Comma Sequences

A comma sequence defines a temperament by means of a sequence of uniquely determined commas at increasing prime limits. If p is the prime limit and n is the number of generators (two for a linear temperament, and so forth) then the first comma of the comma sequence will be in the q-limit, where q is the nth odd prime, and the subsequent commas in prime limits each one prime beyond the last. Hence a linear temperament will have a 5-limit comma as the first comma of the comma sequence, a 7-limit comma as the second comma, and so forth.

The first comma of the comma sequence is the reduced (meaning it is greater than one and not a power) q-limit comma of the temperament with smallest numerator and denominator; usually there is only one q-limit comma to be considered. Subsequent commas of the comma sequence are such that the wedge product of these commas is reduced, in the sense that the greatest common divisor of its coefficients is 1, and such that they are the commas of smallest Tenney height, defined as the product of the numerator and the denominator for the ratio in its reduced form, or equivalently as the one with smallest Tenney norm, which define the reduction of the temperament to that prime limit. The wedge product of the comma sequence defines, via its complement, the wedgie; and requiring that the gcd is 1 is equivalent to requiring that every comma of the system is a product of commas in the comma sequence; in other words, there is no torsion.

If the comma sequence is uniquely determined, it supplies one way of identifying temperaments. Its particular value is that it shows family relationships clearly. If [c_1, c_2, ...c_n] is a comma sequence, we may call a sequence [c_1, ..., c_n, c_{n+1}] in the next prime limit a daughter sequence, and the corresponding temperament a daughter, to which the preceding one is mother. We then have granddaughters, grandmothers, sisters etc. in the obvious way. If the generators of the daughter correspond to the generators of the mother, we can call it legitimate, otherwise illegitimate; if we want to get really cute we could make the illegitimate children all male as a way of telling them apart. We may also extend these relations to cousins, second cousins, great aunts and so forth, legitimate and otherwise.

For instance, the comma sequence for the 11-limit version of meantone called huygens is [81/80, 126/125, 99/98] whereas the sequence for meanpop is [81/80, 126/125, 385/384]. These are therefore sisters, with their mother standard septimal meantone, with sequence [81/80, 126/125] and grandmother 5-limit meantone with sequence [81/80]. The temperament with sequence [81/80, 525/512] is Aunt Flattone, and the one with sequence [81/80, 2401/2400] is the scapegrace Uncle Squares, the illegitimate son of 5-limit Grandma Meantone. Uncle Mothra, with sequence [81/80, 1029/1024] is another illegitimate son of Grandma Meantone, and his legitimate daughter, Cousin Cynder, is therefore a relation also, of course, with sequence [81/80, 1029/1024, 99/98].

Sometimes this definition of the comma sequence causes problems, leading to situations where the family relationships are not defined; a way around those problems, though often with less elegant looking commas, is the Hermite comma sequence. This is obtained by taking a set of commas defining the kernel, converting this to a matrix of monzos, reversing the order of the elements in the rows of the monzos, reducing to Hermite normal form, and reversing once again. So, for instance, from [81/80, 126/125, 385/384] we obtain the matrix of monzos [<-4 4 -1 0|, <1 2 -3 1 0|, <-7 -1 1 1 1|]; reversing the rows gives the matrix [[0 -1 4 -4], [0 1 -3 2 1], [1 1 1 -1 -7]], and Hermite reduction of this gives [[0 0 1 -4 4], [0 1 0 -10 13], [1 0 0 13 -24]]. Reversing the rows again converts back to monzos, and if we also change sign as needed to obtain comma greater than one, we obtain [<-4 4 -1 0|, <-13 10 0 -1 0|, <-24 13 0 0 1|]. The Hermite comma sequence is uniquely defined in all cases, and can always be used to define family relationships for temperaments.