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== Introduction ==
== Introduction ==


This article summarizes all the musical tuning theory needed to understand the how and why of the [[The_Kite_Guitar|Kite Guitar]].
This article summarizes all the tuning theory needed to understand the how and why of the [[The_Kite_Guitar|Kite Guitar]].


There are two main reasons for going ''microtonal'' (using pitches between the ones on standard "Western" instruments). One reason is to get new sounds, either just for experimenting or to play music from different cultures such as Middle Eastern quartertones. The other reason is to improve the sounds we already have by tuning them better. The Kite guitar does both.
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 on a guitar is easy -- just add new frets anywhere, and you get something new! But getting everything in tune is far harder. So most of this article is about harmonic tuning. It just turns out that by getting enough notes to tune everything accurately, we also get many exciting new sounds "for free".
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".


=== Tuning terminology ===
First, some terminology: Our standard tuning divides the octave into 12 equal steps, which is called 12-equal or 12-EDO ('''E'''qual '''D'''ivision of the '''O'''ctave). 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¢.
First, some terminology: Our standard tuning divides the octave into 12 equal steps, which is called 12-equal or 12-EDO (EDO for "equal divisions of the octave"). Precise tuning is measured in ''cents''. One hundred cents equals one semitone in 12-EDO. For example, a minor 3rd is 3 semitones, or 300¢.


==== The harmonic series ====
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 <u>essential</u> for understanding microtonal music theory. For more on this, see the [[wikipedia:Harmonic_series_(music)|wikipedia article]], or  these excellent youtube videos by [https://youtu.be/Wx_kugSemfY Andrew Huang] and [https://youtu.be/i_0DXxNeaQ0 Vi Hart].
An absolute musical pitch can be labeled with a precise frequency. For example, the 5th string on a standard guitar is an "A", and it is tuned to 110Hz. But like most musical tones, the sound is actually a combination of multiple frequencies at once. That one string sound has A-110 along with A-220, E-330, A-440, and so on. Notice the pattern: the lowest frequency (also called the "fundamental") and multiples of that. That pattern is called the ''harmonic series'', and each of the frequencies in it are called ''harmonics''.
 
String instruments, wind instruments, and even the human voice all make ''harmonic'' sounds that follow the harmonic series. In contrast, drums and bells can have other combinations which are ''inharmonic''. Understanding the harmonic series is <u>essential</u> for understanding most microtonal music theory.
 
For more on harmonics, see Andrew Huang's excellent video introduction: https://youtu.be/Wx_kugSemfY and Vi Hart's more in-depth discussion: https://youtu.be/i_0DXxNeaQ0


== Just Intonation (JI) part 1 ==
== Just Intonation (JI) part 1 ==
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== Just Intonation part 2 ==
== 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). And in fact a just major chord has a lower 3rd of 5/4 and an upper 3rd of 6/5, which do indeed add up to a 3/2 fifth.  
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 two different ways.  


Ratios add up 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.
Ratios add up 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.
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== EDOs ==
== EDOs ==


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, or Equal Division of an Octave. 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.
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 key note of any scale. The disadvantage is that the harmonies are no longer perfectly in tune.  
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 key note of any scale. The disadvantage is that the harmonies are no longer perfectly in tune.  
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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¢).
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 <u>and</u> 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: [[wikipedia:Equal_temperament|en.wikipedia.org/wiki/Equal_temperament]].  
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 <u>and</u> 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: [[wikipedia:Equal_temperament|en.wikipedia.org/wiki/Equal_temperament]].  


== The Kite Guitar ==
== The Kite Guitar ==
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