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Message: 9975 - Contents - Hide Contents Date: Tue, 10 Feb 2004 01:00:08 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>> >y latest position is that I can live with log-flat badness with >> appropriate cutoffs. The problem with anything more tricky is that >> we have no data. Not vague historical data, actually no data. >>Three questions regarding this statement. > >1. Why is log-flat badness with cutoffs (on error and complexity) >less tricky than the cutoff functions Paul and I have been looking >at.logflat is unique among badness functions I know of in that it does not favor any region of complexity or error (thus it reveals something about the natural distribution of temperaments) and has zero free variables.>Log-flat badness with cutoffsThe cutoffs are of course completely arbitrary, but can be easily justified and explained in the context of a paper.>2. Assuming for the moment that we have no data, why isn't that >just as much of a problem for log-flat badness with e&c cutoffs >as for any other proposed cutoff relation?Ignoring the cutoffs, logflat does reveal something fundamental about the distribution of temperaments. Whether musically appropriate or not (utterly unfalsifiable assumptions), it gives an unbiased view of ennealimmal vs. meantone, etc.>i.e. How should we decide what cutoffs to use on error, complexity >and log-flat badness?You can tweak them to satisfy your sensibilities as best as possible, same as you're tweaking the moat to factor infinity to satisfy your sensibilities as best as possible.>3. Why don't discussions of the value of various temperaments in >the archives of the tuning list constitute data on this, or at >least evidence?Because nobody here or on the tuning list has the slightest clue about what's musically useful. Nobody has composed more than a few ditties in any of these systems.>> But as long as Dave and Paul were having fun I >> didn't want to say anything. They have a way of coming up with >> neat stuff, though so far their conversation has been >> impenetrable to me. >It doesn't explain what the heck a moat is, for starters. -Carl

Message: 9977 - Contents - Hide Contents Date: Tue, 10 Feb 2004 16:35:07 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>> >he rectangle enclosed by error and complexity bounds. You answered >> that the axes were infinitely far away, but the badness line AB >> doesn't seem to be helping that. >>If you simply bound complexity alone, you get a finite number of >temperaments. Most are complete crap.Above I suggest a rectangle which bounds complexity and error, not complexity alone. In the circle suggestion I suggest a circle plus a complexity bound is sufficient. -Carl

Message: 9978 - Contents - Hide Contents Date: Tue, 10 Feb 2004 19:50:39 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> Thus it's great for a paper for mathematicians. Not for musicians. >> The *contents* of the list is what's great for musicians, not > how it was generated.No; I agree with Graham that we should "teach a man to fish".>>>> Log-flat badness with cutoffs >>>>>> The cutoffs are of course completely arbitrary, but can be easily >>> justified and explained in the context of a paper. >>>> But there are *three* of them! >> ...still trying to understand why the rectangle doesn't enclose > a finite number of temperaments... Which rectangle? > With moats it seems you're pretty-much able to hand pick the list,No way, dude! The decision is virtually made for us. If you can find a wider moat in the vicinity, we'll adopt it.> By 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.

Message: 9979 - Contents - Hide Contents Date: Tue, 10 Feb 2004 20:52:23 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> It was principal components analysis, but the reasoning behind the > implementation was obscure.The reasoning was to draw an elliptical moat.> OK, Carl, so everyone's been sorely underestimating the true > usefulness of 665-equal and 612-equal, yes?Sounds like you are. Not everyone plays live music and has that as their focus, like you.> Dave's exactly right. If we're violating it suddenly at the cutoffs > but nowhere else, we're clearly not conforming to any kind of > psychological badness criterion.And if you are simply drawing squiggly lines on a graph, you are?

Message: 9980 - Contents - Hide Contents Date: Tue, 10 Feb 2004 21:56:27 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Graham Breed Dave Keenan wrote:> I disagree. It's just too hard for non-mathematicians. Unless by > "fish" you mean "go to Graham's web site and use the temperament > finder there" in which case I'm all for it! And this would let us not > worry too much that we may have left some temperament out of the paper > that someone someday may find useful.That's roughly what I meant. Of course, the temperament finder could always do with improving (even mathematicians have trouble understanding it!) and could do with a good user guide -- hence the "teach" part. And I could really do with help with that. We also need to give mathematicians the instructions for writing their own temperament finders for their own websites, or software packages, or idle amusement. Either endeavour would be more worthy of my time than endless discussions about what temperaments to include on a list. But, while I'm here: - log-flat looks like a good place to start - silence is negative infinity in decibels - spherical projection! - can somebody give a friendly explanation of complex hulls? - would k-means have anything to do with the clustering? K-Mean Clustering Tutorial * [with cont.] (Wayb.) Graham

Message: 9981 - Contents - Hide Contents Date: Tue, 10 Feb 2004 01:03:30 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Related how? Via log-flat badness? Meanwhile, error and complexity > are related?If C is the complexity and E is the error for a 7-limit linear temperament which belongs to an infinite list of best examples, then E ~ k C^2. Taking the logs, log(E) ~ 2 log(C) + c. In other words, we have a relationship; one can very roughly be estimated in terms of the other.>> Seeing the plots would be nice. It remains vaporware to me, even if >> Paul has them, if they aren't made available. >> OK, I'll do more of them when I have a chance, but ultimately, I > don't think I want to force any musician to think about what log > (error) means or what log(complexity) means.You are going to explain TOP error and complexity, but this would be too much math??>>> High complexity really isn't such a big deal for some uses. JI can > be>> said to have infinite complexity in a sense (no amount of fifths > and>> octaves will net you a pure major third, etc) which I think shows >> Paul's worry about where it is on the graph is absurd, >> No, it shows the bullshit you're putting into my mouth is absurd, as > I agreed in a recent post.You go on and on about not finding the zero error line, though evidently not finding the infinite complexity line is not a problem. I think that is absurd, but if you want it, you can find it on the projective plane.

Message: 9982 - Contents - Hide Contents Date: Tue, 10 Feb 2004 05:03:50 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: >>> Well Paul and I see it as bringing it in closer touch with reality. >> Convince us. Make a case. Show some loglog plots and prove they make > no sense. Explain why what you are doing does make sense. Is this an > unreasonable request?I seem to have been doing nothing but that for the past two days. The fact that you haven't recognised it as such says to me that we're somehow talking past each other much worse than I thought. We're trying to come up with some reasonable way to decide on which temperaments of each type to include in a paper on temperaments, given that space is always limited. We want to include those few (maybe only about 20 of each type) which we feel are most likely to actually be found useful by musicians, and we want to be able to answer questions of the kind: "since you included this and this, then why didn't you included this". So Gene may have a point when he talks about cluster analysis, I just don't find his applications of it so far to be producing useful results. Our starting point (but _only_ a starting point) is the knowledge we've built up, over many years spent on the tuning list, regarding what people find musically useful, with 5-limit ETs having had the greatest coverage. It may be an objective mathematical fact that log-flat badness gives uniform distribution, but you don't need a multiple-choice survey to know it is a psychological fact that musicians aren't terribly interested in availing themselves of the full resources of 4276-ET ()or whatever it was. Nor are they interested in a 5-limit temperament where 6/5 is distributed. So we add complexity and error cutoffs which utterly violate log-flat badness in their region of application (so why violate log-flat badness elsewhere and make the transition to non-violatedness as smooth as possible. Corners in the cutoff line are bad because there are too many ways for a temperament to be close to the outside of a corner. A moat is a wide and straight (or smoothly curved) band of white space on the complexity-error chart, surrounding your included temperaments. It is good to have a moat so that you can answer questions like "since you included this and this, then why didn't you included this", by at least offering that "it's a long way from any of the included temperaments, on an error complexity plot". The way to find a useful moat is to start with the temperaments you know everyone will want included, and those that almost no one will care about, and check out the space between the two.

Message: 9983 - Contents - Hide Contents Date: Tue, 10 Feb 2004 07:24:05 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> For musicians, I'd make the list 5 for each limit; 10 tops. For >>> people reading a theory paper, 20 would be interesting. >>>> Ridiculous. I've *composed* in about that many temperaments. >> You're not a professional musician, are you?For the right price, I'll compose in your choice of temperament.

Message: 9984 - Contents - Hide Contents Date: Tue, 10 Feb 2004 16:41:13 Subject: Re: Rhombic dodecahedron scale From: Carl Lumma>>> > Voronoi cell for a lattice is every point >>> at least as close (closer, for an interior point) to a paricular >>> vertex than to any other vertex. The Voronoi cells for the >>> face-centered cubic lattice of 7-limit intervals is the rhombic >>> dodecahedron >>>> Something Fuller demonstrated, in his own tongue. > >Right. Fuller? Buckminster. >>> These>>> fill the whole space, like a bee's honeycomb. >>>> Isn't it also the dual to the FCC lattice (hmm, maybe dual isn't >> the right word here...) >>The dual to the fcc lattice is the bcc lattice (body-centered cubic >lattice.) Indeed, sorry.>>> The Delaunay celles of a >>> lattice are the convex hulls of the lattice points closest to a >>> Voronoi cell vertex; in this case we get tetrahedra and octahedra, >>> which are the holes of the lattice, and are tetrads or hexanies. >>> The six (+-1 0 0) verticies of the Voronoi cell >>>> *The* Voronoi cell? Which one do you mean? >

Message: 9985 - Contents - Hide Contents Date: Tue, 10 Feb 2004 19:53:21 Subject: Rhombic dodecahedron scale From: Gene Ward Smith Here is a scale which arose when I was considering adding to the seven limit lattices web page. A Voronoi cell for a lattice is every point at least as close (closer, for an interior point) to a paricular vertex than to any other vertex. The Voronoi cells for the face-centered cubic lattice of 7-limit intervals is the rhombic dodecahedron with the 14 verticies (+-1 0 0), (0 +-1 0), (0 0 +-1), (+-1/2 +-1/2 +-1/2). These fill the whole space, like a bee's honeycomb. The Delaunay celles of a lattice are the convex hulls of the lattice points closest to a Voronoi cell vertex; in this case we get tetrahedra and octahedra, which are the holes of the lattice, and are tetrads or hexanies. The six (+-1 0 0) verticies of the Voronoi cell correspond to six hexanies, and the eight others to eight tetrads. If we put all of these together, we obtain the following scale of 19 notes, all of whose intervals are superparticular ratios: ! rhomb.scl Union of Delauny cells for the rhombic dodecahedron Voronoi cell centered at (0 0 0) 19 ! 21/20 15/14 8/7 7/6 6/5 5/4 4/3 48/35 7/5 10/7 35/24 3/2 8/5 5/3 12/7 7/4 28/15 40/21 2 Here it is in TOP Marvel: ! rhombmarv.scl TOP Marvel version of rhomb.scl 19 ! 85.229563 115.634597 231.269195 268.545555 316.498758 384.180152 499.814749 547.767953 585.044313 615.449347 652.725707 700.678910 816.313507 883.994902 931.948105 969.224465 1084.859062 1115.264096 1200.493659

Message: 9986 - Contents - Hide Contents Date: Tue, 10 Feb 2004 20:57:43 Subject: Re: ! From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Did either of you guys look at the loglog version of the moat-of-23 7- > limit linear temperaments?I have a plot with unlabled axes and a curved red line on it. Obviously, since I don't know what is being plotted, I draw no conclusion.

Message: 9987 - Contents - Hide Contents Date: Tue, 10 Feb 2004 21:53:17 Subject: Re: The same page From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:>>> 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). >> Both are referred to as the "wedgie" by whom?For example, in the original post to Paul Hj. explaining Pascal's triangle. Clearly there, when there's only one val involved, the wedgie can only be a multimonzo, not a multival.>> Ok great. But what's all about this algebraic dual? Is this >> something I can do to matrices, like complement and transpose? >> It's the complement.Oh yeah -- sorry!

Message: 9988 - Contents - Hide Contents Date: Tue, 10 Feb 2004 01:06:06 Subject: 23 "pro-moated" 7-limit linear temps, L_1 complex.(was: Re: 126 7-limit linears) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: >>> I'm hoping paul can easily replot those ET plots loglog. >> When I do so, at least keep in mind that rather than log (complexity), > 2^complexity has actually been proposed as a criterion (i.e., by > Fokker), and that error^2, at least, has gotten much attention as a > measure of pain, while log(error) has gotten none.That it's gotten none is what I'm complaining about. That 2^complexity has been discussed bores me to tears, unless you can explain *why*.

Message: 9989 - Contents - Hide Contents Date: Tue, 10 Feb 2004 06:19:14 Subject: ! From: Gene Ward Smith --- 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? Did you notice how straight the thing looks in loglog coordinates? Your plots make it clear that loglog is the right approach. Look at them!

Message: 9990 - Contents - Hide Contents Date: Tue, 10 Feb 2004 19:56:01 Subject: Re: The same page From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> 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), >> I thought val ^ val -> comma,No, it's dual to the comma.> so val ^ val must not be a val-wedgie.Yes, I meant val ^ val.> What's a val-wedgie? > > Anybody have a handy asci 'units' table for popular wedge products > in ket notation? ie, > > [ val > ^ [ val > -> [[ wedgie >> > < monzo ] ^ < monzo ] -> ?<val] ^ <val] -> <<bival|| [monzo> ^ [monzo> -> ||bimonzo>> 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).> ...etc. >>>>> 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. >> What's linear error?No -- (5-limit linear) error. See the original message.

Message: 9991 - Contents - Hide Contents Date: Tue, 10 Feb 2004 21:03:18 Subject: Re: The same page From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> Forgot 'em, but you seem to have them figured out. Modulo some > slight>> fiddling if you must fiddle, >> I'd like to understand this slight fiddling, and apply this > understanding to the 7-limit linear case (and elsewhere).You can take the val and simply choose the first number in it, the number of steps in an octave. Or, you can normalize it by 1/log2(prime), and take the maximum. Or, you can TOP tune it, normalize that, and take the maximum.>> That's how "wedgie" is defined. >> They're merely duals of one another, but why this definition?It's nicer than the monzo version, as the first part of it gives you the period and the mapping to primes of the generator, and the signs of the mapping are not screwed around with. I can> understand taking the wedge product of monzos much better than I > understand the wedge product of vals.See above. Vals is clearly the correct choice.

Message: 9993 - Contents - Hide Contents Date: Tue, 10 Feb 2004 01:10:01 Subject: Re: Loglog From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> My apologies again, these used log of error, but not log of > complexity. Using log of complexity crammed all the interesting stuff > to the far left to the point of illegibility, in the cases I > originally tried.Sounds like a reason to get rid of most of your points, which are gumming up the works anyway, and look at the good stuff.

Message: 9994 - Contents - Hide Contents Date: Tue, 10 Feb 2004 06:21:03 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- 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.

Message: 9995 - Contents - Hide Contents Date: Tue, 10 Feb 2004 07:39:38 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: > Then first loglog plots I've seen were just now posted by Paul; they > make a *very* strong case for loglog, not at all to my surprise. It > would be interesting now to see linears.Paul posted the linear versions a while ago. And I posted a link to Paul's message introducing them, for Carl earlier today.>> Corners in the cutoff line are bad because there are too many ways > for>> a temperament to be close to the outside of a corner. >> There's only one way to do it, which is to do it. I don't see why > this is any kind of argument. Something on the very edge of your > criterion is by definition marginal, whereever your margin lies. You > can try to avoid this by moats, but that's only going to take you so > far, and if you are not careful (and I've seen no signs of care) into > regions where the justification is dubious. If you want a list, why > not just pick your favorites and put them on it?Because if someone plots them on a graph (whether log or linear), along with some nearby ones we left out, then if the only way to draw a line separating them is to have lots of zigs and zags in it, they will have good reason to complain.>> A moat is a wide and straight (or smoothly curved) band of white > space>> on the complexity-error chart, surrounding your included > temperaments.>> It is good to have a moat so that you can answer questions > like "since>> you included this and this, then why didn't you included this", by > at>> least offering that "it's a long way from any of the included >> temperaments, on an error complexity plot". >> If the moat is gerrymandered, you get that question anyway, don't you? >Sure. But the wider and smoother your moat, the easier you can be let off the hook. :-) Also "gerrymander" is a derogatory term and originally applied only to electoral boundaries redefined to suit the encumbent. We have no need to apologise for choosing boundaries that we know to the best of our combined knowledge to only include the X most useful temperaments. Indeed that's the whole idea. We're never going to agree with everyone, but a good moat will lessen the scope for disagreement.>> The way to find a useful moat is to start with the temperaments you >> know everyone will want included, and those that almost no one will >> care about, and check out the space between the two. >> Right. Then you put them on a loglog plot, and try to draw a straight > line between them, and find to your amazement that it works.No! I'm afraid I've tried, but I can find absolutely no way to make a straight line work for this on a log log plot.> Now you > only have the corners to worry about, and what you are doing is > easier to justify.If it was a straight line, why would I have corners to worry about?> Is this so bad?Now I've tried, and got the results I fully expected from my experience of the kinds of things that happen when you go from linearlinear to loglog.> Why the opposition to even trying?Because I was pretty sure from the above experience and having already looked at it on both linear-linear and log-linear that it would be a waste of time.

Message: 9996 - Contents - Hide Contents Date: Tue, 10 Feb 2004 19:58:27 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> Your plots make it clear you'd better trash the idea of doing moats >> in anything but loglog. >> On the contrary.You did notice the approximately linear arragement, I presume? Does that suggest anything to you?

Message: 9997 - Contents - Hide Contents Date: Tue, 10 Feb 2004 21:05:12 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> Our starting point (but _only_ a starting point) is the knowledge >>>> we've built up, over many years spent on the tuning list, regarding >>>> what people find musically useful, with 5-limit ETs having had the >>>> greatest coverage. >>>>>> You're gravely mistaken about the pertinence of this 'data source'. >>> Even worse than culling intervals from the Scala archive. >>>> How do you know this? >> Assuming a system is never exhausted, how close do you think we've > come to where schismic, meantone, dominant 7ths, augmented, and > diminshed are today with any other system? Carl,I've been saying we have evidence regarding "5-limit ETs". The above are all linear temperaments, not equal temperaments, and one is strictly 7-limit. Also I'm not sure I'm parsing that sentence correctly. I think you're asking how well we have explored any systems other than those linear temperaments you mention. My answer is, "Not very far". But at least you seem to agree that the systems you mention have been somewhat explored, and so we have some evidence of their musical usefulness. The same goes for several _equal_ temperaments.> If you had gone to apply your program in Bach's time, would you have > included augmented and diminished? "Oh, nobody's ever expressed > interest about them on a particular mailing list with about enough > aggregate musical talent to dimly light a pantry, so they must not be > worth mentioning." It is said the musicians of Bach's time did not > accept the errors of 12-tET.You've got the wrong end of the stick here, and are putting words in my mouth. I never proposed using failure-to-be-mentioned-on-the-tuning- lits as a reason to exclude a temperament, that would be ridiculous. I only propose using those that _have_ been mentioned (as useful), and general discussions on desirable properties, as a starting point and then widening the circle roughly equally in all directions from there.> 5-limit ETs being shown musically useful on the tuning list? > Exactly what music are you thinking of? We're fortunate to have > had some great musicians working with new systems -- Haverstick, > Catler, Hobbs, Grady -- but we've chased all of them off the list,Even if that were true, it would not disqualify their past testimony.> and 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.

Message: 9998 - Contents - Hide Contents Date: Tue, 10 Feb 2004 22:00:37 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>>> OK, Carl, so everyone's been sorely underestimating the true >>> usefulness of 665-equal and 612-equal, yes? >>>> Sounds like you are. Not everyone plays live music and has that as >> their focus, like you. >> But are you using these to approximate JI or truly for their inherent > properties?I'm in the middle of working on an ennealimmal piece now. Inherent properties are a major aspect for this kind of thing. 612 is a fine way to tune ennealimmal, though I plan on using TOP for this one. This stuff really is practical if you care to practice it. In terms of commas, we have a sort of complexity of the harmonic relationships they imply--distance measured in terms of the symmetrical lattice norm possibly being more relevant here than Tenney. Past a certain point the equivalencies aren't going to make any differences to you, and there is another sort of complexity bound to think about.

Message: 9999 - Contents - Hide Contents Date: Tue, 10 Feb 2004 01:15:50 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> Well Paul and I see it as bringing it in closer touch with reality.Convince us. Make a case. Show some loglog plots and prove they make no sense. Explain why what you are doing does make sense. Is this an unreasonable request?

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