LIMIT

This is rooted in history. Pythagoras was big on the number 3, and he asserted that musical scales are based on what would later be called the "circle of fifths". In the harmonic series, harmonic #3 is a perfect fifth above harmonic #2, hence the frequency ratio 3/2 is the interval of a perfect fifth. So a "Pythagorean scale" is one based on "stacking" perfect fifths (multiplying or dividing by 3/2), and using octave equivalence (multiplying or dividing by 2).

For example, say the root of our scale is denoted 1/1.

If we go up by a fifth, we get (1/1)*(3/2) = 3/2, the "fifth" of the scale.

If we go down from the root by a fifth, we get (1/1)/(3/2) = 2/3, which is below the root, but by octave equivalence we raise it up an octave (2/3)*(2/1) = 4/3, to get the "fourth" of the scale.

If we go a fifth above the fifth: (3/2)*(3/2) = 9/4, a perfect ninth, but bring it down an octave to put our scale degrees all within one octave: (9/4)/(2/1) = 9/8, the "second" of the scale.

Et cetera: we keep going up by another fifth atop the last one, and adjusting by an octave when necessary, resulting in the "Pythagorean major scale":

1/1 9/8 81/64 4/3 3/2 27/16 243/128

Because all the intervals can be broken down into combinations of octaves and fifths, only the prime numbers 2 and 3 are involved; for example the Pythagorean major seventh 243/128 = 35/27 = (3*3*3*3*3)/(2*2*2*2*2*2*2), where ^ indicates an exponent or power. (A prime number is one that has no factors other than itself and 1.) Since the highest prime involved in the scale is 3, we call this a "3-limit" scale. Similarly, any just interval that involves no factors other than 2 and 3 would be called a "3-limit" interval.

The Pythagorean major third 81/64 = 408c is a nice bright interval that was not considered consonant by Pythagoras; for him the only consonances were octaves, fifths and fourths. Much later in musical history, major and minor thirds and sixths became accepted as consonances, and it was found that there were smoother versions than the Pythagorean ones. Harmonic #5 is a major third above harmonic #4, so 5/4 is a (harmonic) major third, now widely regarded as the most consonant major third. Then the standard just major scale can be understood as taking the Pythagorean root, fifth and fourth, and constructing a harmonic major triad on each; this gives the scale

1/1 9/8 5/4 4/3 3/2 5/3 15/8

Here, all intervals can be seen as combinations of octaves (2/1), fifths (3/2), and/or major thirds (5/4), so only the prime numbers 2, 3, and 5 are involved. For example, the just major seventh 15/8 = 3*5/23 = (3*5)/(2*2*2), which can be thought of as a major third above the perfect fifth (5/4)*(3/2) or as a perfect fifth above the major third (3/2)*(5/4). Here the highest prime in the scale is 5, so this is a "5-limit" scale.

Similarly, any just interval that involves no factors other than 2, 3 and 5 would be called a "5-limit" interval. For example, the interval between harmonic #25 and harmonic #16 is 25/16 = 52/24 = (5/4)*(5/4), a major third above a major third, giving a type of flat minor sixth, a 5-limit interval. Similarly, between #27 and #16 is 27/16 = 33/24 = (9/8)*(3/2), a perfect second above a perfect fifth (or vice versa), giving a Pythagorean major sixth, a 3-limit interval.

Later, historically, some theorists began to consider sevenths as consonant, specifically the harmonic seventh 7/4. So, for example, the dominant seventh type chord

1/1 5/4 3/2 7/4

would be called a 7-limit chord, since the highest prime that appears is 7. Similarly, [a] . . pentatonic harp scale

1/1 8/7 21/16 3/2 7/4

would be called a 7-limit scale. For example, the (-29c) fourth 21/16 = (7/4)*(3/2)/2 is the seventh above the perfect fifth, but brought down by an octave. (By the way, since this latter scale only involves, besides octaves, perfect fifths (3/2) and sevenths (7/4), some of us would say this scale has two harmonic dimensions, the 3 dimension and the 7 dimension; the dominant seventh chord above it has 3 dimensions. This relates to the harmonic lattice idea.)

This whole limit idea assumes that some complicated just intervals can be understood as a combination of simpler just intervals; the simplest intervals that can't be further broken down into other simpler just intervals are the prime intervals (assuming octave equivalence, i.e., ignoring factors of 2): 3/2, 5/4, 7/4, 11/8, 13/8, 17/16, 19/16, etc... Whether this is meaningful to the ear is open to debate. When you hear a just major seventh 15/8 interval, do you hear it as its own special sound, or as a combination of a fifth and a major third?

Limit, n-limit. A pitch system in just intonation whose ratios contain the prime number n and no higher primes is said to be an n-limit system. By usage, certain odd non-primes such as 9, 15, and 21 may also be said to define n-limit systems. [John H. Chalmers, Jr., Divisions of the Tetrachord, Frog Peak Music, Hanover, NH 1993, p209]

The prime limit of a just scale or system is the highest prime factor that appears in all the prime factorizations of the numerators and denominators of the frequency ratios of the system.

(A prime number is one that has no factors other than itself and 1; the first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, ...)

Example: a 12-tone scale

1/1 28/27 9/8 7/6 5/4 4/3 45/32 3/2 14/9 5/3 7/4 15/8

looking at the prime factorizations of all the numerators and denominators:

1 = 1
2 = (2)
3 = (3)
4 = (2)2
5 = (5)
6 = (2)*(3)
7 = (7)
8 = (2)3
9 = (3)2
14 = (2)*(7)
15 = (3)*(5)
27 = (3)3
28 = (2)2*(7)
32 = (2)5
45 = (3)2*(5)

the highest prime that appears is 7, so this is a 7-limit scale.

David Canright
December 11, 2002

Let me add that Partch's idea of "odd-number limit" only makes sense in the context of his "tonality diamond" of criss-crossing overtone & undertone scales, and is not always the same as the "prime limit" in current usage. For example, Partch's "7-limit Diamond" does have the prime limit 7, but his "9-limit Diamond" also has the prime limit 7, since 9 is not a prime. IMHO, it does not make sense in any other context to refer to a "9-limit" scale, for example.

. . . Margo [Schulter] proposes a 17-tone scale (or more), in essence to have a choice of bright Pythagorean thirds & sixths or smoother (approximating 5-limit just) thirds & sixths, while maintaining pure perfect fifths & fourths. This is based on the fortunate coincidence that the Pythagorean and Ptolemaic commas are almost identical: when you have a chain of 12 perfect fifths, octave reduced, you do not end up exactly at your starting point, you overshoot by the Pythagorean comma = difference between 12 fifths and 7 octaves = 531441/524288 = 312/219 = 23.46001038c; the difference between a Pythagorean third 81/64 = 34/26 and a just one 5/4 is the Ptolemaic comma = (81/64)/(5/4) = 81/80 = 21.50628960c; their difference is called the skhisma = 32805/32768 = 38*5/215 = 1.953720787c ~ 2c, close to imperceptible. Another way to understand this is, if you follow a chain of 8 perfect fourths, you get the interval 8192/6561 = 213/38 = 384.3599930c, which is another Pythagorean third which is only 2c flat of the just third 5/4 = 386.3137139c. So in practice, most folks ignore the skhisma, and say that this other Pythagorean third is essentially just...

. . . [The] 72-tone approach approximates this, using tempered fifths (where the 23.5c Pythagorean comma is split equally among the 12 fifths, so each is 2c flat), and a "comma" of 16.667c; you can choose between bright standard tempered thirds (400c) or smoother, nearly just thirds (383.333c ~ 386.3137139c = 5/4). The Pythagorean approach, even tempered, embodies a 3-limit aesthetic; the latter approach embodies a 5-limit aesthetic. (Here I'm stretching the prime-limit idea to cover the harmonic "intent" of a tempered system.) But the 72-tone system ALSO gives good approximations to harmonic sevenths (966.667c ~ 968.8259065c = 7/4) and harmonic elevenths (550.000c ~ 551.3179423c = 11/8). So in harmonic intent, you can think 11-limit, which opens up a whole lotta harmonic ground (four harmonic dimensions: 3, 5, 7, 11)!

[Note: my cents values have so many digits because I'm using a computer, cut & paste; anything beyond the first decimal is only of theoretical interest, not perceptual...]

David Canright

December 11, 2002 21:07