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Message: 3225 - Contents - Hide Contents Date: Mon, 14 Jan 2002 23:17:31 Subject: Re: metric visualization From: paulerlich 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: 3226 - Contents - Hide Contents Date: Mon, 14 Jan 2002 23:52:10 Subject: Re: algorithm sought From: dkeenanuqnetau --- In tuning-math@y..., "clumma" <carl@l...> wrote:> Anyway, Gene, what I was thinking... under the > dyadic definition, all n-limit chords must be > connected on the n-limit lattice, and must have > a Hahn-diameter of 1. See: > > Music (and Music Theory) * [with cont.] (Wayb.) > > There ought to be a geometric way to find these > structures...Does this have to work for temperaments or only rational scales?
Message: 3228 - Contents - Hide Contents Date: Tue, 15 Jan 2002 06:23:05 Subject: Re: algorithm sought From: clumma>> >aul 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 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.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. -Carl
Message: 3229 - Contents - Hide Contents Date: Tue, 15 Jan 2002 21:42:53 Subject: [tuning] Re: badly tuned remote overtones From: paulerlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> > I don't recall ever getting a response to this. > Still interested ... > >>> From: monz <joemonz@y...> >> To: <tuning@y...>; <tuning-math@y...> >> 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 - Error 404 * [with cont.] (Wayb.) pblock.gif >> >> 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@s...> >>> To: <tuning@y...> >>> 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@j...> >>> To: <tuning-math@y...> >>> 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?I thought he specified those in another post.
Message: 3230 - Contents - Hide Contents Date: Tue, 15 Jan 2002 23:10:41 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 5:09 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. >> Right, but . . . did he apply any of them explicitly within any of > the 'constructions of unison vectors' you gleaned from him? > Otherwise, you're just "assuming the answer".I wanted to attempt a more rigorous answer to this. Back in July, when I started all this about Schoenberg, I wrote: Yahoo groups: /tuning-math/messages/516?expand=1 * [with cont.]> Message 516 > From: monz <joemonz@y...> > Date: Wed Jul 18, 2001 5:16am > Subject: lattices of Schoenberg's rational implications > > > ... > > Schoenberg then extends the diagram to include the > following overtones: > > fundamental partials > > F 2...12, 16 > C 2...11 > G 2...12Here, I will further adapt Schoenberg's diagram to make his explanation as clear as possible, by adding the partial-numbers and the fundamentals, which are the two factors which when multiplied together give the relative frequency-number of each note. The fundamentals are F = 4, C = 6, G = 9. d = 12*9 = 108 c = 11*9 = 99 b = 10*9 = 90 a = 9*9 = 81 g = 8*9 = 72 f = 11*6 = 66 f = 16*4 = 64 (f = 7*9 = 63) e = 10*6 = 60 d = 9*6 = 54 d = 6*9 = 54 c = 12*4 = 48 c = 8*6 = 48 b = 5*9 = 45 b = 11*4 = 44 (bb= 7*6 = 42) a = 10*4 = 40 g = 9*4 = 36 g = 6*6 = 36 g = 4*9 = 36 f = 8*4 = 32 e = 5*6 = 30 (eb= 7*4 = 28) d = 3*9 = 27 c = 6*4 = 24 c = 4*6 = 24 a = 5*4 = 20 g = 3*6 = 18 g = 2*9 = 18 f = 4*4 = 16 c = 3*4 = 12 c = 2*6 = 12 f = 2*4 = 8> (eb) (bb) > c d e f g a b c d e f g a b c d > [44] [64] > (28) (42) [66] > 24 27 30 32 36 40 45 48 54 60 63 72 81 90 99 108 > > > ... > > The partial-numbers are also given for the resulting scale > at the bottom of the diagram, showing that 7th/F (= eb-28) > is weaker than 5th/C (= e-30), and 7th/C (= bb-42) is weaker > than 5th/G (= b-45). > > Also note that 11th/F (= b-44), 16th/F (= f-64) and 11th/C > (= f-66) are all weaker still, thus I have included them in > square brackets. These overtones are not even mentioned by > Schoenberg.These are all the unison-vectors implied by Schoenberg's diagram: E 5*6=30 : Eb 4*7=28 = 15:14 B 11*4=44 : Bb 7*6=42 = 22:21 B 5*9=45 : B 11*4=44 = 45:44 B 5*9=45 : Bb 7*6=42 = 15:14 F 16*4=64 : F 7*9=63 = 64:63 F 11*6=66 : F 16*4=64 = 33:32 F 11*6=66 : F 7*9=63 = 22:21 A 9*9=81 :(A 20*4=80) = 81:80 C 11*9=99 :(C 24*4=96) = 33:32 (The high "A" and "C" in parentheses are not explicitly indicated by Schoenberg, but may be inferred from his theory.) So the only 5-limit unison-vector indicated here is the 81:80 syntonic comma, and even that is only inferred but not stated. Its applicability to his theory, as well as that of other 5-limit UVs, must be inferred from a careful study of other explanations in _Harmonielehre_, as I indicated in my last post. The 15:14 arises only in connected with the notes Schoenberg himself placed in parentheses. The other UVs are explicitly indicated by Schoenberg. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3231 - Contents - Hide Contents Date: Tue, 15 Jan 2002 00:17:42 Subject: Re: algorithm sought From: clumma>Does this have to work for temperaments or only rational scales?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. -Carl
Message: 3232 - Contents - Hide Contents Date: Tue, 15 Jan 2002 03:00:45 Subject: Re: Gene's relative connectedness From: Pierre Lamothe (PL) (GWS)> Do a connectedness of a scale relative to S, be a property of the scale or a > property of the relation between the scale and S?The scale and S. So our concepts are not conflictual since the contiguity is a property of a mode in itself. There can be more than one path according to the definitions as given. Another way to define it is to define the graph of a scale relative to S, and then it is connected iff the graph is connected It's not easy to see that sense from your definition since a graph is defined by a set of nodes and a set of vertices. It seems the scale would be the set of nodes while S would be the set of vertices. However in a graph, one vertice link two nodes, but it seems you relie your nodes with a chain of vertices, implying other nodes, outside the scale, to rely nodes. If I understand the connectedness in a graph, that corresponds to a possible way between any two nodes, using a unique vertice between two nodes. The question of infinity is another one; however the rational numbers are an infinite field, and in fact any ordered field (for which you were giving some of the axioms a while back) is infinite I use only finite sets. In the harmoid frame, I work with the finite classes mod <2> and a set of intervals representing these classes (first octave or centered octave or pivots). Even if a relation remains valid at infinity, I never use something requiring the existence of elements outside the perceptible domain. I can plunge a gammoid structure in a group, for instance, to show the link with the periodicity block, but I have access at all that from operations in a finite set. You could say that using logarithm implies the infinity of the real field. Yes, but I don't use the operative properties of that field, restricting to Z-module, in which I restrict to classes mod <2>, and finally I restrict to a finite area well-defined around the unison. Pierre [This message contained attachments]
Message: 3233 - Contents - Hide Contents Date: Tue, 15 Jan 2002 20:55:15 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 5:09 PM > Subject: [tuning-math] [tuning] Re: badly tuned remote overtones > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >>>>> From: paulerlich <paul@s...> >>> To: <tuning-math@y...> >>> 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. >> Right, but . . . did he apply any of them explicitly within any of > the 'constructions of unison vectors' you gleaned from him? > Otherwise, you're just "assuming the answer".Well, on p 176 of _Harmonielehre_ (p 155 of the Carter translation), Schoenberg illustrates the "Circle of 5ths", and explicitly notates the equivalences Cb=B, Gb=F#, and Db=C# for the major keys, and ab=g#, eb=g#, and bb=a# for the minor keys that are +5, +6, and +7 "5ths" (respectively) from the origin C-major/a-minor. This is in Chapter 9, "Modulation", and all thru his discussion of modulation Schoenberg assumes the enharmonic equivalence of keys like this. This is before he ever gets into anything about the enharmonic equivalence of implied 5-limit harmonies. He never explicitly says that a "5th" is always to be interpreted as a 3:2, but his explanation of the basic tones of the scale as overtones of the subdominant, dominant, and tonic implies that he's thinking of "5ths" as either 3-limit intervals or meantone generators, or both. And in the "Table of the Circle of Fifths for C Major (a minor)", on the following page, he lists the keys next to each other going in opposite directions around the circle, so that one column has only the "sharp" keys and the other only the "flat" keys. To me, this implies even more strongly a conception based on a generating "5th", whether it's Pythagorean or meantone. And again, the enharmonic equivalences are explicitly stated. If meantone is assumed instead of Pythagorean, then the enharmonic equivalence illustrated in Schoenberg's "Circle of 5ths" is the diesis 128:125 = [7 0 -3]. So I can see why you might consider *this* to be the "missing link" unison-vector. So, to answer your question directly: Schoenberg equates pairs of enharmonically-equivalent pitches, but no, he never explicitly mentions whether those pairs of pitches are derived via Pythagorean or meantone tuning. So yes, I'm assuming certain intervals as unison-vectors, based as much as possible on the pitch-relationships explicitly detailed by Schoenberg. But I would venture to say, based on what he wrote in _Harmonielehre_, that he expected *both* the Pythagorean comma *and* the diesis (and all of their combinations) to be tempered out.>>>>> 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. >> >> ... >> >> If I assume what is probably the most basic case, and >> plug the Pythagorean comma into that row, >> ??? Why is that the most basic case?This is my thinking: Schoenberg's theory certainly assumes all the "traditional" enharmonic equivalence of the 12-EDO scale -- by "traditional", I mean all of the enharmonic equivalences that may arise in the 3- and 5-limits. To my mind, the 3-limit (linear, 1-D) is both historically and conceptually more basic than 5-limit (planar, 2-D). The notational difference between a "sharp" and what later became its enharmonically equivalent "flat", ocurred first in Pythagorean tuning. And so, along this line of reasoning, the Pythagorean comma is historically and conceptually a more basic enharmonicity than any of the 5-limit examples. However, as implied above, I will also grant the possibility that Schoenberg may have intended the diesis as a unison-vector, and will examine that case below as well. Also, I understand Gene's "notation" a little better now. So, taking this particular matrix as an example, 2 3 5 7 11 unison-vector ~cents [ -2 2 1 0 -1 ] = 45:44 38.90577323 [-19 12 0 0 0 ] = 531441:524288 23.46001038 [ -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 12 ] [ 19 -11 19 0 19 ] [ 28 -16 28 0 27 ] [ 34 -20 34 -1 34 ] [ 41 -24 42 0 41 ] So, for an example of how the unison-vector maps to a homomorphism, the matrix describing the mapping of 45:44 to h12 is: [ 12 ] [ -2 2 1 0 -1 ] [ 19 ] [ 28 ] [ 34 ] [ 41 ] which translates into (12*-2)+(19*2)+(28*1)+(34*0)+(41*-1) = -24 + 38 + 28 + 0 + -41 = 1 So when I look at how all the unison-vectors map to the homomorphisms, I get: homomorphism h12 -h7 h12 h0 h12 unison-vector [ 1 0 0 0 0 ] 45:44 [ 0 1 0 0 0 ] 531441:524288 [ 0 0 1 0 0 ] 33:32 [ 0 0 0 1 0 ] 64:63 [ 0 0 0 0 1 ] 81:80 So now let me try to get this straight. This matrix is telling us that one of three mappings to 12-EDO may be chosen, in which we distinguish either 45:44, 33:32, or 81:80 as pairs of distinct notes. Correct? Plugging 128:125 into the 2nd row instead of the Pythagorean comma, a look at the mapping of unison-vectors gives us exactly the same matrix as above. (But of course, this time the last column of the inverse gives a 9-EDO rather than 12-EDO mapping, so that there are two 12-EDO mappings this time rather than three.) So assuming 128:125 to be a unison-vector, we still may choose between either of two 12-EDO mappings, in which we distinguish either 45:44 or 33:32. With 128:125 as a unison-vector, along with the others we use here, 12-EDO *always* tempers out the syntonic comma 81:80. With the Pythagorean Comma as a UV instead, 12-EDO may or may not temper out the syntonic comma, depending on which homomorphism is chosen.>>> So, now it seems that I've found the inconsistency in >> Schoenberg's mapping of 5 as well. >> Only if you assume the Pythagorean comma, right?Right -- that's the only example I've found so far which changes the mapping of 5. Gene's PB (using 56:55 as a UV) found an inconsistent mapping to 7, and he and I have both found several which map 11 inconsistently.> You need to include the prime-factor 2 for PB calculations > too, if you're to weed out cases of torsion.Ahh! ... now *that's* a useful little tidbit!! Thanks!> Any of the PBs that give you a determinant of 12, if all the unison > vectors are tempered out, implies 12-tET. Geometrically, this will be > modeled by a torus or hyper-torus . . . can you make out the > inflatable torus model in the photocopy of the Hall article I sent > you (sorry the photocopy didn't come out so good -- check your > library for a better version)?The picture in the Hall article is pretty hard to make out ... but there's an identical diagram of a "Chicken-wire Torus" in "Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition" by Jack Douthett and Peter Steinbach, in _Journal of Music Theory_ 42:2 (Fall 1998), on p 248. So I understand how it works and can see it to some degree on this diagram. Of course, the actual *physical* model Hall used is preferable ... where can I get one?! -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3234 - Contents - Hide Contents Date: Tue, 15 Jan 2002 21:47:47 Subject: [tuning] Re: badly tuned remote overtones From: paulerlich --- 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. > >>> From: monz <joemonz@y...> >> To: <tuning-math@y...> >> Sent: Friday, January 11, 2002 1:13 AM >> Subject: [tuning-math] [tuning] Re: badly tuned remote overtones >> >> >> Hi Paul and Gene, >> >> >>>>> From: paulerlich <paul@s...> >>> To: <tuning@y...> >>> 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@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 = 7Well there you go. This shows that Schoenberg's UVs imply a 7-tone system, not a 12-tone system as you claim.>>> ... <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?>>> 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?>> >>>>> From: genewardsmith <genewardsmith@j...> >>> To: <tuning-math@y...> >>> 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?Because it works to get you 12 as one of the five resulting cardinalities.>> 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?Only by replacing 33/32 with 56/55.
Message: 3235 - Contents - Hide Contents Date: Tue, 15 Jan 2002 00:26:53 Subject: Re: algorithm sought From: clumma> Paul H. applied it to temperaments, and scales. I'm using it > to define n-limit chords (rational only). Did I make a mistake?To put it another way: every note in the chord must be connected to every other note by exactly one lattice link. In the brute- force method, the huge majority of chords will later fail the test. Why not generate them directly by searching the lattice? There's got to be some graph theory somewhere that will do this. -Carl
Message: 3236 - Contents - Hide Contents Date: Tue, 15 Jan 2002 09:56:26 Subject: more graph theory terminology From: clumma It seems that undirected graphs with diameter 1 are considered "complete" graphs. It's easy to generate these at any card k. They all have edge connectivity k-1. So the problem of finding all k-card chords in the n-limit may be equivalent to finding all the orientations of the complete order-k graph in the n- limit lattice. And this may be equivalent to finding all the 1-diameter order-k subgraphs of the n-limit lattice (when the lattice is a directed graph). Or maybe directedness isn't what we want to use to differentiate the edges of the lattice. There's also a concept called "weighting" -- weighted graphs are called "networks". -Carl
Message: 3237 - Contents - Hide Contents Date: Tue, 15 Jan 2002 21:48:01 Subject: Re: more graph theory terminology From: clumma>>> >o it seems a 1-diameter subgraph is also called a clique? >>>> Unless the author defines clique to be maximal. The cliques of a >> graph give us consonant chords in the sense I've been using it on >> the masses of asses thread. >>Look up "saturated" in Monz' dictionary -- perhaps this is >relevant?The maximal cliques of the n-limit lattice are the n-limit saturated chords, I think. My problem is to find all the order-k cliques in the lattice. [Hope I'm not making any errors here -- I'm an utter neophyte with the graph theory.] -Carl
Message: 3238 - Contents - Hide Contents Date: Tue, 15 Jan 2002 00:39:50 Subject: Re: algorithm sought From: genewardsmith --- 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 half the sizeof 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 anythingwill be consonant.
Message: 3239 - Contents - Hide Contents Date: Tue, 15 Jan 2002 09:58:23 Subject: Re: more graph theory terminology From: clumma BTW- On the topic of using edge connectivity to evaluate scales... While it is impressive that connectivitiy is sufficient to select the diatonic scale from all 7-tone meantones, the results may not be so impressive when k is closer to card(n) (in the 5-limit, card(n) is only 3, while k=7 for the diatonic scale). After all, we're not claiming that things like harmonics 6-12 are our favorite 11-limit scales (actually, I like this scale a lot, but I seem to be alone in this opinion around here...). A more important property of the diatonic scale may be that it has several weakly-connected, complete subgraphs. As weakly connecteded as they can be, and still be so numerous and in-tune, I could argue. Of course this is nothing new; we've been using "chord coverage" for years, and chords/notes is practically what this list is about. But if we can put this stuff into graph-theory terms, we may be able to use the many powerful existing tools out there to soup-up our searches. -Carl
Message: 3240 - Contents - Hide Contents Date: Tue, 15 Jan 2002 21:53:04 Subject: [tuning] Re: badly tuned remote overtones From: paulerlich --- 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".
Message: 3241 - Contents - Hide Contents Date: Tue, 15 Jan 2002 00:53:19 Subject: Re: algorithm sought From: clumma>> >aul H. applied it to temperaments, and scales. I'm using it >> to define n-limit chords (rational only). Did I make a mistake?Turns out _diameter_ is already graph-theory terminology, and it is the term we want. I'm afraid I didn't keep the URL for the source of: "Let G be a graph and v be a vertex of G. The eccentricity of the vertex v is the maximum distance from v to any vertex. That is, e(v)=max{d(v,w):w in V(G)}." "The diameter of G is the maximum eccentricity among the vertices of G. Thus, diameter(G)=max{e(v):v in V(G)}." The "radius" of G is the minimum eccentricity. -Carl
Message: 3242 - Contents - Hide Contents Date: Tue, 15 Jan 2002 10:35:08 Subject: Re: more graph theory terminology From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:> But if we can put this stuff into graph-theory terms, we may be > able to use the many powerful existing tools out there to soup-up > our searches.You might also reverse the process, and try to see what a graph theory concept means in scale terms. Things such as the chromatic number, domination number, genus and so forth strike me as interesting in that way.
Message: 3243 - Contents - Hide Contents Date: Tue, 15 Jan 2002 21:57:29 Subject: Re: algorithm sought From: paulerlich --- In tuning-math@y..., "clumma" <carl@l...> wrote:>> 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) >> Is there a post where the derivation for this is given? >>> Beyond that we need to decide if 3 stays the same size as 5, 7, >> and 11, or is half as long. >> You can expect me to become completely confused when dealing > with prime limits.Gene is talking about odd limits, just as we are. He's just attempting to capture them in a Euclidean lattice with prime axes. I'm concerned that this won't always work -- Gene, perhaps you could post the implied lengths for 1:3 1:5 3:5 1:7 3:7 5:7 1:9 5:9 7:9 1:11 3:11 5:11 7:11 9:11 etc. (up to whatever odd limit you could see yourself being interested in).
Message: 3244 - Contents - Hide Contents Date: Tue, 15 Jan 2002 10:45:11 Subject: Re: more graph theory terminology From: clumma>> >ut if we can put this stuff into graph-theory terms, we may be >> able to use the many powerful existing tools out there to soup-up >> our searches. >>You might also reverse the process, and try to see what a graph >theory concept means in scale terms. Things such as the chromatic >number, domination number, genus and so forth strike me as >interesting in that way.I bet! There seeems to be no shortage of terms in this field. I didn't actually run across domination number or genus. I did see chromatic number, but didn't stop to read the def. I'll go back and get these -- thanks for the pointers. So it seems a 1-diameter subgraph is also called a clique? Anyway, the problem of finding n-limit chords seems related to something called the Clique Problem. Naturally, the mathematicians are more interested in wether problems can be done or not than (the usually much easier problem of) actually finding the answer. . . "Algorithms for the Maximum Clique Problem A 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." ...'but we're not going to tell you what they are...' ------- Other promising leads I haven't checked yet: http://www1.ics.uci.edu/~eppstein/pubs/Epp-TR-94-25.pdf - Type Ok * [with cont.] (Wayb.) "Approximating Minimum-Size k-Connected Spanning Subgraphs via Matching", Joseph Cheriyan, Ramakrishna Thurimella. -Carl
Message: 3247 - Contents - Hide Contents Date: Tue, 15 Jan 2002 11:11:15 Subject: Re: more graph theory terminology From: genewardsmith --- In tuning-math@y..., "clumma" <carl@l...> wrote:> So it seems a 1-diameter subgraph is also called a clique?Unless the author defines clique to be maximal. The cliques of a graph give us consonant chords in the sense I've been using it on the masses of asses thread.> A 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.
Message: 3249 - Contents - Hide Contents Date: Tue, 15 Jan 2002 03:13:12 Subject: Re: Gene's relative connectedness From: genewardsmith --- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:> or scale and degrees have another sense where it could exist more than one > path between the first and the last degrees of the scale.There can be more than one path according to the definitions as given. Another way to define it is to define the graph of a scale relative to S, and then it is connected iff the graph is connected, for which see http://mathworld.wolfram.com/ConnectedGraph.html * [with cont.]> How to believe that a "relative connectedness" could convey an exact sense, when > it seems that that conveys about nothing?A scale can for example be connected in the 7-limit but not in the 5-limit, so it conveys something.> Do a connectedness of a scale relative to S, be a property of the scale or a property > of the relation between the scale and S?The scale and S.> I suppose Gene seeks to adapt, in his simplest way, my concept on contiguity.No, I seek to apply standard mathematical concepts of path-connectedness ingraphs to this particular situation.> A gammier mode is not relatively connected but absolutely connected sincethe unique > set A for which it has sense to refer the connectedness is strictly determined by the > gammier structure itself.That could be done with RI scales, but is in general too restrictive for myintended applications.> Why I would want to introduce infinity, like your simplest way, while my way is so short?The question of infinity is another one; however the rational numbers are an infinite field, and in fact any ordered field (for which you were giving some of the axioms a while back) is infinite. In any event, I want to consider more than RI scales.
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