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Message: 4300 - Contents - Hide Contents Date: Fri, 15 Mar 2002 18:53:49 Subject: Re: Wedge product definitions from mathworld From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> If you have attempted that somewhere, could you please point me to it.I've started writing (someone needs to) but haven't got far. The plan is to explain wedge products, among other things, in an attempt to explain the theory of it all.> The mathworld definition makes no sense to me whatsoever, since it is > recursive and I can't find where the recursion bottoms out. Can you > please give a definition that only applies to matrices or vectors > (whatever makes sense for temperaments) and not general k-forms, and > that eventually bottoms out into ordinary scalar multiplication *?(1) You consider intervals to be expressed as vectors in the basis {e2,e3,e5...} etc. so that 2^-4 3^4 5^-1 would be -4 e2 + 4 e3 - e5, for instance. (2) The basis vectors ep eq wedge according to ep^eq = - eq^ep (antisymmetric); aside from that everything is associative and distributive. In fact, *any* two intervals u, v will give u^v = - v^u. (3) If you wedge together n intervals, you get a blade; in particular a wedge of n basis vectors is a blade. Linear combinations of n-blades produce n-intervals. If U is an n-interval and V is an m-interval, then U^V = (-1)^nm V^U; it can be calculated by simply working with the basis vectors. (4) If we have a p-limit, so that we have d=phi(p) primes, we get a 2^d sized object, the exterior algebra, which is graded into scalars (dimension d choose 0 = 1) intervals (dimension d choose 1 = d) 2-intervals (dimension d choose 2) and so forth, up to d-intervals, of dimension d choose d = 1, which are pseudo-scalars. (5) Everything I said about intervals is also true of their dual objects, vals. A basis for the vals is v2, v3, v5,... where vp is the p-adic valuation map which sends a rational number q to the exponent of p in the factorization of q. (6) Thus, you get n-vals as well as n-intervals. (7) There is a natural inner product between n-vals and n-intervals, which can be expressed as the determinant of the matrix [vp(eq)] as it runs through the list of primes. For n !=0, the natural inner product would be 0; this defines an inner product between the exterior algebra of vals and the exterior algebra of intervals. (8) There is an isomorphism between the exteror algebra of vals and of intervals (Poincare duality) defined by relating the inner product of a n-val with a n-interval (going to scalars) with the wedge product of (n-d)-vals and n-intervals, going to pseudoscalars, which makes the (n-d)-vals and n-intervals isomorphic. Hence the wedgie one gets by wedging h12^h22 and the wedgie one gets by wedging 64/63^50/49 can be equated, and both considered to be the pajara wedgie. This probably made things worse. :(
Message: 4301 - Contents - Hide Contents Date: Fri, 15 Mar 2002 19:12:20 Subject: Re: Wedge product definitions from mathworld From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> (7) There is a natural inner product between n-vals and n-intervals, > which can be expressed as the determinant of the matrix [vp(eq)] > as it runs through the list of primes.This is murky; I should have said, if h1^h2^...^hn is a val n-blade, and q1^q1^...^qn is an interval n-blade, then form the matrix [hi(qj)], 1 <= i,j <= n, and take the determinant. The particular case when we have wedges of basis vectors leads to vectors with either 0 or 1 as entries and with determinants -1, 0, 1; which can be used to define the inner product in general by linearity.> This probably made things worse. :(When I first presented this I tried to avoid all theory and just do the special cases of 5,7, and 11-limits explicitly. No one seemed to like it, but this is still a possible alternative approach.
Message: 4302 - Contents - Hide Contents Date: Fri, 15 Mar 2002 08:35:53 Subject: Re: Starling temperament mapping From: genewardsmith --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:> Gene, > > Can you please give your proposed canonical form if it differs from this, > and explain the conventions involved.If we are looking at a d-dimensional temperament (ie, d is one less than the number of generators) then we have d+1 columns, and by taking d+1 primes and leaving one out, we can wedge the wedgie with d of the rest of the primes, producing a sans 2, sans 3, sans 5 etc. column. If we have a gcd > 1, we divide out by that, and we make the top number of the column positive by multiplying by -1 if necessary. We now have a mapping matrix which starts out with a diagonal square matrix part on top (since the sans p part will give zeros for all of the primes other than p, leaving only one nonzero entry.) This is my proposed normal form, not to be confused with Smith Normal Form though in fact it looks kind of similar. :) Here's the 126/125 example: 126/125^3^5 = [1 0 0 -1] 126/125^2^5 = [0 -1 0 2] 126/125^2^3 = [0 0 1 3] We change [0 -1 0 2] to [0 1 0 -2], and use these three to get the columns of the mapping matrix: [ 1 0 0] [ 0 1 0] [ 0 0 1] [-1 -2 3] In this case, we have 1s along the diagonal, so the generators are appromiately equal to 2, 3, and 5.
Message: 4303 - Contents - Hide Contents Date: Fri, 15 Mar 2002 20:55:04 Subject: Re: Dave's 23 best 5-limit temperaments From: paulerlich --- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:> On Tue, 12 Mar 2002 05:30:45 -0000, "genewardsmith" > <genewardsmith@j...> wrote: >>> 1990656/1953125 Extends to the 1029/1024^126/125 = >> [9,5,-3,-21,30,-13] system, and needs a name.) >> >> map [[0, 9, 5], [1, 1, 2]] >> >> generators 77.96498962 1200 >> >> keenan 12.03289099 rms 2.983295872 g 6.377042156 >> This looks like a good candidate for the "Starling" name. Besides having > 126/125 as a unison vector, its generator is a prominent melodic interval > in Starling temperament.wouldn't this be a *tempering of* starling? i thought that starling was the *planar* temperament defined by *only* 126/125 vanishing; this here is one *linear* temperament of that (1029/1024 vanishing as well), with a 'fifth' of 702 cents, and a 'major third' of 390 cents. this generator would imply that the most important scales have 15, 16, 31, 46, 77 . . . notes per octave. is this really starling?
Message: 4304 - Contents - Hide Contents Date: Fri, 15 Mar 2002 09:25:37 Subject: Re: Starling temperament mapping From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote: > >> Gene, >>>> Can you please give your proposed canonical form if it differs from this, >> and explain the conventions involved.I've been looking at whether this really works in general, and it seems it doesn't--we don't always get the wedgie back by wedging the rows, because we can get a mulitple (ie, torsion problems again.) But it's so neat I'll try to see if I can save it somehow.
Message: 4305 - Contents - Hide Contents Date: Fri, 15 Mar 2002 22:47:39 Subject: Re: Dave's 23 best 5-limit temperaments From: dkeenanuqnetau --- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:> On Tue, 12 Mar 2002 05:30:45 -0000, "genewardsmith" > <genewardsmith@j...> wrote: >>> 1990656/1953125 Extends to the 1029/1024^126/125 = >> [9,5,-3,-21,30,-13] system, and needs a name.) >> >> map [[0, 9, 5], [1, 1, 2]] >> >> generators 77.96498962 1200 >> >> keenan 12.03289099 rms 2.983295872 g 6.377042156 >> This looks like a good candidate for the "Starling" name. Besides having > 126/125 as a unison vector, its generator is a prominent melodic interval > in Starling temperament.This is the one you suggested calling chrome. Starling? No. As Paul said, its only one possible further tempering of the real Starling planar temperament. But also it's a very uninteresting temperament, having so many generators to the fifth, with errors still as large as 3 cents. With a smallest MOS of 15 notes (having one triad) it's also fairly uninteresting melodically. It's marginally more interesting at 7-limit, but beyond that forget it. Quarter minor thirds is a good enough name for it for me.
Message: 4306 - Contents - Hide Contents Date: Fri, 15 Mar 2002 13:02 +0 Subject: Re: Standardising temperament mappings From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <3.0.6.32.20020314182114.00b2f6f0@xx.xxx.xx> David C Keenan wrote:> 1a. What units to use for generators > > While octaves make good sense within a program for calculating > temperaments, I think it's clear that either cents or rational octaves > will > be better understood by most readers. Since a basis nearly always > includes > a generator whose optimum value can't be exprtessed as a simple rational > fraction of an octave, it seems cents should be the standard unit. We > might > however want to include supplementary information when the cents do not > make it obvious that there are say exactly 19 generators (or periods) to > the octave, even though this information will also be buried in the map > matrix.Yes, that all sounds right. I'm tempted to say that the period size in cents should be the supplement if the "octave" size in cents and number of divisions thereof are known. But it doesn't really matter as both things are there.> So it seems obvious to me that the first generator to be listed for a > linear temperament should be the one that generates only the lowest > prime > (usually 2). We sometimes distinguish this generator by calling it the > "period", particularly when we consider it to be generating an > "interval of > equivalence" (usually at least an approximate octave, but the tritave > 1:3 > also has a following). In this case we consider that the number of > pitches > and their spacing _within_ a single interval-of-equivalence is what > constitutes the scale, and we consider the number of > intervals-of-equivalence provided on any given instrument to be a detail > irrelevant to our calculations.So you mean (periodSize, generatorSize) for this datum?> 1c. How to determine a canonical set of generators. > > This has been partly assumed above. I suggest a generator is first found > that generates only the lowest prime (but note that in some cases the > interval of equivalence may not be related to the lowest prime, e.g. a > tritave-based scale that includes some ratios of 2) and it should be the > smallest such generator, which means that it will always be a > whole-number > division of (the approximation of) that lowest prime. Then the next > generator is only what is needed to generate the next prime (in > conjunction > with the first generator if necessary). i.e. we are aiming for a > triangular > map matrix (an accepted canonical form). But in the end the last > generator > has to pick up all the remaining primes.That assumes the "primes" are in order, and the first identified as an equivalence interval. What should we do for things like tritave-equivalent scales? It does make sense to always list the primes in the same order.> At each step the generator given should be the smallest that will do the > job. So at least in the case of a linear temperament, the second > generator > will always be less than half the first generator (or period). ... > Note that a different choice of basis vector (generators) will lead to a > different mapping matrix and make comparison tedious.I'd prefer to choose a generator less than the period such that the first element of the map is positive. That's much more robust. It means the scale can be identified by the mapping-by-generators and the number of periods to an octave, with the generator and octave sizes as almost free parameters.> There are various ways to express this in matrix form depending on > whether > our vectors are rows or columns. Since it is much easier to deal with > row > vectors in text, I propose we use: > > [If you're viewing this from Yahoo's dumb web interface, you'll need to > click either Reply, or Message index then Expand Messages, to see it > formatted properly.] > > [lg(2) lg(3) lg(5) lg(7)] ~= > > [gen0 gen1 gen2] [p2 p3 p5 p7] > [g2 g3 g5 g7] > [h2 h3 h5 h7]I'd prefer that the other way around, so that the units are always to the right of the values. i.e. [-1 1][lg(2) lg(3)] = 0.585 octaves = 702 cents instead of [lg(2) lg(3)][-1 1] = octaves 0.585 = cents 702 For that to work with matrix multiplication, [lg(2) lg(3)] would have to be transposed, and there's a knock-on effect with the mapping. I much prefer listing octaves first, because then they can be indexed as zero and the octave-equivalent case uses indices from 1.> The most obvious flattening of the above matrix is neither of these. I > suggest we use: > > [[p2 p3 p5 p7] [0 g3 g5 g7]] > > (Why not omit the ","s?)Actually, that's how I used to do it, but I won't re-write the program so it works naturally again. Those ","s are because I'm using Python's default stringification of sequences. Some versions used Numeric arrays, so no commas. I'm going to have to write a proper HTML formatting routine anyway, so I can tackle this as part of that. Should be a good project for the weekend if we're agreed. Graham
Message: 4307 - Contents - Hide Contents Date: Fri, 15 Mar 2002 22:58:28 Subject: Re: Wedge product definitions from mathworld From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> This probably made things worse. :(No. Thanks Gene. It probably got me a tiny bit closer. But I still don't see a definition that tells me how to calculate the wedge product of two vectors with given bases, using operations I'm already familiar with. I don't think there's much point trying to tell me what a wedge invariant (wedgie) is until I understand wedge product. Graham says his python code is harder to understand than the Mathworld page, so I won't even bother looking at it, since I'm not familiar with the python libraries anyway. If you make me an Excel spreadsheet that implements wedge product for say 1D, 2D, 3D and 4D vectors, then I'd probably figure it out. :-)
Message: 4308 - Contents - Hide Contents Date: Fri, 15 Mar 2002 15:00:03 Subject: Re: Standardising temperament mappings From: dkeenanuqnetau --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <3.0.6.32.20020314182114.00b2f6f0@u...> > David C Keenan wrote: > So you mean (periodSize, generatorSize) for this datum? Yes. > That assumes the "primes" are in order, and the first identified as an > equivalence interval. What should we do for things like > tritave-equivalent scales? It does make sense to always list the primes > in the same order.Ah yes. Thanks for reminding me. I forgot to make it explicit that I always want the primes to be ordered from smallest to largest (from left to right or top to bottom). For tritave-equivalent scales having no ratios of even numbers, there's no problem. If it has 2s in it I'd say the canonical matrix should still put the 2s first. However there's nothing to stop you _also_ giving a non-canonical one where you order it 3, 2, 5, 7 to put the interval of equivalence first, so long as you make this explicit. It's probably a good idea to make the "target"(?) vector (list of logs of primes) explicit anyway, particularly when we start doing things like 2, 5, 7.>> At each step the generator given should be the smallest that will do the >> job. So at least in the case of a linear temperament, the second >> generator >> will always be less than half the first generator (or period). ...> I'd prefer to choose a generator less than the period such that the first > element of the map is positive. That's much more robust. It means the > scale can be identified by the mapping-by-generators and the number of > periods to an octave, with the generator and octave sizes as almost free > parameters.I don't understand what you mean by "more robust". Whatever conventions we settle on to give a unique form will allow an octave-based temperament to be identified by the mapping-by-generators and the number of periods to an octave. Why would my proposal of generator less than half period, fail to do this. Your program currently agrees with my proposal. It always gives a generator less than half the period. Are you proposing to change that? I assume you understood I meant "smallest with GCD(row) = 1", so no degeneracy.>> There are various ways to express this in matrix form depending on >> whether >> our vectors are rows or columns. Since it is much easier to deal with >> row >> vectors in text, I propose we use: >> >> [If you're viewing this from Yahoo's dumb web interface, you'll need to >> click either Reply, or Message index then Expand Messages, to see it >> formatted properly.] >> >> [lg(2) lg(3) lg(5) lg(7)] ~= >> >> [gen0 gen1 gen2] [p2 p3 p5 p7] >> [g2 g3 g5 g7] >> [h2 h3 h5 h7] >> I'd prefer that the other way around, so that the units are always to the > right of the values. i.e. > > [-1 1][lg(2) lg(3)] = 0.585 octaves = 702 cents > > instead of > > [lg(2) lg(3)][-1 1] = octaves 0.585 = cents 702 > > For that to work with matrix multiplication, [lg(2) lg(3)] would have to > be transposed, and there's a knock-on effect with the mapping.Well yes, and the effect is that _everything_ has to be transposed. All the vectors will be colums and the matrix will be higher than it is wide. I'm quite used to using row vectors carrying units and coming _before_ the transformation matrix in computer graphics. The vectors may be in pixels or millimeters but it doesn't seem at all strange to put the thing to be transformed first, followed by the desired transformation. As in [x' y' z' 1] = [x y z 1] [ cos(a) sin(a) 0 0] [-sin(a) cos(a) 0 0] [ 0 0 1 0] [ 0 0 0 1] (Rotation of a point about the z axis thru an angle a). This is straight out of a textbook. It seems Gene agrees with me on this. The only problem I have with his format is that he's been puting the period (or fractional-octave generator) last. (and the minimum generator size thing)> I much prefer listing octaves first, because then they can be indexed as > zero and the octave-equivalent case uses indices from 1.At least you and I agree on that.>> The most obvious flattening of the above matrix is neither of these. I >> suggest we use: >> >> [[p2 p3 p5 p7] [0 g3 g5 g7]] >> >> (Why not omit the ","s?) >> Actually, that's how I used to do it, but I won't re-write the program so > it works naturally again.How can this flattening be "natural" to you if you would have the mapping matrix be higher than it is wide? But I guess I shouldn't complain if you agree with me on the flattening. :-)> Those ","s are because I'm using Python's > default stringification of sequences. Fair enough.
Message: 4309 - Contents - Hide Contents Date: Fri, 15 Mar 2002 15:27 +0 Subject: Re: Standardising temperament mappings From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a6t29j+cdq0@xxxxxxx.xxx> dkeenanuqnetau wrote:> I don't understand what you mean by "more robust". Whatever > conventions we settle on to give a unique form will allow an > octave-based temperament to be identified by the mapping-by-generators > and the number of periods to an octave. Why would my proposal of > generator less than half period, fail to do this.The generator might get larger, in which case it would be incorrectly described. If you set the first non-zero element of the generator mapping to be positive, the mapping is uniquely described regardless of the size of the generator. Actually, it looks like I use this convention when comparing temperaments for equivalence, but not for display.> Your program currently agrees with my proposal. It always gives a > generator less than half the period. Are you proposing to change that?The one for generating linear temperaments from equal temperaments does. But the one for generating linear temperaments from unison vectors doesn't. Some generators are a lot bigger than the period because they become larger on optimization. So I'll have to change that program to adjust the mapping after optimization.> I assume you understood I meant "smallest with GCD(row) = 1", so no > degeneracy.Um what if there's contorsion?>> Actually, that's how I used to do it, but I won't re-write the > program so>> it works naturally again. >> How can this flattening be "natural" to you if you would have the > mapping matrix be higher than it is wide? But I guess I shouldn't > complain if you agree with me on the flattening. :-)It's natural for the program because it holds the mapping as a list of pairs. It used to be a pair of lists, which makes it easier to do some multiplications. This way, it can be generated by looping through the primes and adding pairs. Also, it's consistent with how I've always done the mapping by step sizes. So I won't be changing the library's internal format, but can change the display the CGI uses. Graham
Message: 4310 - Contents - Hide Contents Date: Sat, 16 Mar 2002 00:37:31 Subject: Re: Wedge product definitions from mathworld From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> If you make me an Excel spreadsheet that implements wedge product for > say 1D, 2D, 3D and 4D vectors, then I'd probably figure it out. :-)First I'd need to figure out Excel.
Message: 4311 - Contents - Hide Contents Date: Sat, 16 Mar 2002 00:44:31 Subject: Re: Wedge product definitions from mathworld From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> If you make me an Excel spreadsheet that implements wedge product for >> say 1D, 2D, 3D and 4D vectors, then I'd probably figure it out. :- ) >> First I'd need to figure out Excel.why not just restate the explicit formulae for 3d and 4d wedge products?
Message: 4312 - Contents - Hide Contents Date: Sat, 16 Mar 2002 03:46:56 Subject: Re: Wedge product definitions from mathworld From: genewardsmith --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:> why not just restate the explicit formulae for 3d and 4d wedge > products? 3D[u1,u2,u3]^[v1,v2,v3] = [u2*v3-v2*u3,u3*v1-v3*u1,u1*v2-v1*u2] This is the right form for both vals and intervals, and Poincare duality says we simply identify the wedge of two vals with an interval, and the wedge of two intervals with a val. Hence in the 5-limit, we may equate the wedge of two vals h and g with an interval, and write h^g = 2^(h(3)*g(5)-g(3)*h(5)) 3^(h(5)*g(2)-g(5)*h(2)) 5^(h(2)*g(3)-g(2)*h(3)) where h and g are vals. 4D [u1,u2,u3,u4]^[v1,v2,v3,v4] = [u3*v4-v3*u4,u4*v2-v4*u2,u2*v3-v2*u3,u1*v2-v1*u2,u1*v3-v1*u3,u1*v4-v1*u4] Poincare duality then permutes this if we wedge two vals, so that if h and g are vals, we may identify h^g with its Poincare dual, and write it as h^g = [h(2)*g(3)-g(2)*h(3),h(2)*g(5)-g(2)*h(5),h(2)*g(7)-g(2)*h(7), h(5)*g(7)-g(5)*h(7),h(7)*g(3)-g(7)*h(3),h(3)*g(5)-g(3)*h(5)] Such a wedge product, whether from two vals or two intervals, represents a temperament if the intervals are small or the vals are et vals. Calling it T, we may wedge T with a val (viewing it as a wedge product of vals) and get an interval by Poincare duality, or wedge it with an interval (viewing it as a wedge product of intervals) and by Poincare duality, end up with a val. If T = [t1,t2,t3,t4,t5,t6], and u = [u1,u2,u3,u4] then T^u = [t6*u4+t5*u3+t4*u2,t3*u3-t4*u1-t2*u4,t1*u4-t3*u2-t5*u1,t2*u2-t6*u1-t1*u3] Now if u is a val, u = [u1,u2,u3,u4] then T^u is given by the above, and may be changed to an interval by using these values as exponents of the primes by Poincare duality. If u is an interval, u = 2^u1 3^u2 5^u3 7^u4, then permute T by changing it to T' = [t4,t5,t6,t1,t2,t3], and then take T'^u by the above formula; the result will be (Poincare duality) a val.
Message: 4313 - Contents - Hide Contents Date: Sat, 16 Mar 2002 04:25:02 Subject: Re: Wedge product definitions from mathworld From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:>> why not just restate the explicit formulae for 3d and 4d wedge >> products?I might add to this that the inner product in the 4D case of two 2-intervals or 2-vals (defined, again, by Poincare duality) is interesting: [u1,u2,u3,u4,u5,u6].[v1,v2,v3,v4,v5,v6] = u1*v4+u2*v5+u3*v6+u4*v1+u5*v2+u6*v3 The length of a wedgie turns out to be zero according to this formula, which distinguishes wedgies from among all 6-D vectors.
Message: 4314 - Contents - Hide Contents Date: Sat, 16 Mar 2002 20:06 +0 Subject: Gould on Generalized Diatonics From: graham@xxxxxxxxxx.xx.xx I decided to get a copy of Mark Gould's paper "Balzano and Zweifel: Another Look at Generalized Diatonic Scales" so here's a summary. It's starts off explaining the Balzano theory. This is roughly that you build up an MOS by alternating a pair of intervals that add up to the generator. For the normal diatonic, these are the major and minor third. It doesn't actually say "MOS" or "well formed" but that seems to be what we're talking about. There's also the peculiar definition that "diatonic" means the larger size of scale step is more common and "pentatonic" is the other way around. He decides not to use the scare quotes from then on, but I will because I'm old fashioned enough to think that a pentatonic should have 5 notes to the octave. The square lattice Mark mentioned on the main list, which looks like a triangular lattice rotated a bit and reflected, is to show this pair of alternating intervals, rather than 5-limit harmony in general. All "diatonics" mentioned in the paper are shown on similar grids. Mark suggests a different tonic for Zweifel's 11 from 20 scale. That's so the not that changes as you move up the chain of generators is the leading note to the tonic. Next he gives his own criteria for "diatonics", which are like Balzano's but without worrying about there being n(n+1) notes in the parent scale. Then there are some examples of such "diatonics". Now the bit on ratios. He makes the startling statement "For all diatonic scales, the two intervals in the grid approximate to a given pair of just intonation ratios" which is about as surprising as finding water in a river. Beyond identifying traditional diatonic scales with 5:4 and 6:5, he doesn't say what ratios were being approximated by the "diatonics" he gave before. Still, I can reverse engineer these. There's an 11 from 27 scale which is based on a 7:6 and 10:9 adding up to a 9:7. The normal diatonic's taken from 26, 19 and 31-equal. 11 from 31, with 6:5 and 11:9 adding up to 16:11, or 55:66:80 19 from 33 and 23 from 40 I don't know because they're not even 5-limit consistent. 11 from 41 is 9:8 plus 8:7 making 9:7, or a 7:8:9 chord. Mark says "I believe that the diatonic scales derived from the different Cn [n-note equal temperament] scales conform to background formations existing independently of the number of tones in the base Cn scale." So meantone is still meantone whether it's taken from 12, 19, 26 or 31 notes. Apparently Balzano and Zweifel failed to mention this. He goes as far as to say that 7 from 17-equal is still approximating 6:5 and 5:4. Then he notes that the three major and four minor thirds don't add up to a 2:1 if they're tuned justly. From which he concludes "a diatonic scale can be seen not as a reordered segment of a 'cycle of fifths' or its microtonal equivalents, but a compact structure in a grid space of two alternating intervals." No mention of imperfect octaves (stretching by half a syntonic comma will give just 6:5 and 5:4 thirds from a 3:2 generator) or periodicity blocks. Then he gives a 5-limit grid using a Tenney-type notation with a 5-limit diatonic for the nominals. After that he talks about how the concept could be extended to more dimensions, but doesn't give any examples. On interesting thing I noticed is that 72 is allowed by Balzano's n(n+1) criterion. So I thought I'd try and fit this technique to the Miracle scales. Stud loco is fulfils all the criteria for a "diatonic" I can work out. You have to generate it from 3 and 4 notes from 72-equal adding up to a secor. This is a strange primary chord, but fulfils the letter of the specification. Similarly, Blackjack could be generated form a quomma and secor-quomma, but it's a "pentatonic" and fails Balzano's "coherence" test because the secor-quomma is more than twice the size of the quomma, except as a subset of 31-equal. Canasta is a "pentatonic" based on the chord 8:11:15 where the generator is an octave-secor instead of a secor. Decimal has an even number of notes, and so can't be generated by alternating intervals. It could be thought of as a pair of septimal slendros, each generated from 12:16:21 or 16:21:28. You could also say it's generated from 14:15:16 where the two "alternating" intervals are really the same. This kludge would work for all even-numbered MOS. I don't know if all odd-numbered MOS can be written this way, but I suspect so. All you need to do is fill in an approximation to the generator, and the absolute number of generators to each internal interval have to add up to the number of notes you want in the scale. Hopefully this will give the number you want for either kind of generator within the octave. Magic 19 could be generated from 8:9:10. If you take generators larger than the octave, you can always avoid Balzano's incoherence. So probably this is cheating, and we'll have to relax that one to get Blackjack. If the rest of Balzano's criteria were relaxed, who knows what we might find! Graham
Message: 4316 - Contents - Hide Contents Date: Mon, 18 Mar 2002 03:52:48 Subject: Grassmann Algebra, by John Browne From: genewardsmith I've been researching the bibliography for my paper, which I am hoping will form the mathematical basis for what other people do. I'm a proponent of the idea that one-line books and papers should be cited when possible, due to their availability, and I found the above, nearly complete book on multilinear algebra. It makes less demands on the mathematical sophistication of the reader than the standard references of Greub's "Multilinear Algebra", or Bourbaki. Best of all, it is *free* and *on line*. If someone wants to learn more about wedge products (also often, as in this book, called exterior products) then chapter 2 of this book seems like a good place to start--not to mention the rest of it. You could even learn about the algebra of screws. :) Grassmann Algebra Book * [with cont.] (Wayb.)
Message: 4317 - Contents - Hide Contents Date: Tue, 19 Mar 2002 01:06:28 Subject: Wedge product understood at last (was: Grassmann Algebra, by John Browne From: dkeenanuqnetau --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> If someone wants to learnmore about wedge products (also often, as in this book, called exterior products) then chapter 2 of this book seems like a good place to start--not to mention the rest of it. You could even learn about the algebra of screws. :) If anyone is still struggling, as I was until a few minutes ago, to understand what the heck the exterior product is (I prefer this name to "wedge product"), and what the point of it is, then just read the first 6 pages of the Introduction. The result was a great "aha!" for me. It is really quite a beautiful conception. I still say we don't "need" it, but I can certainly see that some things will be described much more elegantly _with_ it. Thanks guys for all your attempts. What I was missing was (a) Its geometrical interpretation "thus enabling us to bring to bear the power of geometric visualization and intuition into our algebraic manipulations." (b) The terms "bivector", "trivector" etc. i.e. The fact that the product of two vectors isn't another vector, and there isn't anything else you have to do to the basis of the result e.g. (e1/\e2, e2/\e3, e3/\e1). I emailed the author, John Browne, (who lives in Melbourne, Australia) and thanked him and told him briefly what we were using it for.
Message: 4318 - Contents - Hide Contents Date: Tue, 19 Mar 2002 16:12 +0 Subject: Re: Wedge product understood at last (was: Grassmann Algebra, by Joh From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <a762uk+mcqd@xxxxxxx.xxx> dkeenanuqnetau wrote:> If anyone is still struggling, as I was until a few minutes ago, to > understand what the heck the exterior product is (I prefer this name > to "wedge product"), and what the point of it is, then just read the > first 6 pages of the Introduction. The result was a great "aha!" for > me.Yes, the introduction already covers a lot of ground. I'm working through the whole thing now.> It is really quite a beautiful conception. I still say we don't "need" > it, but I can certainly see that some things will be described much > more elegantly _with_ it.Once you know the formalism, it's easier to calculate the temperaments using exterior algebra than matrices. More people will already be familiar with matrices, but I suppose most musicians won't be with either. One problem is that you can think of the matrix approach as solving an equation. The alternative model is that the wedgie allows you to tell which intervals are equivalent to a unison, and can be generated from a sample set of such intervals. I'm not sure why it should also give mappings to convert prime coordinates to tempered ones, other than by analogy with matrices.> Thanks guys for all your attempts. What I was missing was > (a) Its geometrical interpretation > "thus enabling us to bring to bear the power of geometric > visualization and intuition into our algebraic manipulations."Yes, it has big geometric and physical applications I didn't know about. The thing I was calling the complement is also called the complement in this book, so that saves me making any changes to my program. Although I could change "Wedgable" to "ExteriorElement". From the regressive product being like an intersection, we should have (h31^h41)v(h19^h41) = h41 So if you set miracle = (h31^h41).complement() magic = (h19^h41).complement() then miracle.complement() v magic.complement() = +/- h41 or equivalently (miracle ^ magic).complement() = +/- h41 Hence we have an answer to the question asked some time ago about how you find the equal temperament common to a pair of linear temperaments. Unfortunately, it doesn't work beyond the 5-limit. Still, I can get at the 7-limit result like this:>>> (~(~magic^{(3,):1}) ^ ~(~miracle^{(3,):1})).invariant()(41, 65, 95)>>> (~(~magic^{(2,):1}) ^ ~(~miracle^{(2,):1}) ^ {(2,):1}).invariant()(41, 65, 0, 115) where I've experimentally redefined the ~ operator to find the complement. The first calculation is (miracle^magic).invariant() for the 5-limit subset. The invariant() method already gets the complement and sign right. The second calculation is the same idea but removing the prime 5 from the reckoning. The two results between them give the correct mapping [41 65 95 115]. Starting from the 11-limit, you'd have to knock out two primes at a time. Graham
Message: 4319 - Contents - Hide Contents Date: Tue, 19 Mar 2002 06:59:40 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> I followed that conversation and, although I have strong convictions > about what was discussed, I just didn't have the time to get involved > in it. My thoughts on this are: > > 1) Johnny is already very familiar with cents, so that is what works > for him. For the rest of us it would take a bit of training to be > able to do the same, and then might we need a calculator to determine > the intervals? When you are writing chords, where do all the cents > numbers go, and how can you read something like that with any > fluency? (But that is for instruments of fixed pitch, which do not > require cents, which brings us to the next point.) > > 2) Tablatures were mentioned in connection with instruments of fixed > pitch, where cents would be inappropriate. I hate tablatures with a > vengeance! Each instrument might have a different notation, and this > makes analysis of a score very difficult. We need a notation that > enables us to understand the pitches and intervals, regardless of > what sort of instrument is used.These were proposed as notations for performers, not composers or analysers. As such I see no great problem with the above, or scordatura.> 3) Gene mentioned that we are most comfortable with whatever is the > most familiar, but a multi-EDO/JI notation such as we are trying to > achieve is going to have something new to learn, no matter what. > It's best to make it as simple and logical as possible, and in my > opinion this is best accomplished by symbols that correspond simply > and directly to tones in each system, maintaining commonality across > those systems as much as possible.I see the commonality thing as the main justification for the kind of notation you and I favour. Otherwise, in any given linear temperament (and an ET is often treated as such) there is a native notation based on the naturals being some MOS of around 5 to 12 notes, which makes analysis easier for that temperament, but only that temperament.> Whether the maximum symbols per note should be one or two is > something that still needs to be resolved, but I think that we should > develop both approaches and see what results. Once that is done, > then we can evaluate each, pro and con. As long as microtonality is > such a niche market, I think that this is one extravagance that we > can afford. Sure.>> No I didn't. So what are the smallest ETs in which this comma >> consistently fails to vanish? >> The comma is 4095:4096 (~0.423 cents). (Has anyone previously found > it and given it a name?)I don't know of a name, but I expect it is listed in John Chalmer's lists of superparticulars at a given prime-limit. Anyone know a URL? I was astonished to learn that such lists seem to be finite.> Multiplying 81/80 by 64/63 gives 36/35 > (~48.770 cents), which comes very close to 1053/1024 (~48.348 cents), > the ratio which is to define the 13 factor. This very small comma > vanishes in ET's 12, 17, 22, 24, 31, 34, 36, 39, 41, 43, 46, 53, 94, > 96, 130, 140, 152, 171, 181, 183, 193, 207, 217, 224, 270, 311, 364, > 388, 400, 494, 525, 581, 612, and 742. > It does not vanish in 19, 27, > 50, 58, 72, 149, 159, or 198, and this seems to be due to > inconsistencies in those ET's.No. Every one of these (except I didn't check 198) is {1,3,5,7,9,13}-consistent.> For those systems under 100 in which > it does not vanish, I don't think that ratios of 13 will be used to > define their notation, so this should not be a problem.Well Gene and I are already using the 13-comma (1024:1053) to notate 27-tET, and it looks like it would be pretty useful in 50-tET too. There are many others under 100-tET where 4095:4096 doesn't vanish. 37-tET is another such, where I was planning to use the 13-comma for notation. 37-tET is {1,3,5,7,13} consistent.>> If you had a ruler with only inch marks, what could you find quicker >> (a) two and a half, less an eighth, or >> (b) two and three eights?So what's your answer? Mine is (a).> Or given these choices, > (a) one and a half, less an eighth, or > (b) one and three eighths, or > (c) three, less one-fourth?My answer is (c), but I don't get it. It isn't the same measurement as the other two. In the analogy, I'm considering the inches to be the naturals (since they are marked), the halves to be the sharps and flats and the eighths (or fourths) to be some comma.> The particular intervals that were mentioned (farther above) in > connection with this issue are ratios of 11 and 13 that I like to > think of as semi- and sesqui-sharps/flats. Making these the large > units of notation (according to the number of stems in the symbol) > rather than full sharps and flats makes the reading of these ratios > much simpler and solves the problem of having to decide whether to > notate a half-sharp/flat or a sharp/flat less a half. In effect, by > halving the units I have turned the halves into wholes, which makes > available choice (c) above, which, in addition to being at least as > simple as choice (a), does not combine an up with a down.Now I get the analogy.> (As to how > I make the distinction between ratios such as 11/9, 39/32, and 16/13, > you will have to wait a while to see; at present I have two different > methods.) > > Anyway, when in doubt I think it will come down to trying it both > ways in order to see what looks better. I mentioned that my latest > approach has two variations, and if I can't determine a clear choice > of one over the other, then I will be presenting both so that we can > decide which is better.Looking forward to it.
Message: 4320 - Contents - Hide Contents Date: Tue, 19 Mar 2002 20:21:51 Subject: Re: Wedge product understood at last (was: Grassmann Algebra, by Joh From: genewardsmith --- In tuning-math@y..., graham@m... wrote:> One problem is that you can think of the matrix approach as solving an > equation.Why is that a problem? You can generalize Cramer's rule with wedges, for that matter.> The thing I was calling the complement is also called the complement in > this book, so that saves me making any changes to my program.I didn't know you knew about compliments. This is all connected to the jabber I giving about Poincare duality, where vals map to compliments of 1-intervals and wedges of vals map to vee products of compliments. By the way, it seems people are adopting the terminology of this book. The wedge product is probably most often called "exterior product" (four syllables rather than one, and I didn't learn it that way, so I like wedge better) but I think "regressive product" goes all the way back to Grassmann--at least, its usually called the "intersection product", but I like "vee product" myself. There's also something called the "Grassmann product" in case this wasn't confusing enough as it is.
Message: 4321 - Contents - Hide Contents Date: Tue, 19 Mar 2002 07:10:22 Subject: Re: Weighting complexity (was: 32 best 5-limit linear temperaments) From: dkeenanuqnetau So Gene, How's the 5-limit list coming along. Also do you have any objections or suggestions for Graham's or my 'standard generators and mapping' proposals. Surely we want canonical forms that are musically meaningful and convenient where possible.
Message: 4322 - Contents - Hide Contents Date: Tue, 19 Mar 2002 16:32:52 Subject: Re: A common notation for JI and ETs From: David C Keenan --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >>> I don't know of a name, but I expect it is listed in John Chalmer's >> lists of superparticulars at a given prime-limit. Anyone know a >> URL? I was astonished to learn that such lists seem to be finite. >> I proved that a while back using Baker's theorem on this list.Good work. Did you (or can you) prove that the smallest on each of John's lists is the smallest there is? Here's John Chalmers list of superparticulars Yahoo groups: /tuning-math/message/1687 * [with cont.] -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 4323 - Contents - Hide Contents Date: Tue, 19 Mar 2002 08:01:47 Subject: Re: A common notation for JI and ETs From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> I don't know of a name, but I expect it is listed in John Chalmer's > lists of superparticulars at a given prime-limit. Anyone know a > URL? I was astonished to learn that such lists seem to be finite.I proved that a while back using Baker's theorem on this list.
Message: 4324 - Contents - Hide Contents Date: Tue, 19 Mar 2002 08:04:13 Subject: Re: Weighting complexity (was: 32 best 5-limit linear temperaments) From: genewardsmith --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> So Gene, > > How's the 5-limit list coming along.It isn't; I've been writing my paper instead.> Also do you have any objections or suggestions for Graham's or my > 'standard generators and mapping' proposals. Surely we want canonical > forms that are musically meaningful and convenient where possible.I modifed my proposal and got it to work, so I suppose I should explain that. The next few days are teaching days, but then we have spring break.
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