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Message: 3200 - Contents - Hide Contents Date: Sat, 12 Jan 2002 18:07:00 Subject: Re: dict/genemath.htm From: manuel.op.de.coul@xxxxxxxxxxx.xxx Gene wrote:>I just spent some time trying to discover what Lumma Stability >was, and failing.If you open the Scala file tips.par in a text editor and search for "stability" you will find a definition. Manuel
Message: 3201 - Contents - Hide Contents Date: Sun, 13 Jan 2002 13:09:21 Subject: Critical Strip Explorer From: monz I found an old post Gene wrote to the sci.math newsgroup a few years ago, discussing the Riemann zeta function and its application to tuning: From: gwsmith@gwi.net (Gene Ward Smith) * [with cont.] (Wayb.) Then I found this really cool applet: Critical Strip Explorer Critical Strip Explorer * [with cont.] (Wayb.) Any more detailed explanations of all this would be most welcome! -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3202 - Contents - Hide Contents Date: Sun, 13 Jan 2002 13:34:59 Subject: all male From: monz Just curious... any ideas on why there are no females among the 58 current members of this list? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3203 - Contents - Hide Contents Date: Sun, 13 Jan 2002 13:43:51 Subject: Re: almost all male From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Sunday, January 13, 2002 1:34 PM > Subject: [tuning-math] all male > > > Just curious... any ideas on why there are no females > among the 58 current members of this list?Oops... my bad. I had only looked at the first "members" page. I see that there are four members on the second page listed as female. Still, that's only 6.90% female vs. 93.10% male. (assuming, of course, that members are being honest about their gender) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3204 - Contents - Hide Contents Date: Sun, 13 Jan 2002 22:07:42 Subject: Re: Critical Strip Explorer From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote: You should look at Yahoo groups: /tuning-math/message/879 * [with cont.] and Yahoo groups: /tuning-math/message/946 * [with cont.] The first of these has a url to a neat applet which is much more relevant to music theory; I still would like to see a musical version of this. I sent email to the fellow who put it up, and he knows nothing about music, so that didn't help.
Message: 3205 - Contents - Hide Contents Date: Sun, 13 Jan 2002 22:42:46 Subject: Re: Some 10-tone, 72-et scales From: dkeenanuqnetau --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>> I started out looking at these as 7-limit 225/224 planar> temperament scales, but decided it made more sense to check the 5 and > 11 limits also, and to take them as 72-et scales; if they are ever > used that is probably how they will be used. I think anyone > interested in the>> 72-et should take a look at the top three, which are all 5-> connected, and the top scale in particular, which is a clear winner. > The "edges" number counts edges (consonant intervals) in the 5, 7, > and 11 limits, and the connectivity is the edge-connectivity in the > 5, 7 and 11 limits. >>>> [0, 5, 12, 19, 28, 35, 42, 49, 58, 65] >> [5, 7, 7, 9, 7, 7, 7, 9, 7, 7] >> Was this Dave Keenan's 72-tET version of my Pentachordal Decatonic?No. The 72-tET version of your Pentachordal Decatonic is this one (pentachordal in two ways). 7 9 7 7|5 7|7 9 7 7 7 7 9 7|7 5|7 7 9 7 But the one you give above, is listed along with it in Yahoo groups: /tuning/messages/27221?expand=1 * [with cont.]
Message: 3206 - Contents - Hide Contents Date: Mon, 14 Jan 2002 02:53:17 Subject: Re: For Joe--proposed definitions From: genewardsmith --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:> I think it would be good not only to separate math definitions but also to avoid using > superficial blending of tuning notions with math concepts in terms like Tone group > and Rational tone group.We already talk about the 7-limit, 11-limit and so forth, which are tone groups, whatever one calls them. Why not give them a name? I don't understand you objection. I suppose we could call them all regular temperaments, with the proviso that rational intonation is a temperament, but I think that would be asking for trouble.> How could you regard an infinite group as a set of intervals or pitches?It's done all the time implicitly. We would say "four octaves of 12 equal temperament" or "five octaves of twelve equal temperament" without regarding them as different temperaments. Keeping things as groups means keeping things simple, and we want things to be simple.> It is important to understand, from the modelizing viewpoint, that any group generated > by more than one rational number, say 2 and 3, would have an infinite number of tones > in the octave. The group structure may not modelize the operative structure on a finite > set of rational intervals.Isn't this like saying there are an infinite number of possible lengths in a meter so we should not regard length measurements as real numbers? This just leads to confusion.> In my opinion, any attempt to understand a categorical perception of the tones in a > musical context without reference to an underlying proper invariant operative structure, > confined to illusion.I don't imagine there are any underlying invariant structures, but intervals can be recognized as transposed, and approximate small integer ratios have a clear significance; modeling that in the simplest way (with morphisms that actually illuminate the situation) leads to finitely generated groups. An isolated experience focusing on sensation> is a regressive perception and attempts to reconstruct an authentic musical experience > with such isolated acoustic impressions appears to me well superficial.I think you start from the ground up, and don't float down from the sky. Simple is good.> Gene, is there the norm of elegance you had in head when you criticized my work at > Christmas day?My defintion is exactly what yours aren't and I wish they were, frankly. A definition is not supposed to be elegant, it is supposed to be precise. If it isn't, it does not do the job it needs to do, which is to convey an exact sense which allows one to understand precisely what is meant, what some is and what it isn't.> I can understand why Manuel said there is no need for a mathematical scale definition.If we are going to define "connected scale", "convex scale", "epimorphic scale" and so forth we first define "scale".> However I would say that at first level, in the reduced minimal sense he uses, the scale > has essentially a mathematical definition as > discrete subset on an ordered setI'm not interested in defining scales on hyperreal numbers or the long line! Real numbers are clearly what we want.> The next level implies the interval notion which requires much more than an ordered set > as starting structure. If > a - b - c - d - ... > is a such discrete ordered set, the successive steps, we would denote > a:b - b:c - c:d - ... > have sense only if there exist a composition law on that set permitting to define also an > order on the intervals. So with the < relation (resp. > or =), depending of the law type > (multiplicative or additive), we have > a:b < c:d <=> bc < ad > a:b < c:d <=> b+c < a+dAll of which makes me think we should at least be working in a totally ordered field, but I don't understand your point. Why all the complicated bells and whistles? We surely do not want to talk about scales which are not contained in the real numbers!
Message: 3207 - Contents - Hide Contents
Date: Mon, 14 Jan 2002 20:20:33
Subject: Gene's relative connectedness
From: Pierre Lamothe
Gene wrote:
Connected scale
A scale is connected with respect to a set of intervals S if for any
two scale degrees a and b, there is a path a=a_0, a_1, ... a_n = b
such that |a_i - a_{i+1}| is an element of S.
Either this definition use precisely his proper terms
Scale
A discrete set of real numbers, containing 0, and regarded as defining tones
in a logarithmic measure, such as cents or octaves, and such that the distance
between sucessive elements of the scale is bounded both below and above
by positive real numbers. The least upper bound of the intervals between
successive elements of the scale is the maximum scale step, and the greatest
lower bound is the minimum scale step. The element of the scale obtained by
counting up n scale steps is the nth degree, by counting down is the –nth
degree; 0 is the 0th degree. The set of positive real numbers which are the tones
so represented is also regarded as the scale.
or scale and degrees have another sense where it could exist more than one
path between the first and the last degrees of the scale. In that second case
the definition of connected scale would not be precise since scale and degree
would not be defined.
In the first case, using strictly the definitions, on could say
A scale whose steps belong to a set S is said
"connected with respect to a set of intervals S".
In the same way,
A women made pregnant by her lover S could be said
"pregnant with respect to her lover S"
while
"non-pregnant with respect to her husband H".
How to believe that a "relative connectedness" could convey an exact sense,when
it seems that that conveys about nothing? I recall what Gene wrote:
A definition is not supposed to be elegant, it is supposed to be precise.If it
isn't, it does not do the job it needs to do, which is to convey an exactsense
which allows one to understand precisely what is meant, what some is and what
it isn't.
Do a connectedness of a scale relative to S, be a property of the scale or a property
of the relation between the scale and S?
I suppose Gene seeks to adapt, in his simplest way, my concept on contiguity. So,
I would like to say him it does'nt convey any idea of relativity.
A gammier mode is not relatively connected but absolutely connected since the unique
set A for which it has sense to refer the connectedness is strictly determined by the
gammier structure itself. But it is very easy to generalize since I have a propension
to use independant axioms. I recall first my two following definitions withthe
correction (k distinct from unison) in contiguity.
Atom definition in an harmoid
a is an atom if
a > u (where u is the unison) and
xy = a has no solution where both (u < x < a) and (u < y < a)
Contiguity axiom is
any interval k is divisible by an atom (k distinct from unison)
or there exist an atom a such that ax = k has a solution
I defined atom in harmoid for it's there it has really sense, but the definition would
remain valid in any finite subset G of rational numbers with standard multiplication,
neutral element 1, and standard order (or a similar additive structure).
In a such subset G respecting the contiguity condition, any element k distinct from the
neutral element is divisible by an atom, say a. Thus, there exist b such that ab = ba = k.
While b is not the neutral element, it remains, like k, divisible by an atom, so there exist
minimally one path, where all steps are atoms, between the neutral element and any k
including sup(G) and min(G), when distinct from the neutral element.
When a such set G is a gammoid, I name these paths modes, reserving the term mode
to denote such (connected) paths when they have the same number of degrees (what
is determined by the congruity condition) in the octave.
To sum up, the contiguity property (maybe the connectedness you seeked to define) is
an essential property of a mode such I conceptualize that, as gammier mode.
Why I would want to introduce infinity, like your simplest way, while my way is so short?
Pierre Lamothe
[This message contained attachments]
Message: 3208 - Contents - Hide Contents Date: Mon, 14 Jan 2002 05:09:12 Subject: Re: For Joe--proposed definitions From: Pierre Lamothe Gene wrote: We already talk about the 7-limit, 11-limit and so forth, which are tone groups, whatever one calls them. Why not give them a name? I don't understand your objection.. If you talked about Tone group and Rational tone group as mathematical structures in which you plunge the few possible tuning values, I would have few to say. But if you define these objects as tuning concepts, without to precise that the group property is not conserved in the restriction to the true tuning values, I say you contribute to propagate a pernicious attitude I have already named numerical fetichism. I suppose we could call them all regular temperaments, with the proviso that rational intonation is a temperament, but I think that would be asking for trouble. Sorry, I don't get it. I suppose it might have sense in temperament language.> How could you regard an infinite group as a set of intervals or pitches?It's done all the time implicitly. We would say "four octaves of 12 equal temperament" or "five octaves of twelve equal temperament" without regarding them as different temperaments. Keeping things as groups means keeping things simple, and we want things to be simple. There is no problem with tempered case since the restriction to the octave conserve the group structure. The problem is essentially in the JI case where the absence of the closure axiom implies that none finite part is a subgroup.> It is important to understand, from the modelizing viewpoint, that any group > generated by more than one rational number, say 2 and 3, would have an infinite > number of tones in the octave. The group structure may not modelize the operative > structure on a finite set of rational intervals.Isn't this like saying there are an infinite number of possible lengths in a meter so we should not regard length measurements as real numbers? This just leads to confusion. We can regard frequency measurements as real numbers and the operative properties on these measurements as those of the real field, in acoustical calculation. But there exist approximatively 3500 internal cilied cells in the cochlea and thus no possibility to perceive really more distinct values. Could you transport the operative properties of the real field in the space of pitch height values? For instance, there exist always another element between any two elements in the real field. Is it the case between perceived pitch values?> In my opinion, any attempt to understand a categorical perception of the > tones in a musical context, without reference to an underlying proper invariant > operative structure, confined to illusion.I don't imagine there are any underlying invariant structures, but intervals can be recognized as transposed, and approximate small integer ratios have a clear significance; modeling that in the simplest way (with morphisms that actually illuminate the situation) leads to finitely generated groups. I suggest for exciting your imagination you read about perception phenomenology. Your simplest way is simply not the way we perceive. Do you know, for instance, that the attribution of a color to an object is not built-in but has to be learned? A born blind recently operated see first colors without to attribute them to objects. We construct our capacity to perceive with our intentional activities. Our aquired abilities don't depend of inaccessible universal mathematical properties but only of those in the space of our real experimentation, by the necessity where we are to integrate, second by second, a sensorial flux renewing constantly aspects of things we intentionaly focus on.> Gene, is there the norm of elegance you had in head when you criticized my > work at Christmas day?My defintion is exactly what yours aren't and I wish they were, frankly. A definition is not supposed to be elegant, it is supposed to be precise. If it isn't, it does not do the job it needs to do, which is to convey an exact sense which allows one to understand precisely what is meant, what some is and what it isn't. I showed to you a short presentation of the chord theorem, in French. The definitions were simple, precise, conveying an exact sense and were elegant. But it was in French, it was not what you wish, and you had none idea about the fertility of this theorem. Is by lazziness, as you mentioned about number like 123.45678901234, that you prefered to criticizise? If you believe really your definition is undoubtly a good definition, I could take time to propagate doubts.> I can understand why Manuel said there is no need for a mathematical scale > definition.If we are going to define "connected scale", "convex scale", "epimorphic scale" and so forth we first define "scale". Yes, but you have, first, to take account that there exist already an implicit definition, for instance, in the scale list of Scala. Secondly, ... I stop here, since you believe you can start afresh, reformulating all concepts as yours.> However I would say that at first level, in the reduced minimal sense he > uses, the scale has essentially a mathematical definition as discrete subset > on an ordered setI'm not interested in defining scales on hyperreal numbers or the long line! Real numbers are clearly what we want. Where do you go with your dirty word? Do you feel bad for I don't use numbers? I talked only about two simple mathematical properties that anybody in tuning List may understand. Real numbers have sense in the temperament treatment but have nothing to do with perceptible pitch height> The next level implies the interval notion which requires much more than an > ordered set as starting structure. If > a - b - c - d - ... > is a such discrete ordered set, the successive steps, we would denote > a:b - b:c - c:d - ... > have sense only if there exist a composition law on that set permitting to > define also an order on the intervals. So with the < relation (resp. > or =), > depending of the law type (multiplicative or additive), we have > a:b < c:d <=> bc < ad > a:b < c:d <=> b+c < a+dAll of which makes me think we should at least be working in a totally ordered field, but I don't understand your point. Why all the complicated bells and whistles? We surely do not want to talk about scales which are not contained in the real numbers! I simply note that to pass of order in an ordered set to the order of what we could name interval between the elements of the ordered set, we need a composition law on the set. I say that we have not to use infinity to position a very little set of tones with its relations. Pierre [This message contained attachments]
Message: 3209 - Contents - Hide Contents Date: Mon, 14 Jan 2002 16:12:19 Subject: Re: metric visualization From: paulerlich --- In tuning-math@y..., "Kees van Prooijen" <kees@d...> wrote:> Paul E. asked me (repeatedly I'm afraid, sorry Paul :-( to clear-up my > pictures in lattice orientation * [with cont.] (Wayb.) > Especially the last one apparently causes confusion because of the mixed > taxicab-euclidean paradigm. > Well, it's just a (lame) attempt to visualize the metrics Paul andI use in> a Euclidean context. So it's _not_ a transformation to a Euclidean metric. > The paradox I mention is in the interpretation of the lattice picture. Equal > distances in the metric are not equal in the picture. (Because then it > _would_ be Euclidean.) So in the last picture I distort the image so that > all distances from the center to the lattice points correspond with the > Euclidean distance we actually see. But this only works for these distances > (relative to the center). > Hope this helps.Thanks Kees. I hope you'll be patient with me, because there's much I'm trying to figure out. Let's drop the Euclidean bit entirely, even from a given reference point. I don't think it helps us at all. It's easy enough to understand a taxicab metric, or whatever you call its triangular generalization. I'm trying to understand the two middle lattices you show on that page. Although "mine" would seem to agree with your own feeling (expressed on your main page) that an appropriate error weighting for evaluating temperaments would be f3 + log(3)/log(5)*f5 + log(3)/log(5)*f(5/3) (I would love for you to explain this feeling if you could) , there seems to be something desirable about "yours" -- for one thing, I have a strong hope that your lattice implies some alternate error weighting (and perhaps a nonlinear formula) that would make my "heuristic" almost exactly perfect. My heuristic is that, if you temper out _one_ unison vector n:d in the 5-limit, and hence end up with a linear temperament, if you then optimize the generator according to the "right" error function, the temperament's (minimized) error is proportional to (n-d)/(d*log(d)) I have found that a simple error function, an equal-weighted RMS over the three consonant intervals, bears out the heuristic to within a factor of 2 for an extremely wide range of unison vectors (one might say all of them). I'm searching for the error function which will make my heuristic almost exactly perfect -- and I suspect "your" "taxicab" metric has a whole lot to do with it. So far, Gene has not helped me at all -- perhaps you can help?? Thanks, Paul
Message: 3210 - Contents - Hide Contents Date: Mon, 14 Jan 2002 17:33:44 Subject: Re: metric visualization From: manuel.op.de.coul@xxxxxxxxxxx.xxx From my experience with taxis in Asia, I can tell that the "taxicab" metric is not always identical to the "city block" metric :-) Manuel
Message: 3211 - Contents - Hide Contents Date: Mon, 14 Jan 2002 09:05:23 Subject: Re: metric visualization From: monz> From: <manuel.op.de.coul@xxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, January 14, 2002 8:33 AM > Subject: Re: [tuning-math] Re: metric visualization > > > > From my experience with taxis in Asia, I can tell that > the "taxicab" metric is not always identical to the > "city block" metric :-)Hah! That's funny. Manuel, I noticed in your "tips.par" file that you refer to the taxicab metric as a "Manhattan" metric. I like that. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3212 - Contents - Hide Contents Date: Mon, 14 Jan 2002 09:26:14 Subject: Re: For Joe--proposed definitions From: monz I've been archiving the dialog between Gene and Pierre over the definitions Gene proposed recently. Here it is: Definitions of tuning terms: Pierre Lamothe cr... * [with cont.] (Wayb.) This was hastily done, and has some aesthetic flaws, such as the difference in font styles between Gene and Pierre. But I'd like everyone to be able to easily follow the whole discussion, and I welcome the input of other voices ... mainly because I myself understand so little of it. :( Here are the original definitions by Gene: Definitions of tuning terms: Gene's tuning mat... * [with cont.] (Wayb.) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3213 - Contents - Hide Contents Date: Mon, 14 Jan 2002 19:34:17 Subject: algorithm sought From: clumma All; I'm looking for a way to generate all chords of a given card. in a given odd limit. The tonality diamond is easy, but I want ASSs too. According to Graham, there aren't that many, so I could just dope them in. Naturally, this is unacceptable. :) Besides, while I trust Graham, I can't divine from his presentation a proof that his method catches all the ASSs, so to speak. So far, I can think of no better method than brute force... building up the chords by 2nds, and removing those that break the odd-limit at each iteration. Can anyone do better? -Carl
Message: 3214 - Contents - Hide Contents Date: Mon, 14 Jan 2002 20:20:46 Subject: Re: algorithm sought From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:> I'm looking for a way to generate all chords of a > given card. in a given odd limit. The tonality > diamond is easy, but I want ASSs too. According > to Graham, there aren't that many, so I could just > dope them in. Naturally, this is unacceptable. :) > Besides, while I trust Graham, I can't divine from > his presentation a proof that his method catches > all the ASSs, so to speak.Brute force sounds like a good way to be sure you are getting everything you want. Is the problem a limit on your computing power?
Message: 3215 - Contents - Hide Contents Date: Mon, 14 Jan 2002 20:40:15 Subject: Re: algorithm sought From: clumma>> >'m looking for a way to generate all chords of a >> given card. in a given odd limit. The tonality >> diamond is easy, but I want ASSs too. According >> to Graham, there aren't that many, so I could just >> dope them in. Naturally, this is unacceptable. :) >> Besides, while I trust Graham, I can't divine from >> his presentation a proof that his method catches >> all the ASSs, so to speak. >>Brute force sounds like a good way to be sure you are >getting everything you want. Is the problem a limit on >your computing power?I don't have a scheme compiler, so, yes, it will be. At limit n, there are on the order of n^2 dyads, and at card k, there will be (n^2)^k untested chords, and the test will cost an average of (k^2)/2, or something. -Carl
Message: 3217 - Contents - Hide Contents Date: Mon, 14 Jan 2002 21:50:24 Subject: Re: metric visualization From: paulerlich --- In tuning-math@y..., "keesvp" <kees@d...> wrote:> Hi Paul, > > Just from the top of my head, try this: > Take the inverse of the last matrix I mention, so you get: > > ( 0.5 * sqrt(3) * log(5) -0.5 * log(5) ) > ( 0 log(3) ) > > or > > ( 0.91024 -0.52553 ) > ( 0.00000 0.57735 ) > > As error function, make a vector of the (signed) errors for 3 and 5, > multiply this with the constructed matrix, take the norm of the > result as error.To verify that I did this right, does the "van-Prooijen-optimal meantone" have a generator of 695.982 cents?> Curious if this will work as you expect.I get 0.00576743 as the error in octaves of this meantone. Now for the Enneadecal temperament, based on unison vector 2^14 * 3^19 * 5^-19 , I get 497.994 cents as the van-Prooijen-optimal generator (period 1/19 oct.), and 0.000152973 octave as the error. Now (81-80)/(81*log(81)) = 0.00280938, and 0.00280938/0.00576743 = 0.487111 Meanwhile, (5^19 - 2^14 * 3^19)/(5^19 * log(5^19)) = 5.314056e-5, and 5.314056e-5/0.000152973 = 0.347385 So, this looks like a "no-go" . . . can you verify that my calculations are correct?> To be exact the non-linear > correction as resulting in the last picture should also be applied (I > think), but I doubt that will very influential.Again, I'm not seeing what the pseudo-Euclidean picture could have to offer. Isn't the taxicab metric already "perfect" -- or am I missing something?
Message: 3218 - Contents - Hide Contents Date: Mon, 14 Jan 2002 21:53:13 Subject: Re: metric visualization From: paulerlich Sorry Kees -- there was an error in my formula -- I'll try again . . . --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> --- In tuning-math@y..., "keesvp" <kees@d...> wrote: >> Hi Paul, >>>> Just from the top of my head, try this: >> Take the inverse of the last matrix I mention, so you get: >> >> ( 0.5 * sqrt(3) * log(5) -0.5 * log(5) ) >> ( 0 log(3) ) >> >> or >> >> ( 0.91024 -0.52553 ) >> ( 0.00000 0.57735 ) >> >> As error function, make a vector of the (signed) errors for 3 and > 5,>> multiply this with the constructed matrix, take the norm of the >> result as error. >> To verify that I did this right, does the "van-Prooijen-optimal > meantone" have a generator of 695.982 cents? >>> Curious if this will work as you expect. >> I get 0.00576743 as the error in octaves of this meantone. Now for > the Enneadecal temperament, based on unison vector > > 2^14 * 3^19 * 5^-19 , > > I get 497.994 cents as the van-Prooijen-optimal generator (period > 1/19 oct.), and 0.000152973 octave as the error. > > Now (81-80)/(81*log(81)) = 0.00280938, and 0.00280938/0.00576743 = > 0.487111 > > Meanwhile, (5^19 - 2^14 * 3^19)/(5^19 * log(5^19)) = 5.314056e-5, and > 5.314056e-5/0.000152973 = 0.347385 > > So, this looks like a "no-go" . . . can you verify that my > calculations are correct? >>> To be exact the non-linear >> correction as resulting in the last picture should also be applied > (I>> think), but I doubt that will very influential. >> Again, I'm not seeing what the pseudo-Euclidean picture could have to > offer. Isn't the taxicab metric already "perfect" -- or am I missing > something?
Message: 3219 - Contents - Hide Contents Date: Mon, 14 Jan 2002 21:56:58 Subject: Re: metric visualization From: paulerlich Kees, when I calculate this:> > ( 0.5 * sqrt(3) * log(5) -0.5 * log(5) ) > ( 0 log(3) )I don't get> ( 0.91024 -0.52553 ) > ( 0.00000 0.57735 )as you said -- instead I get ( 1.3938 -0.80472 ) ( 0 1.0986 ) or, using base-5 logs, ( 0.86603 -0.5 ) ( 0 0.68261 ) Did you have a typo somewhere?
Message: 3221 - Contents - Hide Contents Date: Mon, 14 Jan 2002 22:08:51 Subject: Re: metric visualization From: paulerlich --- In tuning-math@y..., "keesvp" <kees@d...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >>>> Did you have a typo somewhere? >> Sorry Paul, forgot to divide by determinant in symbolic expression. > The numerics should be ok.The two matrices you gave (one with formulae, one with 5-digit numbers) do not agree, even up to a multiplicative constant. So am I to take the matrix with 5-digit numbers? If I do, I get 695.72 cents as the optimal meantone generator, with error 0.0036241 oct.; I get 498.02 cents as the optimal enneadecal generator, with error 9.3852e- 005 . . . now 0.0036241/0.0028094 = 1.29, and 9.3852e-5/5.3141e-5 = 1.7661 . . . so this doesn't seem to work either . . . can you verify my numbers?
Message: 3223 - Contents - Hide Contents Date: Mon, 14 Jan 2002 22:49:20 Subject: Re: metric visualization From: paulerlich --- In tuning-math@y..., "keesvp" <kees@d...> wrote:> I don't know what went wrong, but the right numbers as far as I can > see now (cross my fingers) are: > ( 0.91024 -0.52553 ) > ( 0.00000 0.71746 )This gives the same result as your original, formulaically expressed matrix.> I will take a better look at it tonight.I appreciate it greatly.
Message: 3224 - Contents - Hide Contents Date: Mon, 14 Jan 2002 23:07:49 Subject: Re: algorithm sought From: clumma>> >'m looking for a way to generate all chords of a >> given card. in a given odd limit. The tonality >> diamond is easy, but I want ASSs too. According >> to Graham, there aren't that many, so I could just >> dope them in. Naturally, this is unacceptable. :) >> Besides, while I trust Graham, I can't divine from >> his presentation a proof that his method catches >> all the ASSs, so to speak. >>Brute force sounds like a good way to be sure you >are getting everything you want. Is the problem a >limit on your computing power?Anyway, Gene, what I was thinking... under the dyadic definition, all n-limit chords must be connected on the n-limit lattice, and must have a Hahn-diameter of 1. See: Music (and Music Theory) * [with cont.] (Wayb.) There ought to be a geometric way to find these structures... -Carl
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