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Message: 3250 - Contents - Hide Contents Date: Tue, 15 Jan 2002 12:34 +0 Subject: Re: algorithm sought From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a20hs9+7sd3@xxxxxxx.xxx> Gene:>> I would work in 3-dimensions for the 9-limit, and just make 3 half >> the size of 5 or 7. In other words, >> >> ||3^a 5^b 7^c|| = sqrt(a^2 + 4b^2 + 4c^2 + 2ab + 2ac + 4bc) >> >> would be the length of 3^a 5^b 7^c. Everything in a radius of 2 of >> anything will be consonant. Carl:> Thanks, Gene. I _really_ can't visualize this, but perhaps it > will provide a general method for finding the chords I seek. > Would everything still hold if I used 4 dimensions and kept all > edges the same length? By gods, I can't figure out where you're > getting the coefficients here. And what are the double pipes? > Not abs. -- there's a sqrt on the other side... I confess I > don't know the distance formula for triangular plots. I could > derrive it with trig. . . nope, it's a mess, 'cause there are many > different triangles involved in the different diagonals. So I > guess I would use the standard Euclidean distance, but I need to > know how to get delta(x) and delta(y) off a triagular lattice.The double pipes are simply for the Euclidean distance. I think the triangular generalisation is like this: Set the x axis to be constant. The y axis is then the 5:4 direction. To get distances, first convert to new axes x' = x + y cos(theta) y' = y sin(theta) where theta is the angle between the x and y axis, 90 degrees for a square lattice or 60 degrees for equilateral triangles. The Euclidean distance is simply sqrt((x')**2 + (y')**2). I did work out the general, multidimensional case, but I don't have the result to hand. Probably, something like x' = x + y*cos(theta) + z*cos(theta) y' = y*sin(theta) + z*cos(theta) z' = z*sin(phi) will work. I don't know how to get from theta, the angle between axes, to phi, the angle between the z axis and the x-y plane. Certainly not in general. Oh, for the algorithm, trying all combinations of consonances above the tonic should work. That'll be O(n**m) where m is the number of notes in the chord, but shouldn't be a problem for the kind of numbers we're talking about. A more efficient way would be to use the general method for ASSes I give on the web page, but you said you don't trust that. Graham
Message: 3251 - Contents - Hide Contents Date: Tue, 15 Jan 2002 22:54:23 Subject: Re: metric visualization From: paulerlich --- In tuning-math@y..., "keesvp" <kees@d...> wrote:> Wel, isn't this yahoo interface nice. Here we go again: > > I don't think MAX will work, because that's a symmetric square > metric, and we want a condition on the signs being equal or not.OK -- how about a MAX on the _three_ consonant intervals?> I am now pretty sure the much to small generator is due to the non- > linear effects. The inverse matrix maps the hexagon to interval > space. The euclidean norm looks at circles. So I guess you first > should do the non-linear transformation circle -> hexagon. > That's getting complicated.Yes, and I don't see any reason to use the euclidean norm.> That's why I suggested taking > the 'reversed' metric to begin with:Did not understand that.> Let f3 and f5 be the errors. > If they are of equal sign: > error = max( |f3*log(5)|, |f5*log(3)| ) > else: > error = |f3*log(5)| + |f5*log(3)|This is fine, but what about the geometry of _your_ triangular lattice? _That_ would seem to be the one that would work, while here it seems you're using _my_ triangular lattice.> I'm afraid I'm being much more intuitive than scientific here. In the > normal meantone case it gives regular 1/4 comma anyway as far as I > can see, so I really don't dare to say if it will be interesting in > any way, or even makes sense.I'll check it out later, as well as any other alternative based on _your_ lattice that you might wish to present.
Message: 3252 - Contents - Hide Contents Date: Tue, 15 Jan 2002 14:57:39 Subject: Re: [tuning] Re: badly tuned remote overtones From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, January 15, 2002 1:47 PM > Subject: [tuning-math] [tuning] Re: badly tuned remote overtones > >>> From: monz <joemonz@y...> >> To: <tuning-math@y...> >> Sent: Tuesday, December 25, 2001 3:44 PM >> Subject: [tuning-math] lattices of Schoenberg's rational implications >> >> >> Unison-vector matrix: >> >> 1911 _Harmonielehre_ 11-limit system >> >> ( 1 0 0 1 ) = 33:32 >> (-2 0 -1 0 ) = 64:63 >> ( 4 -1 0 0 ) = 81:80 >> ( 2 1 0 -1 ) = 45:44 >> >> Determinant = 7 >> Well there you go. This shows that Schoenberg's UVs > imply a 7-tone system, not a 12-tone system as you claim.Ack! ... Paul, I'm not really *claiming* that Schoenberg's system implies 12-tone -- I'm just trying to find out more accurately what Schoenberg had in mind. He himself seems to have thought that 12-EDO could give a decent representation of 11-limit harmony. I'm simply trying to reconstruct his thought-process. I admit that I've made statements like "it looks to me like Schoenberg's explanation in _Harmonielehre_ definitely implies a 12-tone periodicity-block" ... but that was probably premature. There's still a lot here left for me to understand.>> ... <snip> ... >> >> But why do I get a determinant of 7 for the 11-limit >> system? Schoenberg includes Bb and Eb as 7th harmonics >> in his description, which gives a set of 9 distinct pitches. >> But even when I include the 15:14 unison-vector, >> In place of which one above?In place of the 45:44, which is the UV with which I have difficulty, because of the inconsistency in Schoenberg's notation: F = 11/C and C = 11/G, but B (not Bb) = 11/F. Here are the matrices I used to make that statement: 3 5 7 11 unison vectors ~cents [ 1 1 -1 0 ] = 15:14 119.4428083 [ 1 0 0 1 ] = 33:32 53.27294323 [-2 0 -1 0 ] = 64:63 27.2640918 [ 4 -1 0 0 ] = 81:80 21.5062896 det = | -7 | 3 5 7 11 unison vectors ~cents [-1 -1 0 0 ] = 16:15 111.7312853 [ 1 0 0 1 ] = 33:32 53.27294323 [-2 0 -1 0 ] = 64:63 27.2640918 [ 4 -1 0 0 ] = 81:80 21.5062896 det = | 5 | Leaving in Schoenberg's notational inconsistency means that we must employ 45:44 in the matrix as well: 3 5 7 11 unison vectors ~cents [ 2 1 0 -1 ] = 45:44 38.90577323 [ 1 0 0 1 ] = 33:32 53.27294323 [-2 0 -1 0 ] = 64:63 27.2640918 [ 4 -1 0 0 ] = 81:80 21.5062896 And again here, det = | -7 | .>> I still get a determinant of -7. And if I use 16:15 >> instead, then the determinant is only 5. >> 15:14 and 16:15 are both clearly semitones. Why would > you use them as UVs?Well, to give you the same kind of answer which you and Gene both gave to me and which didn't help me at all ... because it works to give me 7 or 5 as a cardinality! Seriously, the reason I chose 15:14 is because I could see that by equating the 11th harmonic sometimes with the "perfect 4th" and sometimes with the "augmented 4th", Schoenberg was implying the tempering-out of a semitone. When I saw that the determinant of the matrix directly above (with unison-vectors 33:32, 45:44, 64:63, and 81:80) was -7, it put up a red flag in my mind. It seemed obvious to me that Schoenberg's equivalences implied only a 7-tone PB, i.e., a diatonic scale and not a chromatic one.> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, January 15, 2002 1:53 PM > Subject: [tuning-math] [tuning] Re: badly tuned remote overtones > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> OK, right ... I understand all that. >> >> But where did it come from? Is it the result of adding >> or subtracting two of the already-existing unison-vectors? >> That wouldn't work, would it? ... because all the vectors >> in the matrix have to be independent. >> Any vector that is _not_ the result of adding or subtracting two of > the already-existing unison-vectors would work to create what Gene > calls a "notation" (but is nothing like a musical notation anyone's > ever seen before). In this case, he made a choice (56/55) that makes > the 12-tone system come out nice in this "notation".OK, fair enough. But what does it have to do with Schoenberg's actual theory? I'm having a hard time seeing how 56/55 is implied by anything Schoenberg actually said. Just to satisfy my curiosity, I've made a table of the sum and difference of all pairs of unison-vectors in my original assessment of Schoenberg's _Harmonielehre_ system, i.e. the matrix [ 2 1 0 -1 ] = 45:44 [ 1 0 0 1 ] = 33:32 [-2 0 -1 0 ] = 64:63 [ 4 -1 0 0 ] = 81:80 . Here are the results: 45/44 and 33/32 : [-2 2 1 0 -1 ] = 45/44 38.90577323 - [-5 1 0 0 1 ] = 33/32 53.27294323 ------------------- [ 3 1 1 0 -2 ] = 120/121 -14.36717 [-2 2 1 0 -1 ] = 45/44 38.90577323 + [-5 1 0 0 1 ] = 33/32 53.27294323 ------------------- [-7 3 1 0 0 ] = 135/128 92.17871646 45/44 and 64/63 : [-2 2 1 0 -1 ] = 45/44 38.90577323 - [ 6 -2 0 -1 0 ] = 64/63 27.2640918 ------------------- [-8 4 1 1 -1 ] = 2835/2816 11.64168143 [-2 2 1 0 -1 ] = 45/44 38.90577323 + [ 6 -2 0 -1 0 ] = 64/63 27.2640918 ------------------- [ 4 0 1 -1 -1 ] = 80/77 66.16986503 45/44 and 81/80 : [-2 2 1 0 -1 ] = 45/44 38.90577323 - [-4 4 -1 0 0 ] = 81/80 21.5062896 ------------------- [ 2 -2 2 0 -1 ] = 100/99 17.39948363 [-2 2 1 0 -1 ] = 45/44 38.90577323 + [-4 4 -1 0 0 ] = 81/80 21.5062896 ------------------- [-6 6 0 0 -1 ] = 729/704 60.41206283 33/32 and 64/63 : [-5 1 0 0 1 ] = 33/32 53.27294323 - [ 6 -2 0 -1 0 ] = 64/63 27.2640918 ------------------- [-11 3 0 1 1 ] = 2079/2048 26.00885143 [-5 1 0 0 1 ] = 33/32 53.27294323 + [ 6 -2 0 -1 0 ] = 64/63 27.2640918 ------------------- [ 1 -1 0 -1 1 ] = 22/21 80.53703503 33/32 and 81/80 : [-5 1 0 0 1 ] = 33/32 53.27294323 - [-4 4 -1 0 0 ] = 81/80 21.5062896 --------------------- [-1 -3 1 0 1 ] = 55/54 31.76665363 [-5 1 0 0 1 ] = 33/32 53.27294323 + [-4 4 -1 0 0 ] = 81/80 21.5062896 --------------------- [-9 5 -1 0 1 ] = 2673/2560 74.77923283 64/63 and 81/80 : [ 6 -2 0 -1 0 ] = 64/63 27.2640918 - [-4 4 -1 0 0 ] = 81/80 21.5062896 ------------------- [10 -6 1 -1 0 ] = 5120/5103 5.757802203 [ 6 -2 0 -1 0 ] = 64/63 27.2640918 + [-4 4 -1 0 0 ] = 81/80 21.5062896 ------------------- [ 2 2 -1 -1 0 ] = 36/35 48.7703814 Does anyone have comments on this? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3253 - Contents - Hide Contents Date: Tue, 15 Jan 2002 17:34:47 Subject: Re: more graph theory terminology From: clumma>> > clique in a graph is a set of vertices which are pairwise >> adjacent. The CLIQUE problem is to determine given a graph >> G and an integer k, whether G has a k-clique. Although this >> problem is NP-complete, several practical algorithms exist." >>Graph theory is unfortunately a great source of NP-complete >problems. We might look for a graph theory package which >does this--Mathematica has a bigger one than Maple, so perhaps >I should check that out.It's unclear wether the above author was talking about maximal cliques, or not. Maximal cliques certainly seem harder than what we're after. I have Maple V, from wayback in 1995, which I've never used, and I finally fired it up, and lo and behold, it doesn't seem to run on Windows 2000. It looks like Mathematica actually comes in some affordable flavors. Those are probably the ones without the graph theory package. :) -Carl
Message: 3254 - Contents - Hide Contents Date: Tue, 15 Jan 2002 23:00:03 Subject: [tuning] Re: badly tuned remote overtones From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> Any vector that is _not_ the result of adding or subtracting two of >> the already-existing unison-vectors would work to create what Gene >> calls a "notation" (but is nothing like a musical notation anyone's >> ever seen before). In this case, he made a choice (56/55) that makes >> the 12-tone system come out nice in this "notation". > >> OK, fair enough. But what does it have to do with Schoenberg's > actual theory? I'm having a hard time seeing how 56/55 is implied > by anything Schoenberg actually said. It isn't!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Message: 3255 - Contents - Hide Contents Date: Tue, 15 Jan 2002 17:51:26 Subject: Re: algorithm sought From: clumma Hi Graham,>The double pipes are simply for the Euclidean distance. Thanks! >I think the triangular generalisation is like this: > >Set the x axis to be constant. The y axis is then the 5:4 >direction. To get distances, first convert to new axes > > x' = x + y cos(theta) > y' = y sin(theta) > >where theta is the angle between the x and y axis, 90 degrees >for a square lattice or 60 degrees for equilateral triangles. >The Euclidean distance is simply sqrt((x')**2 + (y')**2).Sweet! I'll see if I can play with this in the coming week.>I did work out the general, multidimensional case, but I don't >have the result to hand. Probably, something like > > x' = x + y*cos(theta) + z*cos(theta) > y' = y*sin(theta) + z*cos(theta) > z' = z*sin(phi) > >will work. I don't know how to get from theta, the angle between >axes, to phi, the angle between the z axis and the x-y plane. >Certainly not in general.Can't it always just be 60 degrees? There is that point where this is no longer the closest packing -- that's bad I presume... Paul once posted something from Mathworld... "The analog of face-centered cubic packing is the densest lattice packing in 4- and 5-D. In 8-D, the densest lattice packing is made up of two copies of face-centered cubic. In 6- and 7-D, the densest lattice packings are cross sections of the 8-D case. In 24-D, the densest packing appears to be the Leech Lattice.">Oh, for the algorithm, trying all combinations of consonances >above the tonic should work. That'll be O(n**m) where m is the >number of notes in the chord, but shouldn't be a problem for the >kind of numbers we're talking about.What's the double-star? Combinations of notes or intervals? Notes gives you CPSs, and intervals don't get all the chords because some chords contain more than one instance of an interval. Plus, order matters, at least for lists of 2nds, so we wind up with the procedure I described when complaining about my lack of a scheme compiler.>A more efficient way would be to use the general method for >ASSes I give on the web page, but you said you don't trust that.I trust it, but I don't understand it. So I can't _really_ trust it. -Carl
Message: 3256 - Contents - Hide Contents Date: Tue, 15 Jan 2002 23:06:15 Subject: Re: algorithm sought From: dkeenanuqnetau --- In tuning-math@y..., "clumma" <carl@l...> wrote:> On finding all the non-magic chords, Graham has a point. > Looking at all the "faces" of the Euler genus has potential. > Actually, I fully expect Dave Keenan to just pull something > out of his hat here.Sorry, nothing in my hat but bird poop.> How's it going Dave! Back from your > trip to the coast (was it?). How was it!It was a coral cay on the Great Barrier Reef. Human population 40, bird population 40,000.> I'm glad you seem > to be able to balance your list involvement lately. Looks > like I've slipped back in to total addiction, for the time > being.Thanks for the encouragment. Just say no. ;-)
Message: 3257 - Contents - Hide Contents Date: Tue, 15 Jan 2002 15:36:31 Subject: Re: [tuning] Re: badly tuned remote overtones From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, January 15, 2002 3:00 PM > Subject: [tuning-math] Re: badly tuned remote overtones > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> Any vector that is _not_ the result of adding or >>> subtracting two of the already-existing unison-vectors >>> would work to create what Gene calls a "notation" >>> (but is nothing like a musical notation anyone's >>> ever seen before). In this case, he made a choice >>> (56/55) that makes the 12-tone system come out nice >>> in this "notation". >> >>>> OK, fair enough. But what does it have to do with >> Schoenberg's actual theory? I'm having a hard time >> seeing how 56/55 is implied by anything Schoenberg >> actually said. > > It isn't!!!!!!!!!!!!!!!!!!!!!!!!!!!!Ah ... I just tried something else, and now there's a tiny flicker of light beaming thru the fog ... So in other words, Gene simply found a unison-vector that made it easy for him to construct a 12-tone PB using some of the unison-vectors supplied by Schoenberg, yes? Well, the other day I did find my own 12-tone PB for Schoenberg, but making use of that again now (with a new twist), I found something very interesting. I wrote: Yahoo groups: /tuning-math/message/2577 * [with cont.]> Message 2577 > From: "monz" <joemonz@y...> > Date: Fri Jan 11, 2002 4:59 am > Subject: Re: badly tuned remote overtones > > > ... I had failed to take into consideration the 5-limit > enharmonicity required by Schoenberg. To construct a > periodicity-block according to his descriptions, one would > have to temper out one of the "enharmonic equivalents". > > We may choose 2048:2025 = > > [2] > [3] * [11 -4 -2] > [5] > > <etc.>I think one would *have* to include a 5-limit "enharmonic unison-vector" here, since Schoenberg explicitly equated A#=Bb, C#=Db, D#=Eb, etc., the usual 12-EDO enharmonically-equivalent stuff. Well, look what happens if one includes 2048:2025 along with *all* the unison-vectors explicitly stated by Schoenberg, in other words, the same matrix Gene used except that it has 2048:2025 instead of 56:55 : matrix 2 3 5 7 11 unison vectors ~cents [ -2 2 1 0 -1 ] = 45:44 38.90577323 [ 11 -4 -2 0 0 ] = 2048:2025 19.55256881 [ -5 1 0 0 1 ] = 33:32 53.27294323 [ 6 -2 0 -1 0 ] = 64:63 27.2640918 [ -4 4 -1 0 0 ] = 81:80 21.5062896 inverse [ 12 7 12 0 -2 ] [ 19 11 19 0 -3 ] [ 28 16 28 0 -5 ] [ 34 20 34 -1 -6 ] [ 41 24 42 0 -7 ] determinant = | 1 | So here, one can see that there are two possible mappings to 12-EDO, and that the only difference between them is the mapping of 11: h12(11)=41 but g12(11)=42. This is *precisely* the inconsistency in Schoenberg's 1911 notation of 11 which I've mentioned many times! So now, does that mean that *this* periodicity-block is the prime candidate for the one Schoenberg probably had in mind? Are there further difficulties, because of the inconsistent mapping of 11? Still curious, -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3258 - Contents - Hide Contents Date: Tue, 15 Jan 2002 18:17:41 Subject: Re: algorithm sought From: clumma Wait a minute. Gene, what are you using to generate all the chords (magic and otherwise) in 31-tET, on the main list? -Carl
Message: 3259 - Contents - Hide Contents Date: Tue, 15 Jan 2002 23:37:47 Subject: Hi Dave K. From: paulerlich Dave, perhaps you can help with this Kees lattice business. It's pure math, but Gene appears to have no interest in it. It seems to me that we should be able, based on Kees's lattice with a 'taxicab' metric, be able to define an error function so that the optimal temperament according to that error function, which tempers out the small interval n:d, should have an error proportional to |n-d|/(d*log(d)) I've shown that our usual RMS error function works for this within about a factor of 2, based on ten examples with an extremely wide range of n and d values. But we should be able to do much better. Also, a 'complexity' measure similar to the 'gens' ones we've been using, but defined appropriately, should turn out to be proportional to log(d) I've shown that our usual RMS 'gens' measure works for this within about a factor of 2, based on the same ten examples. But we should be able to do much better.
Message: 3260 - Contents - Hide Contents Date: Tue, 15 Jan 2002 11:32:28 Subject: Re: badly tuned remote overtones From: monz Apologies for wasting so much bandwidth ... but I sent both of these out this morning with the wrong date, and they might have been missed. ----- Original Message ----- From: monz <joemonz@xxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Monday, January 15, 2001 9:38 AM Subject: Re: [tuning-math] Re: [tuning] Re: badly tuned remote overtones> > I don't recall ever getting a response to this. > Still interested ... > >>> From: monz <joemonz@xxxxx.xxx> >> To: <tuning@xxxxxxxxxxx.xxx>; <tuning-math@xxxxxxxxxxx.xxx> >> Sent: Friday, January 11, 2002 2:04 PM >> Subject: [tuning-math] Re: [tuning] Re: badly tuned remote overtones >> >> >> >> First, I'd like to start this post off with a link to my >> "rough draft" of a lattice of the periodicity-block Gene >> calculated for Schoenberg's theory: >> >> Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) >> >> This shows the 12-tone periodicity-block (primarily 3- and 5-limit, >> with one 11-limit pitch), and its equivalent p-block cousins at >> +/- each of the four unison-vectors. >> >> >> Now to respond to Paul... >> >>>>> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> >>> To: <tuning@xxxxxxxxxxx.xxx> >>> Sent: Friday, January 11, 2002 12:47 PM >>> Subject: [tuning] Re: badly tuned remote overtones >>> >>> >>> You seem to be brushing some of the unison vectors you had >>> previously reported, and from which Gene derived 7-, 5-, and 2-tone >>> periodicity blocks, under the rug. >> >>>> Ah ... so then this, from Gene: ... >>>>> From: genewardsmith <genewardsmith@xxxx.xxx> >>> To: <tuning-math@xxxxxxxxxxx.xxx> >>> Sent: Wednesday, December 26, 2001 3:25 PM >>> Subject: [tuning-math] Re: Gene's notation & Schoenberg lattices >>> >>> ... This matrix is unimodular, meaning it has determinant +-1. >>> If I invert it, I get >>> >>> [ 7 12 7 -2 5] >>> [11 19 11 -3 8] >>> [16 28 16 -5 12] >>> [20 34 19 -6 14] >>> [24 42 24 -7 17] >>> >>>> ... actually *does* specify "7-, 5-, and 2-tone periodicity blocks". >> Yes? >> >>>>> Face it, Monz -- without some careful "fudging", Schoenberg's >>> derviation of 12-tET as a scale for 13-limit harmony is not >>> the rigorous, unimpeachable bastion of good reasoning that >>> you'd like to present it as. >> >>>> Your point is taken, but please try to understand my objectives >> more clearly. I agree with you that "Schoenberg's derviation ... >> is not the rigorous, unimpeachable bastion of good reasoning" etc. >> I'm simply trying to get a foothold on what was in his mind when >> he came up with his radical new ideas for using 12-tET to represent >> higher-limit chord identities. >> >> I've seen it written (can't remember where right now) that without >> the close personal attachment to Schoenberg that his students had, >> it's nearly impossible to understand all the subtleties of his >> teaching. I'm just trying to dig into that scenario a bit, and >> in a sense to "get closer" to Schoenberg and his mind. >> >>>>> The contradictions in Schoenberg's arguments were known at least >>> as early as Partch's Genesis, and he isn't going to weasel his way >>> out of them now :) If 12-tET can do what you and Schoenberg are >>> trying to say it can, it can do _anything_, and there would be >>> no reason ever to adopt any other tuning system. >> >>>> Ahh ... well, I think you've put on finger on the crux of the matter. >> >> Schoenberg consciously rejected microtonality and also made a >> conscious decision to use the 12-tET tuning as tho it *could* do >> "_anything_". >> >> As I've documented again and again, he *did* have a favorable attitude >> towards adopting other tuning systems, but was of the opinion that >> only in the future would the time be right for that. With us now >> living *in* that future, it seems to me that perhaps he was right >> after all. Perhaps it's even possible that Schoenberg's actions >> in adopting the "new version" of 12-tET ("atonality") helped to >> precipitate the current trend towards microtonality and alternative >> tunings. ...? >> >> >> Always curious about these things, >> >> -monz > >----- Original Message ----- From: monz <joemonz@xxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Monday, January 15, 2001 9:59 AM Subject: Re: [tuning-math] [tuning] Re: badly tuned remote overtones> I also never got replies on my questions here, and > am still waiting. I'm particularly curious about > how 56/55 was added as a unison-vector. Thanks. > >>> From: monz <joemonz@xxxxx.xxx> >> To: <tuning-math@xxxxxxxxxxx.xxx> >> Sent: Friday, January 11, 2002 1:13 AM >> Subject: [tuning-math] [tuning] Re: badly tuned remote overtones >> >> >> Hi Paul and Gene, >> >> >>>>> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> >>> To: <tuning@xxxxxxxxxxx.xxx> >>> Sent: Thursday, January 10, 2002 10:10 AM >>> Subject: [tuning] Re: badly tuned remote overtones >>> >>> >>> --- In tuning@y..., "monz" <joemonz@y...> wrote: >>>>>>> The periodicity-blocks that Gene made from my numerical analysis >>>> of Schoenberg's 1911 and 1927 theories are a good start. >>>>>> Well, given that most of the periodicity blocks imply not 12-tone, >>> but rather 7-, 5-, and 2-tone scales, it strikes me that Schoenberg's >>> attempted justification for 12-tET, at least as intepreted by you, >>> generally fails. No? >> >> >>>> I originally said: >> >>>>> From: monz <joemonz@xxxxx.xxx> >>> To: <tuning-math@xxxxxxxxxxx.xxx> >>> Sent: Tuesday, December 25, 2001 3:44 PM >>> Subject: [tuning-math] lattices of Schoenberg's rational implications >>> >>> >>> Unison-vector matrix: >>> >>> 1911 _Harmonielehre_ 11-limit system >>> >>> ( 1 0 0 1 ) = 33:32 >>> (-2 0 -1 0 ) = 64:63 >>> ( 4 -1 0 0 ) = 81:80 >>> ( 2 1 0 -1 ) = 45:44 >>> >>> Determinant = 7 >>> >>> ... <snip> ... >>> >>> But why do I get a determinant of 7 for the 11-limit system? >>> Schoenberg includes Bb and Eb as 7th harmonics in his description, >>> which gives a set of 9 distinct pitches. But even when >>> I include the 15:14 unison-vector, I still get a determinant >>> of -7. And if I use 16:15 instead, then the determinant >>> is only 5. >> >> >>>> But Paul, you yourself said: >> >>>>> From: Paul Erlich <paul@xxxxxxxxxxxxx.xxx> >>> To: <tuning-math@xxxxxxxxxxx.xxx> >>> Sent: Thursday, July 19, 2001 12:43 PM >>> Subject: [tuning-math] Re: lattices of Schoenberg's rational > implications >>> >>>>>> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>>>>> Could anyone out there do some periodicity-block >>>> calculations on this theory and say something about that? >>>>>> It's pretty clear that Schoenberg's theory implies a 12-tone >>> periodicity block. >> >>>> That was quite a while ago ... have you changed your position >> on that? I thought that Gene showed clearly that a 12-tone >> periodicity-block could be constructed out of Schoenberg's >> unison-vectors. >> >> >>>>> From: genewardsmith <genewardsmith@xxxx.xxx> >>> To: <tuning-math@xxxxxxxxxxx.xxx> >>> Sent: Wednesday, December 26, 2001 12:27 AM >>> Subject: [tuning-math] Re: lattices of Schoenberg's rational > implications >>> >>>>>> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>>>> Can someone explain what's going on here, and what candidates >>>> may be found for unison-vectors by extending the 11-limit system, >>>> in order to define a 12-tone periodicity-block? Thanks. >>>>>> See if this helps; >>> >>> We can extend the set {33/32,64/63,81/80,45/44} to an >>> 11-limit notation in various ways, for instance >>> >>> <56/55,33/32,65/63,81/80,45/44>^(-1) = [h7,h12,g7,-h2,h5] >>> >>> where g7 differs from h7 by g7(7)=19. >> >>>> Gene, how did you come up with 56/55 as a unison-vector? >> Why did I get 5 and 7 as matrix determinants for the >> scale described by Schoenberg, but you were able to >> come up with 12? >> >>>>> Using this, we find the corresponding block is >>> >>> (56/55)^n (33/32)^round(12n/7) (64/63)^n (81/80)^round(-2n/12) >>> (45/44)^round(5n/7), or 1-9/8-32/27-4/3-3/2-27/16-16/9; the >>> Pythagorean scale. We don't need anything new to find a >>> 12-note scale; we get >>> >>> 1--16/15--9/8--32/27--5/4--4/3--16/11--3/2--8/5--5/3--19/9--15/8 >>> >>> or variants, the variants coming from the fact that 12 >>> is even, by using 12 rather than 7 in the denominator. >> >>>> Can you explain this business about variants in a little >> more detail? I understand the general concept, having seen >> it in periodicity-blocks I've constructed on my spreadsheet, >> but I'd like your take on the particulars for this case. >> >> >> >> -monz > _________________________________________________________Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3261 - Contents - Hide Contents Date: Tue, 15 Jan 2002 23:42:40 Subject: [tuning] Re: badly tuned remote overtones From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > I think one would *have* to include a 5-limit "enharmonic > unison-vector" here, since Schoenberg explicitly equated A#=Bb, > C#=Db, D#=Eb, etc., the usual 12-EDO enharmonically-equivalent > stuff.Did he do this explicitly within any of the 'constructions of unison vectors' you gleaned from him? And anyway, why not 128:125? Seems simpler . . .> So now, does that mean that *this* periodicity-block > is the prime candidate for the one Schoenberg probably > had in mind?What do you mean, *this* periodicity block? How many notes does it contain? You only reported a determinany of 1, but that wasn't for a PB, that was for a "notation" with one "extra" unison vector relative to what would be needed for a PB.
Message: 3262 - Contents - Hide Contents Date: Tue, 15 Jan 2002 20:38:57 Subject: Re: more graph theory terminology From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:> I have Maple V, from wayback in 1995, which I've never used, > and I finally fired it up, and lo and behold, it doesn't seem > to run on Windows 2000.Email Waterloo and see if they will upgrade you.
Message: 3263 - Contents - Hide Contents Date: Tue, 15 Jan 2002 23:56:48 Subject: Re: algorithm sought From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> ||3^a 5^b 7^c|| = sqrt(a^2 + 4b^2 + 4c^2 + 2ab + 2ac + 4bc) >> >> would be the length of 3^a 5^b 7^c. Everything in a radius of 2 of >> anything will be consonant.> What happens to 9:5 and 9:7?||9/5|| = ||3^2 5^(-)|| = sqrt(4+4+0-4+0+0) = 2 ||9/7|| = ||3^2 7^(-1)|| = sqrt(4+0+4+0-4+0) = 2 Hence both 9/5 and 9/7 are consonant with 1.
Message: 3264 - Contents - Hide Contents Date: Tue, 15 Jan 2002 20:45:58 Subject: Re: algorithm sought From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:> Can't it always just be 60 degrees? There is that point where > this is no longer the closest packing -- that's bad I presume... > Paul once posted something from Mathworld...That's exactly what I've been talking about and presenting for the last few months! For the 5-limit, it is ||3^a 5^b|| = sqrt(a^2 + ab + b^2) For the 7-limit ||3^a 5^b 7^c|| = sqrt(a^2 + b^2 + c^2 + ab + ac + bc) Beyond that we need to decide if 3 stays the same size as 5, 7, and 11, or is half as long.> "The analog of face-centered cubic packing is the densest lattice > packing in 4- and 5-D. In 8-D, the densest lattice packing is made > up of two copies of face-centered cubic. In 6- and 7-D, the densest > lattice packings are cross sections of the 8-D case. In 24-D, the > densest packing appears to be the Leech Lattice."Forget the densest packing, we want the packing corresponding to octave classes of intervals. There's a lot of great mathematics in the above if you ever get interested in pure math, though. :)
Message: 3265 - Contents - Hide Contents Date: Tue, 15 Jan 2002 20:54:15 Subject: [tuning] Re: badly tuned remote overtones From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> I also never got replies on my questions here, and > am still waiting. I'm particularly curious about > how 56/55 was added as a unison-vector. Thanks.I added 56/55 to get what I call a notation, and to give me a scale step vector. It was therefore a computational aid which as a byproduct displayed some systems related to Schoenberg's.
Message: 3266 - Contents - Hide Contents Date: Tue, 15 Jan 2002 16:18:24 Subject: Re: [tuning] Re: badly tuned remote overtones From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, January 15, 2002 3:42 PM > Subject: [tuning-math] [tuning] Re: badly tuned remote overtones > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>> I think one would *have* to include a 5-limit "enharmonic >> unison-vector" here, since Schoenberg explicitly equated A#=Bb, >> C#=Db, D#=Eb, etc., the usual 12-EDO enharmonically-equivalent >> stuff. >> Did he do this explicitly within any of the 'constructions > of unison vectors' you gleaned from him?Well, not specifically *this* interval. But according to his notational usage, *any* of the 5-limit enharmonicities should apply.> And anyway, why not 128:125? Seems simpler . . .OK, Paul, I tried 128:125 in place of 2048:2025, and the inverse I get is: [ 12 7 12 0 -9 ] [ 19 11 19 0 -14 ] [ 28 16 28 0 -21 ] [ 34 20 34 -1 -26 ] [ 41 24 42 0 -31 ] So you're right ... this still shows the inconsistent mapping to 11 in h12(11)=41, g12(11)=42. Naturally, since I only replace one row of the UV-matrix, there's only one column of the inverse that's different (see? ... I really *am* learning this stuff!), and that's the last column. So, I asked before ... what do these other columns mean? What's -h9 showing us that's different from the -h2 in my other matrix? [ -2 ] [ -3 ] [ -5 ] [ -6 ] [ -7 ] And what about that h0 column? What does that mean? I also tried plugging the skhisma into the "5-limit enharmonicity" row of the matrix (other than 81:80, that is). This time, the last column of the inverse reads: [ -5 ] [ -8 ] [ -11 ] [ -14 ] [ -17 ] and all other columns are the same as the two I derived before, but with the signs reversed. If I use the complement of the skhisma, the other four rows are the same as before and the last one is as above but all positive. If I assume what is probably the most basic case, and plug the Pythagorean comma into that row, again I get reversed signs for the first four columns, and a last column of: [ -12 ] [ -19 ] [ -27 ] [ -34 ] [ -41 ] which is identical to the first column except for the mapping of 5, so that (if we use "f") we may say that -f12(-5)=-27 whereas -h12(-5) and -g12(-5) both = 28 . So, now it seems that I've found the inconsistency in Schoenberg's mapping of 5 as well.>> Schoenberg 1911, _Harmonielehre_, p >> (Carter English translation, p 24) >> >> The two tones _E_ and _B_ appear in the first octave >> [of his illustration -- i.e., the second "8ve" of the >> overtone series], but _E_ is challenged by _Eb_, B by _Bb_. >> ... The second octave resolved the question in favor of >> _E_ and _B_.I'm not interested right now in arguing the logic of Schoenberg's statement. He invokes the "decreasing audibility" of the higher overtones in this very paragraph, and then ignores that, in his estimation of which overtones may be perceived more clearly than others. But the point I'm making is that Schoenberg was fudging the notation of both 5 and 11 in his description, and *this* matrix seems to disclose all of that.>> So now, does that mean that *this* periodicity-block >> is the prime candidate for the one Schoenberg probably >> had in mind? >> What do you mean, *this* periodicity block? How many notes does it > contain? You only reported a determinany of 1, but that wasn't for a > PB, that was for a "notation" with one "extra" unison vector relative > to what would be needed for a PB.Yikes! All too true. So then, what relevance does my construction have, if any at all? It seems to me to show the mechanics of Schoenberg's notational inconsistency. Can you or someone else clear up this business about the difference between PBs and "notations"? Is it that prime-factor 2 is left out of PB calculations (assuming "8ve"-equivalency) but must be included for "notation"? Paul, I understand the criticisms you've written about what I'm trying to do here, but I still think the effort is worthwhile. Certainly, Schoenberg's "pantonal" [= atonal] style assumed that any of the 12-EDO pitches could be used equally well as the center of its own tonal universe. It would be informative to see how each of those universes may be modeled conceptually, and how they relate to each other. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3267 - Contents - Hide Contents Date: Tue, 15 Jan 2002 20:56:33 Subject: Re: algorithm sought From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:> Wait a minute. Gene, what are you using to generate all the > chords (magic and otherwise) in 31-tET, on the main list?Pute brute force, using a routine which steps through all the partions of 31 and tests them. In the case of 72, there are five million or so and I will probably need to do something different if I do it.
Message: 3268 - Contents - Hide Contents Date: Tue, 15 Jan 2002 16:26:34 Subject: Re: [tuning] Re: badly tuned remote overtones From: monz> From: monz <joemonz@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, January 15, 2002 4:18 PM > Subject: Re: [tuning-math] [tuning] Re: badly tuned remote overtones > > > If I assume what is probably the most basic case, and > plug the Pythagorean comma into that row, again I get reversed > signs for the first four columns, and a last column of: > > [ -12 ] > [ -19 ] > [ -27 ] > [ -34 ] > [ -41 ] > > which is identical to the first column except for the > mapping of 5, so that (if we use "f") we may say that > -f12(-5)=-27 whereas -h12(-5) and -g12(-5) both = 28 .Oops ... my bad. The last number is missing the minus sign. So it's " -h12(-5) and -g12(-5) both = -28 ". Gene, is there any reason to notate it like this? Wouldn't it be equivalent to reverse all signs and say " f12(5)=27 whereas h12(5) and g12(5) both = 28 "? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3269 - Contents - Hide Contents Date: Tue, 15 Jan 2002 21:09:18 Subject: Re: algorithm sought From: clumma>> >ait a minute. Gene, what are you using to generate all the >> chords (magic and otherwise) in 31-tET, on the main list? >>Pute brute force, using a routine which steps through all the >partions of 31 and tests them. In the case of 72, there are >five million or so and I will probably need to do something >different if I do it.Sorry to cross-list things here, but I assume you get magic chords just because you define the edges in degrees of the tempered system in the beginning... leaving you no easy way to get only the magic chords at the end. The posts have been great, but it's a lot of work to figure out which ones are magic, even with the ratios. On finding all the non-magic chords, Graham has a point. Looking at all the "faces" of the Euler genus has potential. Actually, I fully expect Dave Keenan to just pull something out of his hat here. How's it going Dave! Back from your trip to the coast (was it?). How was it! I'm glad you seem to be able to balance your list involvement lately. Looks like I've slipped back in to total addiction, for the time being. -Carl
Message: 3270 - Contents - Hide Contents Date: Tue, 15 Jan 2002 13:23:47 Subject: Re: badly tuned remote overtones From: monz In addition to still hoping for answers to my previous questions on this thread, I have some more:> Message 2577 > From: "monz" <joemonz@y...> > Date: Fri Jan 11, 2002 4:59 am > Subject: Re: badly tuned remote overtones Yahoo groups: /tuning-math/message/2577 * [with cont.] > > > Now I realize my mistake: I had failed to take into > consideration the 5-limit enharmonicity required by Schoenberg. > To construct a periodicity-block according to his descriptions, > one would have to temper out one of the "enharmonic equivalents". > > We may choose 2048:2025 = > > [2] > [3] * [11 -4 -2] > [5] > > > Plugging that into the unison-vector matrix I had already > derived before: > <etc.>Now here, I've also added a row vector for the "8ve", as Gene had demonstrated: matrix 2 3 5 7 11 unison vectors ~cents [ 1 0 0 0 0 ] = 2:1 0 [ 11 -4 -2 0 0 ] = 2048:2025 19.55256881 [ -5 1 0 0 1 ] = 33:32 53.27294323 [ 6 -2 0 -1 0 ] = 64:63 27.2640918 [ -4 4 -1 0 0 ] = 81:80 21.5062896 inverse [ 12 0 0 0 0 ] [ 19 -1 0 0 2 ] 1 [ 28 -4 0 0 -4 ] * -- [ 34 2 0 -12 -4 ] 12 [ 41 1 12 0 -2 ] OK, so I can see that the first column vector of the inverse gives the 12-EDO homomorphism: 12-EDO note-name degree with C = 1/1 h12(2) = 12 C h12(3) = 19 G h12(5) = 28 E h12(7) = 34 Bb h12(11) = 41 F And this agrees with the two cases of overtones on "C" and "G", where Schoenberg equated the "perfect 4th" with the 11th overtone, but not with the case of the "F" fundamental, where he equated the "augmented 4th" with 11. But now what do all those other column vectors mean? They all start with 0 ... what does that mean? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3271 - Contents - Hide Contents Date: Tue, 15 Jan 2002 13:27:14 Subject: Re: [tuning] Re: badly tuned remote overtones From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, January 15, 2002 12:54 PM > Subject: [tuning-math] Re: badly tuned remote overtones > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>> I also never got replies on my questions here, and >> am still waiting. I'm particularly curious about >> how 56/55 was added as a unison-vector. Thanks. >> I added 56/55 to get what I call a notation, and to give > me a scale step vector. It was therefore a computational > aid which as a byproduct displayed some systems related > to Schoenberg's.OK, right ... I understand all that. But where did it come from? Is it the result of adding or subtracting two of the already-existing unison-vectors? That wouldn't work, would it? ... because all the vectors in the matrix have to be independent. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3272 - Contents - Hide Contents Date: Tue, 15 Jan 2002 17:42:12 Subject: adding/subtracting unison-vectors From: monz Question: What significance is there in the addition or subtraction of unison-vectors? In either case, the result is another unison-vector which may be substituted in the matrix in place of either of those two, correct? Or does it work only for subtraction, or only for addition? Here's the specific example I have in mind: In Message 2159, I wrote:> From: "monz" <joemonz@y...> > Date: Tue Dec 25, 2001 6:44 pm > Subject: lattices of Schoenberg's rational implications Yahoo groups: /tuning-math/message/2159 * [with cont.] > > > 1911 _Harmonielehre_ 11-limit system > > ( 1 0 0 1 ) = 33:32 > (-2 0 -1 0 ) = 64:63 > ( 4 -1 0 0 ) = 81:80 > ( 2 1 0 -1 ) = 45:44 > > Determinant = 7But I see now that adding together the two 11-limit unison-vectors cancels out prime-factor 11, and we're left with one of the standard 5-limit "semitones": 3 5 7 11 ratio ~cents [ 1 0 0 1 ] = 33:32 53.27294323 + [ 2 1 0 -1 ] = 45:44 38.90577323 ---------------- [ 3 1 0 0 ] = 135:128 92.17871646 This is the interval which Rameau called the "mean semitone", and Ellis called the "larger limma". See my webpage: Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) It seems to me that this is perhaps why Schoenberg, in 1911, gave an inconsistent notation for the 11th harmonic. If I subtract these two 11-limit unison-vectors, I get: 3 5 7 11 ratio ~cents [ 1 0 0 1 ] = 33:32 53.27294323 + [ 2 1 0 -1 ] = 45:44 38.90577323 ---------------- [-1 -1 0 2 ] = 121:120 14.36717 What do either of these operations have to do with Schoenberg's theory? -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3273 - Contents - Hide Contents Date: Tue, 15 Jan 2002 21:33:18 Subject: Re: metric visualization From: paulerlich The conditional metric -- isn't this just a triangular-taxicab metric over a suitably defined lattice? P.S. Assuming that, and so using max instead of Euclidean norm, I found some slight variation of your matrix last night, that seemed to get the meantone number to agree with the enneadecal number . . . but today, I've lost it! HELP!! --- In tuning-math@y..., "keesvp" <kees@d...> wrote:> That's exactly what the nonlinear thingy is about, to correct the > result of the linear transformation and euclidean norm. I guess you > could avoid all this by just taking the conditional metric I started > with. > > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> By the way, Kees, by "norm" I assume you meant the 2-norm . . . but >> that would be the Euclidean norm . . . would the appropriate > taxicab>> norm be, instead, the maximum of the entries in the vector? Or what?
Message: 3274 - Contents - Hide Contents Date: Tue, 15 Jan 2002 21:35:21 Subject: Re: algorithm sought From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "clumma" <carl@l...> wrote: >>> Paul H. applied it to temperaments, and scales. I'm using it >> to define n-limit chords (rational only). Did I make a mistake? >> Part of the difficulty for me is, the smallest ASSs are 9-limit, >> and that requires more than 3 dimensions. >> I would work in 3-dimensions for the 9-limit, and just make 3 halfthe size of 5 or 7. In other words,> > ||3^a 5^b 7^c|| = sqrt(a^2 + 4b^2 + 4c^2 + 2ab + 2ac + 4bc) > > would be the length of 3^a 5^b 7^c. Everything in a radius of 2 of > anything will be consonant.What happens to 9:5 and 9:7?
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