Regular tuning and temperament

This article is about regular tunings of musical scales in general. For alternative tunings of the guitar known as "regular", see Regular tuning.

In the traditional theory of music, a regular temperament is an assignment of relative pitches to the notes of the chromatic scale (i.e. a tuning of this scale), ^[1] in which every interval of a perfect fifth ^[2] is tempered ^[3] by the same amount. ^[4] A tuning in which all the perfect fifths remain untempered ^[5] is called "Pythagorean tuning". In this article, unless the context implies otherwise, the unqualified term "fifth" by itself will mean "perfect fifth".

Authorities differ as to which tunings they regard as being included amongst those they refer to as "temperaments", and therefore as to whether or not they include Pythagorean tuning amongst those they refer to as "regular temperaments". Some treat the term "temperament" as a synonym for "tuning", ^[6] others reserve the term "temperament" for tunings that are not any form of just intonation (including Pythagorean tuning), ^[7] and one eminent musicologist, James Murray Barbour, required a temperament to have at least one interval with an irrational frequency ratio. ^[8] In any case, all regular temperaments and Pythagorean tuning are collectively referred to as "regular tunings". ^[9] Tunings and temperaments which include perfect fifths of different sizes are called "irregular".

In the diatonic scales of common practice Western music, any note is either separated from another by a whole number of perfect fifths, or is in the same pitch class as such a note. ^[10] In these scales, therefore, the interval between any pair of notes can be formed either by adding a whole number of fifths to, or subtracting them from, a whole number of octaves, or by subtracting a whole number of octaves from a whole number of fifths. A regular tuning of such scales, therefore, can be constructed from a single chain of fifths, the pitch of any note being determined by that of a single reference note and the common size of all the perfect fifths. There are, however, some more extended scales for which the construction of regular tunings requires two or more chains of fifths offset from each other by intervals which cannot be formed from combinations of whole numbers of fifths and octaves.

A modified concept of regular tuning has been proposed by some modern musical theorists, who define it to be a general system of music intervals produced by combining together intervals of a finite number of fixed sizes, not necessarily including a fifth or an octave. ^[11]

Some examples of regular temperaments are meantone temperaments and equal temperaments.

Description and examples

In the chromatic scale of western music, the perfect fifths are the intervals between any two successive notes in the following list:

F^♭, C^♭, G^♭, D^♭, A^♭, E^♭, B^♭, F, C, G, D, A, E, B, F^♯, C^♯, G^♯, D^♯, A^♯, E^♯, B^♯,

or its extension to include notes with more flats or sharps. An assignment of pitches to these notes, and others in the same pitch classes, is a regular tuning if the intervals between pairs of successive notes in the list are all the same size. The tuning is Pythagorean if the intervals are all just perfect fifths, or is a regular temperament if they are tempered fifths.

Denote the size of the fifth, as a fraction of an octave, by f. ^[12] For the fifths to qualify as tempered, f must be close, but not exactly equal, to its just value of log₂(3/2). ^[13] To preserve the proper pitch ordering of the notes within each octave, ^[14]

C_n < C_n^♯, D_n^♭ < D_n < D_n^♯, E_n^♭ < E_n < F_n < F_n^♯, G_n^♭ < G_n < G_n^♯, A_n^♭ < A_n < A_n^♯, B_n^♭ < B_n < C_n+1,

it must also satisfy the inequalities, 4/7 < f < 3/5. ^[15]

In such a regular tuning, the common size of the intervals E–F and B–C, the diatonic semitone, is 3 − 5f of an octave. The common size of the intervals between the flattened version of a note and the note itself, or between the note itself and its sharpened version, the chromatic semitone, is 7f − 4 of an octave. The common size of the intervals C–D, D–E, F–G, G–A and A–B, the whole tone, is the sum of the diatonic and chromatic semitones, or 2f − 1 of an octave. The diatonic semitone is larger than, smaller than, or equal to, the chromatic semitone according as f is less than, greater than, or equal to, 7/12. In the last case, the diatonic and chromatic semitones are both 1/12 of an octave, exactly half the size of a tone, and the tuning is 12-tone equal temperament.

Unless f = 7/12, as in 12-tone equal temperament, the first five notes in the list above, G^♭, D^♭, A^♭, E^♭, B^♭, will not belong to the same pitch classes as the last five, F^♯, C^♯, G^♯, D^♯, A^♯, and therefore not be enharmonically equivalent to them. If f > 7/12, each sharp is 12f − 7 of an octave sharper than its otherwise enharmonically equivalent flat, whereas, if f < 7/12, it is 7 − 12f of an octave flatter. On a standard musical keyboard, with only the 12 notes of the chromatic scale in each octave, only one note from each of the enharmonically alternative pairs, F^♯/G^♭, C^♯/D^♭, G^♯/A^♭, D^♯/E^♭, A^♯/B^♭, can be assigned to any one of the appropriate black keys. The assignments traditionally chosen were either F^♯, C^♯, G^♯, E^♭ and B^♭, or F^♯, C^♯, A^♭, E^♭ and B^♭. ^[16] The size of the intervals G^♯–E^♭ and C^♯–A^♭ (and of the other diminished sixths, B–G^♭, F^♯–D^♭, D^♯–B^♭ and A^♯–F) is 7 − 11f of an octave. For most of the regular temperaments used in practice in the past, these intervals were too dissonant to perform acceptably as perfect fifths, and were then referred to pejoratively as "wolf intervals". ^[17]

A regular tuning is obtained by selecting the desired size of the perfect fifths ^[18] and constructing one or more chains of fifths, in which the upper end-point of each fifth is the same as the lower end-point of the next one in the same chain. The scale to be tuned comprises all the notes belonging to the same pitch classes as those at the end-points of the fifths in all the chains. If an end-point of one of the fifths in each chain is taken as a reference, the pitches of all the notes of the scale relative to those references can be determined by travelling in either direction by a whole number of fifths along the appropriate chain, starting from its reference, and then moving up or down by a whole number of octaves. When there is more than one chain, the pitches of the notes in one chain relative to those of another are completely determined by any single interval between any pair of notes, of which one belongs to the first chain and the other to the second.

Examples

If a chain of twelve fifths starts from the note A^♭, the notes at the ends of successive fifths in the chain will be:

A^♭, E^♭, B^♭, F, C, G, D, A, E, B, F^♯, C^♯, G^♯.

If the G^♯ at the end of the chain is not in the same pitch class as the A^♭ at its start, then extending the chain upwards will produce notes D^♯, A^♯, E^♯, B^♯, F , ... etc., and extending it downwards will similarly produce notes D^♭, G^♭, C^♭, F^♭, B^♭♭, ... etc. The following tables give the sizes in fractions of an octave of the intervals between the note C at the start of an octave and each of the naturals, sharps and flats within the octave immediately above it when it is tuned with a regular tuning whose fifth is a fraction f of an octave. The columns of the tables are ordered according to the sizes of the corresponding intervals.

Table 1a (4/7 < f < 7/12)

C

C♯

D♭

D

D♯

E♭

E

F♭

E♯

F

F♯

G♭

G

G♯

A♭

A

A♯

B♭

B

C♭

B♯

0

7f – 4

3−5f

2f−1

9f−5

2−3f

4f−2

5−8f

11f–6

1−f

6f–3

4−6f

f

8f–4

3−4f

3f–1

10f–5

2−2f

5f–2

5−7f

12f–6

Table 1b (7/12 ≤ f < 3/5)

C

B♯

D♭

C♯

D

E♭

D♯

F♭

E

F

E♯

G♭

F♯

G

A♭

G♯

A

B♭

A♯

C♭

B

0

12f–7

3−5f

7f – 4

2f−1

2−3f

9f−5

5−8f

4f−2

1−f

11f–6

4−6f

6f–3

f

3−4f

8f–4

3f–1

2−2f

10f–5

5−7f

5f–2

The regular tunings given as examples below have either been proposed or used as a means of determining the relative pitches of the notes in the above sequence.

Notation

f_Jdef= log₂(3/2) , the size, in octaves, of a just (i.e. Pythagorean) perfect fifth, equivalent to about 702 cents.

t_Jdef= log₂(5/4) , the size, in octaves, of a just major third, equivalent to about 386 cents

c_sdef= 4 f_J − t_J − 2 , the size, in octaves, of a syntonic comma, equivalent to about 21.5 cents.

c_ddef= 12 f_J − 7 , the size, in octaves, of a ditonic (i.e. Pythagorean) comma, equivalent to about 23.5 cents.

c_hdef=c_d − c_s, the size, in octaves, of a schisma, equivalent to about 1.95 cents.

12-tone equal temperament

When f = 7/12, as it is in the now standard 12-tone equal temperament, the total span of twelve fifths will be exactly 7 octaves, the notes G^♭, D^♭, A^♭, E^♭, B^♭ will belong to the same pitch classes as F^♯, C^♯, G^♯, D^♯, A^♯ respectively, and the diminished sixths, B–G^♭, F^♯–D^♭, C^♯–A^♭, G^♯–E^♭, D^♯–B^♭ and A^♯–F will be the same size, 7/12 of an octave, as the perfect fifths.

Other equal temperaments

a. When f = m/n, a rational fraction of an octave, where m and n are positive integers with no proper common factor, then n of those fifths will span exactly m octaves, and a chain of that many fifths will generate a scale of n notes per octave, spaced equally apart at intervals of 1/n^th of an octave. The resulting temperament is called "n-tone equal temperament". If m/n ≠ 7/12, the pitch class of each sharp will be different from that of its enharmonically alternative flat, and separated from it by an interval of | 12 m/n − 7 | of an octave. Examples include 17-tone, 19-tone, 22-tone, 31-tone, 41-tone and 53-tone equal temperaments, with fifths of sizes 10/17, 11/19, 13/22, 18/31, 24/41 and 31/53 of an octave respectively.

b. If a whole number, r, of interleaved chains of fifths, of the same size, m/n of an octave, as above, are offset from one another by intervals of 1/(r n) of an octave, they will generate a scale of r n notes per octave, spaced equally apart at that same interval, thus producing an r n-tone equal temperament. Examples include 24-tone, 34-tone and 58-tone equal temperaments, each with two interleaved chains of fifths of sizes 7/12, 10/17 and 17/29 of an octave respectively, and 72-tone equal temperament with 6 interleaved chains of fifths of size 7/12 of an octave.

Pythagorean tuning and schismic temperament

When all the fifths are Pythagorean—that is,  f = f_J —the total span of twelve fifths is larger than 7 octaves by the Pythagorean comma, which is more than a quarter of this tuning's diatonic semitone. ^[19] The diminished sixths, B–G^♭, F^♯–D^♭, C^♯–A^♭, G^♯–E^♭, D^♯–B^♭ and A^♯–F, are flatter than the fifth by this amount, sufficient for them to qualify as wolves.

Apart from the presence of wolves, Pythagorean tuning has the disadvantage that its normal thirds and sixths are quite dissonant. Its major thirds, A^♭–C, E^♭–G, B^♭–D, F–A, C–E, G–B, D–F^♯ and A–C^♯, for instance, are wider than just by a syntonic comma, as are its major sixths, and its normal minor sixths and thirds are correspondingly narrower than just by the same amount, which is too large for any of these intervals to be recognised as consonances. The diminished fourths, E–A^♭, B–E^♭, F^♯–B^♭ and C^♯–F, on the other hand, are narrower than just major thirds by only a schisma, thus making them reasonably consonant. By extending the tuning as far as the notes G and D (for the chromatic scale from A^♭ to C^♯ around the spiral of fifths), or D and A (for the chromatic scale from E^♭ to G^♯ around the spiral of fifths), each note of the chromatic scale can be made to serve as the bass for reasonably good thirds and sixths, as indicated in the following table:

Table 2

Bass

A♭

E♭

B♭

F

C

G

D

A

E

B

F♯

C♯

G♯

minor third (augmented second)

B

F♯

C♯

G♯

D♯

A♯

E♯

B♯

F

C

G

D

A

major third (diminished fourth)

D

A

E

B

F♭

C♭

G♭

D♭

A♭

E♭

B♭

F

C

minor sixth (augmented fifth)

E

B

F♯

C♯

G♯

D♯

A♯

E♯

B♯

F

C

G

D

major sixth (diminished seventh)

G

D

A

E

B

F♭

C♭

G♭

D♭

A♭

E♭

B♭

F

The names given in parentheses in this table are those assigned to the corresponding intervals by traditional musical terminology, where the degree assigned to an interval is determined by the number of letter-named notes included in it.

Alexander Ellis gave the name "skhismic temperament" ^[20] to a Pythagorean tuning whose thirds and sixths were regarded as being the intervals given in the above table, rather than the ones those names would refer to in traditional musical terminology.

Quarter-comma meantone

In quarter-comma meantone, the size of the perfecf fifths,  fdef=f_J − c_s/4 = (t_J + 2)/4, is chosen so that four of them span an interval of two octaves plus a just major third, thus making all the major thirds and minor sixths of the resulting scale perfectly just. The total span of twelve of these fifths is less than 7 octaves by the so-called lesser diesis, which is about 40% of an equal-temperament semitone. The diminished sixths, B–G^♭, F^♯–D^♭, C^♯–A^♭, G^♯–E^♭, D^♯–B^♭ and A^♯–F, are sharper than the fifth by this amount, which is again sufficient for these intervals to qualify as wolves. The title "quarter-comma meantone" derives from the fact that the size of the fifth in this temperament is smaller than its just value by a quarter of the syntonic comma.

Other meantones

If q is a positive rational fraction, a regular temperament with fifths of size  fdef=f_J − q c_s —that is, a multiple q of a syntonic comma smaller than their just size of  f_J —is commonly referred to as "q-comma meantone". Fractions other than 1/4 which have been suggested as possible values for q are 1/3, 1/5, 1/6, 1/8, 1/9, 1/10, 2/7, 2/9, 3/10 and 5/18. ^[21]

Helmholtzian temperament

A regular tuning with a fifth of size  fdef=f_J – c_h/8 , and with the thirds and sixths identified with the intervals specified by the scheme of table 2 is called "Helmholtzian temperament". ^[22] In this temperament, the major thirds and minor sixths, as given by the scheme of table 1, are perfectly just, while the fifths and minor thirds are flatter than just, and the fourths and major sixths sharper than just, by the imperceptible amount of an eighth of a schisma.

More recent alternative concept

In the traditional definition of regular tuning, it is taken for granted that the perfect fifth whose size is being fixed by the tuning is that of the diatonic major scale of common practice Western music. A regular tuning in this sense is thus inextricably tied to the structure of that scale and others closely associated with it. It is a much less useful concept for other styles of music whose scales cannot be straightforwardly derived from the diatonic. ^[23]

Some musical theorists have recently proposed an alternative definition of regular tuning which can be applied to a wider range of musical styles. According to this alternative definition, a regular tuning is a collection of intervals which can be produced by combining together intervals of a finite number of fixed sizes. ^[11] For the purposes of the definition, an interval is regarded as being completely determined by just its size and direction (ascending or descending), independent of any specific range of pitches which it might span, with a descending interval being regarded as the negative of an ascending one of the same size. Thus, the intervals between any of the pairs of notes [C_n,E_n], [D_n,F^♯_n], [E_n,G^♯_n], etc., in 12-tone equal temperament, for example, are considered to be merely separate instantiations of the same interval—an ascending tempered major third of size one-third of an octave—, and the intervals between the pairs [E_n,C_n], [F^♯_n,D_n], [G^♯_n,E_n], etc., to be instantiations of its negative—a descending tempered major third of the same size. An ascending interval in this sense can therefore be represented by a single number that specifies it size, and a descending one by the negative of the number specifying its size.

A regular tuning in this alternative sense is not required to contain octaves or perfect fifths, or to be usable for tuning any of the usual scales of common practice music, and will therefore not necessarily be obtainable as the set of intervals of a regular tuning in the traditional sense. The collection of intervals that can be produced by combining together octaves, just perfect fifths and just major thirds, ^[24] for instance, although a regular tuning according to this alternative definition, cannot be the set of intervals of any tuning that is regular under the traditional definition. ^[25] For the rest of this article, the term "regular tuning", unless otherwise qualified, will be used in this alternative sense.

Intervals that can be combined together to produce all the other intervals of a given regular tuning are called "generators" of the tuning. Any interval of such a tuning must have the form   n₁g₁ + n₂g₂ + … + n_rg_r , where  g₁, g₂, … , and g_r  are generators for it, and  n₁, n₂, … , and  n_r  are integers. The minimum number of non-zero generators needed to produce all the intervals of the tuning is called its "rank", ^[26] and regular tunings of rank 2 are called "linear". The collection of all intervals obtainable by combining together the generators of a regular tuning has the mathematical structure of a free abelian group of the same rank. The theory of these groups can therefore be useful for investigating the properties of regular tunings.

Examples

The intervals of a regular tuning in the traditional sense

Some examples of linear tunings generated by an octave and a perfect fifth, generally tempered (Milne et al., 2007, p.18).

The intervals of any regular tuning in the traditional sense can be obtained by combining together its perfect fifth, a pure octave, and possibly some intervals separating chains of fifths that may be offset from one another. That set of Intervals is therefore a regular tuning in the alternative sense. The set of intervals of an equal temperament—in which just one size of interval is needed to produce all the others—has rank 1. Since the intervals of Pythagorean tuning and meantone and Helmholtzian temperaments can be produced from their fifths and the octave, but not from any intervals of just one size, those sets of intervals have rank 2, and are therefore linear.

p-limit tuning and just intonation

An interval obtainable as a combination of generators whose sizes in octaves are all of the form log₂(q), with all q being prime numbers no greater than some fixed prime p, is called a "p-limit interval", and a regular tuning is called a p-limit tuning if all its intervals are p-limit intervals. ^[27] Thus, since the intervals of Pythagorean tuning are generated by the octave, of size one (i.e. log₂(2)) octave, and the just perfect twelfth, of size log₂(3) octaves, they form a 3-limit tuning. Likewise, the set of intervals generated by the octave, a just perfect twelfth, and a just major seventeenth, of size log₂(5) octaves, constitutes a 5-limit tuning. Since this set of intervals cannot be generated by any smaller collection of generators, it is a regular tuning of rank 3.

In just intonation, the pair of notes forming every consonance has a frequency ratio whose numerator and denominator are small whole numbers. As a consequence, these intervals must belong to a p-limit tuning for some small prime p. Thus, the intervals of Ptolemy's syntonic diatonic scale, for instance, in which the relative pitches of the notes are completely determined by the requirement that all three of its primary triads be perfectly just major chords, form a 5-limit tuning, since they can be generated by intervals of sizes log₂(2), log₂(3), and log₂(5) octaves.

By going one step further, to proper 7-limit tunings, ^[28] it is possible to obtain regular tunings with perfectly just septimal intervals. The regular tuning generated by intervals of sizes log₂(2), log₂(3), log₂(5), and log₂(7) octaves, for instance, is a 7-limit tuning of rank 4 which contains perfectly just harmonic seventh intervals of size log₂(​⁷⁄₄) octaves, as well as perfect fourths and fifths, and major and minor thirds and sixths which are all perfectly just.

Temperament mappings

By definition, a tempered interval is one which is very nearly—but not quite—just. Thus, if   g₁, g₂, … , g_r   are all either tempered or just intervals of a regular tuning, there must be some just intervals,   h₁, h₂, … , h_r , such that   h₁ − g₁, h₂ − g₂, … , h_r − g_r   are all very close to zero. ^[29] Since   h₁, h₂, … , and h_r  are all just, they must belong to a p-limit tuning for some small prime p. If they are all distinct, and there is no interval which can be expressed in two or more different ways as a combination of them, ^[30] then the association of the interval   n₁h₁ + n₂h₂ + … + n_rh_r  to the interval   n₁g₁ + n₂g₂ + … + n_rg_r , for each r-tuple,  n₁, n₂, … , n_r , of integers, defines a mathematical function from the p-limit tuning generated by   h₁, h₂, … , h_r  onto the regular tuning generated by   g₁, g₂, … , g_r . This function, which turns out to be a useful tool for the construction of tunings with desired properties, is called a temperament mapping. ^[31]

Examples

For the intervals of 12-tone equal temperament

The only just interval of 12-tone equal temperament is the octave. Its tempered intervals are its perfect fourth and fifth, and its major and minor third and sixth, whose sizes in octaves are ​⁵⁄₁₂,​⁷⁄₁₂, ​¹⁄₃, ​¹⁄₄, ​³⁄₄, ​²⁄₃, respectively. The corresponding just intervals can all be generated from the octave, just perfect fifth and just major third. ^[32] Thus, the temperament mapping for the intervals of 12-tone equal temperament associates the 5-limit interval   n₁ + n₂f_J + n₃t_J , for any triple of integers,  n₁, n₂, n₃ , with the interval   n₁ + 7/12n₂ + 1/3n₃ .

For a tuning generated by an octave and a fifth

If a regular tuning generated by an octave and a tempered perfect fifth of size  f  octaves, the sizes of its perfect fourth, major and minor thirds, and major and minor sixths, in octaves, are 1 − f ,  4 f − 2 ,   2 − 3 f,   3 f − 1  , and 3 − 4 f, respectively. If  f = f_J − ​^c_s⁄₄  (quarter-comma meantone), the octave, major thirds and minor sixths are perfectly just, but all other consonances are tempered. If  f = f_J − ​^c_s⁄₃ (one-third-comma meantone), the octave, minor thirds and major sixths are perfectly just, and all other consonances are tempered. Otherwise, all consonances except the octave are tempered. ^[33] In any case, the temperament mapping for the intervals of this temperament associates the 5-limit interval  n₁ + n₂f_J + n₃t_J , for any triple of integers,  n₁, n₂, n₃ , with the interval  n₁ + n₂f + n₃(4 f − 2) .

For the intervals of schismic temperament

The just intervals of schismic temperament are its octave and perfect fourth and fifth, and its tempered intervals are its major and minor third and sixth. The sizes of these latter intervals, in octaves, are t_J – c_h, f_J – t_J + c_h, 1 + t_J – f_J – c_h, and 1 + c_h – t_J, respectively. Thus, the temperament mapping for the intervals of schismic temperament associates the 5-limit interval   n₁ + n₂f_J + n₃t_J , for any triple of integers,  n₁, n₂, n₃ , with the interval   n₁ + n₂f_J + n₃ (t_J – c_h) .

For the intervals of Helmholtzian temperament

The just intervals of Helmholtzian temperament are its octave, major third and minor sixth, and its tempered intervals are its perfect fourth and fifth, and its minor third and major sixth. The sizes of these latter intervals, in octaves, are 1 + c_h/8 − f_J, f_J – c_h/8, f_J − t_J − c_h/8, and 1 + t_J + c_h/8 – f_J, respectively. Thus, the temperament mapping for the intervals of Helmholtzian temperament associates the 5-limit interval   n₁ + n₂f_J + n₃t_J , for any triple of integers,  n₁, n₂, n₃ , with the interval   n₁ + n₂ ( f_J – c_h/8 ) + n₃t_J .

Images of the just consonances under the above temperament mappings

Let T₁, T₂, T₃, T₄ be the temperament mappings for the intervals of 12-tone equal temperament, a regular tuning generated by an octave and a fifth, and schismic and Helmholtzian temperaments respectively, as described above. The following table lists the images of each of the 5-limit just consonances under these temperament mappings.

Table 3

just interval	n1, n2, n3	image under the mappingT1 (12-tone equal)	image under the mappingT2 (linear with fifth of size f )	image under the mappingT3 (schismic)	image under the mappingT4 (Helmholtzian)
octave	1, 0, 0	1	1	1	1
perfect fifth	0, 1, 0	​7⁄12	f	fJ	fJ – ch/8
perfect fourth	1, –1, 0	​5⁄12	1 − f	1 – fJ	1 + ch/8 − fJ
major third	0, 0, 1	​1⁄3	4 f − 2	tJ − ch	tJ
minor third	0, 1, -1	​1⁄4	2 − 3 f	fJ + ch − tJ	fJ − tJ −ch/8
major sixth	1, -1, 1	​3⁄4	3 f − 1	1 + tJ − fJ − ch	1 + tJ +  ch/8 − fJ
minor sixth	1, 0, –1	​2⁄3	3 − 4 f	1 + ch − tJ	1 − tJ

Commas, rank, and relations between consonances

The purpose of tempering consonances is to allow a greater number of them to be accommodated within the confines of a fixed scale, at the expense of compromising their purity. In a regular tuning, this manifests itself in the existence of intervals that can be formed from two different combinations of consonances, even though their corresponding just versions are independent.

Examples

In 12-tone equal temperament, an interval comprising 7 octaves can also be expressed as a combination of 12 perfect fifths. In any regular tuning generated by an octave and a tempered fifth (including 12-tone equal temperament), an interval comprising 4 perfect fifths can be expressed as a combination of 2 octaves plus a major third. In both schismic and Helmholtzian temperament an interval comprising 5 octaves can be expressed as a combination of 8 perfect fifths plus a major third.

If  T₁, T₂, T₃, T₄  are the temperament mappings defined above, these relations can be written as:

0 = 12 T₁(f_J) − 7 T₁(1) = T₁(12 f_J − 7) = T₁(c_d)

0 = 4 T₁(f_J) − 2 T₁(1) − T₁(t_J) = T₁(4 f_J − 2 − t_J) = T₁(c_s)
0 = 4 T₂(f_J) − 2 T₂(1) − T₂(t_J) = T₂(4 f_J − 2 − t_J) = T₂(c_s)
0 = 8 T₃(f_J) + T₃(t_J) − 5 T₃(1) = T₃(8 f_J + t_J − 5) = T₃(c_h)

0 = 8 T₄(f_J) + T₄(t_J) − 5 T₄(1) = T₄(8 f_J + t_J − 5) = T₄(c_h) .

That is, these relations correspond to the ditonic comma's being mapped to zero under the temperament mapping of 12-tone equal temperament, the syntonic comma's being mapped to zero under the temperament mappings of any regular tuning generated by an octave and a tempered fifth, and the schisma's being mapped to zero under the temperament mappings of schismic and Helmholtzian temperaments.

General case

If some interval of a regular tuning can be formed from two different combinations of its consonances,  c₁, c₂, ... , c_t , whose corresponding just versions are independent, ^[34] there must exist positive whole numbers,  n₁, n₂, ... , n_t , and  m₁, m₂, ... , m_t , with  n_i ≠ m_i  for any i, such that

 n₁c₁ + n₂c₂ + ... + n_tc_t = m₁c₁ + m₂c₂ + ... + m_tc_t .

Subtracting the right side of this equation from the left gives

 q₁c₁ + q₂c₂ + ... + q_tc_t = 0 ,

where  q_i = n_i − m_i ≠ 0  for  i = 1, 2, ... , t . Such an equation is called a "(linear) relation".

If  T  is the temperament mapping for the regular tuning, and  h₁, h₂, ... , h_t  the just consonances such that  T(h_i) = c_i  for  i = 1, 2, ... , t , the above linear relation can be written as

      0 = q₁T(h₁) + q₂T(h₂) + ... + q_tT(h_t)
= T(q₁h₁ + q₂h₂ + ... + q_th_t) .

Thus,  q₁h₁ + q₂h₂ + ... + q_th_t , which is a non-zero p-limit interval for some small prime p, is mapped to zero by the temperament mapping  T . If all the integers  q₁, q₂, ... ,  and  q_t  are reasonably small in magnitude, then because each of the intervals  h_i  is equal to, or closely approximated by, the corresponding  c_i , the interval  q₁h₁ + q₂h₂ + ... + q_th_t  will be quite small. It is referred to as a " comma ".

The set of all p-limit intervals  h  such that  T(h) = 0  is called the "comma lattice" of the tuning. ^[35] The comma lattice of the general regular tuning described above is itself another a regular tuning whose rank is strictly less than  t. It follows from one of the fundamental theorems of linear algebra ^[36] that the rank of the regular tuning and the rank of its comma lattice must sum to  t.

Examples

The regular tuning formed by the intervals of 12-tone equal temperament has rank 1. Its comma lattice, generated by the syntonic and ditonic commas, has rank 2.

A regular tuning generated by the octave and a tempered perfect fifth whose size in octaves is irrational has rank 2. Its comma lattice, generated by the syntonic comma, has rank 1.

The regular tuning formed by the intervals of schismic temperament and that formed by the intervals of Helmholtzian temperament both have rank 2. They both have the same comma lattice, generated by the schisma, of rank 1.

Finding minimal sets of generators

If a regular tuning has generators  g₁, g₂, … , g_t  that satisfy one or more linear relations, then its rank  r , say, will be strictly less than  t . Although there must then exist a set of  r  generators for the tuning, it may not always be possible to select these from among the unmodified generators in the original set,   { g₁, g₂, … , g_t} . In general, some, at least, of the intervals in a minimal set of  r  generators will have to be constructed as combinations of 2 or more of the originals. ^[37]

Example

Consider the regular tuning generated by the octave, just perfect fifth, and a tempered major third of size τdef=​¹⁸⁄₅ − ​²⁸⁄₅f_J = ​⁴⁶⁄₅ − ​²⁸⁄₅ log₂(3) octaves. ^[38] A temperament mapping T  for this tuning is defined by  T(n₁ + n₂f_J + n₃t_J) = n₁ + n₂f_J + n₃ (​¹⁸⁄₅ − ​²⁸⁄₅f_J) = n₁ + ​¹⁸⁄₅n₃ + (n₂ − ​²⁸⁄₅n₃) f_J . The generators satisfy the relation   0 = 18 − 28 f_J − 5 τ , corresponding to the comma   18 − 28 f_J − 5 t_J , a 5-limit interval of size approximately 13.7 cents. Although this regular tuning has rank 2, none of its original generators can be expressed as a combination of the other two, so no pair of them can serve as generators for it. The fifth part,  φdef=1 − f_J/5 , of a just perfect fourth, however, can be written as  φ = −7 + 11 f_J + 2 τ  (a combination of the original generators), and since   1 = f_J + 5 φ ,  f_J = 1 − 5 φ , and  τ = −2 + 28 φ = −2 f_J + 18 φ , either the octave or the just perfect fifth together with the interval φ constitute a pair of generators for this tuning. ^[39]

General Case

Finding a minimal set of generators for a regular tuning with consonances satisfying some linear relations requires, in general, some fairly advanced, but well-known, techniques of linear algebra. Proofs are omitted from the following outline. ^[40]

If the generators  g₁, g₂, … , g_t  of a regular tuning satisfy the linear relations:

${\displaystyle {\begin{aligned}0&\ =\ a_{11}g_{1}+a_{12}g_{2}+\dots +a_{1t}g_{t}\\0&\ =\ a_{21}g_{1}+a_{22}g_{2}+\dots +a_{2t}g_{t}\\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \dots \\0&\ =\ a_{s1}g_{1}+a_{s2}g_{2}+\dots +a_{st}g_{t}\end{aligned}}}$ ,

and no others apart from linear combinations of these, then the rank, c, say, of the tuning's comma lattice is the rank of the s × t matrix  A  whose entries are the integers  a_ij , and the rank of the tuning itself is  t − c . ^[41] Let  S  be the Smith normal form of  A ,  U₁, U₂  be  s × s  and  t × t  unimodular matrices ^[42] such that  S = U₁A U₂ ,  W  be the   (t − c) × t  submatrix of  U₂⁻¹  comprising its last  t − c  rows,  V  be the  t × (t − c)  submatrix of  U₂  comprising its last  t − c  columns,  g  be the  t × 1  column vector whose entries are  g_i , and let  w = W g .

Since the entries of  W  are integers, the entries,  w₁, w₂, … , w_{t– c} , of  w  are intervals in the tuning generated by  g₁, g₂, … , g_t . But since the entries of  V  are also integers, and  g = V w , it follows that  w₁, w₂, … , w_{t– c}  constitute a minimal set of  t − c  generators for the tuning.

Examples

a.  The matrix of the right side of the relation   0 = 18 g₁ − 28 g₂ − 5 g₃ , satisfied by the generators  g₁def= 1, g₂def=f_J , and  g₃def=τ  of the immediately preceding example, is  ${\displaystyle A\,{\overset {def}{=}}\,\left({\begin{matrix}18&-28&-5\end{matrix}}\right)}$ , which has Smith normal form ${\displaystyle \left({\begin{matrix}1&0&0\end{matrix}}\right)}$ . Postmultiplying  ${\displaystyle A}$  by either of the two unimodular matrices,

${\displaystyle \left({\begin{matrix}0&1&0\\-2&1&-5\\11&-2&28\end{matrix}}\right)}$  or ${\displaystyle \left({\begin{matrix}2&1&5\\0&1&0\\7&-2&18\end{matrix}}\right)}$ ,

will reduce it to its Smith normal form:

${\displaystyle \left({\begin{matrix}1&0&0\end{matrix}}\right)=\left({\begin{matrix}18&-28&-5\end{matrix}}\right)\left({\begin{matrix}0&1&0\\-2&1&-5\\11&-2&28\end{matrix}}\right)=\left({\begin{matrix}18&-28&-5\end{matrix}}\right)\left({\begin{matrix}2&1&5\\0&1&0\\7&-2&18\end{matrix}}\right)}$ .

The inverses of these unimodular matrices are:

${\displaystyle \left({\begin{matrix}18&-28&-5\\1&0&0\\-7&11&2\end{matrix}}\right)}$  and  ${\displaystyle \left({\begin{matrix}18&-28&-5\\0&1&0\\-7&11&2\end{matrix}}\right)}$ , respectively.

Applied to the original generators,  g₁, g₂ , and  g₃ , the first of these produces the pair of generators  g₁  and  −7 g₁ + 11 g₂ + 2 g₃ , while the second produces the pair  g₂  and  −7 g₁ + 11 g₂ + 2 g₃ .

b. (Miracle temperament)   Consider the regular tuning generated by the octave, g₁ , temperered perfect fifth,  g₂ , of size

​⁶⁄₁₉ + ​¹²⁄₁₉f_J − ​⁶⁄₁₉t_J = ​⁶⁄₁₉ + ​¹²⁄₁₉ log₂(3) − ​⁶⁄₁₉ log₂(5)  octaves, ^[43]

tempered major third,  g₃ , of size

​¹²⁄₁₉ − ​¹⁴⁄₁₉f_J + ​⁷⁄₁₉t_J = ​¹²⁄₁₉ − ​¹⁴⁄₁₉ log₂(3) + ​⁷⁄₁₉ log₂(5)  octaves, ^[44]

and two other intervals,  g₄, g₅ , of sizes

​¹⁷⁄₁₉ − ​⁴⁄₁₉f_J + ​²⁄₁₉t_J = ​¹⁷⁄₁₉ − ​⁴⁄₁₉ log₂(3) + ​²⁄₁₉ log₂(5) octaves

and

−​⁴⁄₁₉ + ​³⁰⁄₁₉f_J − ​¹⁵⁄₁₉t_J = −​⁴⁄₁₉ + ​³⁰⁄₁₉ log₂(3) − ​¹⁵⁄₁₉ log₂(5)  octaves,

respectively. These last two intervals are tempered 2.3 cents and 0.6 cents flatter than a just septimal minor seventh and a just undecimal semi-augmented fourth, with freqency ratios ​⁷⁄₄ and ​¹¹⁄₈, respectively.

The generators  g₁, g₂, g₃, g₃, g₄, g₅  satisfy the set of relations

${\displaystyle {\begin{aligned}0&\ =\ 3g_{1}-g_{2}\ \ \ \ \ \ \ \ \ \ -3g_{4}\\0&\ =\ g_{1}\ \ +g_{2}\ \ -g_{3}\ -g_{4}-g_{5}\\0&\ =\ g_{1}\ -2g_{2}-2g_{3}+g_{4}\end{aligned}}}$ ,

which has matrix

${\displaystyle A=\left({\begin{matrix}3&-1&0&-3&0\\1&1&-1&-1&-1\\1&-2&-2&1&0\end{matrix}}\right)}$ 

Postmultiplying  A  by the unimodular matrix

${\displaystyle U=\left({\begin{matrix}0&0&0&1&0\\-1&0&-3&0&6\\1&0&3&1&-7\\1&0&1&1&-2\\-2&-1&-7&-1&15\end{matrix}}\right)}$ 

reduces it to Smith normal form:

${\displaystyle A\,U=\left({\begin{matrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\end{matrix}}\right)}$ .

From this it follows that the tuning has rank 2, and its comma lattice rank 3. The last two rows of the inverse of  ${\displaystyle U}$ ,

${\displaystyle U^{-1}=\left({\begin{matrix}3&-1&0&-3&0\\1&1&-1&-1&-1\\1&-2&-2&1&0\\1&0&0&0&0\\1&-1&-1&0&0\end{matrix}}\right)}$ ,

give the octave,  g₁ , and the interval  g₆def=g₁ − g₂ − g₃ = ​¹⁄₁₉ + ​²⁄₁₉ log₂(3) − ​¹⁄₁₉ log₂(5) octaves as a minimal set of two generators for the tuning. This last interval is called a "secor", after George Secor, who discovered it, and the regular tuning just described is called "miracle temperament". ^[45] The tuning's tempered fifth comprises 6 secors, and its tempered major third one octave minus 7 secors. Its minor sixth is thus tempered to 7 secors.

Notes

1. The relative pitches assigned to this scale are naturally inherited by its subscales—the diatonic major scales and their relative minors and its pentatonic subscales, for instance. All regular temperaments other than 12-tone equal temperament and its derivatives also naturally generate relative pitch assignments for extra notes which can be (and have been) used to create extensions of the chromatic scale to more than 12 notes.
2. Premises of this definition are that octaves are untempered—that is, the frequencies of their end-points are in the ratio of 2 to 1—, that every octave is identically divided by notes selected from the same set of pitch classes and that a "perfect fifth" is the interval between the first and fifth scale degrees of a diatonic major scale, however tuned. Equivalently, it is the interval formed by juxtaposing 4 diatonic semitones and 3 chromatic semitones, or the interval between the notes sounded by two keys separated by 6 others on a standard musical keyboard, not necessarily tuned according to the now standard 12-tone equal temperament, and with the proviso that if the upper note is a flat then so is the lower, and if the lower note is a sharp then so is the upper. The term "perfect" is used here solely to distinguish the interval referred to from augmented and diminished fifths (such as the interval between B and F), and does not imply that the fifth has to be just.
3. That is, has had the frequency ratio between its end-points adjusted to be close to, but not exactly equal to, its just value of 3/2. The purpose of this tempering is commonly to reduce the frequency ratio between the end-points of the major thirds and minor sixths and increase that between the end-points of the minor thirds and major sixths so that they are closer to their just values of 5/4, 5/3, 6/5 and 8/5 respectively. In practice, this means that the frequency ratio between the end-points of the tempered fifths will normally be less than 3/2.
4. Bergholt (1895; 212, p.510; 214, p.637), Lloyd & Boyle (1978, p.163); Panetta (1987, p.15); Lindley & Turner-Smith (1993, p.43); Barbour (2004, pp.xi, 32–44); Donahue (2005, p.39); Duffin (2007, pp.39, 33–46); Blood (2013, Tc-Te); Di Veroli (2013, p.42); Dolata (2016, p.98). Lindley (1984, p.8), Lindley, (2001, p.249) and Jorgensen (1991 , p.776) use the term "regular meantone temperament" to refer to the same concept.
5. That is, the frequency ratio between their end-points has its just value of 3/2.
6. Ellis (1885, pp.431, 433), Bosanquet (1889, p.421–2), and Lloyd & Boyle (1978, p.162), for instance. Bosanquet defines "temperament" to mean "the division of the octave", and takes the division produced by what he calls the "Pythagorean system" to be one instance of this.
7. Duffin (2007, p.38) and Blood (2013, Tc-Te), for example.
8. Barbour (2004, pp.xii, 5). Although Barbour, somewhat inconsistently, seems to include Pythagorean tuning amongst the ones he refers to as "regular temperaments".
9. Bosanquet (1876, p.60, 1889, p.421–2) uses the expression "regular system" rather than "regular tuning" for the same concept.
10. That is, it differs from such a note by a whole number of octaves.
11. See, for example, Milne et. al. (2007 and 2008).
12. That is, the size of each fifth is 1200f cents, and the frequency ratio between its end-points is 2^f.
13. Where  log₂(x) , for any positive  x , is the binary logarithm of  x .
14. Here specified using scientific pitch notation, in which the subscript indicates the octave to which the note belongs, with C₄ being middle C.
15. Blackwood (1985, pp.200–1); Milne et. al. (2007, p.22).
16. Barbour (2004, p.25–44).
17. Donahue (2005, p.215). Sometimes also called "wolf fifths", despite their really being diminished sixths (Jorgensen 1991, p.10; Donahue, 2005, p.23; Barbour, 2004, p.xii).
18. Here taken to be measured in octaves and fractions of an octave. Mathematically, the size of an interval in this sense is the base-2 logarithm of the frequency ratio between its end-points, or equivalently,  s/1200  where  s  is its size in cents.
19. And nearly a quarter of the semitone of 12-tone equal temperament.
20. Ellis (1885, p.435). Now usually spelt "schismic".
21. Barbour (2004, pp.31–44)
22. Ellis (1885, p.435).
23. Erlich (2006)
24. That is, the collection of all 5-limit intervals.
25. Any tuning of a diatonic scale which contains both a just perfect fifth and a just major third must always contain perfect fifths of at least two different sizes. If the major third C–E is perfectly just, for example, then at least one of the fifths C–G, G–D, D–A or A–E cannot be perfectly just.
26. Milne et al. (2007, p.20)
27. Wright (2009, p.137). As thus defined, a p-limit tuning is automatically also a q-limit tuning for any other prime q greater than p.
28. That is, to one which does in fact contain an interval of size log₂(7) octaves.
29. Or equal to zero whenever the interval  g_i  happens to be perfectly just.
30. If this is the case, the intervals   h₁, h₂, … , and h_r  are said to be "independent".
31. Milne et al. (2007, p.20)
32. The just perfect fourth can be obtained by subtracting a just perfect fifth from an octave, the just minor third by subtracting a just major third from a just perfect fifth, the just major sixth by subtracting a just perfect fifth from the sum of an octave and a just major third, and the just minor sixth by subtracting a just major third from an octave.
33. Or will be so far out of tune as to be no longer consonant, if  f  is sufficiently close to one of its extreme limits, ​⁴⁄₇ or ​³⁄₅.
34. For this to be possible, at least one of the consonances  c_i  must be tempered.
35. Milne et. al. (2008, p.2). These authors, however, use this term to refer not to the set of commas itself, but to the isomorphic lattice of t-tuples,   ( q₁, q₂, ... , q_t ) , of integers such that  T(q₁h₁ + q₂h₂ + ... + q_th_t) = 0 
36. Namely, the rank–nullity theorem.
37. The intervals of 12-tone equal temperament, for instance, are generated by its perfect fifth and an octave, but not by either of those two intervals by itself. The equal temperament semitone, which does generate all the intervals of this regular tuning, is a combination of 7 equal-temperament fifths less 4 octaves.
38. This major third, comprising 5​³⁄₅ just perfect fourths less 2 octaves, is approximately 2.7 cents sharper than just.
39. Milne et al. (2007, p.20) describe a procedure for obtaining two intervals that will generate all the intervals of a regular tuning generated by three generators satisfying a single linear relation. When applied to this example, that procedure produces the intervals ​¹⁄₅ (one fifth of an octave) and  ​^f_J⁄₅  (one fifth of a just perfect fifth). While these intervals can indeed be combined together to produce any interval of the tuning, neither of them can be expressed as a combination of whole numbers of copies of the original generators. So, while the tuning generated by these two intervals does include all the intervals of the original, it is a proper superset of this latter, and not identical to it.
40. Which is a slightly more detailed version of that given by Newman (1997, pp.376–7).
41. If the relations are independent, then  A  will be of full rank, and  c = s < t . If desired, the elimination of redundant relations can reduce the problem to this case.
42. A matrix is unimodular when it is invertible and both its own entries and those of its inverse are integers. The Smith normal form and unimodular matrices  U₁, U₂  relating it to the original matix always exist for matrices having integer entries. The matrix U₁ will be unique if the original matrix has full rank, but not otherwise. The matrix U₂ will not be unique.
43. Approximately 1.7 cents flatter than just.
44. Approximately 3.3 cents flatter than just.
45. Secor (1975, 1976, 2006).

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