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Message: 1550 - Contents - Hide Contents Date: Wed, 05 Sep 2001 23:03:45 Subject: Re: Tenney's harmonic distance From: genewardsmith@xxxx.xxx --- 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: 1551 - Contents - Hide Contents Date: Wed, 05 Sep 2001 23:22:00 Subject: Re: Question for Gene From: genewardsmith@xxxx.xxx --- 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: 1552 - Contents - Hide Contents Date: Wed, 05 Sep 2001 01:14:22 Subject: Re: Distance measures cut to order From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:> One of the most interesting mappings done by Wilson is his mapping > onto a Penrose tiling, treating it as a two dimensional > representation of a 5 dimensional space. Wowsers!When nines or fifteens> are treated as independent axes from threes and fives, > interesting 'wormholes' in the lattice start to appear, where > alternative representations of the same pitch class occur in > surprisingly different contexts.These wormholes will appear no matter what metric you use. They might be thought of as universal commas--3^2 is "approximated" by 9.
Message: 1553 - Contents - Hide Contents
Date: Thu, 06 Sep 2001 18:38:19
Subject: Re: about hypothesis and theorem
From: genewardsmith@xxxx.xxx
--- In tuning-math@y..., Pierre Lamothe <plamothe@a...> wrote:
I was looking at your 25/24 and 27/20 example again, and it seems to
me your objection that my conditions on validiy are too weak is well-
taken. I get
[ 5]
v = [ 7]
[11]
for the corresponding val, and 1-6/5-4/3-3/2-5/3-(2) for the block.
The trouble is, the scale steps are not in order! We have v(1)=0,
v(6/5)=1, v(4/3)=3, v(3/2)=2, v(5/3)=4, and it seems we should not
allow such a beast. Perhaps requiring that the scale steps be in
order for a set to be valid would be enough.
Message: 1554 - Contents - Hide Contents Date: Thu, 06 Sep 2001 18:59:25 Subject: Re: Theorem Paul From: genewardsmith@xxxx.xxx --- In tuning-math@y..., genewardsmith@j... wrote:> If {n_i} are a set of generators for the kernel of a valid val v (or > in other words, if they generate the dual group to the group > generated by v) then we call the set valid.This needs to be changed to the following: If n={n_i} are a set of generators for the kernel of a valid val v, and if B is the block defined by n and octave equivalence, and if the elements of B b_i are placed in ascending order by ascending values of v(b_i), then we will call the set valid. The reason why the new condition is essential is that we are creating MOS by finding something close to the v(2)-et. That is a sort of super-MOS, with only one step size and everything as smooth as possible. However, if the steps are not in order it is not a super MOS at all, and we don't *want* to get close to it!
Message: 1555 - Contents - Hide Contents
Date: Thu, 06 Sep 2001 15:46:22
Subject: Re: about hypothesis and theorem
From: Pierre Lamothe
In post 979 <genewardsmith@j...> wrote:
<< We find that h_4 has the property
h_4(25/24)=0 and h_4(27/20)=1 >>
My question concerned an homomorphism in G = <2 3 5> Z^3
such that H(25/24) = 0 and H(27/20) = 0.
Using X = (x,y,z) this homomorphism is
H(X) = 5x + 8y + 14z
and its projection in G/<2>Z is
H(X mod 2) = (3y + 4z) mod 5
-----
The validity condition used in the theorem appears
independant of the primes, requiring only positive vals.
So, in G = <2 3 5> Z^3, forget the basis <2 3 5> and
replace it by B = <2 p1 p2> where p1 and p2 are unknown.
The unison vectors are now v = (i,3,-1) and u = (j,-1,2)
which were 27/20 and 25/24 in the basis <2 3 5>. Here
i and j are unknown, but the periodicity remains 5 and
the represention modulo 2 of the kernel (class 0 or
sublattice generated by u and v) is the same. What may
change is only the ordering of classes within the block.
0 0 0
0 * 0 0
0 0 X X * 0
0 0 X X 0
0 0 * 0
0 0 0
0 0 0 0
The homomorphism, which was H(X) = 5x + 8y + 14z, is now
[x i j]
H(X) = det [y 3 -1] = 5x - (2i + j)y - (i + 3j)z
[z -1 2]
where i and j correspond to the "modality" of the unison
vectors. In the basis <2 3 7> this set of unison vectors
is perfectly valid and the intervals between the elements
of the block are precisely the complete slendro gammier.
I could have chosen a more "pathological" (skewed) case.
This one underline the problem with a weak conception
using periodicity block.
In the basis <2 3 5> the classes modulo 5 of the intervals
between the elements of the block are identical to the
<2 3 7> case with
H(X mod 2) = (3y + 4z) mod 5
0 0 0
0 * 0 0
0 * 3 1 4 2 * 0
0 4 2 0 3 1 0
0 * 3 1 4 2 * 0
0 * 0
0 0 0 0
while these intervals correspond precisely to the Zarlino
gammier which is heptatonic and not pentatonic. Where's the
problem?
[ I will neglect in the following the skewness
of the mesh determined by a particular set of
unison vectors for a given homomorphism. I
want to focus on ordering and one can suppose
it's the simplest block, so having the minimal
complexity product or sonance sum of vectors. ]
Any homomorphism determines a partial ordering structure in
a lattice corresponding to its classes. Indeed, each vector
X is "labeled" by H(X) and the set of vectors is partially
ordered by the total order (... -3 -2 -1 0 1 2 3 ...) in Z.
For each "label" (or class) there exist an infinity of
vectors (or intervals) giving a dense recovering of all the
octaves. So this infinite ordering has nothing to do with
the ordering of the "size" (width) of the intervals (even
if it serves to find temperaments).
The algebra of classes has sense only to give consistency.
What is required in JI is a corresponding partial algebra of
intervals in an minuscule domain around the unison where it
remains possible to perceive difference in sonance quality.
So here's an essential condition in the use of unison vectors
and periodicity block:
---------------------------------------------------------
The order of classes has to correspond to order of widths
for the intervals of the chosen (supposed minimal) block,
and consequently for the intervals between these elements.
---------------------------------------------------------
Comparing the order in the "pathological" case with <2 3 5>
and the valid case with <2 3 7> we have
0 1 2 3 4
1 9/8 15/8 3/2 5/4
1 5/3 4/3 10/9 16/15
1 8/5 9/5 16/15 6/5
1 9/8 21/16 3/2 7/4
1 7/6 4/3 12/7 16/9
1 8/7 9/7 32/21 12/7
If the ordering structure is not considered, the pathological
structure is isomorph to the valid one (for the composition
of the intervals). It's a CS but non ordered. If we add width
ordering to these structures it's no longer isomorph.
-----
Using the 13 elements intervals generated by the "pathological"
block, one can use my methods deriving from gammier theory to
find the corresponding valid set of unison vectors.
Considering the ordered set of these intervals (width order),
we have simply to find the atoms of the set which are those that
cannot be factorized in "inferior" elements distinct of unison
within this set.
These atoms are here 16/15, 10/9 and 9/8. If this set of 13
intervals is consistent, we will find an homomophism such that
H(16/15) = H(10/9) = H(9/8) = 1
which is effectively
H(X) = 7x + 11y + 16z
giving
H(X mod 2) = (4x + 2z) mod 7
0 0 0
0 * 0
* 1 4 2 6 * 0
0 6 3 0 4 1 0
* 1 4 2 6 *
0 * 0
0 0
0 0
The simplest mesh is determined by 81/80 and 25/24
0 0 0
0 * 0
0 2 6 3 * 0
0 0 4 1 5 0
0 *
0 0 0
0 0
0 0
Finally, translating the block to fit within our
13 intervals we have the well-known Zarlino mode.
0 0 0
0 * 0
* 5 2 6 * 0
0 3 0 4 1 0
* *
0 * 0
0 0
0 0
Pierre
Message: 1556 - Contents - Hide Contents Date: Thu, 06 Sep 2001 20:46:10 Subject: Re: Tenney's harmonic distance From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- 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?Look over the harmonic entropy list (are you subscribed yet)? It's a continuous function of interval size, has local minima at the simple ratios, is conceptually very simple . . .> > (2) Do you know how to retune a midi file in such a way that the > pitches are set to anything you choose?Lots of people should be able to help you with this. Herman Miller?
Message: 1557 - Contents - Hide Contents Date: Thu, 06 Sep 2001 20:50:37 Subject: Re: Question for Gene From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- 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 about the previous lattices on that page?>>> 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.Well, they're not equilateral triangles, but they are triangles. The metric is the shortest path along the edges of this triangular graph. Is it not correct to call that a "taxicab metric"?> > 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.Sir, there is an accepted definition of "lattice" that is used in geometry and crystallography. Every point, and its local connections, is congruent to every other point and its local connections . . . something like that. We had this discussion a long time ago on the tuning list.
Message: 1558 - Contents - Hide Contents Date: Thu, 06 Sep 2001 20:50:57 Subject: Re: Tenney's harmonic distance From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Look over the harmonic entropy list (are you subscribed yet)? It's a > continuous function of interval size, has local minima at the simple > ratios, is conceptually very simple . . .I looked at it; I think you need to upload something to the files which defines what the group is about by defining harmonic entropy.
Message: 1559 - Contents - Hide Contents Date: Thu, 06 Sep 2001 20:53:37 Subject: Re: about hypothesis and theorem From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- 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.I think we have to add the condition that the JI block, pre- tempering, is CS. My proof doesn't work otherwise.
Message: 1560 - Contents - Hide Contents Date: Thu, 06 Sep 2001 20:57:51 Subject: Re: Tenney's harmonic distance From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> Look over the harmonic entropy list (are you subscribed yet)? It's > a>> continuous function of interval size, has local minima at the > simple>> ratios, is conceptually very simple . . . >> I looked at it; I think you need to upload something to the files > which defines what the group is about by defining harmonic entropy.There are a couple of posts which tell you how to calculate harmonic entropy . . . let me know if those are clear.
Message: 1561 - Contents - Hide Contents Date: Thu, 06 Sep 2001 22:07:34 Subject: Re: about hypothesis and theorem From: genewardsmith@xxxx.xxx --- In tuning-math@y..., Pierre Lamothe <plamothe@a...> wrote:> > In post 979 <genewardsmith@j...> wrote: > > << We find that h_4 has the property > h_4(25/24)=0 and h_4(27/20)=1 >> > > My question concerned an homomorphism in G = <2 3 5> Z^3 > such that H(25/24) = 0 and H(27/20) = 0. > > Using X = (x,y,z) this homomorphism is > > H(X) = 5x + 8y + 14zActually, this is the homomorphism for 48/25 and 27/20. We can find the right one by taking the determinant of [ x y z] [-2 3 -1] [-3 -1 2], which is 5x + 7y + 11z.> Any homomorphism determines a partial ordering structure in > a lattice corresponding to its classes. Indeed, each vector > X is "labeled" by H(X) and the set of vectors is partially > ordered by the total order (... -3 -2 -1 0 1 2 3 ...) in Z.I just told Paul this definition of lattice we needed worry about and now you go and use it. :)> So here's an essential condition in the use of unison vectors > and periodicity block: > --------------------------------------------------------- > The order of classes has to correspond to order of widths > for the intervals of the chosen (supposed minimal) block, > and consequently for the intervals between these elements. > ---------------------------------------------------------This seems to be what I have just proposed.> Using the 13 elements intervals generated by the "pathological" > block, one can use my methods deriving from gammier theory to > find the corresponding valid set of unison vectors.Where is gammier theory described? Welcome back, Pierre!
Message: 1562 - Contents - Hide Contents Date: Thu, 06 Sep 2001 22:25:01 Subject: Re: Tenney's harmonic distance From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> There are a couple of posts which tell you how to calculate harmonic > entropy . . . let me know if those are clear. Which posts?
Message: 1563 - Contents - Hide Contents Date: Thu, 06 Sep 2001 22:28:58 Subject: Re: Tenney's harmonic distance From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> There are a couple of posts which tell you how to calculate > harmonic>> entropy . . . let me know if those are clear. > > Which posts?350 . . . follow the thread from there.
Message: 1564 - 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: 1565 - Contents - Hide Contents Date: Thu, 06 Sep 2001 07:39:23 Subject: Re: Theorem Paul From: genewardsmith@xxxx.xxx --- 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: 1566 - Contents - Hide Contents Date: Thu, 06 Sep 2001 21:01:31 Subject: Re: about hypothesis and theorem From: Pierre Lamothe In post 988 <genewardsmith@j...> wrote: << I just told Paul this definition of lattice we needed worry about and now you go and use it. :)>>In this context it's just funny it looks it's me who picked the idea of another concerning the ordering condition. But generally I began to think it's a problem for me to diffuse ideas very slowly in a such forum. I'm now hesitating to post on many subject, waiting to have time to write first in my website. I had begun, for example, many graphical studies (in vectorial form) about JI relations relatively to MIRACLE, Canasta and Blackjack scales. These images criticize, for example, ideas like the use of an absolute convexity on a linear temperament. A such convexity has only to reflect a multilinear convexity and I consider there is a flaw around the conception of the scales mentioned. Do I have to show now what I have and discuss that or to wait I will have time to write an article where I will attack vigourously with these ideas? I don't know yet. I judge only I want to be credited for my works. ----- About the homomorphism I had calculated. I have always the good functions but errors in calculation. :) I don't know where I picked the wrong values. I had prepared many examples before to choose the best one illustrating the necessity to use the ordering condition. I copied bad. Using my formula [x i j] H(X) = det [y 3 -1] = 5x - (2i + j)y - (i + 3j)z [z -1 2] obviously, with u = 27/20 and v = 25/24, we have i = -2 and j = -3 H(X) = 5x + 7y + 11z H(X mod 2) = (2y + z) mod 5 This homomorphism determines the same sublattice as the first one and is simply inversed relatively to it (so not detected) (x,y,z) --> 5x + 8y + 14z (x,y,z) mod 2 --> (3y + 4z) mod 5 Near the unison we have 1 4 3 0 2 rather than 2 0 3 4 1 ----- <genewardsmith@j...> wrote: << Where is gammier theory described? >> It's a typical example of the dissemination of my ideas. I did'nt take time to write a condensed paper. You could find snatches only in French on my website or a bit in bad English on some of my hundred posts on the tuning lists. Just seek for "Pierre" or "Lamothe" or "Lamonthe". ----- My visit on the list was only to talk about this ordering condition I had already mentionned in the appendice of a precedent message to J. Gill. This appendice talked about another condition concerning the complexity ordering which is assured when using only integers in the basis. Pierre
Message: 1567 - Contents - Hide Contents Date: Thu, 06 Sep 2001 08:20:11 Subject: Re: about hypothesis and theorem From: genewardsmith@xxxx.xxx --- 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.
Message: 1568 - Contents - Hide Contents Date: Thu, 06 Sep 2001 21:14:18 Subject: Re: Question for Gene From: Pierre Lamothe In post 984 <genewardsmith@j...> wrote: <<> 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.and Paul respond: << Sir, there is an accepted definition of "lattice" that is used in geometry and crystallography. Every point, and its local connections, is congruent to every other point and its local connections . . . something like that. We had this discussion a long time ago on the tuning list.>>We don't have this problem in French for we use "treillis" in the fist sense of partial ordering where two elements have always inferior and superior "bornes", and "réseau" in the sense of discrete Z-module. Personally, I use both the "treillis" (in melodic representation) and the "réseaux" (in harmonic representation). I have the problem to be understood with the unique term "lattice".
Message: 1569 - Contents - Hide Contents Date: Thu, 06 Sep 2001 21:17:09 Subject: Re: Tenney's harmonic distance From: Herman Miller On Thu, 06 Sep 2001 20:46:10 -0000, "Paul Erlich" <paul@xxxxxxxxxxxxx.xxx> wrote:>> (2) Do you know how to retune a midi file in such a way that the >> pitches are set to anything you choose? >>Lots of people should be able to help you with this. Herman Miller?I used to do it by hand, setting the pitch bend in Cakewalk, but now I mainly use Graham Breed's Midiconv program, which puts in the pitch bends according to a scale file you can edit. With pitch bends you have to be careful not to have overlapping notes, and I found while doing the Warped Canon project that certain timbres respond to pitch bend messages even after the note off, so I had to alternate channels. And sometimes, as in "Transformation" for instance (17-TET), I'll use Midiconv to tune the 12 most common notes and then tune the others by hand. You can get Midiconv from this link on Graham Breed's page: http://www.microtonal.co.uk/progs/Midiconv.zip - Type Ok * [with cont.] (Wayb.) Midiconv can also split one channel into multiple channels to get around the problem with overlapping notes. -- see my music page ---> ---<The Music Page * [with cont.] (Wayb.)>-- hmiller (Herman Miller) "If all Printers were determin'd not to print any @io.com email password: thing till they were sure it would offend no body, \ "Subject: teamouse" / there would be very little printed." -Ben Franklin
Message: 1570 - Contents - Hide Contents Date: Fri, 07 Sep 2001 18:35:54 Subject: Re: Distance measures cut to order From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote: >>> One of the most interesting mappings done by Wilson is his mapping >> onto a Penrose tiling, treating it as a two dimensional >> representation of a 5 dimensional space.Was this posted a while ago? I don't see this post in today's archives.> > Wowsers!Well, it's a two dimensional _slice_ through a 5-dimensional space, as you probably know . . .> > When nines or fifteens>> are treated as independent axes from threes and fives, >> interesting 'wormholes' in the lattice start to appear, where >> alternative representations of the same pitch class occur in >> surprisingly different contexts. >> These wormholes will appear no matter what metric you use. They might > be thought of as universal commas--3^2 is "approximated" by 9.These are not what I call "wormholes".
Message: 1571 - Contents - Hide Contents Date: Fri, 07 Sep 2001 20:56:15 Subject: Re: about hypothesis and theorem From: Paul Erlich --- In tuning-math@y..., Pierre Lamothe <plamothe@a...> wrote:> I had begun, for example, many graphical studies (in vectorial > form) about JI relations relatively to MIRACLE, Canasta and > Blackjack scales. These images criticize, for example, ideas > like the use of an absolute convexity on a linear temperament. > A such convexity has only to reflect a multilinear convexity > and I consider there is a flaw around the conception of the > scales mentioned. Do I have to show now what I have and discuss > that or to wait I will have time to write an article where I will > attack vigourously with these ideas? I don't know yet. I judge > only I want to be credited for my works.I am very interested in any such criticisms, and welcome them warmly, but as you know, all theoretical edifices must have at their foundation, in my opinion, _perceptual_ conditions, not purely _mathematical_ ones. In my view, the operation of creating a "good" periodicity block, and then tempering out some or all of its defining intervals, is an eminently natural musical operation, logically prior to "higher-level" musical considerations such as the choice of a tonic, etc. All concepts, such as pitch, interval, etc., are considered to be perceptual entities from the beginning, and are only "mathematized" as necessary for ease in manipulation. In your gammier theory, by contrast, I am unable so far to discern any such foundation; instead I see some appeal to perceptual properties of intervals, applied seemingly incongruously to pitches, as well as an appeal to outmoded and ahistorical just rationalizations of various world scale systems. Please understand this this is only my opinion and nothing could benefit us more than a hearty exchange of conflicting viewpoints. Personally, I think Blackjack, let along Canasta, have too many notes to be heard and conceptualized in their entirety, the way diatonic scales and their Middle-Eastern cousins are, and perhaps my decatonic scales can be. But for the problem at hand, which was to provide Joseph Pehrson with a manageable subset of 72-tET to tune up on his keyboard, that would provide a maximum number of audibly just harmonies, they can't be beat, and I don't think the gammier theory will have much in addition to say on this question.
Message: 1572 - Contents - Hide Contents Date: Fri, 07 Sep 2001 23:02:52 Subject: Re: Distance measures cut to order From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Was this posted a while ago? I don't see this post in today's > archives.Yahoo waited a few days before letting it appear.
Message: 1573 - Contents - Hide Contents Date: Fri, 07 Sep 2001 19:20:06 Subject: Re: about hypothesis and theorem From: Pierre Lamothe Paul Sorry, I don't want to begin a discussion now. Maybe it will be possible to compare our "perceptual foundation" in future. Until date you can keep confortably :) the opinion I use maths with less sense about music than what is in discussion on the tuning lists. I leave words for a while. I give you only some images. It's made for the eyes. "Que ceux qui ont des yeux pour voir . . ." <gammoids * [with cont.] (Wayb.)> Pierre
Message: 1574 - Contents - Hide Contents Date: Fri, 07 Sep 2001 02:05:02 Subject: Re: Tenney's harmonic distance From: genewardsmith@xxxx.xxx --- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:> I used to do it by hand, setting the pitch bend in Cakewalk, but now I > mainly use Graham Breed's Midiconv program, which puts in the pitch bends > according to a scale file you can edit.How do get Midiconv to input a midi file and output an ascii file of pitch values which I can edit?
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