![Regular divisibility lattice](https://upload.wikimedia.org/wikipedia/commons/b/b6/Regular_divisibility_lattice.svg)
Regular numbers are numbers that evenly divide powers of
60 (or, equivalently, powers of
30). Equivalently, they are the numbers whose only prime divisors are
2,
3, and
5. As an example, 60
2 = 3600 = 48 × 75, so as divisors of a power of 60 both 48 and 75 are regular.
These numbers arise in several areas of mathematics and its applications, and have different names coming from their different areas of study.
* In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, these numbers are called 5-smooth, because they can be characterized as having only 2, 3, or 5 as their
prime factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s. This is a specific case of the more general -
smooth number
In number theory, an ''n''-smooth (or ''n''-friable) number is an integer whose prime factors are all less than or equal to ''n''. For example, a 7-smooth number is a number whose every prime factor is at most 7, so 49 = 72 and 15750 = 2 × 32 × ...
s, the numbers that have no prime factor greater
* In the study of
Babylonian mathematics
Babylonian mathematics (also known as ''Assyro-Babylonian mathematics'') are the mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the centuries following the fall of Babylon in 539 BC. Babyl ...
, the divisors of powers of 60 are called regular numbers or regular sexagesimal numbers, and are of great importance in this area because of the
sexagesimal
Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
(base 60) number system that the Babylonians used for writing their numbers, and that was central to Babylonian mathematics.
* In
music theory
Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the "rudiments", that are needed to understand music notation (ke ...
, regular numbers occur in the ratios of tones in
five-limit just intonation
In music, just intonation or pure intonation is the tuning of musical intervals
Interval may refer to:
Mathematics and physics
* Interval (mathematics), a range of numbers
** Partially ordered set#Intervals, its generalization from numbers to ...
. In connection with music theory and related theories of
architecture
Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and constructing building ...
, these numbers have been called the harmonic whole numbers.
* In
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, regular numbers are often called Hamming numbers, after
Richard Hamming
Richard Wesley Hamming (February 11, 1915 – January 7, 1998) was an American mathematician whose work had many implications for computer engineering and telecommunications. His contributions include the Hamming code (which makes use of a Ha ...
, who proposed the problem of finding computer
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
s for generating these numbers in ascending order. This problem has been used as a test case for
functional programming
In computer science, functional programming is a programming paradigm where programs are constructed by Function application, applying and Function composition (computer science), composing Function (computer science), functions. It is a declar ...
.
Number theory
Formally, a regular number is an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
of the form
, for nonnegative integers
,
, and
. Such a number is a divisor of
. The regular numbers are also called 5-
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebraic ...
, indicating that their greatest
prime factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
is at most 5. More generally, a -smooth number is a number whose greatest prime factor is at
The first few regular numbers are
Several other sequences at the
On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...
have definitions involving 5-smooth numbers.
Although the regular numbers appear dense within the range from 1 to 60, they are quite sparse among the larger integers. A regular number
is less than or equal to some threshold
if and only if the point
belongs to the
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
bounded by the coordinate planes and the plane
as can be seen by taking logarithms of both sides of the inequality
.
Therefore, the number of regular numbers that are at most
can be estimated as the
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
of this tetrahedron, which is
Even more precisely, using
big O notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
, the number of regular numbers up to
is
and it has been conjectured that the error term of this approximation is actually
.
A similar formula for the number of 3-smooth numbers up to
is given by
Srinivasa Ramanujan
Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis ...
in his first letter to
G. H. Hardy
Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
.
Babylonian mathematics
![Table de division et de conversion des fractions - Louvre - AO 6456](https://upload.wikimedia.org/wikipedia/commons/a/ab/Table_de_division_et_de_conversion_des_fractions_-_Louvre_-_AO_6456.JPG)
In the Babylonian
sexagesimal
Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
notation, the
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
of a regular number has a finite representation. If
divides
, then the sexagesimal representation of
is just that for
, shifted by some number of places. This allows for easy division by these numbers: to divide by
, multiply by
, then shift.
For instance, consider division by the regular number 54 = 2
13
3. 54 is a divisor of 60
3, and 60
3/54 = 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 + 6/60
2 + 40/60
3, also denoted 1:6:40 as Babylonian notational conventions did not specify the power of the starting digit. Conversely 1/4000 = 54/60
3, 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. These tables existed relatively unchanged throughout Babylonian times.
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 and the broken tablet
Plimpton 322
Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. It has number 322 in the G.A. Plimpton Collection at Columbia University. This tablet, believed to have been written about 1800 BC, has a table ...
has been interpreted by
Neugebauer Neugebauer is a German surname. Notable people with the surname include:
*Alfred Neugebauer, Austrian film actor
* Fritz Neugebauer, Second President of the National Council of Austria
*Gerry Neugebauer, American astronomer
* Hans E J Neugebauer, G ...
as listing
Pythagorean triple
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
s
generated by
and
both regular and less than 60.
Music theory
In
music theory
Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the "rudiments", that are needed to understand music notation (ke ...
, the
just intonation
In music, just intonation or pure intonation is the tuning of musical intervals
Interval may refer to:
Mathematics and physics
* Interval (mathematics), a range of numbers
** Partially ordered set#Intervals, its generalization from numbers to ...
of the
diatonic scale
In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, ...
involves regular numbers: the
pitches in a single
octave
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
of this scale have frequencies proportional to the numbers in the sequence 24, 27, 30, 32, 36, 40, 45, 48 of nearly consecutive regular numbers. Thus, for an instrument with this tuning, all pitches are regular-number
harmonic
A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
s of a single
fundamental frequency
The fundamental frequency, often referred to simply as the ''fundamental'', is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. In ...
. This scale is called a 5-
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
tuning, meaning that the
interval between any two pitches can be described as a product 2
i3
j5
k of powers of the prime numbers up to 5, or equivalently as a ratio of regular numbers.
5-limit musical scales other than the familiar diatonic scale of Western music have also been used, both in traditional musics of other cultures and in modern experimental music: 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
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
's
tonnetz
In musical tuning and harmony, the (German for 'tone network') is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739. Various visual representations of the ''Tonnetz'' can be used to show traditiona ...
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
Grid, The Grid, or GRID may refer to:
Common usage
* Cattle grid or stock grid, a type of obstacle is used to prevent livestock from crossing the road
* Grid reference, used to define a location on a map
Arts, entertainment, and media
* News g ...
. 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
In articulatory phonetics, a consonant is a speech sound that is articulated with complete or partial closure of the vocal tract. Examples are and pronounced with the lips; and pronounced with the front of the tongue; and pronounced wit ...
. However the
equal temperament
An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...
of modern pianos is not a 5-limit tuning, and some modern composers have experimented with tunings based on primes larger than five.
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
and each such pair defines a
superparticular ratio
In mathematics, a superparticular ratio, also called a superparticular number or epimoric ratio, is the ratio of two consecutive integer numbers.
More particularly, the ratio takes the form:
:\frac = 1 + \frac where is a positive integer.
Thu ...
that is meaningful as a musical interval. These intervals are 2/1 (the
octave
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
), 3/2 (the
perfect fifth
In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so.
In classical music from Western culture, a fifth is the interval fro ...
), 4/3 (the
perfect fourth
A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth () is the fourth spanning five semitones (half steps, or half tones). For example, the ascending interval from C to ...
), 5/4 (the
just major third
Just or JUST may refer to:
__NOTOC__ People
* Just (surname)
* Just (given name)
Arts and entertainment
* ''Just'', a 1998 album by Dave Lindholm
* "Just" (song), a song by Radiohead
* "Just", a song from the album ''Lost and Found'' by Mudvayne ...
), 6/5 (the
just minor third
Just or JUST may refer to:
__NOTOC__ People
* Just (surname)
* Just (given name)
Arts and entertainment
* ''Just'', a 1998 album by Dave Lindholm
* "Just" (song), a song by Radiohead
* "Just", a song from the album ''Lost and Found'' by Mudvayne ...
), 9/8 (the
just major tone
Just or JUST may refer to:
__NOTOC__ People
* Just (surname)
* Just (given name)
Arts and entertainment
* ''Just'', a 1998 album by Dave Lindholm
* "Just" (song), a song by Radiohead
* "Just", a song from the album ''Lost and Found'' by Mudvayne ...
), 10/9 (the
just minor tone
Just or JUST may refer to:
__NOTOC__ People
* Just (surname)
* Just (given name)
Arts and entertainment
* ''Just'', a 1998 album by Dave Lindholm
* "Just" (song), a song by Radiohead
* "Just", a song from the album ''Lost and Found'' by Mudvayne ...
), 16/15 (the
just diatonic semitone
A semitone, also called a half step or a half tone, is the smallest interval (music), musical interval commonly used in Western tonal music, and it is considered the most Consonance and dissonance#Dissonance, dissonant when sounded harmonically ...
), 25/24 (the
just chromatic semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.
It is defined as the interval between two adjacent no ...
), and 81/80 (the
syntonic comma
In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) ( ...
).
In the Renaissance theory of
universal harmony, musical ratios were used in other applications, including the
architecture
Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and constructing building ...
of buildings. In connection with the analysis of these shared musical and architectural ratios, for instance in the architecture of
Palladio
Andrea Palladio ( ; ; 30 November 1508 – 19 August 1580) was an Italian Renaissance architect active in the Venetian Republic. Palladio, influenced by Roman and Greek architecture, primarily Vitruvius, is widely considered to be one of th ...
, the regular numbers have also been called the harmonic whole numbers.
Algorithms
Algorithms for calculating the regular numbers in ascending order were popularized by
Edsger Dijkstra
Edsger Wybe Dijkstra ( ; ; 11 May 1930 – 6 August 2002) was a Dutch computer scientist, programmer, software engineer, systems scientist, and science essayist. He received the 1972 Turing Award for fundamental contributions to developing progra ...
. 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
,
, and
, where
is any Hamming number.
* Therefore, the sequence
may be generated by outputting the value 1, and then
merging
Merge, merging, or merger may refer to:
Concepts
* Merge (traffic), the reduction of the number of lanes on a road
* Merge (linguistics), a basic syntactic operation in generative syntax in the Minimalist Program
* Merger (politics), the comb ...
the sequences
,
, and
.
This algorithm is often used to demonstrate the power of a
lazy functional programming language
In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions that m ...
, 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
Generative may refer to:
* Generative actor, a person who instigates social change
* Generative art, art that has been created using an autonomous system that is frequently, but not necessarily, implemented using a computer
* Generative music, mus ...
solutions might be non-trivial.
In the
Python programming language
Python is a high-level, general-purpose programming language. Its design philosophy emphasizes code readability with the use of significant indentation.
Python is dynamically-typed and garbage-collected. It supports multiple programming p ...
, lazy functional code for generating regular numbers is used as one of the built-in tests for correctness of the language's implementation.
A related problem, discussed by , is to list all
-digit sexagesimal numbers in ascending order, as was done for
by Inakibit-Anu, the
Seleucid
The Seleucid Empire (; grc, Βασιλεία τῶν Σελευκιδῶν, ''Basileía tōn Seleukidōn'') was a Greek state in West Asia that existed during the Hellenistic period from 312 BC to 63 BC. The Seleucid Empire was founded by the ...
-era scribe of tablet AO6456. In algorithmic terms, this is equivalent to generating (in order) the subsequence of the infinite sequence of regular numbers, ranging from
to
.
See for an early description of computer code that generates these numbers out of order and then sorts them; Knuth describes an ad hoc algorithm, which he attributes to , for generating the six-digit numbers more quickly but that does not generalize in a straightforward way to larger values of
. describes an algorithm for computing tables of this type in linear time for arbitrary values of
.
Other applications
show that, when
is a regular number and is divisible by 8, the generating function of an
-dimensional extremal even
unimodular lattice
In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in ''n''-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamen ...
is an
th power of a polynomial.
As with other classes of
smooth number
In number theory, an ''n''-smooth (or ''n''-friable) number is an integer whose prime factors are all less than or equal to ''n''. For example, a 7-smooth number is a number whose every prime factor is at most 7, so 49 = 72 and 15750 = 2 × 32 × ...
s, regular numbers are important as problem sizes in computer programs for performing the
fast Fourier transform
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in th ...
, a technique for analyzing the dominant frequencies of signals in
time-varying data. For instance, the method of requires that the transform length be a regular number.
Book VIII of
Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
's ''
Republic
A republic () is a "state in which power rests with the people or their representatives; specifically a state without a monarchy" and also a "government, or system of government, of such a state." Previously, especially in the 17th and 18th c ...
'' involves an allegory of marriage centered on the highly regular number 60
4 = 12,960,000 and its divisors (see
Plato's number
Plato's number is a number enigmatically referred to by Plato in his dialogue the ''Republic'' (8.546b). The text is notoriously difficult to understand and its corresponding translations do not allow an unambiguous interpretation. There is no rea ...
). Later scholars have invoked both Babylonian mathematics and music theory in an attempt to explain this passage.
[; .]
Certain species of
bamboo
Bamboos are a diverse group of evergreen perennial flowering plants making up the subfamily Bambusoideae of the grass family Poaceae. Giant bamboos are the largest members of the grass family. The origin of the word "bamboo" is uncertain, bu ...
release large numbers of seeds in synchrony (a process called
masting
Mast is the fruit of forest trees and shrubs, such as acorns and other nuts. The term derives from the Old English ''mæst'', meaning the nuts of forest trees that have accumulated on the ground, especially those used historically for fattening do ...
) 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. It has been hypothesized that the biological mechanism for timing and synchronizing this process lends itself to smooth numbers, and in particular in this case to 5-smooth numbers. Although the estimated masting intervals for some other species of bamboo are not regular numbers of years, this may be explainable as measurement error.
Notes
References
*.
*.
*.
*.
*.
*.
*.
*
*.
*.
*.
*.
*.
*.
*}.
*.
*
*. Errata in ''CACM'' 19(2), 1976. Reprinted with a brief addendum in ''Selected Papers on Computer Science'', CSLI Lecture Notes 59, Cambridge Univ. Press, 1996, pp. 185–203.
*
*.
*.
*.
*.
*.
*
*.
*.
*
*
*.
External links
Table of reciprocals of regular numbers up to 3600from the web site of Professor David E. Joyce, Clark University.
RosettaCodeGeneration of Hamming_numbers in ~ 50 programming languages
{{Classes of natural numbers
Babylonian mathematics
Integer sequences
Functional programming
Mathematics of music