Regular number

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A Hasse diagram of divisibility relationships among the regular numbers up to 400. The vertical scale is logarithmic. Inspired by similar diagrams by Kurenniemi [1].

In the study of Babylonian mathematics, a regular number or regular sexagesimal number is a whole number which divides a power of 60. In number theory, these numbers are called 5-smooth, because they can be characterized as having only 2, 3, or 5 as prime factors. They are a specific case of what are called k-smooth numbers, those sets of numbers that have no prime factors greater than k. In music theory, regular numbers occur in the ratios of tones in just intonation, also called 5-limit tuning for this reason. In computer science, regular numbers are often called Hamming numbers, after Richard Hamming, who first considered computer algorithms for generating these numbers in order.^[1]

The first few regular numbers are

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, ... (sequence A051037 in OEIS).

Several other sequences at OEIS have definitions involving 5-smooth numbers [2]. Although the regular numbers appear dense within the range from 1 to 60, they are quite sparse among the larger integers. A regular number n = 2ⁱ3^j5^k ≤ N if and only if the point (i,j,k) belongs to the tetrahedron bounded by the coordinate planes and the plane , so the number of regular numbers that are at most N can be estimated as the volume of this tetrahedron,

[edit] Babylonian mathematics

In the Babylonian sexagesimal notation, the reciprocal of a regular number has a finite representation, thus being easy to divide by. Specifically, if n divides 60^k, then the sexagesimal representation of 1/n is just that for 60^k/n, shifted by some number of places.

For instance, suppose we wish to divide by the regular number 54 = 2¹3³. 54 is a divisor of 60³, and 60³/54 = 4000, so dividing by 54 in sexagesimal can be accomplished by multiplying by 4000 and shifting three places. In sexagesimal 4000 = 1×3600 + 6×60 + 40×1, or (as listed by Joyce) 1:6:40. Thus, 1/54, in sexagesimal, is 1/60 + 6/60² + 40/60³, also denoted 1:6:40 as Babylonian notational conventions did not specify the power of the starting digit. Conversely 1/4000 = 54/60³, so division by 1:6:40 = 4000 can be accomplished by instead multiplying by 54 and shifting three sexagesimal places.

The Babylonians used tables of reciprocals of regular numbers, some of which still survive (Sachs, 1947). These tables existed relatively unchanged throughout Babylonian times (Aaboe, 1965).

Although the primary reason for preferring regular numbers to other numbers involves the finiteness of their reciprocals, some Babylonian calculations other than reciprocals also involved regular numbers. For instance, tables of regular squares have been found (Aaboe, 1965) and the broken cuneiform tablet Plimpton 322 has been interpreted by Neugebauer as listing Pythagorean triples generated by p, q both regular and less than 60.^[2]

[edit] Music theory

In music theory, the just intonation of the diatonic scale involves regular numbers: the ratios of the tones in a single octave of this scale form the sequence 24, 27, 30, 32, 36, 40, 45, 48 of nearly-consecutive regular numbers. Thus, for an instrument with this tuning, all tones are regular-number harmonics of a single fundamental frequency. This scale is called a 5-limit tuning, meaning that all intervals can be described as powers of the prime numbers up to 5, that is, 2, 3, and 5.

Some music theorists have stated more generally that regular numbers are fundamental to tonal music itself, and that tone ratios based on primes larger than 5 cannot be harmonic.^[3] However the well temperament of modern pianos is not a 5-limit tuning, and some modern composers have experimented with tunings based on primes larger than five. 5-limit musical scales other than the familiar diatonic scale of Western music have been used both in traditional musics of other cultures and in modern experimental music: Honing and Bod (2005) list 31 different 5-limit scales, drawn from a larger database of musical scales. Each of these 31 scales shares with diatonic just intonation the property that all intervals are ratios of regular numbers. Euler's tonnetz provides a convenient graphical representation of the tones in any 5-limit tuning, by factoring out the octave relationships (powers of two) so that the remaining values form a planar grid.

In connection with the application of regular numbers to music theory, it is of interest to find pairs of regular numbers that differ by one. There are exactly ten such pairs (x, x+1)^[4] and each such pair defines a superparticular ratio (x + 1)/x that is meaningful as a musical interval. These intervals are 2/1 (the octave), 3/2 (the perfect fifth), 4/3 (the perfect fourth), 5/4 (the just major third), 6/5 (the just minor third), 9/8 (the just major tone), 10/9 (the just minor tone), 16/15 (the just diatonic semitone), 25/24 (the just chromatic semitone), and 81/80 (the syntonic comma).

[edit] Computation of regular numbers

Algorithms for calculating the regular numbers in ascending order were popularized by Edsger Dijkstra. Dijkstra (1981) attributes to Hamming the problem of building the infinite ascending sequence of all 5-smooth numbers; this problem is now known as Hamming's problem, and the numbers so generated are also called the Hamming numbers. Dijkstra's ideas to compute these numbers are the following:

• The sequence of Hamming numbers begins with the number 1.

• The remaining values in the sequence are of the form 2h, 3h, and 5h, where h is any Hamming number.

• Therefore, the sequence H may be generated by outputting the value 1, and then merging the sequences 2H, 3H, and 5H.

This algorithm is often used to demonstrate the power of a lazy functional programming language, because this algorithm can be implemented as described above, using a constant number of arithmetic operations per generated value, while a similarly efficient implementation for a strict functional or imperative programming language is non-trivial.^[5]

In the Python programming language, lazy functional code for generating regular numbers is used as one of the built-in tests for correctness of the implementation. [3]

[edit] Other applications

Heninger, Rains, and Sloane (2005) show that, when n is a regular number and is divisible by 8, the generating function of an n-dimensional extremal even unimodular lattice is an nth power of a polynomial.

As with other classes of smooth numbers, regular numbers are important as problem sizes in fast Fourier transform codes. For instance, the method of Temperton (1992) requires that the transform length be a regular number.

Book VIII of Plato's Republic involves an allegory of marriage centered around the highly regular number 60⁴ = 12,960,000 and its divisors. Later scholars have invoked both Babylonian mathematics and music theory in an attempt to explain this passage.^[6]

[edit] Notes

1. ^ However, see Gingerich (1965) for an early description of computer code that generates these numbers out of order and then sorts them.
2. ^ See Conway and Guy (1996) for a popular treatment of this interpretation. Plimpton 322 has other interpretations, for which see its article, but all involve regular numbers.
3. ^ Asmussen (2001), for instance, states that "within any piece of tonal music" all intervals must be ratios of regular numbers, echoing similar statements by much earlier writers such as Habens (1889). In the modern music theory literature this assertion is often attributed to Longuet-Higgins (1962), who used a graphical arrangement closely related to the tonnetz to organize 5-limit tones.
4. ^ Halsey and Hewitt (1972) note that this follows from more general results of Størmer (1897), and provide a proof for this case; see also Silver (1971).
5. ^ See, e.g., Hemmendinger (1988) or Yuen (1992).
6. ^ Barton (1908); McClain (1974).

[edit] References

[edit] External links

• Table of reciprocals of regular numbers up to 3600 from the web site of Professor David E. Joyce, Clark University