A proposed definition for block

I think I've found a definition for "block" which is both simple and
covers what we would want covered; I'd be interested to hear
criticism either that something is called a block which shouldn't be,
or is not which should be.

Let <b_2,..., b_k> be a (k-1)-tuple of positive rational numbers in
the p-limit, where k=pi(p). Let N be the (k-1)xk matrix whose ith row
corresponds to the factorization of b_i into primes, and suppose the
(k-1)x(k-1) minors of N (that is, the determinants of the (k-1)x(k-1)
square submatricies of N) are relatively prime, so that their gcd is
1, and that the first minor, obtained by excluding the first column,
is n <>0. Let M be the square matrix obtained by adding a top row
[1,0,0,…,0], corresponding to 2, to N, whose determinant n is
therefore not zero, so that M^(-1) is defined.

We define a norm on the k-dimensional real vector space R^k, which we
denote ||v|| where v = [v_1,...,v_k], by taking the product
w = v M^(-1), with coordinates [w_1, …, w_k], and setting
||v|| = max(|w_1|,..., |w_k|). This makes R^k into a normed vector
space and the notes of the p-limit into a lattice. If S is any set of
p-notes (i.e., of positive rational numbers in N_p), then we define
the *diameter* dia(S) of S to be max_{i,j \in S}(||s_i – s_j||) over
all pairs of elements s_i and s_j in S. We define S to be a block for
the set <b_i> if S is maximal subject to the condition that its
diameter be less than one. By maximal I mean no element can be added
to S without the new set having a diameter of at least one.

We could add conditions to <b_i> to make the blocks we obtain more
reasonable—-in particular, the condition that it can be completed to
a valid basis for N_p.

--- In tuning-math@y..., genewardsmith@j... wrote:
> I think I've found a definition for "block" which is both simple
and
> covers what we would want covered; I'd be interested to hear
> criticism either that something is called a block which shouldn't
be,
> or is not which should be.

Can you give some examples? I'm not following all the implications.

P.S. I'm not sure if you're implying a certain "metric" here,
specifically one in octave-equivalent space . . . if you are, I'd
warn you that I tend to be adamant about using a "triangular" rather
than a "rectangular" lattice, so that 15:8 is a longer distance than
5:3 . . . otherwise, I get very upset :)

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:

> Can you give some examples? I'm not following all the implications.

I'll give some examples; I had it in mind to continue with the "N"
series of scales in any case.

> P.S. I'm not sure if you're implying a certain "metric" here,
> specifically one in octave-equivalent space . . . if you are, I'd
> warn you that I tend to be adamant about using a "triangular"
rather
> than a "rectangular" lattice, so that 15:8 is a longer distance
than
> 5:3 . . . otherwise, I get very upset :)

There is a metric (on notes, not just on equivalence classes), since
a normed vector space entails a metric. A real normed vector space V
is a real vector space together with a norm map || ||:V --> R, such
that for v in V we have

(1) ||v|| >= 0

(2) ||v|| = 0 <==> v = 0

(3) If c is a scalar, then
||c v|| = |c| ||v||

(4) ||u + v|| <= ||u|| + ||v||

Any normed vector space is a metric space in which the norm map is a
continuous function. An example would be the taxicap (L1) norm,
leading to your favorite taxicap metric.

The metric in this case is what it has to be in order for the
definition to work, and it depends on the choice of unison vectors.
There's no use getting mad at it, whatever it may be in any
particular case. :)