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Message: 950 - Contents - Hide Contents Date: Mon, 03 Sep 2001 21:01:03 Subject: Re: Question for Gene From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> Gene, could you please comment on this: >> It looks to me like he's trying to do something which doesn't quite > work, which is to analyze a taxicab metric in Euclidean terms.What if we forget about the Euclidean part -- can you shed any light on the "paradox" in pure taxicab terms?
Message: 951 - Contents - Hide Contents Date: Mon, 03 Sep 2001 21:02:23 Subject: Re: Distance measures cut to order From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> Rather than screwing around trying to transform taxicab metrics, I > would suggest starting off with a Euclidean metric which works the > way you want it to work.Goodness no! I definitely want a taxicab metric.
Message: 952 - Contents - Hide Contents Date: Mon, 03 Sep 2001 21:06:36 Subject: Re: Black magic From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> I've already mentioned that if we take the N-et, and set r = N/ln(2), > and then calculate > > r' = (r+G+1/8)/ln(r) > > we get an adjusted tuning after setting N' = ln(2) r'. Here G > represents the nearest Gram point, which is round(g(r)), where > > g(r) = r ln(r) - r - 1/8 > > and "round" rounds to the nearest integer. This strikes me as almost > black magic, it's so easy. Another piece of the same magic is this: > define a function > > tend(N) = 180 (g(r) - round(g(r))), > > where again r = N/ln(2), and the "180" makes tend read out in degrees > from -180 to 180. Tend gives the tendency of an et, being positive > for ets with a sharp tendency, and negative for flat ets. We have for > example: > > N tend(N) > > 7 -23 > 10 8 > 12 13 > 15 42 > 19 -40 > 22 22 > 27 75 > 31 -22 > 34 40 > 41 -11 > 46 15 > 53 -3 > 58 67 > 72 -55 > 99 54 > > When using these to create MOS of M steps out of N, it is better that > the tendencies of M and N agree. Thus 19, 31, and 41 are reasonable > fits to the flat 72, while 22, 46 (and 21, where we have tend(21) = > 14) are less apt, and 58 is downright awkward. On the other hand, > when adding two ets to get an et, then it is better if the tendencies > are opposite, where they tend to cancel. For instance both 22+31 and > 19+34 lead to the neutral 53, whereas adding the slightly sharp 12 to > the distinctly sharp 15 leads to the very sharp 27. > > Both the meantone and the 72 systems tend towards flatness, and it > might be interesting to look to the sharp systems (such as the 15 out > of 27 system I mentioned) for something a little different. 22 out of > 46,That's the Shrutar. Are you familiar with my 10-out-of-22 and 14-out-of-26 (or 14-out-of-38) systems? Blackwood's 10- out-of-15? The diminished scale (8-out-of-12)?> > I haven't had any zeta feedback--does any of this make sense?Can you describe "tendency" a little more precisely for us ignoramuses?
Message: 953 - Contents - Hide Contents Date: Mon, 03 Sep 2001 21:37:31 Subject: Re: Symmetric 5-limit and 7-limit distance measures From: genewardsmith@j... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:>> If we take everything in a sphere of radius 1 around [0, 0] we get >> >> 1 - 6/5 - 5/4 - 4/3 - 3/2 - 5/3 - (2),> What happened to 8/5? This _should_ be the 5-limit Tonality Diamond.What happened is that I forgot to put it in. Maybe I should have run a computer program. :)>> More interesting is what happens when we >> center at [1/2, 1/2, -1/2]: >> >> q7(v - [1/2, 1/2, -1/2]) <= 1/2 >> >> gives us >> >> 1 - 15/14 - 5/4 - 10/7 - 3/2 - 12/7 - (2). >> That's the hexany. You should familiarize yourself with CPS(Combination Product Set) scales . . . the hexany is the 2)4> (1,3,5,7) hexany, meaning it's the set of numbers you get when youtake products of 2 numbers at a time out of {1,3,5,7}. I've been thinking of it as an octahedron for the last quarter- century, but by whatever name it is a fundamental doodad in the 7- limit. Does anyone but me think of the hexany as a regular octahedron, and the 7-limit note-classes as forming the (unique) semiregular 3D honeycomb, with cells consisting of tetrahedra and octahedra, and vertex figure the cubeoctahedron? Or is the hexany a purely combinatorial idea, s your definition suggests? The> 3)6 (1,3,5,7,9,11) and 3)6 (1,3,7,9,11,15) are called Eikosanies,and are similarly symmetrical in the equilateral-triangular> lattice.Hmmm. In the first place, the hexany is not in an equilateral- triangle lattice, but a 3D semiregular lattice, at least in my metric. In the second place, you have both 3 and 9 in the above list, so it can't be entirely symmetrical. So is the 2)5 (1,3,5,7,9) dekany, when 9 gets its own axis -- Dave Keenan created a splendid rotating dekany that> actually plays the notes as the dekany rotates in 4-dimensional space.3 and 9 have separate generators, you figure?
Message: 954 - Contents - Hide Contents Date: Mon, 03 Sep 2001 22:06:20 Subject: Re: Symmetric 5-limit and 7-limit distance measures From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>>> If we take everything in a sphere of radius 1 around [0, 0] we get >>> >>> 1 - 6/5 - 5/4 - 4/3 - 3/2 - 5/3 - (2), >>> What happened to 8/5? This _should_ be the 5-limit Tonality Diamond. >> What happened is that I forgot to put it in. Maybe I should have run > a computer program. :) >>>> More interesting is what happens when we >>> center at [1/2, 1/2, -1/2]: >>> >>> q7(v - [1/2, 1/2, -1/2]) <= 1/2 >>> >>> gives us >>> >>> 1 - 15/14 - 5/4 - 10/7 - 3/2 - 12/7 - (2). >>>> That's the hexany. You should familiarize yourself with CPS> (Combination Product Set) scales . . . the hexany is the 2)4>> (1,3,5,7) hexany, meaning it's the set of numbers you get when you> take products of 2 numbers at a time out of {1,3,5,7}. > > I've been thinking of it as an octahedron for the last quarter- > century, but by whatever name it is a fundamental doodad in the 7- > limit. Does anyone but me think of the hexany as a regular > octahedron,Everyone does. You should have been on the tuning list for the last year. Erv Wilson drew this octahedron in the 60's, I believe.> and the 7-limit note-classes as forming the (unique) > semiregular 3D honeycomb, with cells consisting of tetrahedra and > octahedra,I've been drawing those for at least 10 years now. Didn't you see, for example, the lattices I just posted for Rami Vitale's scale (ASCII) and Justin White's scale (.gif) on the tuning list?> and vertex figure the cubeoctahedron?I think George Olshevsky explained "vertex figure" to us. And the cuboctahedron is the shape of the 7-limit Tonality Diamond, etc, etc. . . .> Or is the hexany a > purely combinatorial idea, s your definition suggests?Combinatorial _and_ geometrical.> > The>> 3)6 (1,3,5,7,9,11) and 3)6 (1,3,7,9,11,15) are called Eikosanies,> and are similarly symmetrical in the equilateral-triangular >> lattice. >> Hmmm. In the first place, the hexany is not in an equilateral- > triangle lattice, but a 3D semiregular lattice, at least in my > metric.What I mean is that the lattice has a lot of equilateral triangles in it. Of course it has a lot of squares in it too, but I'm just trying to distinguish it, in layman's terms, from the Cartesian square lattice, which is common too (Euler and Fokker come to mind).> In the second place, you have both 3 and 9 in the above list, > so it can't be entirely symmetrical.9 gets its own axis, and is _not_ assumed to equal 3*3.> > So is the 2)5 (1,3,5,7,9) dekany, when 9 gets its own axis -- Dave > Keenan created a splendid rotating dekany that>> actually plays the notes as the dekany rotates in 4-dimensional > space. >> 3 and 9 have separate generators, you figure?Well, one can think of it either way, but the figure is more symmetrical when one puts 3 and 9 on different axes. The problem is that there are a couple of 9-limit consonances that don't show up as direct connections. But with the alternative, having only axes for prime numbers, one doesn't have direct connections for _any_ of the ratios of 9. One solution is to use a taxicab metric, along with "wormholes" (search the tuning list archives).
Message: 955 - Contents - Hide Contents Date: Mon, 03 Sep 2001 22:58:50 Subject: Re: Black magic From: genewardsmith@j... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:>> Both the meantone and the 72 systems tend towards flatness, and it >> might be interesting to look to the sharp systems (such as the 15 out >> of 27 system I mentioned) for something a little different. 22 out of >> 46,> That's the Shrutar. Are you familiar with my 10-out-of-22 and 14-out-of-26 (or 14-out-of-38) systems? Blackwood's 10-> out-of-15? The diminished scale (8-out-of-12)?As you've probably figured out by now, there's lots I'm not familiar with--in fact, most of what I know is what I worked out for myself 20- 30 years ago, after reading Helmholtz and something called "Music, a Science and an Art" by a JI advocate whose name I can't recall. It all really started in grade school, where a music teacher came into class and told us about white keys and black keys, and how there were seven notes to the scale and twelve to the octave. When I asked "Why?", she said "That's just the way it is." This sort of answer never makes me happy. However, let me guess: Shrutar 2n mod 23, pattern 22222222223 * 2 Paul 10 out of 22, 2n mod 11, pattern 22223 * 2 (Similarly, 12 out of 22, 2n mod 11, pattern 222221 * 2) Blackwood 10 out of 15, 2n mod 3, pattern 21 * 5 Etc. Is that right?> Can you describe "tendency" a little more precisely for us ignoramuses?It measures the proportional distance to the nearest Gram point from the N-et. What it actually does is give a high reading if 3, 5, 7 etc. pile up in one direction or another--tending to be sharp or flat, and a low reading if they are less consistent, with some sharp and some flat. In some sense all odd primes are considered, but things are heavily weighted in favor of the first few.
Message: 956 - Contents - Hide Contents Date: Mon, 03 Sep 2001 23:35:34 Subject: Re: Symmetric 5-limit and 7-limit distance measures From: genewardsmith@j... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> I've been drawing those for at least 10 years now. Didn't you see,for example, the lattices I just posted for Rami Vitale's> scale (ASCII) and Justin White's scale (.gif) on the tuning list?I did, and it looked suspiciously regular (and hence familiar) to me; that's why I suggested you could do a better job of drawing it.> What I mean is that the lattice has a lot of equilateral trianglesin it. Of course it has a lot of squares in it too, but I'm just> trying to distinguish it, in layman's terms, from the Cartesiansquare lattice, which is common too (Euler and Fokker come> to mind).Do people ever look at the reciprocal lattices--the hexagons in the 5- limit, and in the 7-limit the bee honeycomb of rhombic dodecahedra, which is what you get by squashing the spheres in a regular close- packing together. These are lattices of chords.> Well, one can think of it either way, but the figure is moresymmetrical when one puts 3 and 9 on different axes. I've never found an ideal solution to this either, though you can certainly make the 3 half as big as 5,7,9,11 and 13.
Message: 957 - Contents - Hide Contents Date: Tue, 04 Sep 2001 00:26:03 Subject: Scala From: genewardsmith@j... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> That's the Shrutar. Are you familiar with my 10-out-of-22 and 14-out-of-26 (or 14-out-of-38) systems? Blackwood's 10-> out-of-15? The diminished scale (8-out-of-12)?Now that I have a sound card, I want to creat midi files, and have just downloaded Scala. It seems rather formidable, but I noticed a lot of Paul Erlich .sla files in a list of files there. Is there a FAQ or something to lead a person through this? Most of the FAQs seem to want to tell me how to do math!
Message: 958 - Contents - Hide Contents Date: Tue, 04 Sep 2001 02:05:49 Subject: Re: Distance measures cut to order From: genewardsmith@j... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> --- In tuning-math@y..., genewardsmith@j... wrote:>> Rather than screwing around trying to transform taxicab metrics, I >> would suggest starting off with a Euclidean metric which works the >> way you want it to work.> Goodness no! I definitely want a taxicab metric.Is this so your worms will have holes? I searched on "wormhole" and came up with nothing, but I wonder what taxicab metrics can do for you that a well-chosen Euclidean metric could not also do.
Message: 959 - Contents - Hide Contents Date: Tue, 04 Sep 2001 20:16:37 Subject: Re: Black magic From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>>> Both the meantone and the 72 systems tend towards flatness, and > it>>> might be interesting to look to the sharp systems (such as the 15 > out>>> of 27 system I mentioned) for something a little different. 22 > out of >>> 46, >>> That's the Shrutar. Are you familiar with my 10-out-of-22 and 14-> out-of-26 (or 14-out-of-38) systems? Blackwood's 10->> out-of-15? The diminished scale (8-out-of-12)? >> As you've probably figured out by now, there's lots I'm not familiar > with--in fact, most of what I know is what I worked out for myself 20- > 30 years ago, after reading Helmholtz and something called "Music, a > Science and an Art" by a JI advocate whose name I can't recall. It > all really started in grade school, where a music teacher came into > class and told us about white keys and black keys, and how there were > seven notes to the scale and twelve to the octave. When I > asked "Why?", she said "That's just the way it is." This sort of > answer never makes me happy. > > However, let me guess: > > Shrutar 2n mod 23, pattern 22222222223 * 2Not quite . . . it's altered to get omnitetrachordality (the two instances of '3' are placed a 3:2 apart).> > Paul 10 out of 22, 2n mod 11, pattern 22223 * 2That's the symmetrical version . . . the omnitetrachodal version is altered, with the two '3's a 3:2 apart.> > (Similarly, 12 out of 22, 2n mod 11, pattern 222221 * 2)On my keyboard, I use an altered version, with the two '1's appearing a 3:2 apart (between E and F and between B and C).> > Blackwood 10 out of 15, 2n mod 3, pattern 21 * 5 Right. > > Etc. Is that right? >>> Can you describe "tendency" a little more precisely for us > ignoramuses? >> It measures the proportional distance to the nearest Gram point from > the N-et. What it actually does is give a high reading if 3, 5, 7 > etc. pile up in one direction or another--tending to be sharp or > flat, and a low reading if they are less consistent, with some sharp > and some flat. In some sense all odd primes are considered, but > things are heavily weighted in favor of the first few.Very, very interesting! I'm familiar with Riemann's zeta function from Schroeder's _Number Theory in Science and Communication_ but that's about it.
Message: 960 - Contents - Hide Contents Date: Tue, 04 Sep 2001 20:22:37 Subject: Re: Symmetric 5-limit and 7-limit distance measures From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> I've been drawing those for at least 10 years now. Didn't you see,> for example, the lattices I just posted for Rami Vitale's>> scale (ASCII) and Justin White's scale (.gif) on the tuning list? >> I did, and it looked suspiciously regular (and hence familiar) to me; > that's why I suggested you could do a better job of drawing it. >>> What I mean is that the lattice has a lot of equilateral triangles> in it. Of course it has a lot of squares in it too, but I'm just>> trying to distinguish it, in layman's terms, from the Cartesian> square lattice, which is common too (Euler and Fokker come >> to mind). >> Do people ever look at the reciprocal lattices--the hexagons in the 5- > limit, and in the 7-limit the bee honeycomb of rhombic dodecahedra, > which is what you get by squashing the spheres in a regular close- > packing together. These are lattices of chords.So far, plotting notes has been more useful -- it encapsulates all the information in terms of the simplest elements musicians must deal with: individual tones.>>> Well, one can think of it either way, but the figure is more> symmetrical when one puts 3 and 9 on different axes. > > I've never found an ideal solution to this either, though you can > certainly make the 3 half as big as 5,7,9,11 and 13.Let's take a step back. If we don't assume octave equivalence, then the Tenney lattice, with a taxicab metric, is quite wonderful. There's an axis for each prime, and distance between rungs along each is log(p). Then the Tenney "harmonic distance" (HD) between two notes turns out to be log(n*d) where n/d is the ratio between the notes. That's quite wonderful -- the harmonic entropy of the simpler ratios n/d turns out to behave very much like log(n*d). Now when we assume octave equivalence . . . and use 9-limit or higher . . . things get hairy.
Message: 961 - Contents - Hide Contents Date: Tue, 04 Sep 2001 20:23:34 Subject: Re: Scala From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> That's the Shrutar. Are you familiar with my 10-out-of-22 and 14-> out-of-26 (or 14-out-of-38) systems? Blackwood's 10->> out-of-15? The diminished scale (8-out-of-12)? >> Now that I have a sound card, I want to creat midi files, and have > just downloaded Scala. It seems rather formidable, but I noticed a > lot of Paul Erlich .sla files in a list of files there. > > Is there a FAQ or something to lead a person through this? Most of > the FAQs seem to want to tell me how to do math!A FAQ for Scala? You can ask Manuel Op de Coul directly; he's on the Tuning List and very helpful.
Message: 962 - Contents - Hide Contents Date: Tue, 04 Sep 2001 20:30:19 Subject: Re: Distance measures cut to order From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:>> --- In tuning-math@y..., genewardsmith@j... wrote: >>>> Rather than screwing around trying to transform taxicab metrics, > I>>> would suggest starting off with a Euclidean metric which works > the>>> way you want it to work. >>> Goodness no! I definitely want a taxicab metric. >> Is this so your worms will have holes? I searched on "wormhole" and > came up with nothing,Probably this discussion happened before the list moved off the Mills server.> but I wonder what taxicab metrics can do for > you that a well-chosen Euclidean metric could not also do.A lot of things. First, consider the Tenney lattice I just described. Second, see my post on this list where I conjectured that unison vectors with numbers of size S and difference between numerator and denominator D imply an amount of tempering proportional to D/(S^2). This depends on the Kees van Prooijen lattice with a taxicab metric. Lots of other reasons too, probably best enunciated by Paul Hahn back in the Mills days.
Message: 963 - Contents - Hide Contents Date: Tue, 04 Sep 2001 22:32:37 Subject: Re: Distance measures cut to order From: Carl Lumma>> >s this so your worms will have holes? I searched on "wormhole" and >> came up with nothing, >>Probably this discussion happened before the list moved off the >Mills server.The discussion you're referring to happened while the list was hosted on onelist, and those messages are in the Yahoo! archive. The problem is that Yahoo!'s search only goes back arbitrarily far... I tried this search three times, and got searches from message #'s 27xxx-27847, 26xxx-27847, and 22xxx-27847, all showing no matches. Unexpectedly, when searching my personal archive to verify the date of the thread, I did find a Mills post containing the term "wormholes": Date: Sat, 30 May 1998 11:43:43 -0400 From: Daniel Wolf <DJWOLF_MATERIAL@c...> To: "INTERNET:tuning@e..." <tuning@e...> Subject: TUNING digest 1431: Wilson's nines Message-ID: <199805301143_MC2-3EA2-A973@c...> Very often Erv Wilson will put his nines on a separate axis from threes. A look at his Xenharmonikon cover art will provide several examples. (All back issues of XH are available from Frog Peak Music). 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. 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. Although I have not followed up on this work for some years, I was getting interesting results in using the Penrose tilings as a control mechanism over random walks over the lattice. While treating nine as two steps on the three axis will certainly be the more efficient means of avoiding redundancies, there can be musically compelling grounds for preserving such redundancies or ambiguities. -Carl
Message: 964 - Contents - Hide Contents Date: Wed, 05 Sep 2001 03:15:54 Subject: Tenney's harmonic distance From: genewardsmith@j... I did a search on this, and found this old posting, to which I respond. --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> I don't think you should! I think n*d is a much better measure of > complexity than one you'd derive from your lattice formula. For > simple ratios, harmonic entropy is proportional to log(n*d). And log > (n*d) is known as Tenney's Harmonic Distance, since it's the city > block distance in his lattice. Thus I think his (octave-specific) > lattice is much better than yours for depicting dissonance.Let's see what happens if we try to make this work in a Euclidean framework. In the 5-limit, we want the following distances from the origin: d(2) = ln(2), d(3) = ln(3), d(5) = ln(5), d(3/2) = ln(6), d(5/2) = ln(10) and d(5/3) = ln(15). The corresponding quadratic form is u^2 + v^2 + w^2 - 2uv - 2uw - 2vw, where u = ln(2)x, v = ln(3)y, w = ln(5)z. The matrix for the corresponding bilinear form is [ 1 -1 -1] [-1 1 -1] [-1 -1 1], which is not positive definite, having eigenvalues of -1, 2, and 2. This is therefore a Lorentzian metric, like the geometry of space- time, which does seem a little goofy--should the consonance of 30 really be imaginary? You can pick eigenvalue coordinates, and collapse the -1 part belonging to [1 1 1] out of the picture and get something positive definite in two dimensions, but this collapses 30 down to 1, which doesn't seem any better. On the other hand if you stick with the obvious, namely u^2+v^2+w^2 then you get 5/3 the same size as 15, which is what the taxicab metric gave you, and a measurement for consonance which seems generally in line with what I think you want: d(2) = ln(2), d(3) = ln(3), d(5) = ln(5), d(3/2) = sqrt(ln(2)^2 + ln(3)^2) = d(6), d(5/2) = sqrt(ln(2)^2 + ln(5)^2) = d(10), d(5/3) = sqrt(ln(2)^2 + ln(5)^2) = d(15). Is there some reason not to use this?
Message: 966 - Contents - Hide Contents
Date: Wed, 05 Sep 2001 20:21:25
Subject: Theorem Paul
From: genewardsmith@j...
Recall that a val was defined to be an element in the dual group to a
note group; if in particular we take the p-limit note group N_p, with
elements we think of concretely as row vectors with integer
coordinate values, then vals are column vectors with integer values.
We make the following definitions: consider the note group N_p where
we take as usual the generating set to be the primes up to p, so that
2 is represented by [1, 0, ..., 0], 3 by [0,1, 0, ..., 0] and so
forth. If a val for notes in this basis has all of its coordinate
values positive, and if the coordinate values have no common
divisor, we will say the val is *valid*. Another way to express this
is to say v is valid if v(q) is positive for all primes q<=p and if
gcd({q_i})=1, where {q_i} is the set of primes up to p.
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. We then have the
following:
(Theorem Paul) Let n={n_i} be a valid set of generators associated to
the val v in N_p, where p>3, and let m be n minus one generator. Let
B be a block of v(2) notes in an octave, defined by n and octave
equivalence. Let M be the group generated by m, and let u be a val
which together with v generates the dual M` to M. Then there exists a
nonnegative integer t such that if w is the val t*v + u we have a
v(2) out of w(2) MOS which tempers out M from everything in the block
B.
Proof: Since v is valid, there must be an odd prime q such that
gcd(v(2), v(q))=1. Hence v(q) generates the cyclic group Z/v(2)Z of
integers reduced modulo v(2). Consider the ratios v(q)/v(2) and
w(q)/w(2). As t approaches infinity, w(q)/w(2) approaches v(q)/v(2).
Let us pick a positive t large enough so that all the coordinate
values of w (ie its values w(q_i) on all the primes up to p) are
positive, and w(2) is larger than v(2). If such a t is large enough,
we also have
|v(q)/v(2) - w(q)/w(2)| < 1/v(2)^2,
and v(q)/v(2) is a semiconvergent to w(q)/w(2) and hence v(q) is a
generator for a v(2) out of w(2) MOS. Since w is in M`, all of the
elements of M (and hence in particular of m) are sent to 0 by w--ie,
they are tempered out. Therefore block B is tempered by w to the v(2)
out of w(2) MOS obtained above, QED.
This is quite a weak condition for vals which are intended to
represent ets, and hence a strong version of Theorem Paul, but
nothing more seems to be needed. In practice we could confine our
attentions to valid sets of generators arising from an h_n et for
some integer n.
Message: 967 - Contents - Hide Contents Date: Wed, 05 Sep 2001 21:32:02 Subject: Re: Distance measures cut to order From: Paul Erlich --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:>>> Is this so your worms will have holes? I searched on "wormhole" and >>> came up with nothing, >>>> Probably this discussion happened before the list moved off the >> Mills server. >> The discussion you're referring to happened while the list was > hosted on onelist, and those messages are in the Yahoo! archive. > The problem is that Yahoo!'s search only goes back arbitrarily > far... I tried this search three times, and got searches from > message #'s 27xxx-27847, 26xxx-27847, and 22xxx-27847, all > showing no matches.Keep clicking "Next" . . . a whole slew of wormhole messages will be found. I'm surprised you and Gene both said you couldn't find them!
Message: 968 - Contents - Hide Contents Date: Wed, 05 Sep 2001 21:36:26 Subject: Re: Tenney's harmonic distance From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> I did a search on this, and found this old posting, to which I > respond. > > --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> I don't think you should! I think n*d is a much better measure of >> complexity than one you'd derive from your lattice formula. For >> simple ratios, harmonic entropy is proportional to log(n*d). And log >> (n*d) is known as Tenney's Harmonic Distance, since it's the city >> block distance in his lattice. Thus I think his (octave-specific) >> lattice is much better than yours for depicting dissonance. >> Let's see what happens if we try to make this work in a Euclidean > framework. In the 5-limit, we want the following distances from the > origin: d(2) = ln(2), d(3) = ln(3), d(5) = ln(5), d(3/2) = ln(6), > d(5/2) = ln(10) and d(5/3) = ln(15). The corresponding quadratic form > is > > u^2 + v^2 + w^2 - 2uv - 2uw - 2vw, > > where u = ln(2)x, v = ln(3)y, w = ln(5)z. > > The matrix for the corresponding bilinear form is > > [ 1 -1 -1] > [-1 1 -1] > [-1 -1 1], > > which is not positive definite, having eigenvalues of -1, 2, and 2. > This is therefore a Lorentzian metric, like the geometry of space- > time, which does seem a little goofy--should the consonance of 30 > really be imaginary? You can pick eigenvalue coordinates, and > collapse the -1 part belonging to [1 1 1] out of the picture and get > something positive definite in two dimensions, but this collapses 30 > down to 1, which doesn't seem any better. > > On the other hand if you stick with the obvious, namely u^2+v^2+w^2 > then you get 5/3 the same size as 15, which is what the taxicab > metric gave you, and a measurement for consonance which seems > generally in line with what I think you want: > > d(2) = ln(2), d(3) = ln(3), d(5) = ln(5), > d(3/2) = sqrt(ln(2)^2 + ln(3)^2) = d(6), > d(5/2) = sqrt(ln(2)^2 + ln(5)^2) = d(10), > d(5/3) = sqrt(ln(2)^2 + ln(5)^2) = d(15). > > Is there some reason not to use this?d(3/2) is between ln(3) and ln(4)! Not good.
Message: 969 - Contents - Hide Contents Date: Wed, 05 Sep 2001 21:40:58 Subject: Re: Theorem Paul From: Paul Erlich I'll have to come back to this later, but I don't like the idea (in the last line) of restricting our attention to MOSs that are within some ET. Also, did you make use of the hyperparallelepiped construction? If you don't, you don't necessarily get an MOS . . . Now do you have a quick way of determining the generator of the linear temperament, given n-1 commatic unison vectors?
Message: 970 - Contents - Hide Contents Date: Wed, 05 Sep 2001 22:00:43 Subject: Re: Tenney's harmonic distance From: genewardsmith@j... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> d(3/2) is between ln(3) and ln(4)! Not good.You probably want to stick with your taxicab if you don't like this; but remember that coordinate transformations transform taxicab spaces to other taxicab spaces, you can't treat any of them like Euclidean spaces.
Message: 971 - Contents - Hide Contents Date: Wed, 05 Sep 2001 22:14:03 Subject: Re: Theorem Paul From: genewardsmith@j... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> I'll have to come back to this later, but I don't like the idea (in > the last line) of restricting our attention to MOSs that are within > some ET.I made that assumption and some others simply to make the proof easy, in the lazy way normal to mathematicians. There isn't any reason to restrict yourself to these in general, but I thought the point was to prove the theorem true. Of course, if what I have stated isn't what what you wanted proven (and I've been having problems with that, as you know) then the proof won't give you all you want. Also, did you make use of the hyperparallelepiped> construction?You get a hyperparallepiped as the walls of the block defined by the generators {n_i} together with 2.
Message: 972 - Contents - Hide Contents Date: Wed, 05 Sep 2001 22:33:52 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: >>> d(3/2) is between ln(3) and ln(4)! Not good. >> You probably want to stick with your taxicab if you don't like this; > but remember that coordinate transformations transform taxicab spaces > to other taxicab spaces, you can't treat any of them like Euclidean > spaces.You better believe it! So, any comments on the questions I asked?
Message: 973 - Contents - Hide Contents Date: Wed, 05 Sep 2001 22:36:21 Subject: Re: Theorem Paul From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> I'll have to come back to this later, but I don't like the idea (in >> the last line) of restricting our attention to MOSs that are within >> some ET. >> I made that assumption and some others simply to make the proof easy, > in the lazy way normal to mathematicians. There isn't any reason to > restrict yourself to these in general, but I thought the point was to > prove the theorem true. Of course, if what I have stated isn't what > what you wanted proven (and I've been having problems with that, as > you know) then the proof won't give you all you want.I guess I'd ideally like my original proof to be used as an outline, with conditions tightened up where needed. But I'll try to take a close look at yours at a later point in time.> > Also, did you make use of the hyperparallelepiped >> construction? >> You get a hyperparallepiped as the walls of the block defined by the > generators {n_i} together with 2.So you did make use of this? I'll have to look more closely later . . .
Message: 974 - Contents - Hide Contents Date: Wed, 05 Sep 2001 22:43:56 Subject: Re: Distance measures cut to order From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:>> --- In tuning-math@y..., genewardsmith@j... wrote: >>>> Rather than screwing around trying to transform taxicab metrics, > I>>> would suggest starting off with a Euclidean metric which works > the>>> way you want it to work. >>> Goodness no! I definitely want a taxicab metric. >> Is this so your worms will have holes? I searched on "wormhole" and > came up with nothingGene, you have to keep clicking "Next" in the search dialog. It goes backwards . . . somewhere around message 4000, they start popping up.
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