it's extremely complex, with unisons appearing only on every 5th string. The open-string interval would be 33\104.
Out of all 7 edos, 41-edo is the best candidate for the Kite guitar. It's not too complex, and it's the most accurate except for 104-edo. While 19-edo and 22-edo guitars have the advantage of less complexity, there is no reason to tune them in 3rds. The standard EADGBE tuning has the advantages of familiarity and increased range. And when tuned thusly, there is no reason to consider them Kite guitars. They are simply microtonal guitars that happen to support Laquinyo/Magic.
For all these edos, and for various non-edo tunings, the intervals within the immediate area of 1/1 are roughly the same. But the intervals up the neck are quite different. Consider 41-edo:
The off zone (white, gray and purple) between the two rainbow zones is the result of omitting every other fret. On a 19-edo guitar, the higher rainbow zone would move one fret to the left. Each pair of gray intervals would fuse together, e.g. v8 and ^8 would merge into P8. The off zone would become a new rainbow zone. The same thing would happen with a 22-edo guitar, except that the upper rainbow zone would move one fret to the right.
Thus 19-edo and 22-edo guitars have no off zone, and one can play a scale freely up and down the neck. The 41-edo Kite guitar's off zone complicates playing melodies. One can only play a scale in a fairly narrow range of the neck. When one runs out of strings, to continue the scale an octave higher one has to jump up 13 frets, a little more than a 5th. In practice, when two guitarists play together, one guitarist typically solos 13 frets above the other.
On a 19-edo or 22-edo guitar, the jump is much smaller, only 6 or 7 frets respectively. This is because the unison on a 19-edo guitar in relative tab is (-1,+6), and for 22-edo (-1,+7). The octave always appears at (+3,+1), and additionally appears at this location plus or minus any number of unisons. Thus the octave appears on every string. On a 41-edo guitar, the unison is (-2,+13), and the octave appears only on every other string.
With a 60-edo or 63-edo Kite guitar, the unison is at (-3,+19) and (-3,+20) respectively. The off zone becomes wider, and the required jump is 19 or 20 frets, almost a full octave. This is so large that in certain keys there may not be enough frets to provide a second rainbow zone. On an 85-edo Kite guitar, the unison is at (-4,+27), the off zone is even wider, and one must jump up a minor 10th. For 104-edo, the unison is at (-5,+33), and the jump is 33 frets, a perfect 12th. Guitars tuned to these larger edos are not very playable other than as a solo instrument, because two guitarists are stuck in the same range. But a non-edo tuning of Laquinyo/Magic is even worse, because there are no unisons, and the octave doesn't appear anywhere up the neck. One can only play about two octaves of a scale, no matter how many frets one has.
In conclusion, 41-edo is clearly the best tuning for the Kite guitar.
Generalizing the Kite guitar concept to other EDOs
The Kite guitar can be thought of as a higher-accuracy tuning of either a 19edo or a 22edo guitar. Other edos can be improved via the "rule of three". Take a common guitar edo. Double the edo and either add or subtract 3. (More precisely, move in the scale tree one step along the ripple line of the 2\3 kite.) Add if the 5th is flat of 3/2, otherwise subtract. Then omit every other fret. You get something resembling the original edo, but better tuned:
- 14edo --> half of 31edo
- 15edo --> half of 27edo
- 16edo --> half of 35edo
- 17edo --> half of 31edo
- 19edo --> half of 41edo (Kite-fretting)
- 20edo --> half of 37edo
- 21edo --> half of 45edo
- 22edo --> half of 41edo (Kite-fretting)
- 24edo --> this method doesn't work
- 31edo --> half of 65edo
The guitar must be tuned in 3rds not 4ths, otherwise the wolf 5th is easier to play than the good one.
The rule of three works because the 5th 3/2 is roughly two-thirds of an octave. So we can add or subtract 2 to the 5th and 3 to the octave, to get to a nearby node in the scale tree. If the original edo has a 5th of m\n, the new edo has a 5th of (2m±2)\(2n±3). We choose between adding and subtracting so that the tuning of 3/2 is improved. However if the 5th is already very accurate, as in edos 12 and 24, both options worsen 3/2, and the rule of three doesn't work.
For 5/4, the formula is m\n --> (2m±1)\(2n±3). Add or subtract to match what we did with 3/2. 5/4 is quite nearly one-third of an octave. Thus the tuning of prime 5 isn't affected too much. In geometric terms, the relevant section of the ripple line of the 1\3 kite is nearly vertical. In the list of edos above, by chance 5/4 is mostly improved. A few edos such as 14 and 17 tune 5/4 so poorly that doubling the edo greatly improves it. But this improved 5/4 is made nearly inaccessible by the omitting of frets. Thus 17edo in effect becomes 31c-edo or 31cc-edo.
To sum up, the tuning of prime 3 is always improved. The tuning of prime 5 is usually slightly improved. If it's worsened, it's only slightly worse. Primes 7 and 11 may get better or worse, it's sort of random. Overall, the tuning is improved.
One could also triple the edo, add/subtract 3, then omit 2/3 of the frets (skip-2 fretting). 19edo becomes one-third of 60edo and 22edo becomes one-third of 63edo, both Kite-Guitar-like frettings.
Instead of three, there could be a rule of five, since 3/2 is roughly three-fifths of an octave, and 7/4 is very nearly four-fifths. It wouldn't affect prime 7 much, but might worsen prime 5. Using ripple lines of the 3\5 pentatonic kite, the 5th changes from m\n to (2m±3)\(2n±5). 7/4 changes from m\n to (2m±4)\(2n±5). 17edo --> 29d-edo and 19edo --> 43d-edo.