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Message: 10100 - Contents - Hide Contents Date: Wed, 11 Feb 2004 21:05:55 Subject: Re: Rhombic dodecahedron scale From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" >> >>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" >>> >>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" >>>>>> wrote: >>>>> 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: >>>>>>>>> I know I'm lagging behind, but I need to ask where the > remaining >> 5>>>> notes come from (14 + 5). Thanks >>>>>> Okay -heres what I know for sure. The 19 tones include >> 3,5,7,15,21,35>>> hexany, all divided by 5 and 7. This makes 11 tones, leaving 8. I >>> can't find any pattern to the 8 remaining however. (Are these the > 8>>> tetrads?). I also discovered that the 19 tones are every > combination>>> of -1, 0 and 1 except for (1,1,1) (-1,-1,-1) triples and every >> double>>> of 1,1,0 and -1,-1,0. I guess what I am saying is that I > understand>>> hexanies but don't know what makes a tetrad. Thanks >>> >>> Paul >>>> There are two types of tetrad. 1:3:5:7 is one, and 105:35:21:15 = 1/ >> (1:3:5:7) is the other. >> I know - but how does this translate to Gene's fractions. Are the > eight tetrads (+-1,+-1,+-1)? But the problem with that is that (1,1,1) > for example doesn't appear in the list (105)These are relative proportions, not absolute figures.

Message: 10102 - Contents - Hide Contents Date: Wed, 11 Feb 2004 13:08:39 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? >>Sure; in the 5-limit it gives the periodicity block, and so on. >>>>> 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. >>Can I see an example of what you have in mind?Above! For all operations one would want to do. With templates for dual and every other damn thing that can be done to a vector by flipping signs, rearranging elements, and other trivial operations. If I could do any better than this I'd make the thing myself! -Carl

Message: 10103 - Contents - Hide Contents Date: Wed, 11 Feb 2004 20:13:47 Subject: Re: The same page 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 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. >> With one val, the wedgie by definition is that val. The only special > case I know is 5-limit linear temperaments, where using the comma as a > wedgie seems a better plan than sticking with the definition.Why not admit both versions of the wedgie in all instances? They're so similar it's hard to see why one would make a big deal out of it. And personally, I have a far better intuitive grasp of monzo-wedgies than val-wedgies, but since we only care about the absolute values, converting from one to the other is trivial.

Message: 10104 - Contents - Hide Contents Date: Wed, 11 Feb 2004 23:38:51 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 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_.Try thinking like a computer composer, who could well desire to keep track of things more easily by scoreing in terms of reasonably small integers.

Message: 10105 - Contents - Hide Contents Date: Wed, 11 Feb 2004 19:36:32 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>>>>> >umans 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). >>>>>> So Carl. You really think it's possible that some human musician >>> could find the temperament where 3/2 vanishes to be a useful >>> approximation of 5-limit JI (but hey at least the complexity is >>> 0.001)? And likewise for some temperament where the number of >>> generators to each prime is around a google (but hey at least the >>> error is 10^-99 cents)? >>>> This is a false dilemma. The size of this thread shows how hard >> it is to agree on the cutoffs. >>Well yeah but we're probably within a factor of 2 of agreeing. >Another species could disagree with us by orders of magnitude.This was addressed to "So Carl". Am I not human?>So you do want cutoffs on error and complexity?I think we want roughly the same things. Except I want to answer questions like those I just mentioned (which among other things investigate making complexity comparable across harmonic limit and dimensionality), and why Paul's creepy complexity gives the numbers it does, before continuing. And I maintain that a survey of the tuning list would be a cataclysmic scientific error. But with reasonable pain axes, such as cents**2 and 2**notes, finding the widest reasonably- convex moat that encloses the desired number of temperaments for each case (limit, dimensionality) would seem to be a good idea and sufficient to eliminate the need for a survey.>But cutoffs utterly violate log-flat badness in the regions >outside of them.I have no problem with smoothing the cutoff region.>> Can you name the temperaments that fell outside of the top 20 >> on Gene's 114 list? > >Yes.Eep! Sorry, I meant the ones that you want that fell outside Gene's top 20/114. -Carl

Message: 10106 - Contents - Hide Contents Date: Wed, 11 Feb 2004 20:14:14 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: >>> False, and I don't appreciate the sarcastic tone of this either. >

Message: 10107 - Contents - Hide Contents Date: Wed, 11 Feb 2004 21:06:10 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> It's precisely as complex in terms of the chord relationships >> involved, so long as you stay below the 9-limit. >> Why do you say this? Is this some mathematical result, or your > subjective feeling? My ears certainly don't seem to agree.It's a non-subjective musical result. The number of intervals involved, or the number of chord changes for chords sharing a note, or sharing two notes, is going to be exactly the same. This is why you can map them 1-1. Getting to a 3/2 from a 1 is two steps for chords sharing two notes, and one step for chords sharing a note (since they share a 3/2.) Exactly the same is true of any other 7-limit consonance, such as 7/4. You go from one major tetrad to another in two steps of chords sharing an interval, no more, no less.>>>> 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. >> I think the Tenney lattice is pretty ideal for this, because > progressing by simpler consonances is more comprehensible and thus > allows for longer chord progressions with the same subjective > complexity.The Tenney lattice is no good for this, since I am assuming octave equivalence. The octave-class Tenney lattice could be argued for, but chords sharing notes or intervals seems far more basic to me so far as chords go. We can start from chords and then get back to the notes.

Message: 10108 - Contents - Hide Contents Date: Wed, 11 Feb 2004 23:47:55 Subject: Re: Rhombic dodecahedron scale From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:> I notice a pattern in these fractions: The difference of primes > between num. and den. is 0 or 1, no prime goes beyond p^1. Is there > any significance to the fact that besides (0,0,0) there are 6 scale > members with difference of 0 and 12 with difference of 1? Can someone > put this in geometric terms? ThanksIn geometric terms you have symmetrical scales defined by taking everything inside a ball around a fixed center. If the center is the unison, you get scales of size 1, 13, 19, 43, ... in that way. In analytic terms, the generating function for the above problem is a series (1+2q+2q^4+2q^9+ ... + 2q^n^2 + ...)^3 = 1+12q+6q^2+24q^3+..., where the coefficient on the q^n term is the number of 7-limit note-classes at a distance of sqrt(n) from the unison. Similar generating functions can be defined for distance from the center of tetrads, hexanies, or the midpoint of the 1-3 interval, with corresponding scales.

Message: 10109 - Contents - Hide Contents Date: Wed, 11 Feb 2004 00:42:29 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> 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. >> Please elaborate on this point of view -- I'm not seeing it.You can look at meantone as something which gives nice triads, as a superior system because it has fifths for generators, as a nice deal because of a low badness figure. Or, you can say, wow, it has 81/80, 126/125 and 225/224 all in the kernel, and look what that implies. The last point of view has nothing directly to do with error and complexity, though the relationship is a close one when analyzed. As you move to lower-error systems, your interest in the error per se falls off, and complexity from the point of view of a vast conceptual keyboard not too interesting--but oh, those commas! That's where the action is in some ways, and more so as we increase the prime limit and we have commas up the wazoo. Ennealimmal is not just low-error, it has commas which are still in the useful range. Then, finally, they get so complex they become pointless.> Could you do me a favor and attempt to speak to me as a human being, > and not deal with me like a chess opponent, trying to look several > moves ahead so that you can defeat me?I washed out of the first round of the US correspondence championship. It's my brother who is the grandmaster.

Message: 10110 - Contents - Hide Contents Date: Wed, 11 Feb 2004 19:37:53 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>> >Dave Keenan wrote:]>>> For me there are three candidates on the table at the moment. log-log >>> circles or ellipses, log-log hyperbolae, and linear-linear >>> nearly-straight-lines. >>>> Can we keep log-flat on the table for the moment? >>If you mean, log-flat with no other cutoffs, then no.I mean log-flat with *some* kind of cutoffs. -Carl

Message: 10111 - Contents - Hide Contents Date: Wed, 11 Feb 2004 20:15:11 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??> I've suggested in the > past adjusting for it, crudely, by dividing by pi(lim).Huh? What's that?

Message: 10112 - Contents - Hide Contents Date: Wed, 11 Feb 2004 21:09:57 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> Alternatively, 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.Not the one you were referring to here.

Message: 10113 - Contents - Hide Contents Date: Wed, 11 Feb 2004 23:49:13 Subject: Re: acceptace regions From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:>> We might try in analyzing or plotting 7-limit linear temperaments a >> transformation like this: >> >> u = 4 - ln(complexity) - ln(error) >> v = 12 - 4 ln(complexity) - ln(error)In the terms Paul, Carl and I have been using, this is a cutoff relation max[ln(complexity)/4 + ln(error)/4, 4 * ln(complexity)/3 + ln(error)/12] < 1 I wish you'd told us what units you're assuming for error here. I can't possibly consider this because I don't know whether it's cents or octaves? ... Only teasing, ... to make a point ;-) So this is a pair of lines that together take a triangular bite out of the lower left edge of the sheet of temperaments. I'm guessing they are designed to depart at equal but opposite angles from a log-flat line tangent to their corner. It seems we may be moving towards some kind of agreement. :-)>> We can obtain a fine list simply by taking everything in the first >> quadrant and leaving the rest. Morover, while the cornet here is > not>> sharp, if we want to smooth it Yes. Definitely.> we can easily accomodate such a desire>> by taking everything above a hyperpola uv = constant in the first >> quadrant--in other words, use uv as a goodness function, and insist >> on a goodness higher than zero. >> >> Think the resulting list is too small? Try moving the origin >> elsewhere, by setting >> >> u' = A - ln(complexity) - ln(error) >> v' = B - 4 ln(complexity) - ln(error) >> >> Still unhappy? I think the slopes of -1 and -4 I use work well, but >> you could try changing slopes *and* origins in order to better get >> what you think is a moat, or are willing to claim is one. >> >> I think a uv plot of 7-limit linears would be interesting. I'd also >> like some kind of feedback, so I don't get the feeling I am talking >> to myself here.OK. But I don't think it will help to do a u v plot. I'd prefer to see it on the existing log log plot, and I'd really like to see if you can come up with one of these hyperbolic-log beasties that gives the same list as Pauls red curve. This is exciting. :-)

Message: 10114 - Contents - Hide Contents Date: Wed, 11 Feb 2004 00:48:15 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Gene Ward Smith --- 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. >> Then why the hell do we need a badness bound?To keep the utter crap at bay, and allow us not to try to publish a list of 1000000 "temperaments". Did you see the vast clouds of darkness on Paul's plots?

Message: 10115 - Contents - Hide Contents Date: Wed, 11 Feb 2004 17:31:36 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, 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. etc. -Carl

Message: 10116 - Contents - Hide Contents Date: Wed, 11 Feb 2004 19:40:39 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>>>> >ave doesn't seem to want the macros which would >>>> be necessary for the scale-building stuff. >>To me, in the context of the current highly mathematical discussion, >this said to me that you think macros are necessary (i.e. you can't do >without them) for scale-building stuff. > >I think this is obviously wrong since you can show how to build a >scale using meantone which is not a macrotemperament. > >But since I now learn that you apparently only meant "desirable" >rather than "necessary" in the strict logical sense,Hate to nitpick now that we understand each other, but it has nothing to do with strict logic, but rather *what* one wants to do. Try this again:>> Do *what* without them? Build any decent scale (the above sense)? >> Or run any kind of decent scale-building program (the sense in >> which I said "necessary")?>you should note >that I long ago agreed to neutral thirds and pelogic being on the >5-limit list. Surely they are macro enough for your purposes.Herman just got through posting on tuning how beep is a great temperament for scale-building. -Carl

Message: 10117 - Contents - Hide Contents Date: Wed, 11 Feb 2004 20:15:29 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! >

Message: 10118 - Contents - Hide Contents Date: Wed, 11 Feb 2004 21:09:24 Subject: Re: The same page 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 not admit both versions of the wedgie in all instances? >> The wedgie then no longer corresponds 1-1 with temperaments, as there > are two of them.So the correspondence is 1-1-1. Why is that a problem?

Message: 10119 - Contents - Hide Contents Date: Wed, 11 Feb 2004 23:58:46 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 Carl Lumma <ekin@l...> wrote:>>>> 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?I'd like to know what you mean by micro-biased. It may well be true, but I'd like to know. At the moment I fell you should be calling me "centrally biased" or some such. 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.

Message: 10120 - Contents - Hide Contents Date: Wed, 11 Feb 2004 13:12:09 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>> > 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. >>Can you give an example of the latter?Fix the origin at 1 cent and 1 note, and the complexity < whatever you want. 100 notes? 20 notes? Or just grow the radius until you enclose the number of temperaments you want to list. I originally thought to include only the upper-right quadrant, but a circle all the way around the origin might be nice to see. -Carl

Message: 10121 - Contents - Hide Contents Date: Wed, 11 Feb 2004 17:36:19 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>> >ave doesn't seem to want the macros which would >> be necessary for the scale-building stuff. >>What are macros?Again, I'm amazed that this well-worn terminology isn't effective here. AKA exos?>Why can't you do scale-building stuff without them?I don't know that it can't, but they're certainly fertile for scale-building. See another recent message for my response to the rest. :) -Carl

Message: 10122 - Contents - Hide Contents Date: Wed, 11 Feb 2004 20:16:37 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> Years ago, when you first made be aware of this fact, I was seduced >> by it, to Dave's dismay. Did you forget? Now, I'm thinking about it >> from a musician's point of view. Simply put, music based on >> constructs requiring large numbers of pitches doesn't seem to be able >> to cohere in the way almost all the world's music does. Of all >> people, I'm suprised Carl is now throwing his investigations along >> these lines by the wayside. > > I'm not.Then why are you suddenly silent on all this?

Message: 10123 - Contents - Hide Contents Date: Wed, 11 Feb 2004 13:13:19 Subject: Re: 23 "pro-moated" 7-limit linear temps From: Carl Lumma>> >ou can look at meantone as something which gives nice triads, as a >> superior system because it has fifths for generators, as a nice deal >> because of a low badness figure. Or, you can say, wow, it has 81/80, >> 126/125 and 225/224 all in the kernel, and look what that implies. >>Having 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? Yes, I do care very much about this. -Carl

Message: 10124 - Contents - Hide Contents Date: Wed, 11 Feb 2004 17:38:10 Subject: Re: ! From: Carl Lumma>You 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!>Then you have the moat surrounding that, with no people in it. >Then you have the rest of the world with the rest of the people >in it. Oooo!!! -Carl

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