A regular temperament is any
tempered system of
musical tuning
In music, there are two common meanings for tuning:
* #Tuning practice, Tuning practice, the act of tuning an instrument or voice.
* #Tuning systems, Tuning systems, the various systems of Pitch (music), pitches used to tune an instrument, and ...
such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. For instance, in
12-TET
12 equal temperament (12-ET) is the musical system that divides the octave into 12 parts, all of which are equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the 12th root of 2 (\sqrt 2/math> ≈ 1.05946). That resul ...
, the system of music most commonly used in the Western world, the generator is a tempered fifth (700 cents), which is the basis behind the
circle of fifths
In music theory, the circle of fifths (sometimes also cycle of fifths) is a way of organizing pitches as a sequence of perfect fifths. Starting on a C, and using the standard system of tuning for Western music (12-tone equal temperament), the se ...
.
When only two generators are needed, with one of them the octave, this is sometimes called a "linear temperament". The best-known example of linear temperaments are
meantone temperaments, where the generating intervals are usually given in terms of a slightly flattened fifth and the octave. Other linear temperaments include the
schismatic temperament
A schismatic temperament is a musical tuning system that results from tempering the schisma of 32805:32768 (1.9537 cents) to a unison. It is also called the schismic temperament, Helmholtz temperament, or quasi-Pythagorean temperament.
Construct ...
of
Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz (; ; 31 August 1821 – 8 September 1894; "von" since 1883) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The ...
and
miracle temperament
George Secor (November 8, 1943 – March 2, 2020) was an American musician, composer and music-theorist from Chicago. He was the discoverer of miracle temperament and eponym of the secor.
As an inventor, Secor and Hermann Pedtke's ''Motorola Scal ...
.
Mathematical description
If the generators are all the
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...
s up to a given prime ''p'', we have what is called ''p''-
limit just intonation
In music, just intonation or pure intonation is a musical tuning, tuning system in which the space between notes' frequency, frequencies (called interval (music), intervals) is a natural number, whole number ratio, ratio. Intervals spaced in thi ...
. Sometimes some
irrational number
In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...
close to one of these primes is substituted (an example of
tempering) to favour other primes, as in twelve tone
equal temperament
An equal temperament is a musical temperament or Musical tuning#Tuning systems, tuning system that approximates Just intonation, just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequency, frequencie ...
where 3 is tempered to 2
to favour 2, or in
quarter-comma meantone
Quarter-comma meantone, or -comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma with ...
where 3 is tempered to 2 to favor 2 and 5.
In mathematical terminology, the products of these generators define a
free abelian group
In mathematics, a free abelian group is an abelian group with a Free module, basis. Being an abelian group means that it is a Set (mathematics), set with an addition operation (mathematics), operation that is associative, commutative, and inverti ...
. The number of independent generators is the
rank of an abelian group
In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group ''A'' is the cardinality of a maximal linearly independent subset. The rank of ''A'' determines the size of the largest free abelian group contained in ''A''. If '' ...
. The rank-one tuning systems are
equal temperament
An equal temperament is a musical temperament or Musical tuning#Tuning systems, tuning system that approximates Just intonation, just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequency, frequencie ...
s, all of which can be spanned with only a single generator, though they don't have to be integer-based equal temperaments. The non-octave scales of
Wendy Carlos
Wendy Carlos (born Walter Carlos; November 14, 1939) is an American musician and composer known for electronic music and film scores.
Born and raised in Rhode Island, Carlos studied physics and music at Brown University before moving to New Y ...
, such as the
Alpha scale
The (alpha) scale is a non-octave-repeating musical scale invented by Wendy Carlos and first used on her album ''Beauty in the Beast'' (1986). It is derived from approximating just intervals using multiples of a single interval, but without re ...
, use one generator that does not stack up to the octave. A rank-two temperament has two generators; hence, meantone is a rank-2 temperament. For the case of
quarter-comma meantone
Quarter-comma meantone, or -comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma with ...
, these may be chosen as
and
.
In studying regular temperaments, it can be useful to regard the temperament as having a
map
A map is a symbolic depiction of interrelationships, commonly spatial, between things within a space. A map may be annotated with text and graphics. Like any graphic, a map may be fixed to paper or other durable media, or may be displayed on ...
from ''p''-limit just intonation (for some prime ''p'') to the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of tempered intervals. To properly classify a temperament's dimensionality one must determine how many of the given generators are independent, because its description may contain redundancies. Another way of considering this problem is that the rank of a temperament should be the rank of its
image
An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
under this map.
For instance, a harpsichord tuner it might think of quarter-comma meantone tuning as having three generators—the octave, the just major third (5:4) and the quarter-comma tempered fifth—but because four consecutive tempered fifths produces a just major third, the major third is redundant, reducing it to a rank-two temperament.
Other methods of
linear
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a '' function'' (or '' mapping'');
* linearity of a '' polynomial''.
An example of a linear function is the function defined by f(x) ...
and
multilinear algebra
Multilinear algebra is the study of Function (mathematics), functions with multiple vector space, vector-valued Argument of a function, arguments, with the functions being Linear map, linear maps with respect to each argument. It involves concept ...
can be applied to the map. For instance, a
map's kernel (otherwise known as "nullspace") consists of ''p''-limit intervals called
commas, which are a property useful in describing temperaments.
External links
*
* A. Milne, W. A. Sethares, and J. Plamondon,
''Isomorphic Controllers and Dynamic Tuning— Invariant Fingering Over a Tuning Continuum'' Computer Music Journal, Winter 2007
*
ttp://lumma.org/tuning/gws/regular.html Smith, Gene Ward, ''Regular Temperaments''* Barbieri, Patrizio
Enharmonic instruments and music, 1470-1900 (2008) Latina, Il Levante Libreria Editrice
{{Musical tuning
Musical temperaments