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Message: 10075 - Contents - Hide Contents Date: Tue, 10 Feb 2004 21:40:50 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>>> --- 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". >>>> 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. >> This is a music *theory* paper, so presenting the bare minimum of > math to actually *derive* our results is appropriate. True, only > heavy theorists will probably want to reproduce the calculations. But > we want to leave referees, at least, with fairly complete confidence > that what we're doing is correct.Fair enough. A sketch that a mathematician can follow. But that seems a long way from "teach a man to fish". The musicians will only be relying on the lists we give (and Graham's finder). ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 10076 - Contents - Hide Contents Date: Wed, 11 Feb 2004 17:04:25 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>Humans seem to find a particular region of complexity and error >attractive and have a certain approximate function relating error and >complexity to usefulness. Extra-terrestrial music-makers (or humpback >whales) may find completely different regions attractive.This seems to be the key statement of this thread. I don't think this has been established. If it had, I'd be all for it. But it seems instead that whenever you cut out temperament T, somebody could come along and do something with T that would make you wish you hadn't have cut it. Therefore it seems logical to use something that allows a comparison of temperaments in any range (like logflat). Then no matter what T is, we can say... "You could have used U, which is in the same range but better." ...or... "T's the best in that range. Bravo!" ...The worst that could happen would seem to be... "T falls outside the range we established for our paper, sorry." ...in which case the reader could perform his own analysis in the above way. With a cooked acceptance region, however, the following could happen... "Oh, T. It didn't meet our guesses about human cognition, but YMMV." -Carl

Message: 10078 - Contents - Hide Contents Date: Wed, 11 Feb 2004 20:57:37 Subject: Re: loglog! From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>> 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. >>>>>>>> The complexity measures cannot be compared across different >>>> dimensionalities, any more than lengths can be compared with areas >>>> can be compared with volumes. >>>>>> Not if it's number of notes, I guess. >>>> What's number of notes?? > > Complexity units.It's only that (or very nearly that) in the ET cases. So it the below still relevant?>>> I've suggested in the >>> past adjusting for it, crudely, by dividing by pi(lim). >>>> Huh? What's that? >

Message: 10079 - Contents - Hide Contents Date: Wed, 11 Feb 2004 22:20:05 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote: a week starting next week (Tue to Fri).> Why can't you understand that the mathematical fact that temperaments > come out with a straight edge on a log-log plot has absolutely no > bearing on which of them will be found musically useful to humans.Does this mean you are simply going to blow me off and ignore the hard work I have done trying to accomodate both you and Paul?> I can certainly understand Paul's impatience.Can you understand my impatience with the fact that neither you nor Paul seems willing even to listen to me?

Message: 10080 - Contents - Hide Contents Date: Wed, 11 Feb 2004 12:58:01 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>> >lternatively, then why doesn't the badness bound alone enclose a >> finite triangle? >>Not only is it, like the rectangle, infinite in area on the loglog >plot, since the zero-error line and zero-complexity lines are >infinitely far away, but it actually encloses an infinite number of >temperaments.Huh; I thought I just saw you and Gene agreeing that a badness bound alone does return a finite list of temperaments. -Carl

Message: 10081 - Contents - Hide Contents Date: Wed, 11 Feb 2004 17:07:36 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>> >nd the complexity < whatever >> you want. 100 notes? 20 notes? >>Why would you need a complexity bound in addition to the circle? The >circle, being finite, would only extend to a certain maximum >complexity anyway . . .To determine its radius. -C.

Message: 10082 - Contents - Hide Contents Date: Wed, 11 Feb 2004 19:12:54 Subject: Re: ! From: Carl Lumma>>> >ou know what a moat is right? >>>> Obviously not! :( >>>>> You have the castle (the circle is its >>> outer bound) with people (temperaments) inside. >>>> Then it's the same as a circle! >>No. A circle is infinitesimally thin.In 3-D we have ball, but I'm not aware of a corresponding term for 2-D. Disk?>A moat has real thickness.Oh, ... yes I'm aware my proposal doesn't include a region of safety. I think it's nice but it isn't a must-have for me.>Of course it doesn't have to be >circular, but continuing in that vein ...Again, you'll have to bear with my tendency for hyperbole. I don't consider specifically circular curvature to be part of the circle suggestion. This goes without saying for me, but I can see how it might not for person X. I think maximum hyperbole is the ideal way to use natural languages, but I really should learn to back it up with examples.>You could draw the smallest circle that encloses all the lucky >temperaments and then you could draw another one outside that which >is the largest circle that still encloses the same ones and no >others. The space between them is the moat. You can then give a >quantitative measure of the size of the moat as the percentage >difference between the radii of the two circles. Good idea. >The term "moat" came to mind because the temperaments sometimes look >like constellations and in the Niven and Pournelle books "The Mote in >Gods Eye" and "The Moat around Murcheson's Eye", "the Moat" is a vast >region of space with no stars.I've only ever read The Integral Trees, which was Niven-only IIRC (I was just a lad). Oh, and a funny analysis of Superman's powers when I was a bit older. Anyway "moat" makes sense because it is a region of safety. If only you had explained (or I noticed) this sooner! -C.

Message: 10083 - Contents - Hide Contents Date: Wed, 11 Feb 2004 20:05:22 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: >>> Again: The horizontal axis, as always, is *complexity*. The vertical >> axis, as always, is *error*. >> I don't want complexity and error, I want log(complexity) and > log(error), and labeling the axes if possible is always a good plan. > > We've already established we're on the>> same page on those. It's easy to see, by the tick marks, if either or >> both of the axes is scaled logarithmically. >> I prefer knowing to guessing. > > The red line is our>> proposed moat. And again, the 7-limit 'linear' temperaments are >> indexed as follows (I show the first three numbers in the val- wedgie, >> since you feel they are the most important): >> Putting in the names would have made it a hell of a lot clearer than > numbers I had no clue about. I'm looking at a Windows file in Linux, > so the name comes out as "7lin23~1.gif"; I hope the actual name means > something I can understand. It does. > The convex hull of your selected > temperaments looks by eyeballing to be 20, 17, 23, 21, 22, 12 which > corresponds to blackwood, augmented, porcupine, supermajor seconds, > nonkleismic, and miracle. This ought to correspond to what I found by > doing the same thing using Maple, but I'm not sure it does. I'm not > impressed with the idea of arbitarily cooking the books to keep > miralce on but leave ennealimmal off,If you followed the discussion, you'd see that the only reason this happened was that your list of 126 was too thin in the high- complexity region to find a moat; otherwise ennealimmal might well have made it in. Rather than assuming ill intentions on the part of everyone else, try slowing down and taking a few extra moments to survey the writings of those who are painstakingly working on this stuff, rather than firing off a knee-jerk reaction upon a sorely incomplete impression.> incidentally, if this is part of > the plan. The slope from blackwood to augmented is not much different > than the slope from augmented to porcupine, and it seems you could > start off simply with the slope of -1, and then drop it off gradually. > That would already began to make more sense out of this;Why is that?> I don't think > imagining you see a moat curve when there are in fact lots of ways to > draw the curve and they look quite different is a very good start.But if it's done quantitatively? Dave said he was unable to find a better moat according to his *quantitative* measure; perhaps it's possible to verify this computationally?> Here is TOP complexity times TOP error for the above: > > Blackwood: 46.9 > Augmented: 48.8 > Porcupine: 45.8 > Supermajor seconds: 31.3 > Nonkleismic: 23.8 > Miracle: 13.3 > > I checked my big list of 32201 temperaments, and found exactly four > where the error times the complexity was less than 50, and the > complexity was less than 10: blackwood, augmented, diminished, and > dominant seventh. You could simply start out like this, and roll down > gradually, and you'd already be making a hell of a lot more sense.In terms of having your inclusion list be insensitive to a fair range of variation in the inclusion criterion? If so, I'm all ears.

Message: 10084 - Contents - Hide Contents Date: Wed, 11 Feb 2004 22:21:31 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> How do you know that? What if your chords are 9-limit chords (either > complete or saturated)?Obviously they aren't, because I said we are staying below the 9-limit.

Message: 10085 - Contents - Hide Contents Date: Wed, 11 Feb 2004 13:02:55 Subject: Re: ! From: Carl Lumma>> >he circle rocks, dude. It penalizes temperaments equally for >> trading too much of their error for complexity, or complexity >> for error. Look >> at the plots, and the first things you hit are 19, 12, and 53. >> And 22 in the 7-limit. Further, my suggestion that 1cents = zero >> should satisfy Dave's micro fears. Or make 0 cents = zero. It >> works either way. >>It does? Look at the graph! How can you make 0 cents = zero when >it's infinitely far away?I thought I had a way to fudge it by adding a constant later, but I can't remember it at the moment.>And what about the position of the origin on the >*complexity* axis??I already answered that.>P.S. The relative scaling of the two axes is completely arbitrary,Howso? They're both base2 logs of fixed units. You mean c is arbitrary in y = x + c? Selecting an "origin" effectively fixes c. -Carl

Message: 10086 - Contents - Hide Contents Date: Wed, 11 Feb 2004 17:09:30 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>>> >aving 81/80 in the kernel implies you can harmonize a diatonic >>> scale all the way through in consonant thirds. Similar commas >>> have similar implications of the kind Carl always seemed to care >>> about. >>>> Don't you mean 25:24? >>No, 81;80. 25;24 in the kernel doesn't give you either a diatonic >scale or 'consonant thirds'.Oh, in the kernel means tempered out (right?) giving neutral thirds. So it isn't immediately obvious why 81:80 throws 5:4 and 6:5 on the same scale degree (always thirds?). . . -Carl

Message: 10087 - Contents - Hide Contents Date: Wed, 11 Feb 2004 19:18:26 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>>>> >e have a choice -- derive badness from first principles or cook >>>> it from a survey of the tuning list, our personal tastes, etc. >>>>>> What first principles of the human psychology of the musical use >>> of temperaments did you have in mind? >>>> Since I'm not aware of any, and since we don't have the means to >> experimentally determine any, I suggest using only mathematical >> first principles >>But badness is clearly a psychological property,No it isn't! What evidence do we have that badness means anything musical at all?>what have mathematical first principles got to do with it?What _don't_ they have to do with? For folks into "digital physics" like me, nothing.>> , or very simple ideas like... >> >> () For a number of notes n, we would expect more dyads in the >> 7-limit than the 5-limit. >> >> () I expect to find a new best comma after searching n notes >> in the 5-limit, n(something) notes in the 7-limit. >>These sound reasonable, but I don't see how to use them to determine >psychologically reasonable cutoff for lists of the temperaments most >likely to be musically useful.I suspect that answering them would present possibilities for psy.-reasonable cutoffs. -Carl

Message: 10088 - Contents - Hide Contents Date: Wed, 11 Feb 2004 20:05:58 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:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> >> wrote:>>> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: >>>>>>> The 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. >>>> That doesn't seem to be true. There are lots of low-complexity >> temperaments with arbitrarily high error. >> "Lots" is not the same as "infintely many". If you bound complexity, > you bound the size of the numbers in a wedgie, and hence bound the > number of wedgies.I realized this last night. Thanks for bearing with my stupidity.

Message: 10089 - Contents - Hide Contents Date: Wed, 11 Feb 2004 22:51:28 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote: ...> But, while I'm here: > > - log-flat looks like a good place to startWe _did_ start there, some years ago, and have been there ever since. What I don't understand is the resistance to moving on, and finding something that's approximately representative of human musical interest in these things. And if you mean it's a good function to modify to obtain the latter, then that's fine. Every smooth cutoff function that Paul or Carl or I have proposed so far (and almost certainly any that might be proposed in the future) are tangent to log-flat badness. What more could you ask?> - silence is negative infinity in decibelsYes, what Paul said is technically wrong. But that doesn't alter the fact that we don't experience mistuning pain as anything like log(cents). Cents is already log(frequency).> - spherical projection!I think we can safely forget about that one.> - can somebody give a friendly explanation of complex hulls?I think that was _convex_ hull. That's just the smallest convex shape that encloses a set of points - a polygon with the "outermost" points as its vertices. I think it goes without saying that we would never exclude a temperament that was inside the convex hull of the included temperaments. Although not as formalised (yet), moats can be considered as taking the convex hull idea a little further, by including not only those _inside_ the convex hull but also those _close_ to the outside of it, and insisting that the hull is not only convex, but smooth.> - would k-means have anything to do with the clustering? > > K-Mean Clustering Tutorial * [with cont.] (Wayb.)Sorry. Don't know anything about these.

Message: 10090 - Contents - Hide Contents Date: Wed, 11 Feb 2004 00:25:19 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> I'm in the middle of working on an ennealimmal piece now. Inherent >> properties are a major aspect for this kind of thing. >> You're using a full basis for the kernel? And it's audible? (Real > questions, not rhetorical or riddles.)I'm not sure what your question means. So far I've been sticking to the 45 note DE, and obviously doing that makes a clear audible difference. However, experience has shown that comma pumps on 2401/2400 or 4375/4374 are not so long that they fail to be comprehensible. The audibility of the differnce between the starting and ending note if you temper is another matter; it's not a hell of a big change, and I don't hear it myself.>> 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. >> How so? You really think a progression by perfect fifths is as > complex as a progression by ratios of 7?It's precisely as complex in terms of the chord relationships involved, so long as you stay below the 9-limit.>> 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. >> I thought this was the only kind. Can you elaborate?If |a b c d> is a 7-limit monzo, the symmetrical lattice norm (seminorm, if we are including 2) is sqrt(b^2 + c^2 + d^2 + bc + bd + cd), and this may be viewed as its complexity in terms of harmonic relationships of 7-limit chords. How many consonant intervalsteps at minimum are needed to get there is another and related measure.

Message: 10091 - Contents - Hide Contents Date: Wed, 11 Feb 2004 20:08:44 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: >>> Unfortunately, we can't publish an infinitely long paper. >> You don't get an infinite list in this way for slopes less than the > critical exponent.Right, exactly as I said, but as I also said, you get a lot of emphasis on high-error temperaments, and I know Dave would be very unhappy with that.> Can you think about my hyperbola plan and tell me if I should > bother?I can't understand your reasoning there. I'd love to consider it if you would elaborate on your thinking.

Message: 10092 - Contents - Hide Contents Date: Wed, 11 Feb 2004 21:03:23 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>> 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? >>>>>>>> We don't care, since we're including *all* the systems with error >>>> and complexity no worse than *any* of these systems, as well as >>>> miracle. And that's quite a few! >>>>>> But you can still make the same kind of error. >>> >>> -Carl >> >> How so? >> 1. The process of expansion into temperament space might not be > finished in the 5-limit.Sure, but we can show a graph of where the temperaments occur in the space, and of course provide the necessary math, so that further expansion can be explored by the reader.> 2. If we don't know anything about 7-limit music, listing all > temperaments at least as "good" (never mind how we determine that) as > the ones used to date in 5-limit music might not mean anything.It might not. And we're not suggesting any "goodness" measure which is applicable to both 5-limit and 7-limit systems of any respective dimensionalities. But we are suggesting something similar be used in each of the Pascal's triangle of cases, which seems logical. If it's wrong, it's wrong, and there goes the premise of our paper. But it's a theory paper, not an edict. I think if the criteria we use are easily grasped and well justified, we will have done a great job publishing something truly pioneering and valuable as fodder for experimentation.

Message: 10093 - Contents - Hide Contents Date: Wed, 11 Feb 2004 23:03:31 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> What I don't understand is the resistance to moving on, and finding > something that's approximately representative of human musical > interest in these things.What I don't understand is why when I try to do that I get either ignored or berated.

Message: 10094 - Contents - Hide Contents Date: Wed, 11 Feb 2004 17:12:48 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>> >hat I don't understand is the resistance to moving on, and finding >> something that's approximately representative of human musical >> interest in these things. >>What I don't understand is why when I try to do that I get either >ignored or berated.Gene, please understand that at least a few of us would love nothing more than to be able to follow your posts, but it isn't necessarily any easier to do it than it is for you to follow our posts. In fact, it's probably harder. -Carl

Message: 10095 - Contents - Hide Contents Date: Wed, 11 Feb 2004 20:11:03 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:>> --- 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. >> I rely on you for that. Can you possibly believe my track record for > working out the logic of a proposal is not a bad one, and that if I am > saying something it might be worth thinking about?I've been thinking about it for years, and mostly supporting it. It's just that I think Dave and Graham should both be in on this, and we were going to lose Dave entirely if we didn't at least try to address his objections. I'm hoping this process will continue, whenever Dave gets back.

Message: 10096 - Contents - Hide Contents Date: Wed, 11 Feb 2004 21:05:06 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> I'm not. >>>> Then why are you suddenly silent on all this? >> Huh? I've been posting at a record rate.Not on this subject of cognitive limits that used to occupy you so.>>> It is well known that Dave, for example, is far more >>> micro-biased than I! >> >> ? >> What's your question?What does micro-biased mean, on what basis do you say this about you vs. Dave, and what is its relevance here?

Message: 10097 - Contents - Hide Contents Date: Wed, 11 Feb 2004 23:19:08 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>>>>> I'm in the middle of working on an ennealimmal piece now. > Inherent>>>> properties are a major aspect for this kind of thing. >>>>>> You're using a full basis for the kernel? And it's audible? (Real >>> questions, not rhetorical or riddles.) >>>> I'm not sure what your question means. So far I've been sticking to >> the 45 note DE, and obviously doing that makes a clear audible >> difference. However, experience has shown that comma pumps on >> 2401/2400 or 4375/4374 are not so long that they fail to be >> comprehensible. The audibility of the differnce between the starting >> and ending note if you temper is another matter; it's not a hell of > a>> big change, and I don't hear it myself. >> I think it's Dave's turn to ponder this.I feel like Shaw in Monty Python's 'Oscar Wilde' sketch. :-) Monty Python's Oscar Wilde Sketch * [with cont.] (Wayb.) I'm afraid I lost the thread and don't have time to pick it up. I'm guessing ennealimmal is so complex and so close to 7-limit JI that most musicians would happily use the two interchangeably without noticing. Is that something we could test?>>>> 612 is a fine >>>> way to tune ennealimmal, though I plan on using TOP for this > one.I don't see how the fact that x/612ths of an octave is a fine way to tune the ennealimmal generator has any bearing on the musical usefulness of 612-ET _as_an_ET_. One can't hear the difference between ennealimmal tuned as a subset of 612-ET and tuned with an ever-so-slightly different generator that is an irrational fraction of an octave.

Message: 10098 - Contents - Hide Contents Date: Wed, 11 Feb 2004 17:16:27 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>>>>> >t is well known that Dave, for example, is far more >>>>> micro-biased than I! >>>> >>>> ? >>>>>> What's your question? >>>> What does micro-biased mean, on what basis do you say this about you >> vs. Dave, and what is its relevance here? >>I'd like to know what you mean by micro-biased. It may well be true, >but I'd like to know.Of all the amazing things I've seen on these lists, the failure of both you and Paul to understand the meaning of "micro-biased" is possibly the most amazing.>At the moment I fell you should be calling me "centrally biased" or >some such.Obviously you did understand it!>I don't want to include either the very high error low >complexity or very high complexity low error temperaments that a >log-flat cutoff alone would include.Yes, you are apparently centrally biased. You should like circles in that case. :) -Carl

Message: 10099 - Contents - Hide Contents Date: Wed, 11 Feb 2004 20:11:57 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> I haven't seen any cluster analysis yet! >>>> It was principal components analysis, but the reasoning behind the >> implementation was obscure. >> I have no idea what you're talking about.Well, I didn't have much grasp of it either. And I do prinicipal components analysis all the time!

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