
LIMIT 

"LIMIT" in Definition of Tuning Terms by Joseph L. Monzo 1. prime limit: A pitch system in Just Intonation where all ratios are of integers containing no prime factors higher than n is said to be an "nlimit" system. [from Paul Erlich, private communication] 2. odd limit: the nlimit is the set all ratios of all odd numbers no greater than n, i.e., those ratios with odd factors no larger than n.
[from Paul Erlich, adapted from Harry Partch, "Genesis of a Music"] . . . . . . . . . . . . . . . . . . . . . . . . . Limit, nlimit. A pitch system in just intonation whose ratios contain the prime number n and no higher primes is said to be an nlimit system. By usage, certain odd nonprimes such as 9, 15, and 21 may also be said to define nlimit systems. [John H. Chalmers, Jr., Divisions of the Tetrachord, Frog Peak Music, Hanover, NH 1993, p209] 

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


Let me add that Partch's idea of "oddnumber limit" only makes sense in the context of his "tonality diamond" of crisscrossing overtone & undertone scales, and is not always the same as the "prime limit" in current usage. For example, Partch's "7limit Diamond" does have the prime limit 7, but his "9limit 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 "9limit" scale, for example. . . . Margo [Schulter] proposes a 17tone scale (or more), in essence to have a choice of bright Pythagorean thirds & sixths or smoother (approximating 5limit 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] 72tone 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 3limit aesthetic; the latter approach embodies a 5limit aesthetic. (Here I'm stretching the primelimit idea to cover the harmonic "intent" of a tempered system.) But the 72tone 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 11limit, 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...]
December 11, 2002 21:07 

December
4, 2003
