Regular temperament is any tempered system of musical tuning 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, 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 is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths. (This is strictly true in the standard 12-tone equal temperament system — using a different system requires one interval of ...

. When only two generators are needed, with one of them the octave, this is called "linear temperament". The best-known example of a linear temperaments is

meantone temperament Meantone temperament is a musical temperament, that is a tuning system, obtained by narrowing the fifths so that their ratio is slightly less than 3:2 (making them ''narrower'' than a perfect fifth), in order to push the thirds closer to pure. M ...

, 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. Construc ...

Hermann von Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Associatio ...

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 of the

prime number 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 up to a given prime ''p'', we have what is called ''p''-

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 ...

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 ...

. Sometimes some

irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...

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 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 ...

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 (81:80 ...

where 3 is tempered to 2 to favor 2 and 5. In mathematical terminology, the products of these generators define a free abelian group. The number of independent generators is the rank of an abelian group. The rank-one tuning systems are

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 best known for her electronic music and film scores. Born and raised in Rhode Island, Carlos studied physics and music at Brown University before moving ...

, such as the

Alpha scale The α (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 r ...

, 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. In studying regular temperaments, it can be useful to regard the temperament as having a

map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...

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 is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...

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 Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...

and

multilinear algebra Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of ''p' ...

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