Regular number
The regular numbers are also called 5-smooth, indicating that their greatest prime factor is at most 5.
[4] Although the regular numbers appear dense within the range from 1 to 60, they are quite sparse among the larger integers.
[5] In the Babylonian sexagesimal notation, the reciprocal of a regular number has a finite representation.
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.
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.
[7] These tables existed relatively unchanged throughout Babylonian times.
[6] One tablet from Seleucid times, by someone named Inaqibıt-Anu, contains the reciprocals of 136 of the 231 six-place regular numbers whose first place is 1 or 2, listed in order.
Noting the difficulty of both calculating these numbers and sorting them, Donald Knuth in 1972 hailed Inaqibıt-Anu as "the first man in history to solve a computational problem that takes longer than one second of time on a modern electronic computer!"
For instance, tables of regular squares have been found[6] and the broken tablet Plimpton 322 has been interpreted by Neugebauer as listing Pythagorean triples
[10] Fowler and Robson discuss the calculation of square roots, such as how the Babylonians found an approximation to the square root of 2, perhaps using regular number approximations of fractions such as 17/12.
[11] Thus, for an instrument with this tuning, all pitches are regular-number harmonics of a single fundamental frequency.
This scale is called a 5-limit tuning, meaning that the interval between any two pitches can be described as a product 2i3j5k of powers of the prime numbers up to 5, or equivalently as a ratio of regular numbers.
Each of these 31 scales shares with diatonic just intonation the property that all intervals are ratios of regular numbers.
[12] Euler's tonnetz provides a convenient graphical representation of the pitches in any 5-limit tuning, by factoring out the octave relationships (powers of two) so that the remaining values form a planar grid.
[12] Some music theorists have stated more generally that regular numbers are fundamental to tonal music itself, and that pitch ratios based on primes larger than 5 cannot be consonant.
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).
[16] In the Renaissance theory of universal harmony, musical ratios were used in other applications, including the architecture of buildings.
[17] Algorithms for calculating the regular numbers in ascending order were popularized by Edsger Dijkstra.
Dijkstra's ideas to compute these numbers are the following: This algorithm is often used to demonstrate the power of a lazy functional programming language, because (implicitly) concurrent efficient implementations, using a constant number of arithmetic operations per generated value, are easily constructed as described above.
Similarly efficient strict functional or imperative sequential implementations are also possible whereas explicitly concurrent generative solutions might be non-trivial.
In algorithmic terms, this is equivalent to generating (in order) the subsequence of the infinite sequence of regular numbers, ranging from
[8] See Gingerich (1965) for an early description of computer code that generates these numbers out of order and then sorts them;[20] Knuth describes an ad hoc algorithm, which he attributes to Bruins (1970), for generating the six-digit numbers more quickly but that does not generalize in a straightforward way to larger values of
[8] Eppstein (2007) describes an algorithm for computing tables of this type in linear time for arbitrary values of
[22] As with other classes of smooth numbers, regular numbers are important as problem sizes in computer programs for performing the fast Fourier transform, a technique for analyzing the dominant frequencies of signals in time-varying data.
For instance, the method of Temperton (1992) requires that the transform length be a regular number.
Later scholars have invoked both Babylonian mathematics and music theory in an attempt to explain this passage.
[24] Certain species of bamboo release large numbers of seeds in synchrony (a process called masting) at intervals that have been estimated as regular numbers of years, with different intervals for different species, including examples with intervals of 10, 15, 16, 30, 32, 48, 60, and 120 years.
Although the estimated masting intervals for some other species of bamboo are not regular numbers of years, this may be explainable as measurement error.
