Regular temperament is any tempered

Musical temperament

In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament system...

 system of musical tuning

Musical tuning

In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...

 such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. The classic example of a regular temperament is meantone temperament

Meantone temperament

Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...

, where the generating intervals are usually given in terms of a slightly flattened fifth and the octave.

The best-known example of a linear temperaments is meantone, but others include the schismatic temperament

Schismatic temperament

The schismatic temperament is a musical tuning system that results from tempering the schisma of 32805:32768 to a unison. It is also called the schismic temperament or Helmholtz temperament.-Comparison with other tunings:...

 of Hermann von Helmholtz

Hermann von Helmholtz

Hermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions to several widely varied areas of modern science...

 and miracle temperament

Miracle temperament

In music, miracle temperament is a regular temperament discovered by George Secor which has as a generator an interval, called the secor, that serves as both the 15:14 and 16:15 semitones. Because 15:14 and 16:15 are equated, their ratio 225:224 \left is tempered out, and two secors give an 8:7...

.

Mathematical description

If the generators are all of the prime numbers up to a given prime p, we have what is called p-limit

Limit (music)

In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...

 just intonation

Just intonation

In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

. Sometimes some irrational number close to one of these primes is substituted (an example of tempering

Temperament

In psychology, temperament refers to those aspects of an individual's personality, such as introversion or extroversion, that are often regarded as innate rather than learned...

) to favour other primes, as in twelve tone equal temperament

Equal temperament

An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...

 where 3 is tempered to 2^19/12 to favour 2, or in quarter-comma meantone

Quarter-comma meantone

Quarter-comma meantone, or 1/4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. This method is a variant of Pythagorean tuning...

 where 3 is tempered to 2·5^1/4 to favor 2 and 5.

In mathematical terminology, the products of these generators defines a free abelian group

Free abelian group

In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...

. The number of independent generators is the rank of an abelian group

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 A is torsion-free then it embeds into a vector space over the...

. The rank one tuning systems are equal temperament

Equal temperament

An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...

s, all of which can be spanned with only a single generator. A rank two temperament has two generators.

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

Map (mathematics)

In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...

 from p-limit just intonation (for some prime p) to the set of tempered intervals. To properly classify a temperament's dimensionality it must be determined 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

Image (mathematics)

In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...

 under this map.

For instance, for a harpsichord tuner it might be normal to 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

Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

 and multilinear algebra

Multilinear algebra

In mathematics, multilinear algebra 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-vectors and multivectors with Grassmann algebra.-Origin:In a vector space...

 can be applied to the map. For instance, a kernel

Kernel (mathematics)

In mathematics, the word kernel has several meanings. Kernel may mean a subset associated with a mapping:* The kernel of a mapping is the set of elements that map to the zero element , as in kernel of a linear operator and kernel of a matrix...

 of it would consist of p-limit intervals called commas

Comma (music)

In music theory, a comma is a minute interval, the difference resulting from tuning one note two different ways. The word "comma" used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning...

, 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
• Holmes, Rich, Microtonal scales: Rank-2 2-step (MOS) scales
• Smith, Gene Ward, Regular Temperaments
• Barbieri, Patrizio. Enharmonic instruments and music, 1470-1900. (2008) Latina, Il Levante Libreria Editrice