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Message: 9025 - Contents - Hide Contents Date: Thu, 08 Jan 2004 05:51:41 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> I understand that functions of the type f(x) -> x^2 + c are shaped >>> like parabolas, but x isn't a generator size here, it's the sum of >>> errors resulting from a generator size. If I took out the ^2 it >>> might be shaped like anything; >>>> Huh? x + c is shaped like anything? >> Traditionally a line, but in this case x is actually this other > function, the summed errors from these arbitrary external just > ratio things.No, silly goose :), the squaring is done *before* the summing! If it's done *after* the summing, it has no effect (since the location of the lowest value of a positive function is also the lowest possible value of the function squared, and if you don't start with a positive function, you're doing something wrong).> As I move the generator size from 0-600 cents and > pump it through say the meantone map, the errors could go up and > down several times for all I know.The error of each interval will be a straight line. The errors squared will be parabolas. The sum of a set of parabolas is a parabola, since the sum of any number of functions of order 2 is a function of order 2 -- since you're squaring some linear functions, then adding, you'll have quadratic terms, linear terms, and constant terms to add, and that's all.

Message: 9026 - Contents - Hide Contents Date: Thu, 08 Jan 2004 13:31:47 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Carl Lumma>>> >here should be no angles defined here, just as there are none in >>> the Tenney lattice. >>>> Why "should"? They have to be, or it won't work. >>Why won't it? My Tenney, non-octave-equivalent way doesn't need >angles defined. You can choose any set of angles you want, and still >embed the result in Euclidean space, but that doesn't even matter -- >what matters are the taxicab distances ONLY.While I'm certainly hoping taxicab proves sufficient, isn't it possible that you'll need angles when more than one comma is involved? -Carl

Message: 9027 - Contents - Hide Contents Date: Thu, 08 Jan 2004 00:49:24 Subject: GOT From: Gene Ward Smith Inspired by TOP, I've run a few geometric complexity weighted optimizations; if you adhere to the philosophy that lower-complexity ratios should be weighted more, the results seem reasonable.

Message: 9028 - Contents - Hide Contents Date: Thu, 08 Jan 2004 21:44:51 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>> There should be no angles defined here, just as there are none in >>>> the Tenney lattice. >>>>>> Why "should"? They have to be, or it won't work. >>>> Why won't it? My Tenney, non-octave-equivalent way doesn't need >> angles defined. You can choose any set of angles you want, and still >> embed the result in Euclidean space, but that doesn't even matter - - >> what matters are the taxicab distances ONLY. >> While I'm certainly hoping taxicab proves sufficient, isn't it > possible that you'll need angles when more than one comma is >involved?Yes, we'll need the *correct* angle-like concept. Then again, I attempted to get around the whole angle issue with my heron's formula application, which almost worked!

Message: 9029 - Contents - Hide Contents Date: Thu, 08 Jan 2004 01:10:56 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Paul Erlich wrote: >>> Interesting! And is that truly the only one that matters? >> The size/complexity tells you the best value for the suitably weighted > minimax in any temperament in which this comma vanishes. If a comma > exists such that its size/complexity is equal to the optimim minimax > error in a given linear temperament, and the comma is in the linear > temperament's kernel, then the two temperaments must be identical.I can't follow that right now.> I'm not sure if such a comma will always exist,Wow -- now that's an interesting question to consider.> but provided it does > it's the only one you need for TOPS.My intuition says it doesn't exist.> It doesn't even have to be made up > of integers, so long as it's a linear combination of commas that >are. ??? > It's a generalization of the proof of the TOP meantone being stretched > quarter-comma. All the factors that get tempered the same way as 2 will > be stretched by the same amount. But if the octaves don't get tempered > at all, some factors will be tempered in one method but not the other.There's something that seems strange about your octave-equivalent method. The comma is supposed to be distributed uniformly (per unit length, taxicabwise) among its constituent "rungs" in the lattice. But it seems that 81:80 = 81:5 would involve 1, not 0, rungs of 5 in the octave-equivalent lattice. But the octave-equivalent lattice can't be embedded in euclidean space, so this completely falls apart??

Message: 9030 - Contents - Hide Contents Date: Thu, 08 Jan 2004 06:55:44 Subject: for gene From: Paul Erlich Yahoo groups: /tuning_files/files/Erlich/gene2... * [with cont.]

Message: 9031 - Contents - Hide Contents Date: Thu, 08 Jan 2004 22:19:39 Subject: A few Tenney-optimal examples From: Gene Ward Smith Here are few examples of what I'm claiming to be the Tenney-opitmized tunings. It seems to amout to a correct definition for the so-called "canonical" tuning. Meantone 5-limit: [1200., 1896.578429, 2786.313713] 7-limit: [1200., 1896.882590, 2787.530363, 3368.825906] Miracle 5-limit: [1201.262692, 1901.955001, 2786.313713] 7-limit: [1201.262692, 1901.955001, 2786.313713, 3370.223971] Orwell 5-limit: [1200.479095, 1901.955001, 2786.313713] 7-limit: [1200., 1901.955001, 2784.876428, 3373.662858] Kleismic 5-limit: [1200., 1901.955001, 2784.962501] 7-limit: [1201.351212, 1901.955001, 2786.313713, 3353.679924] It would be interesting to have Paul's analysis.

Message: 9032 - Contents - Hide Contents Date: Thu, 08 Jan 2004 01:12:18 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> Hence you can impose a weighted >> minimax over all intervals within a given prime limit. >> Aha! So why then isn't the prime limit also superfluous?It is, unless you want to control the dimension of your temperament.

Message: 9033 - Contents - Hide Contents Date: Thu, 08 Jan 2004 07:07:21 Subject: TOP and normed vector spaces From: Gene Ward Smith A (real) normed vector space is a real vector space with a norm (duh :)) If u and v are two vectors, and ||u|| denotes the norm, then 1. ||u|| >= 0 2. ||u|| = 0 iff u = 0 (the zero vector, not the number) 3. If c is a scalar, ||c u|| = |c| ||u|| 4. ||u + v|| <= ||u|| + ||v|| A normed vector space is a metric space, with metric d(u, v) = ||u - v|| An example of a normed vector space is the p-limit Erlich space, where the norm is || |u2 u3 u5 ... up> || = |u2|+log2(3)|u3|+ ... + log2(p)|up| The p-limit intervals live inside this space and form a lattice. A linear functional on a real vector space is a linear mapping from the space to the real numbers. It is like a val, but its coordinates can be any real number. An example is the size functional, SIZE = <1 log2(3) ... log2(p)| As Joe is fond of pointing out, this functional maps every point of the Erlich space, not just lattice points, to a real number. The space of linear functionals is the dual space. It has a norm induced on it defined by ||f|| = sup |f(u)|/||u||, u not zero We may change basis in the Erlich space by resizing the elements, so that the norm is now || |v2 v3 ... vp> || = |v2| + |v3| + ... + |vp| This is called the L1 norm. Each of the basis elements now represents something of the same size as 2, but that should not worry us. It is a standard fact that the dual space to L1 is the L infinity norm, and vice-versa. This means (with this resizing of our basis vectors) that the correct norm for linear functionals is || <f2 f2 ... fp| || = Max(|f2|, |f3|, ..., |fp|) A tuning map T is a type of linear functional. If c is a comma which T tempers out, then T(c) = 0. If we have a set of commas C which are tempered out, then this defines a subspace Null(C) of the space of linear functionals, such that for any T in Null(C), T(c)=0 for each comma tempered out. For points in this this subspace, there will be a minimum distance to SIZE, and using the same proceedure we use to get a unique minimax we can find a unique minimal distance point TOP at this minimum distance from SIZE (which is <1 1 1 ... 1| in the basis we are now using.) One neat thing about this is that it generalizes immediately to other normed vector spaces containing complete p-limit (meaning, 2 is included as a prime number) lattices. In particular, there is a geometric complexity version of TOP.

Message: 9034 - Contents - Hide Contents Date: Thu, 08 Jan 2004 22:20:39 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> 81:5 involves three rungs of 3:1 and one rung of 3:5. > > Oh yeah. >>> For the 5-odd >> limit, these rungs are of equal length,Wait a minute -- we're obviously talking about different things here! I read this too quickly. 3:1 is a ratio of 3, and 3:5 is a ratio of 5, so the latter should be longer!!> and so that error has to be>> shared between them. That leaves 3:1 and 3:5 having an equal > amount of>> temperament, and so 1:5 must be untempered. > > Ha!I retract that "Ha!" . . .

Message: 9035 - Contents - Hide Contents Date: Thu, 08 Jan 2004 01:19:08 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>>>> I never understood this process, >>>>>>>> Solving a system of linear equations? >>> >>> Uh-huh. >>>> Well, the easiest way to understand is to solve one equation for one >> variable, plug that solution into the other variables so that you've >> eliminated one variable entirely, and repeat until you're done. >> I remember this technique from Algebra, but I didn't think it would > be applicable here, since I assumed the variables wouldn't be > independent in that way.They're not, you actually have an extra equation.> What do these equations look like? For meantone,prime2 = period; prime3 = period + generator; prime5 = 4*generator. You can throw out any equation -- say the first. so generator = .25*prime5, prime3 = period + .25*prime5, period = prime3 - .25*prime5.>>> Why are you assuming octave repetition, what does this assumption >>> amount to? >>>> That you'll have the same pitches in each (possibly tempered) octave. >>>>> If 2 is in the map, one of the generators had better well generate >>> it. If it isn't in the map, assuming octave repetition seems like >>> a bad idea to me. >>>> Any recent cases where you'd prefer not to see 2 in the map? >> I'd always prefer to see it, but why assume?Agreed. But it's a "default", a "convention", that many would assume.

Message: 9036 - Contents - Hide Contents Date: Thu, 08 Jan 2004 07:08:29 Subject: another for gene From: Paul Erlich Yahoo groups: /tuning_files/files/Erlich/gene3... * [with cont.]

Message: 9037 - Contents - Hide Contents Date: Thu, 08 Jan 2004 22:28:13 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Graham Breed Paul Erlich wrote:> What's the worst comma for 12-equal in the 5-limit?[0 28 -19> or 22876792454961:19073486328125 TOPping it gives a narrow octave of 0.99806 2:1 octaves.> Me too -- but the lengths aren't compatible in a Euclidean space. > Remember the whole big "wormholes" discussion from years ago?Yes, I remember all about the wormholes, and they don't have anything to do with this. You only need them for odd limits.> We can either embed a lattice, with a taxicab distance, into > Euclidean space, or we can't. But just because we can, doesn't mean > we should use Euclidean distance! NONONONONONO!You could try taxicab distance, I'm not sure it'd work right. But you can also use Euclidian distance, and it looks like a more straightforward way to me.> Why won't it? My Tenney, non-octave-equivalent way doesn't need > angles defined. You can choose any set of angles you want, and still > embed the result in Euclidean space, but that doesn't even matter -- > what matters are the taxicab distances ONLY.When did it become *your* way? The problem that either triangular or angular lattices solve doesn't arise in octave specific lattices, as we've always known. But Euclidian metrics can still be useful. From what I remember/understood, geometric complexity was one. Graham

Message: 9038 - Contents - Hide Contents Date: Thu, 08 Jan 2004 01:20:31 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> I wrote:>> The size/complexity tells you the best value for the suitably weighted >> minimax in any temperament in which this comma vanishes. If a comma >> exists such that its size/complexity is equal to the optimim minimax >> error in a given linear temperament, and the comma is in the linear >> temperament's kernel, then the two temperaments must be identical. >> Actually, it's more complicated than that. After finding the planar > temperament, you then need to adjust intervals that didn't get tempered > so that they work with the correct linear temperament family.Phew! I thought I had gone crazy! Thanks for clarifying.

Message: 9039 - Contents - Hide Contents Date: Thu, 08 Jan 2004 07:15:45 Subject: Re: TOP and normed vector spaces From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> An example of a normed vector space is the p-limit Erlich space,This is the Tenney space.> We may change basis in the Erlich space by resizing the elements, so > that the norm is now > > || |v2 v3 ... vp> || = |v2| + |v3| + ... + |vp|how are the v's defined?> and using the same proceedure we use to get > a unique minimax we can find a unique minimal distance point TOP at > this minimum distance from SIZEnot following . . .> One neat thing about this is that it generalizes immediately to other > normed vector spaces containing complete p-limit (meaning, 2 is > included as a prime number) lattices. In particular, there is a > geometric complexity version of TOP.What's better about it? I think Tenney complexity is a better guide to consonance, to tuning sensitivity, and even to musical complexity of a chain of intervals.

Message: 9040 - Contents - Hide Contents Date: Thu, 08 Jan 2004 22:25:41 Subject: Reply to Gene (was: Re: non-1200: Tenney/heursitic meantone temperament) From: Paul Erlich Gene's proposed canonical meantone: 5-limit: [1200., 1896.578429, 2786.313713] Let's evaluate: Interval...Approx....|Error|....Comp=log2(n*d)...|Error|/Comp 2:1........1200.00.....0.............1...............0 3:1........1896.58....5.38..........1.58............3.41 So already you've exceeded the maximum weighted error of my proposal by a factor of 2!> Interval...Approx....|Error|....Comp=log2(n*d)...|Error|/Comp > 2:1........1201.70....1.70...........1..............1.70 > 3:1........1899.26....2.69..........1.58............1.70 > 4:1........2403.40....3.40...........2..............1.70 > 5:1........2790.26....3.94..........2.32............1.70 > 3:2.........697.56....4.39..........2.58............1.70 > 6:1........3100.96....0.99..........2.58............0.38 > 8:1........3605.10....5.10...........3..............1.70 > 9:1........3798.53....5.38..........3.17............1.70 > 10:1.......3991.96....5.64..........3.32............1.70 > 4:3.........504.13....6.09..........3.58............1.70 > 12:1.......4302.66....0.70..........3.58............0.20 > 5:3.........890.99....6.64..........3.91............1.70 > 15:1.......4689.52....1.25..........3.91............0.32 > 16:1.......4806.79....6.79...........4..............1.70 > 9:2........2596.83....7.08..........4.17............1.70 > 18:1.......5000.22....3.69..........4.17............0.88 > 5:4.........386.86....0.55..........4.32............0.13 > 20:1.......5193.65....7.34..........4.32............1.70 > 8:3........1705.83....7.79..........4.58............1.70 > 24:1.......5504.36....2.40..........4.58............0.52 > 25:1.......5580.52....7.89..........4.64............1.70 > 6:5.........310.70....4.94..........4.91............1.01 > 10:3.......2092.69....8.33..........4.91............1.70 > 30:1.......5891.22....2.95..........4.91............0.60 > 32:1.......6008.49....8.49...........5..............1.70 > 36:1.......6201.92....1.99..........5.17............0.38 > 8:5.........814.84....1.15..........5.32............0.22 > 40:1.......6395.35....9.04..........5.32............1.70 > 9:5........1008.27....9.33..........5.49............1.70 > 45:1.......6588.78....1.44..........5.49............0.26 > 16:3.......2907.53....9.49..........5.58............1.70 > 48:1.......6706.06....4.10..........5.58............0.73 > 25:2.......4378.82....6.19..........5.64............1.10 > 50:1.......6782.21....9.59..........5.64............1.70 > 27:2.......4496.09....9.77..........5.75............1.70 > 54:1.......6899.49....6.38..........5.75............1.11 > 12:5.......1512.40....3.24..........5.91............0.55 > 15:4.......2286.12....2.15..........5.91............0.36 > 20:3.......3294.39...10.03..........5.91............1.70 > 60:1.......7092.92....4.65..........5.91............0.79 > 1296:5..... > and so on. Thinking about a few of these example spacially should > help you see that the weighted error can never exceed > > cents(81/80)/log2(81*80) = 1.70 > > for ANY interval. > > Is there a just (RI) interval in this meantone? The idea of duality > leads me to guess 81*80:1 = 6480:1 . . . > > 6480:1....15194.10....0.03 > > almost, but no cigar.

Message: 9041 - Contents - Hide Contents Date: Thu, 08 Jan 2004 07:20:15 Subject: Re: another for gene From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: The guy who really needs to comment on this is Dave. Is there some combination of green and blue lines you can live with as a cutoff, Dave?

Message: 9042 - Contents - Hide Contents Date: Thu, 08 Jan 2004 22:29:46 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Paul Erlich wrote: >>> What's the worst comma for 12-equal in the 5-limit? >> [0 28 -19> or 22876792454961:19073486328125Wow. How did you find that?> TOPping it gives a narrow octave of 0.99806 2:1 octaves.Shall I proceed to calculate Tenney-weighted errors for all (well, a bunch of) intervals? I hope you're onto something!>> Me too -- but the lengths aren't compatible in a Euclidean space. >> Remember the whole big "wormholes" discussion from years ago? >> Yes, I remember all about the wormholes, and they don't have anything to > do with this. You only need them for odd limits.I thought that's what you were talking about in the thread where I brought them up! Odd limit, right?>> We can either embed a lattice, with a taxicab distance, into >> Euclidean space, or we can't. But just because we can, doesn't mean >> we should use Euclidean distance! NONONONONONO! >> You could try taxicab distance, I'm not sure it'd work right. But you > can also use Euclidian distance, and it looks like a more > straightforward way to me. >>> Why won't it? My Tenney, non-octave-equivalent way doesn't need >> angles defined. You can choose any set of angles you want, and still >> embed the result in Euclidean space, but that doesn't even matter -- >> what matters are the taxicab distances ONLY. >> When did it become *your* way?Did someone publish it before? It's currently not Gene's way, anyway.> The problem that either triangular or > angular lattices solve doesn't arise in octave specific lattices, as > we've always known. But Euclidian metrics can still be useful. From > what I remember/understood, geometric complexity was one.I've been trying to convince Gene otherwise, and he said something about minor thirds being shorter than major thirds there . . .

Message: 9043 - Contents - Hide Contents Date: Thu, 08 Jan 2004 07:20:45 Subject: Temperament agreement From: Dave Keenan Continued from the tuning list. Paul:>With my (Tenney) complexity and (all-interval-Tenney-minimax) error >measures?With these it seems I need to scale the parameters to k=0.002 p=0.5 and max badness = 75 where badness = complexity * exp((error/k)**p) I'd be very interested to see how that compares with your other cutoff lines. These errors and complexities don't seem to have meaningful units. Complexity used to have units of generators per diamond and error used to have units of cents, both things you could relate to fairly directly.

Message: 9044 - Contents - Hide Contents Date: Thu, 08 Jan 2004 22:46:34 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Graham Breed Paul Erlich wrote:> Wait a minute -- we're obviously talking about different things here! > I read this too quickly. 3:1 is a ratio of 3, and 3:5 is a ratio of > 5, so the latter should be longer!!They're both 5-odd limit intervals, and so they each have a 5-odd limit complexity of 1. Graham

Message: 9045 - Contents - Hide Contents Date: Thu, 08 Jan 2004 07:33:24 Subject: Re: TOP and normed vector spaces From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> We may change basis in the Erlich space by resizing the elements, > so>> that the norm is now >> >> || |v2 v3 ... vp> || = |v2| + |v3| + ... + |vp| >> how are the v's defined?For a rational number, vp is the p-adic valuation of number q, that is,the exponent in the factorization of q into primes. For other points in the Tenney space it's just a coordinate.>> and using the same proceedure we use to get >> a unique minimax we can find a unique minimal distance point TOP at >> this minimum distance from SIZE >> not following . . .Remember, we have a way of measuring distance between tuning maps. Hence, given a tuning map SIZE and a subspace of tuning maps Null(C), we can find those at a minimum distance from SIZE.>> One neat thing about this is that it generalizes immediately to > other>> normed vector spaces containing complete p-limit (meaning, 2 is >> included as a prime number) lattices. In particular, there is a >> geometric complexity version of TOP. >> What's better about it?What's better about it is that it is Euclidean, which is convenient in many ways. For instance, the version of TOP here would simply be an orthogonal projection. I think Tenney complexity is a better guide> to consonance, to tuning sensitivity, and even to musical complexity > of a chain of intervals.Have you even thought for five seconds about the geometric complexity map with 2 included before coming to this conclusion? In the 5-limit, it would be || |u2 u3 u5> || = sqrt(u2^2+log2(3)^2 u3^2+ log2(5)^2 u5^2+u2u3+u2u5+log2(3)^2 u3u5)) So why is this so very, very much worse?

Message: 9046 - Contents - Hide Contents Date: Thu, 08 Jan 2004 22:45:47 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Paul Erlich wrote: >>> Wait a minute -- we're obviously talking about different things here! >> I read this too quickly. 3:1 is a ratio of 3, and 3:5 is a ratio of >> 5, so the latter should be longer!! >> They're both 5-odd limit intervals, and so they each have a 5-odd limit > complexity of 1.Well then we are talking about different things. I'm talking about "expressibility" as the distance measure.

Message: 9047 - Contents - Hide Contents Date: Thu, 08 Jan 2004 07:51:31 Subject: Re: TOP and normed vector spaces From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>>> We may change basis in the Erlich space by resizing the elements, >> so>>> that the norm is now >>> >>> || |v2 v3 ... vp> || = |v2| + |v3| + ... + |vp| >>>> how are the v's defined? >> For a rational number, vp is the p-adic valuation of number q, that > is,the exponent in the factorization of q into primes. For other > points in the Tenney space it's just a coordinate.There are no other points in the Tenney space. Anyway, I lost the train of thought.>>> and using the same proceedure we use to get >>> a unique minimax we can find a unique minimal distance point TOP > at>>> this minimum distance from SIZE >>>> not following . . . >> Remember, we have a way of measuring distance between tuning maps.In the dual space?> Hence, given a tuning map SIZE and a subspace of tuning maps Null (C), > we can find those at a minimum distance from SIZE.Hmm . . .>>> One neat thing about this is that it generalizes immediately to >> other>>> normed vector spaces containing complete p-limit (meaning, 2 is >>> included as a prime number) lattices. In particular, there is a >>> geometric complexity version of TOP. >>>> What's better about it? >> What's better about it is that it is Euclidean, which is convenient > in many ways.Do we really need this convenience? Can't we work with the taxicab metric?

Message: 9048 - Contents - Hide Contents Date: Thu, 08 Jan 2004 22:50:03 Subject: Re: non-1200: Tenney/heursitic meantone temperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:>> Paul Erlich wrote: >>>>> Hmm . . . by *all* the errors, I meant for lots and lots of >>> intervals, like I did. >>>> Oh, well, here's the 9-limit with a few bonuses: >> >> 3:1 0.002827 >> 5:1 0.000000 >> 5:3 0.001930 >> 7:1 0.000903 >> 7:3 0.000693 >> 7:5 0.000903 >> 9:1 0.002827 >> 9:5 0.002827 >> 9:7 0.002027 >> 15:1 0.001147 >> 27:1 0.002827 >> 27:5 0.002827 >> So you're dividing by expressibility here? Interesting . . . !Graham, it sure doesn't look like you're using Euclidean distance here!!!

Message: 9049 - Contents - Hide Contents Date: Thu, 08 Jan 2004 08:20:44 Subject: Re: TOP and normed vector spaces From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:>> For a rational number, vp is the p-adic valuation of number q, that >> is,the exponent in the factorization of q into primes. For other >> points in the Tenney space it's just a coordinate. >> There are no other points in the Tenney space. Anyway, I lost the > train of thought.Then why did you correct me to "Tenney space"? Presumably, I knew what space I wanted even if I didn't have the right name for it. There *are* other points in my space, and what you seem to be talking about is a lattice.>>>> and using the same proceedure we use to get >>>> a unique minimax we can find a unique minimal distance point > TOP >> at>>>> this minimum distance from SIZE >>>>>> not following . . . >>>> Remember, we have a way of measuring distance between tuning maps. >> In the dual space?Correct. Tuning maps are points in the dual space, and that is a normed vector space, and hence a metric space, so we know the distance between two tuning maps. One, SIZE, is the JI tuning map. We want a tuning map in the subspace Null(C) as close as possible to SIZE.> Do we really need this convenience? Can't we work with the taxicab > metric?I compared the two, and taxicab does seem to work better. The Euclidean version thinks minor thirds are slightly better than major thirds, for instance.

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