This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

Previous Next Last

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950

950 - 975 -

Message: 975 - Contents - Hide Contents

Date: Wed, 05 Sep 2001 23:03:45

Subject: Re: Tenney's harmonic distance

From: genewardsmith@j...

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


> You better believe it! So, any comments on the questions I asked?


I'll look at it again, but I have some questions also:

(1) Can you define harmonic entropy in terms of your taxicab metric, 
or if not in any terms you like?

(2) Do you know how to retune a midi file in such a way that the 
pitches are set to anything you choose?

Message: 976 - Contents - Hide Contents

Date: Wed, 05 Sep 2001 23:22:00

Subject: Re: Question for Gene

From: genewardsmith@j...

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


> In particular, I'm assuming a city-block or taxicab metric. Is Kees 
> observing that in his final lattice? It looks like he isn't.


Kees is talking about circles and transforming as if in a Euclidean 
space, so you aren't on the same wavelength.


> What else can you say?


I'm not sure what your triangular lattice metric is. A taxicab needs 
two lines to run along; you can make these 120 degrees to each other 
but you can't get an array of equilateral triangles out of it.

By "lattice", mathematicians usually mean one of two things. The 
first has to do with partial orderings and need not concern us, the 
second defines a lattice as a discrete subgroup of R^n whose quotient 
group is compact. I'm not always sure what people mean when they say 
lattice in this neighborhood.

Message: 977 - Contents - Hide Contents

Date: Thu, 06 Sep 2001 03:26:37

Subject: about hypothesis and theorem

From: Pierre Lamothe

Hi Paul and tuning-math members

I was surprised to find intense (and abstract) activity on the List after
my vacation. It takes a while before I have leisure to read all that. I
regret to have not the possibility to participate. However I would like
simply to ask a question permitting to see it misses probably a condition.

Let u and v be the vectors 25/24 and 27/20 in the lattice <2 3 5> Z^3 whose
generic element is (2^x)(3^y)(5^z). The vectors u and v determine (with the
octave) the "pathologic" periodicity block <1 9/8 5/4 3/2 15/8> supposed
valid (in the theorem) since it corresponds to the homomorphism

   H(x,y,z) = 5x + 8y + 14z

Could you show how the hypothesis, the definitions, the conditions of
validity and the theorem would be applied in this case? Could you exhibit a
generator and a scale?

Pierre

Message: 978 - Contents - Hide Contents

Date: Thu, 06 Sep 2001 07:39:23

Subject: Re: Theorem Paul

From: genewardsmith@j...

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


> Now do you have a quick way of determining the generator of the 
> linear temperament, given n-1 commatic unison vectors?


Let's see if this helps:

Recall that a notation for the note group N_p was a k-tuple of vals 
[u_1, ..., u_k], where k = pi(p) is the number of primes up to p, and 
where the kxk square matrix we get by writing the vals as column 
vectors is unimodular, meaning it has determinant +-1. We may call 
this the *val matrix* for the notation; corresponding to it is a 
*basis matrix* which is the matrix inverse of the val matrix. The 
rows of the basis matrix are the basis notes of the notation, and it 
may also be written as a k-tuple of rational numbers
(q_1, ..., q_k) where if q_i = [e_1, e_2, ..., e_k] we also write it 
multiplicitively as the rational number q_i= 2^e_1 * 3^e_2 ... p^e_k.

We then have for any prime r <= p

r = q_1^u_1(r) * q_2^u_2(r) * ... q_k^u_k(r),

so that anything which can be written as the product of the first k 
primes can also be written as the product of q_1, ..., q_k; that is, 
both are a basis for the note group N_p.

I just downloaded Graham's midiconv program, and he is doing this 
sort of thing in his tun files. For instance, in 12from31.tun we find 
the matrix

[-3 -1  2]
[ 7  0 -3]
[-4  4 -1]

which is a basis matrix (since it is unimodular.) Inverting it we get 
the val matrix

[12  7 3]
[19 11 5]
[28 16 7],

which is the notation [h_12, h_7, h_3]. If you look at Graham's file 
you will see he is using this notation.

Every note in N_5 can be expressed in terms of this notation as

q = (25/24)^h_12(q) * (128/125)^h_7(q) * (81/80)^h_3(q)

just as it can also be written

q = 2^v_2(q) * 3^v_3(q) * 5^v_5(q),

where v_2, v_3 and v_5 are the 2-adic, 3-adic and 5-adic valuations 
of number theory.

Suppose now we want to temper out 81/80, so that we will write the 
approximation to q, ~q, as

~q = a^h_12(q) * b^h_7(q).

Finding a basis for this temperament means the same as tuning the 
above basis, which we may do in various ways, e.g., least squares. If 
we like we may assume ~2 = 2, in which case we really need to specify 
only one value, since the other than be found from

2  =  a^h_12(2) * b^h_7(2) = a^12 * b^7.

To take another example, consider the basis matrix defined by the 
5-tuple (176/175, 385/384, 8019/8000, 441/440, 540/539), which in 
matrix form is

[ 4  0 -2 -1  1]
[-7 -1  1  1  1]
[-6  6 -3  0  1]
[-3  2 -1  2 -1]
[ 2  3  1 -2 -1].

The inverse of this matrix is

[72   58 -31   53  46]
[114  92 -49   84  73]
[167 135 -72  123 107]
[202 163 -87  149 129]
[249 201 -107 183 159],

which is the notation [h_72, h_58, -h_31, h_53, h_46]. If we remove 
any one element from the basis 5-tuple, and take octave equivalence 
in its place, we get a JI block whose number of notes is abs(h(2)) 
for the val corresponding to the basis element we removed. For 
instance, by taking out 540/539, which is in the kernel of all the 
vals but h_46, which has instead h_46(540/539)=1, we get a block of 
46 notes. We may temper this in various ways by removing other 
val/basis pairs, getting equal, linear etc. temperaments. Thus for 
instance by tuning  ~q = a^h_72(q) (for instance, in the usual way!) 
we get the 46 block expressed in the 72-et. If we tune 
~q = a^h_72(q) * b^h_53(q), we get a linear temperament, and so forth.

We also have for example that ker(h_72) is generated by all the basis 
vectors except 176/175, where h_72(176/175)=1. Just as each val is 
associated to the group it generates (of rank one) and hence to the 
dual group, i.e. the kernel, of corank one (in this case, that would 
be rank four), every basis note q_i generates a rank one group, whose 
dual group null(q_i) is of corank one (in this case four again.) 

While null(q_i) is of corank one and has an infinity of elements, if 
we list only valid vals of the form u_n for integers n we get a 
finite list, which is an interesting thing to consider for any 
comma-like interval. For instance, 128/125 is associated in this way 
to multiples of 3 through 42, excluding h_6 which is invalid. In the 
same way, 25/24 is associated to 3,4,7,10,13 and 17; and 81/80 to 
5,7,12,19,26,31,43,45,50,55,67,69,74,81,88,98,105 and 117. If we 
place some limit based on a measure of goodness when we do this we of 
course can get an even smaller list.

Message: 979 - Contents - Hide Contents

Date: Thu, 06 Sep 2001 08:20:11

Subject: Re: about hypothesis and theorem

From: genewardsmith@j...

--- In tuning-math@y..., Pierre Lamothe <plamothe@a...> wrote:


> Could you show how the hypothesis, the definitions, the conditions 
of
> validity and the theorem would be applied in this case? Could you 
exhibit a
> generator and a scale?


We find that h_4 has the property h_4(25/24)=0 and h_4(27/20)=1. We 
then look at vals of the form t*h_5 + h_4, and when t=1 we get

    [ 9]
g = [13]
    [20].

Note that this is *not* h_9, which has coordinate values 9, 14 and 21.
However, 7/5 is a semiconvergent to 13/9, 11/5 is a semiconvergent to 
20/9 and for that matter 1/5 is a semiconvergent to 2/9. We get a 
scale of pattern 22221, 5 steps in a 9-et. It may not do a very good 
job of representing your "pathological" block, but then 27/20 is not 
much of a comma. If you want to exclude this kind of thing we need to 
change the statement of the theorem, but then we must ask what, 
exactly, people want to prove.

Previous Next Last

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950

950 - 975 -