One way to understand Just Intonation is in terms of the harmonic series; every interval used in Just Intonation can be found somewhere in the harmonic series. By definition, the harmonic series is that sequence of frequencies which is all whole-number multiples of any particular fundamental frequency (see Figure (GIF) or EPS version). Thus, since any just interval is expressible as a frequency ratio of two whole numbers, that interval is also the interval between those same two harmonics. For example, the ratio 7/5 is the interval from the fifth harmonic to the seventh. So by becoming familiar with the harmonic series as a musical scale, one also comes to know all the just intervals included. (To put it in strictly numerical terms, a familiarity with the whole numbers also includes a familiarity with the proportions of whole numbers, i.e., the rational numbers.)
Of course, the harmonic series is infinite, but the most common just intervals are found among the lowest harmonics. As one goes farther up the series one finds more exotic intervals, until at some point the new intervals are no longer musically meaningful. This limit is not fixed, but is a matter of aural familiarity and personal taste. (For this article, we'll stop arbitrarily at 16.)
In this article we will journey up the harmonic series step by step, discussing the special properties of each harmonic in turn. I'll also mention the intervals included in the harmonics up to that point. As we go up the series, we will be tracing musical history in that the simplest harmonies were the first to be used and those higher up came later. (Unfortunately, Western music effectively stopped at harmonic 5.)
To make this more meaningful, it will help to have an example of the harmonic series that you can hear and explore. The harmonic series occurs naturally in a variety of physical situations, but the most accessible example is that of a stretched string. (To hear intervals between harmonics requires two strings tuned in unison.) The best for clarity are high tension metal strings, such as on a steel string guitar, but one can use whatever string instrument is handy. The technique is to touch the string lightly (don't press down) at a harmonic node and pluck (or bow) the string near the far end. (To minimize damping you should remove the touch at the node as soon as the harmonic sounds.) To hear the second harmonic, the node is at half the length of the string; the third harmonic has a node at one-third the string length; the fourth at one-fourth, etc.
1: The first harmonic is the fundamental, which is the sound of the open string. This is the reference tone, the root of the harmonic series. When referring to any harmonic as an interval, the fundamental is implied as the other tone of the interval. Thus the interval of the first harmonic (relative to the fundamental) is just unison, or 1, or 1/1. This is clearly the most consonant interval, so much so that it hardly counts as an interval: that is, a unison is often not heard as two distinct tones. Nonetheless, unisons can be very powerful.
2: The second harmonic is twice the frequency of the first, and so makes the interval of 2/1, or an "octave." Such terms as "octave," "fifth," etc., imply diatonic music and even temperament, so I prefer to express intervals by frequency ratio, for precision and to eliminate unwanted connotations. However, the interval 2/1 is so ubiquitous that it needs a short, expressive name, and I propose "double" (rather than "octave") to express the frequency doubling (rather than the eighth tone of the scale).
The double is clearly the second most consonant interval. Indeed, it is an astounding and mysterious psychoacoustic fact (known as "octave equivalence") that all known musical cultures consider a tone twice the frequency of another to be, in some sense, the same tone as the other one (only higher). Because two tones in the relation 2/1 do represent the same harmonic identity, one can increase or decrease an interval by a double without essentially changing its harmonic meaning. Its size and sound are drastically changed, but the interval still retains its harmonic identity even after a shift of a double. Because of "octave equivalence," and for simplicity, it is conventional to deal only with intervals between unison and double (between 1 and 2) and to shift any other interval by doubles to bring it into this range. Thus, when looking at a ratio (as an interval), factors of 2 in the numerator or denominator don't affect its harmonic sense.
3: The third harmonic (3/1 relative to the fundamental) does not sound like the same tone as the first two, so represents a new harmonic entity, a "perfect fifth" (3/2) above the double of the fundamental. It is not clear whether the third most consonant interval is 3/1 or 3/2, nor is it clear what "consonance" means beyond this point, so I'll stop using that term. What is clear is that the interval 3/2 is very strong and easily recognized.
To Pythagoras, the number 3 meant divinity, and the musical interval 3/2 represented musical perfection. He built his whole musical system on this one interval (with the 2/1, of course). For example, combining 3/2 with itself gives the tone a 3/2 above the 3/2, or a 9/4, or equivalently (a double down) a 9/8. Continuing in this way to extend both up and down by 3/2's gives the "cycle of fifths." (Actually it's more like a spiral, since it never gets back to where it started but instead generates an infinity of intervals involving only powers of 3 and 2.) This idea also appears in ancient Chinese music, and by a slight compromise (closing the cycle) gives the Western 12-tone equal temperament. The 3/2 is at the basis of traditional Western harmony as the most common modulation interval.
4: The fourth harmonic is just the double of the double, and so is effectively a repetition of the fundamental. This is the first composite, i.e., it can be broken down (factored) into simpler intervals (4 = 2x2). This is in direct contrast to harmonic 3, which is prime: it can't be constructed of simpler intervals. This distinction is important, for if we take the harmonics up to some point as a basis for combinations to generate a scale or system of intervals, only the primes are relevant, since the composites are already combinations of the primes. So if we stop here with the fourth harmonic, we can still get only Pythagorean intervals, through combinations of the primes 2 and 3. (And as we said before, 2 hardly counts.)
Up to this point, we have 1, 2, 3, 4, a four-tone scale that includes those four intervals and also 3/2 and 4/3. There are twice as many tones in the second double (2, 3) as in the first (1), and this is generally true of the harmonic series: each time you go up another double, you include twice as many tones as the last double.
5: The fifth harmonic is a prime harmony, a "major third" (5/4) above the second double of the fundamental. (It is rather ironic that 5 is a "third" while 3 is a "fifth," but at least 7 is a "seventh," as we will see.) While 3/2 is assoclated with power, 5/4 is more expressive of sentiment and emotion. To my ear, these qualities of the primes show up to some extent in combined intervals; for example, the included interval 5/3 (a "major sixth") seems to convey some "fiveness" and some "threeness." This would imply that the character of any just interval includes expression of the prime factors that make it up. Of course, all this is highly subjective.
In early times the interval 5/4 was considered discordant; its acceptance as consonant came centuries after 3/2 gained popularity. Then it became firmly established as an essential component of traditional triadic harmony. In fact, traditional harmony stopped at 5; standard just diatonic scales involve only the primes 2, 3, and 5. This is what is meant by "5-limit": the highest prime used is 5.
6: This is just the double of 3, and so is nothing new except that it makes a "minor third" (6/5) above the 5.
7: The seventh harmonic is another prime, a 7/4 (a "bluesy flat seventh") above the second double of the fundamental. Since it is prime, the intervals it makes with each of the lower harmonics are new, but considering "octave equivalence" we get only three: 7/6 (a "bluesy minor third"), 7/5 (a "diminished fifth"), and 7/4. As you may have gathered by now, I feel the prime 7 conveys blueness. (However, other people think of different intervals as blue.)
Harmonics 4, 5, 6, and 7 form a just "dominant seventh" chord, but the "dominant seventh" chord of equal temperament does not convey the same sound. Thus even though traditional harmony talks of "sevenths" it cannot express a 7/4. (The tempered approximation is 31 cents sharper and suggests instead other just "sevenths," such as 16/9.) So the harmonic 7 is the first major extension to traditional harmony, and in combinntlon gives a lot of new intervallic territory to explore.
8: This is the third double of the fundamental, not a new identity. It gives the new included intervals 8/7 and 8/5.
9: The first odd composite, a "major second" (9/8) above the third double of the fundamental. Because 9 is not the double of anything, it sounds like something new, and Harry Partch considered it a new identity. This is reasonable in the context of a simple harmonic scale (e.g., harmonics 5 through 9). But in terms of generating scales through combinations of a limited set of harmonics (which Partch did), 9 is already a combination (3x3) so in that sense gives nothing new. This sometimes causes confusion, leading people to speak of "9-limit" scales even though 9 is composite, not prime.
The new included intervals are 9/8, 9/7, and 9/5.
10: Just the double of 5, giving the new included intervals 10/9 and 10/7.
11: A new prime harmonic, an 11/8 (a "quarter-tone sharp fourth") above the third double of the fundamental. This interval is radically different from anything in standard harmony; there is no way to even suggest it on a regular piano (although it is well represented in the "quarter-tone" system). Many people find this interval very dissonant and unpleasant. I find it harmonious in a striking way, suggestive of something beyond, or of questioning. It may be an acquired taste, though some people appreciate it on first hearing.
Partch chose 11 as his limit in order to include some really revolutionary intervals, as a much bolder step than 7-limit. And in the context of his hexad, 4, 5, 6, 7, 9, 11, it sounds like it really fits. Even so, it's way out there.
The eleventh harmonic gives the new included intervals 11/10, 11/9, 11/8, 11/7, and 11/6.
12: The second double of 3, giving the included intervals 12/11 and 12/7.
13: Another prime, a 13/8 (a "neutral sixth") above the third double of the fundamental. This is another way-out interval, although apparently less dissonant than 11/8. To my ear, the character of 13 is a "beyondness" somewhat similar to that of 11.
I chose 13 as my prime limit mainly to use the elght-tone harmonic scale 8 through 15. But also, while it is a large step from 7-limit to 11-limit, it seems only a small step from 11-limit to 13-limit. (Then, too, I m partial to the number 13 because it s a Fibonacci number, and 13/8 is a Fibonacci approximation to the Golden Mean.)
Being prime, 13 gives lots of new included intervals: 13/12, 13/11, 13/10, 13/9, 13/8, and 13/7.
14: The double of 7, giving the included intervals 14/13, 14/11, and 14/9.
15: The fifteenth harmonic is a major seventh (15/8) above the third double of the fundamental. It is the first non-square odd composite, so 15/8 can be thought of as a 5/4 above a 3/2, or vice versa. As an odd harmonic, it can be considered an identity in Partch's sense. It gives these included intervals: 15/14, 15/13, 15/11, and 15/8.
16: Here we reach the fourth double of the fundamental (with the included intervals 16/15, 16/13, 16/11, and 16/9), and the end of our tour. (Can you get this high on the string you re using for demonstration? It s difficult.)
The point of all this is that the harmonic series gives a handy frame of reference in trying to understand Just Intonation. By knowing the harmonic series, up to some point, as a scale, one also knows the just intervals included to that point. And more complicated ratios can be understood in terms of the prime factors involved. In light of this, the prime harmonics are seen to be the most important ones to know, because all other just intervals can be analyzed as combinations of prime harmonies. And because the primes cannot be analyzed in terms of each other, each prime represents an independent dimension of Just Intonation.