Normal lists

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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 every normal form 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.

Defactored Hermite Form

This is Dave Keenan and Douglas Blumeyer's "Canonical form" for a temperament, formed from defactoring the matrix (aka removing contorsion) prior to putting it into Hermite form.

We may write a list of [math]k[/math] vals as an [math]k×d[/math] matrix, where the rows of the matrix are the vals, and [math]d[/math] is the dimensionality of the system[1]. To get the Defactored Hermite form, we do the following:

  1. First, defactor it (aka, make sure it is saturated). [2]. Note that if the matrix was not full-rank, this will result in the elimination of some rows[3]. We now have an [math]r×d[/math] 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 form of [1 0 -4 -13] 0 1 4 10], corresponding to generators of ~2/1 and ~3/1. This also happens to be the same as its Hermite form, given a basis of vals which doesn't have any enfactoring/contorsion.

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 k×m matrix, A+, and multiply this from the left by the row vector of JIP, J0 = [1 log23 log25 … log2p].

[math]G_\text{F} = J_0 A^+[/math]

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

The "mapping" (though not the "map 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.

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're 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 \\ \colorbox{pink}{$-$}162.7368 \\ \end{array} & → & \left[ \begin{array} {rrr} 1 & 2 & 3 \\ 0 & \colorbox{yellow}{$-$}3 & \colorbox{yellow}{$-$}5 \\ \end{array} \right] & \begin{array} {rrr} +1198.5948 \\ \colorbox{yellow}{+}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 is similar to the positive generator form, but the matrix is further normalized such 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 positive generator 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 JI preimages of the generators such as ~3/1 and ~3/2, which may not always be obvious or unambiguous, and which can be tricky to find. 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 example the transversal generators for the positive generator form of septimal meantone are 20253807/9765625 and 3/1, not 2/1 and 3/1, so it's not obvious that 20253807/9765625 is very close to 2/1 and it wouldn't be pleasant to reduce 3/1 by that. However in cents these are 1201.34 and 1898.56, so it's quite obvious there that you need to reduce 1898.56 by 1201.34 once to get it between 0 and 1201.34.

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

Minimal generator form

The minimal generator form (or mingen form) is a form specific to rank-2 temperaments, where the matrix is normalized such that the generator is positive and no greater than half the period.[4][5]

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 size 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.

Normal interval lists

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.

Defactored Hermite form

Given a matrix of p-limit intervals, we can find its defactored Hermite form in much the same way as a mapping. The only difference is that the matrix must be antitransposed once at the beginning of the operation and once again at the end.

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 normalized list 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 normalize [81/80, 126/125] 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. 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. The above normal interval list, for example, characterizes septimal meantone, defining the normal comma list of septimal meantone.

Note that the defactored Hermite form of the comma list requires the list to be 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/is enfactored, so the former is Beep's 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 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 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 ratio form

The minimal ratio form shows ratio-wise the simplest comma list sufficient to define the temperament, where the ratios are rated by the product of the numerator and the denominator, i.e. the Benedetti height.

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

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:

Notes

  1. Calling the prime-counting function, written π(x), on the prime limit will give us this number. For examples, π(2) = 1, π(3) = 2, π(5) = 3, π(7) = 4, π(11) = 5, etc.
  2. Historically, this step was not explicitly recognized as necessary for normal forms. It is quite likely that the vast majority of normal forms found on the wiki are not contorted/enfactored, but specifically defining this canonical form to include this requirement is an important step toward ensuring that, which will prevent redundant temperaments from being catalogued. In various domains, normal forms are often required to be unique, however, canonical forms are required to be unique even more often that normal forms are; according to Wikipedia: Canonical form, 'the distinction between "canonical" and "normal" forms varies from subfield to subfield. In most fields, a canonical form specifies a unique representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness.' This is the rationale behind defining "canonical" as opposed to merely "normal". To be more specific, The HNF does provide a unique representation of matrices, i.e. from a perspective of pure mathematics, and so you will certainly find throughout mathematical literature that HNF is described as providing a unique representation, and this is correct. However, when applied to the RTT domain, i.e. to mappings, the HNF sometimes fails to identify equivalent mappings as such. And the critical flaw with HNF is its failure to defactor matrices - meaning that a "contorted" mapping matrix has a different Hermite normal form than a non-contorted one with the same kernel - and this is because dividing rows is not a permitted elementary row operation when computing the HNF. See: https://math.stackexchange.com/a/685922 The canonical form as described here does defactor matrices, and therefore it delivers a truly canonical result.
    There is also a rarely mentioned Hermite Canonical Form, or HCF, described here: http://home.iitk.ac.in/~rksr/html/03CANONICALFACTORIZATIONS.htm, which sort of combines the HNF's normalization constraint and the RREF's reduced constraint (all pivots equal 1, all other entries in pivot columns are 0, both above and below the pivot), but we didn't find it useful because due to its constraint that all pivots be 1, it does not preserve periods that are genuinely unit fractions of an octave (at first glance, when a pivot is not equal to 1, it might trigger you to think that the mapping is enfactored. But temperaments can legitimately have generators that divide primes evenly, such as 5-limit Blackwood, [5 8 0] 0 0 1], which divides the octave into 5 parts. So any form that enforces pivots all be 1's, such as HCF and RREF, would fail this criteria.) It also doesn't qualify as an echelon form, which becomes apparent only when you use it on rank-deficient matrices, because it doesn't require the rows of all zeros to be at the bottom; instead it (bizarrely, though maybe it's related to how the SNF requires all pivots exactly along the main diagonal) requires the rows to be sorted so that all the pivots fall on the main diagonal.
  3. Note that canonicalizing a mapping does not remove trailing dimensions with only zeros.
    In the case of a mapping, this would take the form of an extra column of all zeros to the right of any non-zero entries, or in other words, an unmapped prime higher than other mapped prime. For example you could have [1 0 -4 0] 0 1 4 0] which is just 5-limit meantone but represented in the 7-limit even though prime 7 is not used.
    And for a comma-basis the form this would take is rotated 90 degrees: a row of all zeros below all other nonzero entries, e.g. [4 -4 1 0].
    The reason these additional zeros should be preserved and these temperaments be treated as different from their untrimmed counterparts is made clear when we consider the difference in the duals. For a comma-basis, the extra dimension implies the presence of extra generators that are unbound to the other generators. For example, a basis for the anti-null-space of [4 -4 1], or in other words its mapping, as we know well is [1 0 -4] 0 1 4]. But that is not a basis for the anti-null-space of [4 -4 1 [math]\color{red}0[/math]]; the mapping for that comma-basis would have to be [1 0 -4 [math]\color{red}0[/math]] 0 1 4 [math]\color{red}0[/math]] [math]\color{red}0[/math] [math]\color{red}0[/math] [math]\color{red}0[/math] [math]\color{red}1[/math]].
  4. This is somewhat like octave reduction combined with octave inversion, because you can't just add or subtract half octaves until it's between 0 and 600 cents; you have to add or subtract octaves until it's between -600 and +600 cents, then multiply by -1 if it's negative.
  5. You could always find a smaller and smaller generator by going negative, so this assumes positive generators.