Normal lists

Since a regular temperament can be represented by multiple equivalent val lists (aka mappings) or monzo lists (aka comma bases), it can be helpful — e.g. when comparing or cataloguing temperaments — to choose a single one of these equivalent lists to use as its unique identifier. A set of rules that are consistently able to narrow the full set of equivalent lists down to a single list for each temperament may be called a normal form, and accordingly, a list which uniquely identifies a temperament in this way may be called a normal list.

Because several different normal forms have been developed, each temperament has several different normal lists: one for each form. These normal lists are not all necessarily different; sometimes some or all of them may be the same.

Hermite normal form

Hermite normal form, or HNF for short, is an important normal form that was defined for integer matrices in the mathematical field of linear algebra. An integer matrix is simply a 2D array of integers, and so we can easily think of lists of vals or commas as integer matrices and therefore leverage HNF in regular temperament theory.

HNF by itself is not used as a normal form for regular temperament theory. However, it is part of the definition of most normal forms here, so it is important to have a basic understanding of it.

There are slightly different definitions of HNF in use, and if you are using a computer program to compute it, you should take care that the same normal monzo or val list is finally achieved. The definition used by the Wikipedia article on Hermite form, probably the most standard, works as follows.

An n by m integral matrix H is in HNF if when we define a function F such that F (i) = 0 if all of the entries in the i-th column of H are 0, and otherwise F (i) is equal to the row number of the first nonzero entry in the i-th column, checking up from the bottom, i.e. from the n-th row, we have

1. If i > j, H[i, j] = 0 (H is upper triangular.)
2. F (i) is a function of the column number i.
3. F (i) = 0 if and only if all of the entries in the i-th column are 0.
4. F is an increasing function of the column number i, and becomes strictly increasing after F (i) becomes positive.
5. If k > F (i) > 0 then H[k, i] = 0; that is, F (i) is the row of the first nonzero entry in the i-th column, counting up from the bottom.
6. If F (i) > 0 then H[F (i), i] > 0; that is, the first nonzero entry in the i-th column, counting up from the bottom, is positive.
7. If F (i) > 0 and i < j then H[F (i), i] > H[F (i), j] ≥ 0; that is, the first nonzero entry in the i-th column, counting up from the bottom, is greater than any of the rest along that row, which however are all non-negative.

There is some redundancy in the statement of these conditions, but that does no harm.

For more information, diagrams, an alternative articulation of the same definition, and comparisons with related integer matrix forms, see: Matrix echelon forms #HNF.

Normal val lists

First we will review the set of normal forms that are defined for val lists (mappings).

Defactored Hermite form

This is the "canonical form" for a temperament that was developed by Dave Keenan and Douglas Blumeyer, formed from defactoring the matrix (aka removing contorsion) prior to putting it into Hermite form.

We may write a list of vals (mapping) as a [math](k,d)[/math]-shaped matrix (read "k by d", i.e. with [math]k[/math] rows and [math]d[/math] columns), where the rows of the matrix are the vals (maps), and [math]d[/math] is the dimensionality of the system. To get the defactored Hermite form, we do the following:

1. First, defactor it (aka make sure it is saturated). . Note that if the matrix was not full-rank, this will result in the elimination of some rows. We now have a [math](r,d)[/math]-shaped matrix, with [math]r[/math] rows where [math]r[/math] is the rank.
2. Then, put this result into HNF.

For example, septimal meantone has the defactored Hermite form of [⟨1 0 -4 -13] ⟨0 1 4 10]}, corresponding to generators of ~2/1 and ~3/1.

The HNF step is not necessary to defactor the mapping, but it is important for canonicalizing into a convenient form.

The key advantage of the defactored Hermite form is its purity and simplicity, while sidestepping many of the issues with contorsion/enfactoring matrices. If your primary need is uniquely identifying temperaments, this is the ideal choice. The remaining normal forms each introduce further constraints on the sizes of the generators, which can be nice and convenient if that matters for your use case, but otherwise is unnecessary.

Positive generator form

Even though by using the HNF the defactored Hermite form ensures that the pivot (first nonzero entry) of each mapping row is a positive number, this does not necessarily mean that the corresponding generators are all positive in pitch. For example, the defactored Hermite form of porcupine is the matrix [⟨1 2 3] ⟨0 3 5]}. The second column of this matrix tells us it takes 2 of the first generator and 3 of the second generator to reach its approximation of 3/1. But as we can tell from the first column of this matrix, it takes only 1 of the first generator and nothing else to reach its approximation of 2/1. Therefore, if we move by 2 of the first generator, we are already at this temperament's approximation of 4/1, and so if we still need to move by 3 of the second generator to reach its approximation of 3/1, then the second generator must be negative. Indeed, it is about 163 cents downward in pitch. Negative generators like this can be surprising and confusing, and so the positive generator form was developed to address this concern.

To obtain this form, we first need to know whether each generator is positive or negative in pitch. Many methods are available for finding this information, but the one which is the easiest (and therefore the one this form is defined as using) is to find the Frobenius generators of the temperament. To break this down, we find the Moore–Penrose pseudoinverse of the mapping V, which we notate V⁺, and multiply this from the left by the row vector of JIP, J = [1 log₂3 log₂5 … log₂p].

[math]G = JV^+[/math]

If the i-th entry in the result is negative, change the signs of every entry in the corresponding row of the mapping.

Note: the "mapping" (though not the "mapping to lattice") listed on temperament pages of this wiki are in this form.

The generators in defactored Hermite form of septimal meantone is positive already, so its positive generator form is the same as its defactored Hermite form, [⟨1 0 -4 -13], ⟨0 1 4 10]], corresponding to generators of ~2/1 and ~3/1. An example of positive generator form that is different from the defactored Hermite form is the porcupine temperament, elaborated below.

Positive generator form example

The defactored Hermite form mapping matrix for porcupine is

[math] \left[ \begin{array} {rrr} 1 & 2 & 3 \\ 0 & 3 & 5 \\ \end{array} \right] [/math]

Its pseudoinverse is

[math] \left[ \begin{array} {rrr} \frac{34}{35} & -\frac{3}{5} \\ \frac{1}{7} & 0 \\ -\frac{3}{35} & \frac{1}{5} \\ \end{array} \right] [/math]

Since we are in the 5-limit here, we want the 5-limit JIP:

[math] \left[ \begin{array} {rrr} \log_2{2} & \log_2{3} & \log_2{5} \\ \end{array} \right] [/math]

Multiplying those:

[math] \begin{array} {l} \left[ \begin{array} {rrr} \log_2{2} & \log_2{3} & \log_2{5} \\ \end{array} \right] & × & \left[ \begin{array} {rrr} \frac{34}{35} & -\frac{3}{5} \\ \frac{1}{7} & 0 \\ -\frac{3}{35} & \frac{1}{5} \\ \end{array} \right] & = & \left[ \begin{array} {rrr} 0.998829 & -0.135614 \\ \end{array} \right] \end{array} [/math]

Those results are in octaves. Multiplying by 1200 gives them in cents: 1198.5948 and -162.7368. So we can recognize these two values as porcupine's octave period and its neutral second generator. The concern here is that this mapping form results in the generator being a neutral second downward, i.e. it has a negative value when measured logarithmically such as in octaves or cents as seen here.

The solution is simply to change the signs of the corresponding row of the matrix, like so:

[math] \begin{array} {lllllllll} \left[ \begin{array} {rrr} 1 & 2 & 3 \\ 0 & 3 & 5 \\ \end{array} \right] & \begin{array} {rrr} +1198.5948 \\ \style{background-color:#F2B2B4;padding:5px}{-}162.7368 \\ \end{array} & → & \left[ \begin{array} {rrr} 1 & 2 & 3 \\ 0 & \style{background-color:#FFF200;padding:5px}{-}3 & \style{background-color:#FFF200;padding:5px}{-}5 \\ \end{array} \right] & \begin{array} {rrr} +1198.5948 \\ \style{background-color:#FFF200;padding:5px}{+}162.7368 \\ \end{array} \end{array} [/math]

And the cents value of the generator flips accordingly.

Equave-reduced generator form

The equave-reduced generator form has the matrix modified from the defactored Hermite normal form so that each generator is equave-reduced, where the equave can be found as the formal prime represented by the first column of the matrix (which is usually the octave). For more information, see Octave reduction #Generalization.

Consider the case of septimal meantone. As we know, its defactored Hermite normal form is [⟨1 0 -4 -13] ⟨0 1 4 10]} which corresponds to generators of ~2/1 and ~3/1. In this case, as is typical, the formal prime represented by the first column of the matrix is 2, and so the equave is the octave. Therefore, all generators must be octave-reduced. But our second generator is ~3/1, which is not octave-reduced. We must alter the mapping in such a way that this row represents a generator of ~3/2 instead. We can do that here by adding the second row of the mapping to the first: [⟨1 1 0 -3] ⟨0 1 4 10]}. So that is septimal meantone's equave-reduced generator form, corresponding to generators of ~2/1 and ~3/2.

Probably the most reliable way to achieve equave-reduced generator form in general, however, is not to work with a generator preimage transversal such as ~3/1 and ~3/2. Instead the Frobenius generators may be used, as described in the positive generator form section just above, and reduction can be accomplished by calling modulo on their cents.

For a general discussion of how to manipulate the sizes of generators in this way, see Generator size manipulation.

Positive equave-reduced generator form

The positive equave-reduced generator form is similar to the equave-reduced generator form, but starts from the positive generator form instead of the defactored Hermite normal form.

Minimal generator form

The minimal generator form (or mingen form) is a form specific to rank-2 temperaments, where the generator is positive and no greater than half the period.

Graham Breed's temperament finder uses this form for all rank-2 temperaments. Septimal meantone in minimal generator form is [⟨1 2 4 7], ⟨0 -1 -4 -10]], corresponding to generators of ~2/1 and ~4/3.

Beyond rank-2, the mingen form of a temperament is no longer unique. You can always get smaller and smaller generators. This is why on Graham Breed's temperament finding tool, beyond rank-2 he simply uses the Hermite Normal Form.

Consider the example in the diagram given here: Generator form manipulation #Beyond rank-2. We begin with [⟨1 2 0 -1] ⟨0 -1 6 10] ⟨0 0 -1 -2]} with generators of 1200.6¢, 499.841¢, and 214.024¢, which therefore already satisfies the condition that each generator is less than half the previous generator. But we can transform it into [⟨1 2 2 3] ⟨0 -1 1 0] ⟨0 0 -1 -2]} which has a third generator of 116.013¢ instead. This is accomplished by adding row 3 to row 2 five times, which decreases generator 3 by the size of five times row 2, from 214.024¢ by 5 × 499.841 = 2499.205¢ to -2285.18¢; and then subtracting row 3 from row 1 twice, which increases generator 3 by the size of two times row 1, from -2285.18¢ by 2 × 1200.6¢ = 2401.2¢ to 116.013¢. And we can get that generator even smaller if we had instead moved up by 499.841 twice to 1213.71¢ and then down by 1200.6¢ once to 13.109¢ (that's a final mapping of [⟨1 2 -1 -3] ⟨0 -1 8 14] ⟨0 0 -1 -2]}.

You could find smaller and smaller generators if you wanted, by essentially finding increasingly small "commas" between the other generators' sizes (e.g. 5 × 1200.6¢ versus 12 × 499.841¢ is a difference of only 4.908¢) and then shifting generators by those commas.

This problem also precludes the possibility of a definitive maximum generator which is still less than half of the previous generator.

Reduced row echelon form (RREF)

Main article: Wikipedia: Row echelon form

If you have a routine which will compute the reduced row echelon form of a matrix with rows consisting of vals using rational number arithmetic, this can be computed and any rows consisting of the zero val stripped off. The result is a unique identifier for the abstract temperament which is closely related to the normal val list. The intervals of the abstract temperament may be found in the same way, by applying the mappings (which are fractional vals) to monzos; or put in another way, by matrix multiplication of the monzos by the reduced row echelon form matrix.

For example, if we feed [⟨22 35 51 62], ⟨31 49 72 87], ⟨84 133 195 236]] into a reduced row echelon form routine, we obtain [⟨1 0 3 1], ⟨0 1 -3/7 8/7], ⟨0 0 0 0]]. Stripping off the zero val in the final row, we get E = [⟨1 0 3 1], ⟨0 1 -3/7 8/7]]. The monzo for 7/6 is [-1 -1 0 1⟩, and E[-1 -1 0 1⟩ = [0 1/7]. Multiply by [1 0 0 0⟩, the monzo for 2, and the result is E[1 0 0 0⟩, which is [1 0]. We have in fact 7-limit orwell temperament, with period 2 and generator approximately 7/6.

For more information, see RREF.

Normal interval lists

Because regular temperaments, where the prime mappings are known but not the specific tuning of the generators, are fully characterized by their kernel, the group of intervals they map to the unison, they can also be characterized by the regular interval list of a set of generators (called commas or unison vectors) for the kernel. A similar set of normal forms are defined for interval lists. The defactored Hermite form and positive forms parallel those for vals. However, the normal form defined for intervals which has "minimal" in its name is quite different conceptually than the normal form defined for vals which has "minimal" in its name. Also, there is no notion of an equave-reduced form for intervals.

In the case of interval lists, the most common format they are presented in is as ratios, not vectors, e.g. [81/80, 64/63] rather than [[-4 4 1 0⟩ [-6 2 0 1⟩]. So you may need to convert ratios to vectors and back when working with these forms.

Antitransposed defactored Hermite form

Given a matrix of p-limit intervals, we can find its antitransposed defactored Hermite form in much the same way as we find the defactored Hermite form of a mapping. The only difference is that they must be put in an "antitranspose sandwich", or in other words, antitransposed once at the beginning, and then antitransposed again at the end.

For example, suppose we have the comma basis for septimal meantone:

[math] \left[ \begin{array} {rrr} -4 & 1 \\ 4 & 2 \\ -1 & -3 \\ 0 & 1 \\ \end{array} \right] [/math]

Note that the interval vectors are columns when put into matrix form like this.

So now we antitranspose, or in other words, transpose the matrix but instead of across its main diagonal (top-left to bottom-right) as with the traditional transpose, across its antidiagonal (top-right to bottom-left).

[math] \begin{array} {l} \left[ \begin{array} {rrr} \style{background-color:#46C5DD;padding:5px}{-4} & \style{background-color:#FFF200;padding:5px}{1} \\ \style{background-color:#FFF200;padding:5px}{4} & \style{background-color:#F2B2B4;padding:5px}{2} \\ \style{background-color:#C6DC67;padding:5px}{-1} & \style{background-color:#F2B2B4;padding:5px}{-3} \\ \style{background-color:#C6DC67;padding:5px}{0} & \style{background-color:#F2B2B4;padding:5px}{1} \\ \end{array} \right] & → & \left[ \begin{array} {rrr} \style{background-color:#F2B2B4;padding:5px}{1} & \style{background-color:#F2B2B4;padding:5px}{-3} & \style{background-color:#F2B2B4;padding:5px}{2} & \style{background-color:#FFF200;padding:5px}{1} \\ \style{background-color:#C6DC67;padding:5px}{0} & \style{background-color:#C6DC67;padding:5px}{-1} & \style{background-color:#FFF200;padding:5px}{4} & \style{background-color:#46C5DD;padding:5px}{-4} \\ \end{array} \right] \end{array} [/math]

This has the effect of both reversing the entries within each interval, as well as reversing the order of the intervals themselves. The purpose of these reversals is so that when the HNF tries to put all the zeros in the bottom-left corner, it gravitates them toward where we want them: the higher primes, and commas that will be earlier in the list after the second antitranspose.

Now we can defactor and HNF this as if it were a mapping.

[math] \left[ \begin{array} {rrr} 1 & 0 & -10 & -13 \\ 0 & 1 & -4 & 4 \\ \end{array} \right] [/math]

Finally we antitranspose again:

[math] \begin{array} {l} \left[ \begin{array} {rrr} \style{background-color:#F2B2B4;padding:5px}{1} & \style{background-color:#F2B2B4;padding:5px}{0} & \style{background-color:#F2B2B4;padding:5px}{-10} & \style{background-color:#FFF200;padding:5px}{-13} \\ \style{background-color:#C6DC67;padding:5px}{0} & \style{background-color:#C6DC67;padding:5px}{1} & \style{background-color:#FFF200;padding:5px}{-4} & \style{background-color:#46C5DD;padding:5px}{4} \\ \end{array} \right] & → & \left[ \begin{array} {rrr} \style{background-color:#46C5DD;padding:5px}{4} & \style{background-color:#FFF200;padding:5px}{13} \\ \style{background-color:#FFF200;padding:5px}{-4} & \style{background-color:#F2B2B4;padding:5px}{-10} \\ \style{background-color:#C6DC67;padding:5px}{1} & \style{background-color:#F2B2B4;padding:5px}{0} \\ \style{background-color:#C6DC67;padding:5px}{0} & \style{background-color:#F2B2B4;padding:5px}{1} \\ \end{array} \right] \end{array} [/math]

And there is our canonical comma basis.

The set of elements of the original list generates a finitely generated free abelian subgroup of the positive rationals under multiplication, and therefore of any p-limit group it lives inside. The list in normal form contains a minimal set of ratios, in an ordering of nondecreasing prime limit which is parsimonious in its use of higher limits. For example, if we put [81/80, 126/125] into normal form we obtain [80/81, 57344/59049]. The first interval is 5-limit, which is as small as possible. The second is 7-limit, which must be the case because the group these two generate is 7-limit. However, it uses only 2, 3 and 7 in its prime factorization, parsimoniously rejecting 5 as the next highest prime limit.

Note that the antitransposed defactored Hermite form of the comma list involves the list being defactored (e.g. torsion to be removed). For example, both [25/27, 35/36] and [25/27, 49/48] characterize beep. But the latter has torsion (i.e. is enfactored), so the former is beep's antitransposed defactored Hermite form.

Normal interval lists can also be used to characterize the just intonation subgroups on which subgroup temperaments are defined and using which subgroup scales may be constructed. On the pages chromatic pairs, subgroup temperaments and just intonation subgroups can be found many examples; the subgroup lists are given in a form where generators of the subgroup are separated by periods so as to flag the fact that the list defines a subgroup. An example would be the Barbados subgroup, 2.3.13/5.

Positive ratio form

Similar to the situation with mappings, even though by using the HNF the antitransposed defactored Hermite form ensures that the first nonzero entry of each comma's prime-count vector is a positive number, this does not necessarily mean that the corresponding commas are all positive in pitch.

For example, the antitransposed defactored Hermite form of meantone's comma basis is [[4 -4 1⟩]. HNF has ensured that the first number is positive. But the vector [4 -4 1⟩ represents the ratio 80/81, which is less than unity; it is still the meantone comma, but it is the meantone comma downward in pitch.

Having negative commas like this can be surprising or confusing, and so the positive ratio form addresses this concern. To correct any negative comma, simply replace it with its reciprocal. In vector form, this can be done by changing the signs on every number; meantone can be put into positive ratio form as [[-4 4 -1⟩].

The positive ratio form of septimal meantone is [81/80, 59049/57344].

Minimal form

Working from the positive ratio form, the minimal form shows the simplest comma list sufficient to define the temperament, where the commas are rated by a complexity measure such as the Tenney height, in which case it is called the Tenney-minimal form. It is obvious why this form is useful, but unfortunately, it is not easy to find since it is equivalent to the shortest basis problem, an NP-hard problem. The LLL reduction algorithm is a good approximation. Otherwise, it is typically solved heuristically by hand.

Note: This is the form shown in the "comma lists" of each temperament on this wiki.

Integer reduced row echelon form (IRREF)

Another important normal form for integer matrices is what Kite Giedraitis has dubbed the IRREF, the integer reduced row echelon form. It is the reduced row echelon form but with integer entries, found by multiplying each row of the matrix by the least common multiple of all denominators in that row. It differs from the Hermite normal form in that each pivot is the only nonzero entry in its column. For a monzo list, it has the advantage of limiting the appearance of the N highest primes to only one comma each (where N is the codimension), isolating each prime's effect on the pergen, but has the disadvantage that the commas tend to have high odd limits, and the comma list may have torsion.

Sometimes the IRREF is identical to the HNF. For more information, see IRREF.

Maple code

Below is Maple code for finding the normal interval and val list, given an interval list or a val list. Note that this code does not deal with torsion/enfactoring, so it assumes your lists have already been defactored.

Code

log2 := proc(x) evalf(ln(x) / ln(2)) end:

# transpose of listlist w
transpos := proc(w)
    local u;
    u := Matrix(w);
    u := LinearAlgebra[Transpose](u);
    convert(u, listlist)
end:

# pseudoinverse of listlist w
pseudo := proc(w)
    local u;
    u := Matrix(w);
    u := LinearAlgebra[MatrixInverse](u, method='pseudo');
    convert(u, listlist)
end:

psu := proc(w) transpos(pseudo(w)) end:

# log2 of first n primes
pril := proc(n)
    local i, u;
    u := NULL;
    for i from 1 to n do
        u := u, log2(ithprime(i))
    od;
    [u]
end:

# reverse of list
revlist := proc(l)
    local i, v, e;
    e := nops(l);
    for i from 1 to e do
        v[i] := l[e - i + 1]
    od;
    convert(convert(v, array), list)
end:

orp := proc(w, p) padic[ordp](w, p) end:

# rank of p-limit of q
pim := proc(q)
    local r, i, p;
    r := 1;
    i := 0;
    while not (r = q) do
        i := i + 1;
        p := ithprime(i);
        r := r * p ^ orp(q, p)
    od;
    if i = 0 then RETURN(0) fi;
    i
end:

# prime limit of rational number q
plim := proc(q)
    ithprime(pim(q))
end:

# converts rational number q to monzo of length n
rat2monz := proc(q, n)
    local v, i;
    for i from 1 to n do
        v[i] := ordp(q, ithprime(i))
    od;
    convert(convert(v, array), list)
end:

# converts monzo to rational number
monz2rat := proc(m)
    local i, t;
    t := 1;
    for i from 1 to nops(m) do
        t := t * ithprime(i) ^ m[i]
    od;
    t
end:

# hermite normal form of listlist l
herm := proc(l)
    local M;
    M := Matrix(l);
    convert(convert(HermiteForm[Z](M), array), listlist)
end:

# normal interval list from list of intervals l
nori := proc(l)
    local i, p, u, v, w;
    p := 1;
    for i from 1 to nops(l) do
        p := max(p, plim(l[i]))
    od;
    u := [];
    for i from 1 to nops(l) do
        u := [op(u), revlist(rat2monz(l[i], p))]
    od;
    v := herm(u);
    for i from 1 to nops(l) do
        u := revlist(v[i]);
        u := monz2rat(u);
        w[i] := u
    od;
    u := [];
    for i from 1 to nops(l) do
        v := w[i];
        if v < 1 then v := 1 / v fi;
        if not v=1 then u := [op(u), v] fi
    od;
    revlist(u)
end:

norv := proc(l)
    # normal val list from list of vals l
    local u, v, w, i, n, a;
    u := herm(l);
    n := rnk(u);
    v := NULL:
    for i from 1 to n do
        v := v, u[i]
    od;
    v := [v];
    u := pseudo(v);
    w := pril(nops(l[1]));
    a := op(matmul(w, u));
    u := NULL;
    for i from 1 to n do
        if a[i] < 0 then u := u, -v[i] fi;
        if a[i] >= 0 then u := u, v[i] fi
    od;
    [u]
end: