Kite Guitar explanation for non-microtonalists

Introduction

This article summarizes all the tuning theory needed to understand the how and why of the Kite Guitar.

There are two main reasons for going microtonal. One is to get new sounds (intervals, actually) such as barbershop 7ths or Middle Eastern quartertones, or experimental ones that no one's ever heard before. Another is to improve the sounds we already have by tuning them better. The Kite Guitar does both.

Getting new sounds is easy -- just add new frets anywhere, and you get something new! But getting everything in tune is much harder. So most of this article is about that. But it turns out that by getting enough notes to tune everything accurately, we also get many exciting new sounds "for free".

First, some terminology: Our standard tuning divides the octave into 12 equal steps, which is called 12-equal or 12-ET (Equal Temperament) or 12-EDO (Equal Division of the Octave). Microtonal music is anything that deviates significantly from that. Intervals are measured in cents. One hundred cents equals a semitone. For example, a 12-EDO minor 3rd is 3 semitones, or 300¢.

A musical pitch is actually a frequency. In fact, it's multiple frequencies at once. For example, A below middle-C is 220hz, but it's also 440 hz, 660 hz, 880 hz, etc. These higher frequencies are called harmonics, and they make a harmonic series. Every string and wind instrument including the voice has these harmonics present in every note. Understanding the harmonic series is essential for understanding microtonal music theory. For more on this, see the wikipedia article, or these excellent youtube videos by Andrew Huang and Vi Hart.

Just Intonation part 1

Just intonation (often abbreviated as JI) is based on the idea that musical intervals are in essence frequency ratios. Any two frequencies in a 2-to-1 ratio are an octave apart, e.g. A-220 and A-440. Thus an octave is in essence the ratio 1:2 or 2/1. Any two frequencies in a 3-to-2 ratio are a fifth apart, e.g. A-220 and E-330. The ratio needn't be exact. A-220 and E-331 make a 331/220 ratio. But the ear "rounds it off" to 3/2, and hears it as an ever so slightly sharp fifth.

In theory, every interval is (or is close to) some sort of ratio, but that ratio might be very complex, like say 37/23. In practice, ratios are only musically meaningful when the two numbers are reasonably sized. The upper limit on the size of the numbers is hotly debated, but it's certainly at least 10.

Here's something many musicians don't know: Choose any two numbers from 1 to 10, and you've made a recognizable musical interval. For example, choose 4 and 5, and you get 5/4, a just major 3rd. Choose 5 and 6, and you get a minor 3rd. Here's all the ratios that use the numbers 1 through 6:

• 1/1 = unison

• 2/1 = octave

• 3/2 = perfect 5th
• 3/1 = perfect 12th

• 4/3 = perfect 4th
• 4/2 = same as 2/1
• 4/1 = double octave

• 5/4 = major 3rd
• 5/3 = major 6th
• 5/2 = major 10th
• 5/1 = major 10th plus an octave

• 6/5 = minor 3rd
• 6/4 = same as 3/2
• 6/3 = same as 2/1
• 6/2 = same as 3/1
• 6/1 = perfect 12th plus an octave

2/1 and 3/2 have very small numbers, and as a result are easily tuned by ear. One can hear the harmonics coinciding or not, causing interference beats if they don't. (Interference beats are that rapid wah-wah-wah you hear when your guitar is a little out of tune.) More complex ratios like 8/5 or 9/4 are harder to tune by ear. The idea behind just intonation is to use only simple ratios in all one's chords, and avoid all interference beats. This makes harmonies much smoother.

For example, the 12-EDO major 3rd is 14¢ sharp of the 5/4 ratio. That's close enough that it still sounds like 5/4 and not some other ratio. But it's off far enough that flattening it down to 5/4 does indeed make it sound calmer and more relaxed. And a major chord sounds smoother with the 3rd slightly flattened. There are subtle interference beats in a 12-EDO chord that go away in JI. This is why it's so hard to tune a guitar by ear, because 12-EDO makes it impossible to get rid of all the beating.

Sounds great, right? So why isn't all music in just intonation? Because there are so many possible ratios that the "universe" of possible notes is mind-boggling. For example, 9/8 is 204¢, a major 2nd. But 10/9 at 182¢ is also a major 2nd. Sometimes one is appropriate, sometimes another is. And if you use the wrong one, you get a "wolf" interval -- an interval that's a "comma" of only 22¢ from a simple ratio. And you get those ugly interference beats you were trying to avoid.

So you need both major 2nds available. And to stay in tune, your melody often has to use both, making small pitch shifts of a comma. Thus 12 notes per octave isn't enough to do JI justice. You really need 20 or so. That's not a problem for choirs and string quartets, and the better ones do indeed sing/play in just intonation. But on a guitar, you need two frets only 20¢ apart. That's so close together that it's hard to physically fit your finger between them. Also, JI guitars often have partial frets that don't span the full fretboard, as shown below. The problem with those is that when you bend a string, you can go past the end of the fret. So JI guitars tend to be hard to play.

Another drawback is the lack of symmetry. In 12-EDO, every note of the 12 can be the root of any chord. Thus there's 12 major chords, 12 minor chords, 12 dom7 chords, etc. And every note can be the tonic of any scale. There's 12 major scales, 12 minor scales, etc. Anything you can do in one key, you can do in any key. Very handy when transposing a song to suit the vocalist's range. But even with 20 notes, a JI guitar can't play in all 12 keys. Often it can only play in a few. And a moderately complex song that modulates to several keys may be impossible to play.

But even with its drawbacks, there's no question that just intonation sounds pleasant, and getting one's harmonies better in tune is a powerful motivation for going beyond 12-EDO. For more about just intonation, see www.kylegann.com/tuning.html.

Just Intonation part 2

Whereas musical intervals add up (major 3rd + minor 3rd = perfect 5th), ratios multiply together (5/4 x 6/5 = 30/20 = 3/2). Since 5/4 = M3, 6/5 = m3 and 3/2 = P5, the two equations are saying the same thing in two different ways.

Ratios multiply together not only within a chord, but also when two chords have common notes. Consider a I - V progression in C. The G note is 3/2 from C, and the B in the G chord is 5/4 above this. 3/2 x 5/4 = 15/8, so the interval from C to B is 15/8. Thus two simple chords can produce a complex ratio.

Now consider a I - IV - V progression, e.g. C - F - G. What's the interval from the F note to the B note? From F to C and then to G and then to B is 3/2 times 3/2 times 5/4 = 45/16. Then subtract an octave by dividing by 2/1 to get 45/32. This very complex ratio is the result of much simpler ratios that occur in the progression as a whole. While it's almost impossible to tune 45/32 directly by ear, it's easy to tune it in the context of this chord progression.

Thus large numbers that factor into smaller numbers are not as complex as they appear. So rather than limiting the size of the numbers, one might limit the size of the factors. It just so happens that every number can be factored into prime numbers (2, 3, 5, 7, 11...) in only one way. So it makes sense to limit the size of the primes used. This is called the prime limit, or limit for short.

Why limit ourselves to only certain ratios? Because JI is so complex that you need to limit things in some way. And in fact every musical culture or genre tends to use a certain prime limit, and this prime limit has a huge effect on the sound.

Since the Renaissance, Western music is 5-limit. All our example ratios so far have been 5-limit. Historically, the prime limit of Western music has steadily increased. In the Middle Ages, ratios only used primes 2 and 3. In the Renaissance, prime 5 was added. Many modern theorists argue that the complex harmonies of jazz, blues and other forms of 20th century music imply prime 7.

7-limit JI, or "jazzy JI", has ratios such as 7/6 and 7/4. They do sound different. To ears accustomed to 12-EDO, they sound flat. But paradoxically, even though the individual notes sound off, often they make a chord sound better. For example, the dom7 chord is noticeably smoother when the minor 7th is heavily flattened. You can hear this for yourself by detuning your guitar. Tune the B string 14¢ flat and the high E string 31¢ flat, and play a G7 chord as x-x-0-0-0-1. Listen to the sound of the chord, not the individual notes. Now play the exact same chord as 10-10-9-10-x-x. Hear the difference?

Unfortunately, detuning the guitar like this improves only the G7 chord, and ruins most other chords. To get this sweet chord in all the keys, you need way more than 12 notes per octave.

I personally find 7-limit JI new and exciting, and barbershoppers love it! Admittedly it's strange, and you may or may not like it at first. But on the Kite guitar, 7-limit comes "for free" as a result of getting all the 5-limit intervals more in tune.

The next prime after 7 is 11. Ratios like 11/6, 11/9 and 12/11 make neutral intervals midway between major and minor. They give melodies a middle eastern sound. For example, Maqam Bayatiis a minor scale with a neutral 2nd and a neutral 6th. There's also 11/8, a 551¢ 4th. Hearing it for the first time is disorienting, because we're used to classifying a 4th as either perfect or augmented. But 11/8 is midway between, making it both and neither. It also falls midway between the major 3rd and the 5th, making for interesting melodies that sound like a cross between major and lydian. Again, you may or may not like these sounds. But many people do, and it's there along with everything else.

EDOs (Equal Divisions of an Octave)

JI ratios are one way to approach tuning. Another way is to take the octave and divide it up into equal-sized steps, making an EDO. Our standard tuning is 12-EDO. Instead of 12, one could have any number of steps. Guitars have been made in many EDOs. Above about 24-EDO, the frets become too close to play comfortably.

The advantage of guitar-sized EDOs is the simplicity. The "universe" of possible notes is a managable size. Unlike just intonation, melodies don't have small pitch shifts of a comma. Another advantage is the symmetry. Unlike just intonation, every note can be the tonic of any scale. The disadvantage is that the harmonies are no longer perfectly in tune.

In general, the larger the EDO, the more in tune it is. The smaller the EDO, the more playable it is. 12-EDO is a great compromise. It happens to approximate certain simple ratios very well. For example, by sheer coincidence, the ratio 3/2 is almost exactly seven twelfths of an octave. It's only 2¢ off. Four twelfths of an octave is pretty close to 5/4, but audibly sharp by 14¢. All 5-limit intervals come from combining 3/2 and 5/4 together, so all 5-limit intervals are about 12-16¢ off.

We tolerate this slight mistuning in exchange for the convenience of having only 12 notes to deal with. But 12-EDO fails to tune 7-limit JI well. A ratio like 7/6 = 267¢ doesn't really exist in 12-EDO, because the nearest interval is 300¢, which sounds much more like 6/5 (316¢).

To get 5/4 more in tune and keep 3/2 in tune, the EDO has to get larger than 12. EDOs such as 19 and 22 do approximate 3/2 reasonably well, and 5/4 better than 12-EDO. But neither 19-EDO nor 22-EDO tunes 7-limit JI very well. For that, the EDO must get even larger. No EDO tunes primes 3, 5 and 7 well until 31-EDO. And prime 3 is worse in 31-EDO than in 12-EDO. The smallest EDO that improves 3, 5 and 7 over 12-EDO is 41-EDO. 53-EDO and 72-EDO are also famous for being very accurate. But a really big EDO like these paradoxically becomes more like JI. There are lots of notes, and you can get everything really in tune, but the sheer complexity is overwhelming. More about EDOs here: EDOs and here: en.wikipedia.org/wiki/Equal_temperament.

This youtube video The Mathematical Problem with Music, and How to Solve It is a nice explanation of 5-limit JI and 12-equal, as well as historical tunings like pythagorean and meantone temperament.

The Kite Guitar

We've seen that pure JI is impractical on a guitar, and we want the simplicity and transpose-ability of an EDO. But it's simply not possible to fit enough frets on the guitar to get everything really in tune.

Fortunately, there's another way to get more notes besides adding frets: detune the strings. A guitar has a built in redundancy, because a note appears in more than one place on the fretboard. The open 1st string note (middle-E) also appears on the 2nd string at fret 5, the 3rd string at fret 9, 4th at fret 14, etc. If you tune every other string half a fret sharp, every other middle-E becomes a new note. Same for every note, and you now have twice as many notes (24-EDO). The downside is that E appears in fewer places and it's sometimes harder to reach. Before, a major 3rd was one string over, one fret back. Now, there's a half-augmented 3rd there, and all your major chords sound very weird! The major 3rd is still on the guitar, but 4 frets away where it's hard to reach. The perfect 4th and 5th are also inaccessible, because the nearby ones have been replaced with strange half-augmented or half-diminished 4ths and 5ths. So tuning your guitar this way gives you something new, but you lose a lot of what you had before.

The Kite guitar adds notes both ways. There are almost twice as many frets, and every other string is detuned by a half-fret. The Kite guitar uses 41-EDO, a very accurate EDO. Omitting half the frets makes such a large EDO quite playable. It feels like and plays like an EDO half the size, e.g. 19-EDO or 22-EDO. The downside is that half the notes are hard to reach. But by an amazing coincidence, in 41-EDO, and only in 41-EDO, these are all dissonant intervals! For example, 41-EDO has good octaves and 5ths, but it also has octaves and 5ths that are ~30¢ sharp or flat of the good ones, that sound awful! Those intervals are moved safely out of the way. Those faraway notes in another context will be exactly the notes you want. It works out that in those contexts, your hand will naturally move to that part of the fretboard, and those notes will become the easily accessible ones. in other words, the layout of the Kite guitar automatically filters out the "wrong" notes, without you even having to think about it!

Unfortunately, the standard EADGBE tuning simply won't work. Because then those slightly sharp/flat octaves and 5ths become all too accessible, and show up in the A and E barre chord shapes. Instead, the guitar is tuned in major 3rds. (There are also some open tunings, but those limit your ability to modulate.)

So the bad news is, you can't simply pick up a Kite guitar and start playing it. There's a learning curve. You have to learn new chord shapes. The good news is, there are fewer chord shapes to learn than you might expect. Here's why: in EADGBE, the G-B interval is different from the other intervals. As a result, the C, D, E, G and A major chords all have different shapes. But the Kite guitar is isomorphic, meaning same-shape, and there's only one shape to learn for all those chords. Because the intra-string interval is always the same.

There's a few other drawbacks. Obviously the closer fret spacing is somewhat less playable (although no worse than a mandolin or ukelele). Omitting half the frets makes finding notes a little harder. Also the major-3rds tuning reduces the overall range of the guitar. Unless you're using an open tuning, or playing with another guitarist, 6 strings is somewhat limiting, and 7 or 8 is best. And of course, there's a learning curve in training your ears to hear all these new sounds. But that's the fun part!

Finally, there's sometimes subtle pitch shifts of a comma. These are the inevitable result of getting everything more in tune. As mentioned, a piece often requires both 9/8 and 10/9. On the Kite guitar, one uses whichever is appropriate at the moment. Sometimes one must use 9/8 immediately before or after 10/9, resulting in a pitch shift of a half-fret, about 30¢. Something similar can happen with 5/3 and 27/16, or with 7/4 and 16/9, etc. The good news is that like watching a magician's trick, casual listeners are usually completely fooled and don't notice the pitch shifts.

So there are disadvantages, but the advantages are enormous. Chords are only a few cents away from JI, and sound great! And there are so many harmonic options. There are four main kinds of 3rds: large major, small major, large minor and small minor. There are likewise four 6ths and four 7ths. There's more of everything: two major chords, two minor chords, two dim7 chords, three augmented chords, four dom7 chords, etc.

The Kite guitar also gives you lots of melodic options. Going up one fret takes you up about 60¢. This is the perfect size -- barely large enough to feel like a small minor 2nd and not a quartertone. In other words, in the right context, two notes a fret apart can feel like two distinct notes of a scale, and not two microtonal versions of the same note. And yet 60¢ is barely small enough so that the ear can be fooled by pitch shifts of half a fret (30¢).

60¢ is also small enough that two frets (120¢) still feels like a minor 2nd, although a large one. Three frets is a small major 2nd and four frets is a large one. Many melodic pathways from one note to another. And there's more! The next string up has other 2nds in between these. There's a mid-sized minor 2nd of 1.5 frets and a mid-sized major 2nd of 3.5 frets. Right between them is the middle-eastern-sounding 11-limit neutral 2nd of 2.5 frets. All these 2nds are available for heptatonic scales. Or you can use the large major 2nd and the small minor 3rd to make an African-sounding near-equipentatonic scale. Or you can play exotic octotonic, nonotonic and decatonic scales.

Naming all 41 notes in all 41 keys, and all the intervals, scales and chords they make, is no small feat. Kite's ups and downs notation manages it by adding only two symbols to the standard notation. Any notes or chords without these new symbols are as usual. From C to G is still a 5th, a D chord is still D F# A, etc. So all that music theory you spent years learning still holds true. Ups and downs are simply added in. The notes just above/below C are called ^C and vC (up-C and down-C). The intervals slightly wider or narrower than a major 3rd are called ^M3 and vM3 (upmajor 3rd and downmajor 3rd). Chords are named e.g. E^m and vGv7 (E upminor and down-G down-7). Everything has a straightforward logical name.

In summary, the Kite guitar offers so much. You can play "normal" music and it sounds cleaner. Complex jazz chords become much less dissonant. You can play barbershop. You can play middle eastern. You can get experimental. You gain so much, and lose so little!