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Message: 9950 - Contents - Hide Contents Date: Tue, 10 Feb 2004 07:15:20 Subject: Re: loglog! From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:> Yahoo groups: /tuning-math/files/Paul/et5loglo... * [with cont.] > > Ok, easy! No moat needed, at least for ETs. Just draw a > circle around the origin and grow the radius until it would > include something that exceeds a single bound -- a "TOP > notes per 1200 cents" bound. For ETs at least. Choose a > bound according to sensibilities in the 5-limit, round it > to the nearest ten, and use it for all limits.That's great, Carl, but in loglog land the origin is arbitrary.

Message: 9951 - Contents - Hide Contents Date: Tue, 10 Feb 2004 16:26:16 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>>>> >..still trying to understand why the rectangle doesn't enclose >>>> a finite number of temperaments... >>> >>> Which rectangle? >>>> The rectangle enclosed by error and complexity bounds. >>Yes, that would enclose a finite number of temperaments.Then why the hell do we need a badness bound? Alternatively, then why doesn't the badness bound alone enclose a finite triangle?>>>> My thoughts are that in the 5-limit, we might reasonably have a >>>> chance of guessing a good list. But beyond that, I would cry >>>> Judas if anyone here claimed they could hand-pick anything. So, >>>> my question to you is: can a 5-limit moat be extrapolated upwards >>>> nicely? >>>>>> Not sure what you mean by that. >>>> Which part? Can the equation/coordinates that defines your fav. >> moat be taken from a 5-limit plot and slapped onto a 7-limit one? >>No. The units are not the same. Well, depends which 5-limit one and >which 7-limit one you mean.See my prev. message about notes units for complexity. Error units ought to be the same!>Remember, we're dealing with a Pascal's >triangle, with one scenario for each number in the triangle, where >the number itself tells you the number of elements in the wedgie, the >rrow number is the number of primes, and the column number is the >codimension.I never knew that or forgot it! -Carl

Message: 9952 - Contents - Hide Contents Date: Tue, 10 Feb 2004 19:45:53 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> But there's less to tweak -- we just find the thickest moat than >> encloses the systems in the same ballpark as the ones we know we >> definitely want to include. This seems a lot less arbitrary than >> tweaking *three* parameters to satisfy one's sensibilities as best > as >> possible. >> Your plots make it clear you'd better trash the idea of doing moats > in anything but loglog.On the contrary.

Message: 9953 - Contents - Hide Contents Date: Tue, 10 Feb 2004 20:43:00 Subject: Re: The seven-limit lattices From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> 1. The 14 notes of the stellated hexany. Do these correspond to the > sides of the cubeoctohedran? (Or the points on a D3 face centered > lattice)...They correspond to the 14 verticies of a rhombic dodecahedron, which I'd better add to the page.> 2. Where in the cubeoctohedran are the tetrahedra and octahedra? > Could you please give an example of a hexany...I'm not sure what your question means re the cubeoctahedron; it has 12 verticies corresponding to the 12 intervals nearest the unison, and forming the 7-limit consonances with it. We've got a 14-note scale of the rhombic dodecahedron/stellated hexany type, and a 13-note scale of 13 notes in a ball of radius one around the unison. {3,5,7,15,21,35} is the canonical hexany from Wilson's combinatorial point of view. Dividing it by 7 and reducing to an octave gives [1, 15/14, 5/4, 10/7, 3/2, 12/7]; dividing by 5 instead gives [1, 21/20, 6/5, 7/5, 3/2, 7/4]; these are the sorts of hexanies one finds (and I found) geometrically, when looking at the ones with the unison as a lattice point.> 3. I understand the change of of basis in paragraph 2. However > in paragraph 3, "In this new coordinate system" you have (000),(100), > (010) and (001). How are these related to the vertices above?That's a mistake; I'll correct it.> 4. Paragraph 7. Where do the negatives in the (1/2, 1/2, 1/2) terms > come from?Those are the basis elements for the lattice of mappings.

Message: 9954 - Contents - Hide Contents Date: Tue, 10 Feb 2004 21:47:03 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> Except you suddenly depart from the lay of the land in two places? >> Why this suddenness? What powerful psychological force operates at >> these two points? >> Lack of interest in tempering out 3/2.But not 4/3?> However, you don't need to set > any error bound if this doesn't worry you, and don't need any > complexity bound at all. We discussed this extensively in the past, > back in the days when you didn't like the idea of getting a finite > list, which this would do for the right badness exponent or slope.Unfortunately, we can't publish an infinitely long paper.>> As I showed, the curve in question, on a log-log plot, looks clearly >> unlike a line. >> Can you tell me where this plot is, with axes clearly labeled, so the > rest of us have a clue? See Yahoo groups: /tuning-math/message/9479 * [with cont.]By your preceding comments, you must already 'have' the plot. Now that you 'know', can you tell me how there could still have been any question as to what the axes represent, since on all the graphs, there never have been any exceptions to x=complexity, y=error, I repeatedly pointed out this fact, and we've already established that we're on the same page for these quantities? You'll have to forgive me if I feel like you're latching on to any possible excuse, real or imagined, to denigrate my work, and that of Dave as well. I've been working extremely hard to wrap my brain around your nearly impenetrable posts for years, going way out on a limb to defend you against your detractors, and hailing you as the savior of our cause. Now I'm working hard to satisfy everyone's needs in this discussion, including yours. If I leave out a detail that could be ambiguous to someone who's very far away from this discussion, I would expect that you, of all people, would be able to tolerate this. If you can't operate on the principle of 'trying to understand' rather than 'trying to dismiss', then you have no right to expect others to, and then your work here, I'm sorry to say, would be the first to fall by the wayside.

Message: 9955 - Contents - Hide Contents Date: Tue, 10 Feb 2004 00:46:04 Subject: Re: Loglog From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> I checked the files I saved of the graphs being posted, and found > no>> loglog examples. I then went over to tuning-files, and found one >> example, >> You missed quite a few then, likeIn none of the cases where the axes are labled does it say any of these are loglog plots. Moreover, they are very, very diffult to make any sense of, because they are too busy, with far too many points and contour lines all over the place.> Yes, for you. No comments, just general derision?I made a comment, which is that the axes are unlabled and I don't know what I am looking at.>> I can't tell by looking at it what the logs >> are logs of, however. Clarifying this would be nice. >> Complexity and error -- what else could it be? Some of the graphs > above are even labeled ;)I have no idea what it is unless you tell me. As I've already pointed out, none of the graphs with labels say they are loglog plots.>> It would also be >> nice if, having created all these loglog images, they were made >> available to the rest of us. >> Yes, I've tried to be nice like this, and will continue to do so.You seem awfully snippy about it. Can you simply point out any examples of plots of temperaments where the axes are in terms of logs of TOP complexity and error?

Message: 9956 - Contents - Hide Contents Date: Tue, 10 Feb 2004 04:34:15 Subject: Re: The same page From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Just to be sure we are on it, in terms of defintions of compexity and > error, here is my page.Where are ETs?> 5-limit, comma = n/d > > Complexity is log2(n*d),Yes, but this can also be expressed in other ways, for example if <<a1 a2 a3|| is the val-wedgie (dual to the comma), it can be expressed as |a1/p3| + |a2/p5| + |a3/p2p5|> so log(complexity) is loglog(n*d). Error is > distance from the TOP tuning to the JIP, or in other words the max of > the absolute values of the errors for 2, 3 and 5 in TOP tuning, > divided by log2(2), log2(3) and log2(5) respectively.It also can be expressed as log(n/d)/log(n*d) (*1200).> Log(error) is > the log of this. Loglog plots compare loglog(n*d) with log(error).i.e., log(log(n*d)) with log(log(n/d)/log(n*d)).> > 7-limit linear > > Complexity is the Erlich magic L1 norm; if <<a1 a2 a3 a4 a5 a6|| is > the wedgie, val-wedgie, yes. > then complexity is > |a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7|.Note the similarity. It really should be one formula for all cases.> Error is the distance from the JIP of the 7-limit TOP > tuning for the temperament;Or same as 5-limit linear error but with an additional term for 7

Message: 9957 - Contents - Hide Contents Date: Tue, 10 Feb 2004 07:18:45 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> I don't have a problem with that. I still think the simplest curves > through moats that are in the right ballpark will be of the form > > (err/k1)^p + (comp/k2)^p < x where p is 1 or slightly less than 1.The ets Paul just got through plotting are lying more or less along straight lines. I don't see any way to make a sensible moat unless your line follows the lay of the land, so to speak.

Message: 9958 - Contents - Hide Contents Date: Tue, 10 Feb 2004 19:45:14 Subject: Re: ! From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>> Why should we want to multiply instead of add? >> Oh, for God's sake Paul-have you looked at your own plots?Of course I have.> Did you > notice how straight the thing looks in loglog coordinates?Yes, this has been clear to me for years.> Your plots > make it clear that loglog is the right approach. Look at them!Geez, you must really be thinking like a mathematician and not a musician.

Message: 9959 - Contents - Hide Contents Date: Tue, 10 Feb 2004 20:44:31 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> ...still trying to understand why the rectangle doesn't enclose >>> a finite number of temperaments... >> >> Which rectangle? >> The rectangle enclosed by error and complexity bounds.Yes, that would enclose a finite number of temperaments.> You answered > that the axes were infinitely far away,In log-log, this is true, and thus you can't *visually* verify that there are a finite number of temperaments enclosed, as you can in the linear-linear plot.>>> My thoughts are that in the 5-limit, we might reasonably have a >>> chance of guessing a good list. But beyond that, I would cry >>> Judas if anyone here claimed they could hand-pick anything. So, >>> my question to you is: can a 5-limit moat be extrapolated upwards >>> nicely? >>>> Not sure what you mean by that. >> Which part? Can the equation/coordinates that defines your fav. moat > be taken from a 5-limit plot and slapped onto a 7-limit one?No. The units are not the same. Well, depends which 5-limit one and which 7-limit one you mean. Remember, we're dealing with a Pascal's triangle, with one scenario for each number in the triangle, where the number itself tells you the number of elements in the wedgie, the rrow number is the number of primes, and the column number is the codimension.

Message: 9960 - Contents - Hide Contents Date: Tue, 10 Feb 2004 21:48:10 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> Prove it. Give log-log plots for your proposed moats, >> I already did that for one case, pointed it out twice, and asked for > your comments.I have saved a plot I don't know how to interpret.>> It's possible we could come to some kind of consensus >> if you would attempt to treat people with something better than the >> contempt you have shown lately. >> I can take your attitude in no other way, unless you either ignored > completely or have an abominably low level of respect for the > discussions Dave and I posted on the topic.Dave has been treating the rest of us with respect and discussing things. I'd like the same from you.> Let's start over. If I'm willing to tolerate a certain level of > error, and a certain level of complexity, why wouldn't I be willing > to tolerate both together?From some points of view complexity doesn't even matter, so the whole premise we've all been operating under can be questioned by someone who is interested in the character of the commas in the kernel, not what complexity they give. As for your question, are you arguing *for* straight line error and complexity bounds, because I don't see where else you can possibly go with it? Tolerating both together is exactly what Dave doesn't tolerate, and I thought you agreed with that.

Message: 9961 - Contents - Hide Contents Date: Tue, 10 Feb 2004 00:52:47 Subject: Re: A post with pending questions From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> No, I ask a lot of questions, and they're almost never rhetorical. > Now, if we could only rewind a couple of years and start over . . .Ooops. :) I automatically circular file anything without even thinking about it if I think it is a guessing game.

Message: 9962 - Contents - Hide Contents Date: Tue, 10 Feb 2004 04:34:59 Subject: Re: The same page From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:>> Complexity is the Erlich magic L1 norm; if >> <<a1 a2 a3 a4 a5 a6|| is the wedgie, then complexity is >> |a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7|. >> Log complexity is log of this. Error is the distance from >> the JIP of the 7-limit TOP tuning for the temperament; log >> (complexity) and log(error) are logs of complexity and >> error, so defined. >> What are p3, p5, etc.? > > -Carllog2(3), log2(5), etc.

Message: 9963 - Contents - Hide Contents Date: Tue, 10 Feb 2004 07:21:39 Subject: Re: Loglog From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:>> No, the good stuff lies along lines, making the whole moat business >> both much easier and far more logical--in case that matters to anyone. >> On the loglog plot. The good stuff looks to me like a bite taken out > of the lower left side of the sheet of temperaments. Teeth marks and > all ;-).I do see some snaggle teeth, like 171 on the 7-limit plot. Should 171 be included? If you want to go that high on complexity, obviously yes. If you don't, obviously no. Yhe dreaded complexity bound strikes again.

Message: 9964 - Contents - Hide Contents Date: Tue, 10 Feb 2004 16:28:29 Subject: Re: The same page From: Carl Lumma>>>> >nybody have a handy asci 'units' table for popular wedge >>>> products in ket notation? ie, >>>> >>>> [ val > ^ [ val > -> [[ wedgie >> >>>> < monzo ] ^ < monzo ] -> ? >>>>>>>> [monzo> ^ [monzo> -> ||bimonzo>> >> >> Great, so what happens when the monzos are commas being >> tempered out? >>That's what they always represent here.Yes of course, but in that case, what does the bimonzo give us? Anything musical?>> A chart running over comma useful things would help our >> endeavor tremendously. >>What would you like to see?A dummy chart for what I need to wedge in order to get what I care about about temperaments.>>> In 3D (e.g., 5-limit), for linear temperaments the bival is dual >>> to the monzo, and for equal temperaments the bimonzo is dual to >>> the val. >>> >>> In 4D (e.g., 7-limit), for linear temperaments the bival is dual >>> to the bimonzo, and both are referred to as the "wedgie" (though >>> Gene uses the bival form). >>>> Ok great. But what's all about this algebraic dual? Is this >> something I can do to matrices, like complement and transpose? >>The wedgie is a sort of vector, not a matrix. The dual involves >reversing the order of the entries, and flipping some of the signs. >There was an extensive thread here recently explaining this.Ok, so the form for that should be on the chart. -Carl

Message: 9965 - Contents - Hide Contents Date: Tue, 10 Feb 2004 19:48:03 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>>> My objection was not to limits on them per se, but to acceptance >>>>>> regions shaped like this (on a log-log plot). >>>>>> >>>>>> err >>>>>> | >>>>>> | (a) >>>>>> |---\ >>>>>> | \ >>>>>> | \ >>>>>> | \ (b) >>>>>> | | >>>>>> | | >>>>>> ------------ comp >>>>>> >>>>>> as opposed to a smooth curve that rounds off those corners marked >>>>>> (a) and (b). >>>>>>>>>> Aha, now I understand your objection. But wait, what's stopping >>>>> this from being a rectangle? Is the badness bound giving the >>>>> line AB? >>>> >>>> Yes. >>>>>>>>> If so, it looks like a badness cutoff alone would give a >>>>> finite region... >>>>>>>> No, because the zero-error line is infinitely far away on a loglog >>>> plot. >>>>>> Can you illustrate this? >>>> How can I illustrate infinity? >>>>> It looks like the zero-error line is >>> three dashes away on the above loglog plot. :) >>>> Since you're smiliing, I'll assume you "got it". >> No, I was just cracking wise. :( >Well, again, the zero-error line is infintely far down. No matter how you set up your log-log plot, and no matter how big it is, the zero- error line will never be on it.

Message: 9966 - Contents - Hide Contents Date: Tue, 10 Feb 2004 20:47:23 Subject: Re: The same page From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Anybody have a handy asci 'units' table for popular wedge products >>> in ket notation? ie, >>> >>> [ val > ^ [ val > -> [[ wedgie >> >>> < monzo ] ^ < monzo ] -> ? >>>>> [monzo> ^ [monzo> -> ||bimonzo>> > > Great, so what happens when the monzos are commas being > tempered out?That's what they always represent here.> A chart running over comma useful things would help our > endeavor tremendously.What would you like to see?>> In 3D (e.g., 5-limit), for linear temperaments the bival is dual to >> the monzo, and for equal temperaments the bimonzo is dual to the val. >> >> In 4D (e.g., 7-limit), for linear temperaments the bival is dual to >> the bimonzo, and both are referred to as the "wedgie" (though Gene >> uses the bival form). >> Ok great. But what's all about this algebraic dual? Is this > something I can do to matrices, like complement and transpose?The wedgie is a sort of vector, not a matrix. The dual involves reversing the order of the entries, and flipping some of the signs. There was an extensive thread here recently explaining this.

Message: 9968 - Contents - Hide Contents Date: Tue, 10 Feb 2004 01:00:26 Subject: Re: Beep isn't useless.... From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:>> The Canon was ideal for 5-limit as there was no such doubt. >> >> Can't we find some "classic" 7-limit-JI piece that demonstrates a > lot>> of different 7-limit consonances and cadences, and warp that? >> Cadences are too tied to particular scales and temperaments.Agreed. But the rest would be good.

Message: 9969 - Contents - Hide Contents Date: Tue, 10 Feb 2004 04:44:50 Subject: Re: The same page From: Paul Erlich Sorry Gene, I thought I was responding to Carl, otherwise I wouldn't have posted all that math that must be obvious to you. --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> Just to be sure we are on it, in terms of defintions of compexity > and>> error, here is my page. >> Where are ETs? >>> 5-limit, comma = n/d >> >> Complexity is log2(n*d), >> Yes, but this can also be expressed in other ways, for example if > > <<a1 a2 a3|| > > is the val-wedgie (dual to the comma), it can be expressed as > > |a1/p3| + |a2/p5| + |a3/p2p5| >>> so log(complexity) is loglog(n*d). Error is >> distance from the TOP tuning to the JIP, or in other words the max > of>> the absolute values of the errors for 2, 3 and 5 in TOP tuning, >> divided by log2(2), log2(3) and log2(5) respectively. >> It also can be expressed as log(n/d)/log(n*d) (*1200). > >> Log(error) is>> the log of this. Loglog plots compare loglog(n*d) with log(error). >> i.e., log(log(n*d)) with log(log(n/d)/log(n*d)). >> >> 7-limit linear >>>> Complexity is the Erlich magic L1 norm; if <<a1 a2 a3 a4 a5 a6|| is >> the wedgie, > > val-wedgie, yes. >>> then complexity is >> |a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7|. >> Note the similarity. It really should be one formula for all cases. >>> Error is the distance from the JIP of the 7-limit TOP >> tuning for the temperament; >

Message: 9970 - Contents - Hide Contents Date: Tue, 10 Feb 2004 07:22:04 Subject: Re: Loglog From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:>> No, the good stuff lies along lines, making the whole moat business >> both much easier and far more logical--in case that matters to anyone. >> On the loglog plot. The good stuff looks to me like a bite taken out > of the lower left side of the sheet of temperaments. Teeth marks and > all ;-).I do see some snaggle teeth, like 171 on the 7-limit plot. Should 171 be included? If you want to go that high on complexity, obviously yes. If you don't, obviously no. Yhe dreaded complexity bound strikes again.

Message: 9971 - Contents - Hide Contents Date: Tue, 10 Feb 2004 16:32:35 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>> >nd only Haverstick could be said to have worked in a "5-limit ET" >> (and it's a stretch). We've got Miller, Smith and Pehrson left, >> with the promising Erlich and monz stuck in theory and/or 12-tET >> land. We're so far from any kind of form that would allow us to >> make statements about musical utility that it's laughable. >>And why would you limit this information to those who have posted? We >have also heard about composers who never go near the tuning lists. >Darreg, Blackwood, Negri and Hanson come immediately to mind.Of course I see your point, and you *can* make a case for it. But I am convinced the experience we have so far might in fact tell us *nothing* about the future. Therefore I am inclined, as usual, to derive everything from first principles or a well-controlled experiment. -Carl

Message: 9972 - Contents - Hide Contents Date: Tue, 10 Feb 2004 19:47:06 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Do you mean TOP error for an equal >>> temperament, >>>> Of course that's what he means. >> For 7-limit ets, how do you decide which comma to use?What comma to use? There's no comma to use: n and d run over *all* ratios n/d, not just one or more commas.

Message: 9973 - Contents - Hide Contents Date: Tue, 10 Feb 2004 20:49:37 Subject: Re: The seven-limit lattices From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" > <paul.hjelmstad@u...> wrote: >>> 1. The 14 notes of the stellated hexany. Do these correspond to the >> sides of the cubeoctohedran? (Or the points on a D3 face centered >> lattice)... >> They correspond to the 14 verticies of a rhombic dodecahedron, which > I'd better add to the page. >>> 2. Where in the cubeoctohedran are the tetrahedra and octahedra? >> Could you please give an example of a hexany... >> I'm not sure what your question means re the cubeoctahedron; it has 12 > verticies corresponding to the 12 intervals nearest the unison, and > forming the 7-limit consonances with it. We've got a 14-note scale of > the rhombic dodecahedron/stellated hexany type, and a 13-note scale of > 13 notes in a ball of radius one around the unison.The 7-limit diamond -- this, too, is depicted in my paper.

Message: 9974 - Contents - Hide Contents Date: Tue, 10 Feb 2004 21:50:18 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: >>> I think the regular plot will be easier to explain than the log- log >> plot. >> Are you going to actually explain it, or just sweep that under the > rug?Yes, excuse me for the brutal honesty, but my track record for successfully explaining things to people is just a bit better than yours. The next few months are going to be insanely brutal as we try to work together putting together an explanation for people to read.

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