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Message: 1850 - Contents - Hide Contents Date: Thu, 11 Oct 2001 18:16:09 Subject: Re: Does Miracle--11 count? From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> Gene -- the complexity is actually only 10, not 13 -- 11 notes in > the>> chain are enough to give a complete tetrad. >> Oops--so that's the definition people use?Well, that's Graham's complexity.
Message: 1851 - Contents - Hide Contents Date: Thu, 11 Oct 2001 18:19:27 Subject: Re: Searching for interesting 7-limit MOS scales From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <9q2f10+h9nk@e...> > Paul wrote: >>>>> #22: 16 notes; commas 1029:1024, 50:49; chroma 36:35 >>>>>> 36-tET, 26-tET . . . curious >>>> This is quite an interesting one . . . after 26, the next proper MOS >> is at 110 . . . score one for 26-tET! >> You're very keen to take ETs here. Why don't you work out what ETs are > consistent with each unison vector, like at > <4 5 6 9 10 12 15 16 18 19 22 26 27 29 31 35 36... * [with cont.] (Wayb.)> and see how often each comes > up?Well, if I can have a single guitar which will work for a lot of these good MOS scales, that saves money, plus I can switch from one to another withing a single composition.> > And in this case, you take the different sets > > (-10, 1, 0, 3), 1029:1024 > 5 10 15 16 26 31 36 41 46 56 57 62 72 77 82 87 ... > > (1, 0, 2, -2), 50:49 > 4 6 10 12 16 18 22 26 > > and check the intersection > > 5 10 15 16 26 31 41 ... > 4 6 10 12 16 18 22 26 >Not sure what that tells me.
Message: 1852 - Contents - Hide Contents Date: Thu, 11 Oct 2001 18:26:48 Subject: Re: Does Miracle--11 count? From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >>> Curiously, no kleismic temperament appears in >> >> 26 29 41 46 58 72 80 87 94 111 113 121 130 149... * [with cont.] (Wayb.) >> or >> 29 41 58 72 80 87 94 111 121 130 149 159 183 1... * [with cont.] (Wayb.) >> I didn't find the 17/46 I mentioned on the seven list; that turns out > not to be significantly different than the 19-27 linear termperament, > with a span for the tetrad of 13 (is that the complexity?)Yup. This temperament is #16 in my recent list.> The 19 out > of 46 and 27 out of 46 scales with this generator might be reasonable > things for your quest to consider.Yes -- if you look back at my recent comments, you'll see that I thought 27-tET is far from ideal for this 19-tone MOS, but I may be willing to live with it, since a 27-tET guitar fingerboard will allow so many other possibilities as well, and 46-tET, while attractive for other reasons, is too tight on a guitar for me.
Message: 1853 - Contents - Hide Contents Date: Thu, 11 Oct 2001 18:28:01 Subject: Re: Does Miracle--11 count? From: Paul Erlich Graham wrote,> I think it might have made the 13-limit list if it had been included.Hmm . . . are there things missing from the 5-, 7-, 9-, and 11-limit lists for similar reasons?> It > should certainly be in the MOS list, as it's worse than this: > > complexity measure: 23 (29 for smallest MOS) > highest error: 0.008301 (9.961 cents) > > in both measures.What's your MOS list? I'm very confused now.
Message: 1854 - Contents - Hide Contents Date: Thu, 11 Oct 2001 21:26 +0 Subject: Re: Does Miracle--11 count? From: graham@xxxxxxxxxx.xx.xx Paul wrote:> Graham wrote, >>> I think it might have made the 13-limit list if it had been > included. >> Hmm . . . are there things missing from the 5-, 7-, 9-, and 11-limit > lists for similar reasons?Meantone-31 is missing from the 11-limit list, because there are no other ETs it's consistent with. It's inaccurate and not unique, but might make the MOS list as it fits into 19. Ah, it can be generated from 31 and 43 31/74, 503.3 cent generator basis: (1.0, 0.419405836425) mapping by period and generator: [(1, 0), (2, -1), (4, -4), (7, -10), (11, -18)] mapping by steps: [(43, 31), (68, 49), (100, 72), (121, 87), (149, 107)] unison vectors: [[-4, 4, -1, 0, 0], [1, 2, -3, 1, 0], [4, 0, -2, -1, 1]] highest interval width: 18 complexity measure: 18 (19 for smallest MOS) highest error: 0.009185 (11.022 cents) Temperaments without two consistent ETs aren't likely to be that accurate or unique, but they can be simple. I might write a script that can generate temperaments from all the possible generators for an ET, then run it for all consistent ETs. Hopefully all linear temperaments of note will have at least *one* consistent equal representative.> What's your MOS list? I'm very confused now.The smallest MOS is taken as the complexity measure, instead of the number of generators for a complete chord. Dave Keenan asked for it a while back, and the script keeps churning them out so I keep uploading them. They're a .mos suffix instead of .txt and there are also .key keyboard mappings and .micro microtemperaments. The template is limitN.whatever where N now goes as odd numbers from 5 to 21. Graham
Message: 1855 - Contents - Hide Contents Date: Thu, 11 Oct 2001 23:09:18 Subject: Re: Breedsma, kalisma, ragisma, schisma From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote:> --- In tuning-math@y..., <manuel.op.de.coul@e...> wrote: >>> By the way, taking the Breedsma, kalisma, ragisma and schisma >> as unison vectors gives a truly big 11-limit PB of 342 tones. >> Using the xenisma in addition gives the same PB. >> One can extend this to an 11-limit notation by adding 385/384, > 441/440 or 540/539; unfortunately, none of these seem to have names.I recommend naming 385/384 after Dave Keenan . . . it figures particularly heavily in his many postings about microtemperament. Keenan's kleisma? 896/891 has come up a lot in my own investigations -- the best Fokker PB fit to Partch's scale seems to differ from it merely by a few 896/891 deviations, and the shrutar tuning Dave Keenan worked out for me involves tempering some pitches by fractions of the diaschisma, and other pitches by fractions of the 896/891. No idea what we'd call it . . . undecimal comma is taken . . .
Message: 1856 - Contents - Hide Contents Date: Thu, 11 Oct 2001 02:11:08 Subject: Re: Does Miracle--11 count? From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Gene -- the complexity is actually only 10, not 13 -- 11 notes in the > chain are enough to give a complete tetrad.Oops--so that's the definition people use?
Message: 1857 - Contents - Hide Contents Date: Thu, 11 Oct 2001 05:27:48 Subject: Re: Does Miracle--11 count? From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> Curiously, no kleismic temperament appears in > > 26 29 41 46 58 72 80 87 94 111 113 121 130 149... * [with cont.] (Wayb.) > or > 29 41 58 72 80 87 94 111 121 130 149 159 183 1... * [with cont.] (Wayb.)I didn't find the 17/46 I mentioned on the seven list; that turns out not to be significantly different than the 19-27 linear termperament, with a span for the tetrad of 13 (is that the complexity?) The 19 out of 46 and 27 out of 46 scales with this generator might be reasonable things for your quest to consider.
Message: 1858 - Contents - Hide Contents Date: Thu, 11 Oct 2001 10:22 +0 Subject: Re: Does Miracle--11 count? From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9q2hsr+gnl9@xxxxxxx.xxx> Paul wrote:> Curiously, no kleismic temperament appears in > > 26 29 41 46 58 72 80 87 94 111 113 121 130 149... * [with cont.] (Wayb.) > or > 29 41 58 72 80 87 94 111 121 130 149 159 183 1... * [with cont.] (Wayb.) > > Graham found at least 10 ways to do "better" in 13-limit and 15-limit.27-equal isn't consistent in the 11-limit. So I don't think this temperament is even being considered in that program. It can be calculated like this:>>> import temper >>> et27 = temper.PrimeET(27, temper.primes[:5]) >>> et58 = temper.PrimeET(58, temper.primes[:5]) >>> et27.basis[4]=94 >>> et27+et5822/85, 310.3 cent generator basis: (1.0, 0.25859397023437097) mapping by period and generator: [(1, 0), (-1, 10), (0, 9), (1, 7), (-3, 25), (5, -5)] mapping by steps: [(58, 27), (92, 43), (135, 63), (163, 76), (201, 94), (215, 100)] unison vectors: [[-1, 5, 0, 0, -2, 0], [9, -1, 0, 0, 0, -2], [5, 2, -5, 0, 1, 0], [0, -2, 0, 5, -1, -2]] highest interval width: 25 complexity measure: 25 (27 for smallest MOS) highest error: 0.006590 (7.909 cents) I think it might have made the 13-limit list if it had been included. It should certainly be in the MOS list, as it's worse than this: complexity measure: 23 (29 for smallest MOS) highest error: 0.008301 (9.961 cents) in both measures. Graham
Message: 1859 - Contents - Hide Contents Date: Thu, 11 Oct 2001 10:22 +0 Subject: Re: Searching for interesting 7-limit MOS scales From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9q2f10+h9nk@xxxxxxx.xxx> Paul wrote:>>> #10: 14 notes; commas 245:243, 50:49; chroma 25:24 >>>> 22-tET, 36-tET . . . what's this? >> Interesting . . . a 7:9 generator in a half-octave . . . > > Hey Graham . . . why does #11 open in your output with '5/6' while > the otherwise identical #1 opens with '1/7'?One gives a 12 note periodicity block, the other 14. I don't know why it's 5/6 instead of 1/6. I used to always take the smallest generator, but the definition of "smallest" can be different for different periodicity blocks. So I use an arbitrary rule which works in this case (the mappings are the same) but doesn't always when the period isn't an octave. Graham
Message: 1860 - Contents - Hide Contents Date: Thu, 11 Oct 2001 11:20 +0 Subject: Re: Searching for interesting 7-limit MOS scales From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9q2f10+h9nk@xxxxxxx.xxx> Paul wrote:>>> #22: 16 notes; commas 1029:1024, 50:49; chroma 36:35 >>>> 36-tET, 26-tET . . . curious >> This is quite an interesting one . . . after 26, the next proper MOS > is at 110 . . . score one for 26-tET!You're very keen to take ETs here. Why don't you work out what ETs are consistent with each unison vector, like at <4 5 6 9 10 12 15 16 18 19 22 26 27 29 31 35 36... * [with cont.] (Wayb.)> and see how often each comes up? And in this case, you take the different sets (-10, 1, 0, 3), 1029:1024 5 10 15 16 26 31 36 41 46 56 57 62 72 77 82 87 ... (1, 0, 2, -2), 50:49 4 6 10 12 16 18 22 26 and check the intersection 5 10 15 16 26 31 41 ... 4 6 10 12 16 18 22 26 Graham
Message: 1861 - Contents - Hide Contents Date: Fri, 12 Oct 2001 20:49:42 Subject: Squares of triangles of triangles From: genewardsmith@xxxx.xxx Numbers of the form n^2/(n^2-1) factor as n^2/(n-1)(n+1), and so it makes sense they show up on these lists. Triangular numerators are similar, we have [n(n+1)/2]/[n(n+1)/2 - 1] = n(n+1)/(n-1)(n+1). The mystery of the squares of triangles of triangles is explained by the fact that from tt(n) = n(n+1)(n^2+n+2)/8, the triangle of a triangle function, we get tt(n)^2/(tt(n)^2-1) = n^2(n+1)^2(n^2+n+2)^2 / (n-1)(n+1)(n^2-n+2)(n^2+n+4)(n^2+3n+4). I think it would be worthwhile to explore this sort of thing further.
Message: 1862 - Contents - Hide Contents Date: Fri, 12 Oct 2001 13:45:33 Subject: Re: Breedsma, kalisma, ragisma, schisma From: manuel.op.de.coul@xxxxxxxxxxx.xxx>I recommend naming 385/384 after Dave Keenan . . . it figures >particularly heavily in his many postings about microtemperament. >Keenan's kleisma?That's fine with me.>896/891: No idea what we'd call it . . . undecimal comma is taken . .I vote undecimal semicomma. It's close in size to Rameau's and Fokker's semicomma, about half a syntonic comma. Manuel
Message: 1863 - Contents - Hide Contents
Date: Sat, 13 Oct 2001 10:20:45
Subject: From generators to vals
From: genewardsmith@xxxx.xxx
I've remarked how easy it is to move from the notation defined by two
vals to an octave-generator system of notation; it is worth remarking
that we can go in the opposite direction easily also, and in a
canonical way.
If g/n is the generator in lowest terms, we have two adjacent
fractions in the Farey sequence Fn next to it, p1/q1 and p2/q2, such
that p1/q1 < g/n < p2/q2 and (p1+p2)/(q1+q2) = g/n; since p1/q1 is
adjacent to p2/q2 in the Farey sequences before we reach Fn, we have
q1p2 - q2p1 = 1, so that the matrix
[q1 q2]
[p1 p2]
has determinant 1, and hence is invertible. This is the matrix which
transforms from the n,g system of coordinates to one based on two
vals; we can find the vals by transforming the vectors for 2, 3, etc
using the above matrix.
For instance, given the 4/19 generator, we have 1/5 < 4/19 < 3/14,
and this generator is the same as the 5-14 system. If we write 3 in
the octave-generator form, it is [2,-2], since 2 + (-2)*(4/19) =
30/19. Then
[14 5]
[2 -2] [ 3 1] = [22 8],
so that if g5 is the 5 val and g14 is the 14 val, we get g5(3) = 8
and g14(3) = 22. In this way we find g14(5) = 32, g4(5) = 12 and
g14(7) = 39, g4(7) = 14; hence g5 = h5, but since h14(5) = 33 we
don't have g14 = h14. The real point, of course, is that g14+g5 = h19.
In the same way, we can find that the minor third, or 5/19 system, is
the 4-15 system, that the 2/19 generator is the 9-10 system, and that
the meantone, as we might have expected, works out to be 7-12. We may
also do this when the interval of repetition is a fraction of an
octave, so that from 1/4<2/7<1/3 we get that the 8/28 generator is
the 12-16 system.
We may also express the same system in terms of the comma group dual
to the val group, so that the 5/19 system is the 4-15 system is the
system of 49/48 and 126/125. From there we can pick an appropriate
chroma, such as 25/24 or 28/27 (which is almost exactly a 19-tone
step) and get a block which the system approximates.
Message: 1864 - Contents - Hide Contents
Date: Mon, 15 Oct 2001 17:44 +0
Subject: It's the 15th!
From: graham@xxxxxxxxxx.xx.xx
That means the new issue of Perspectives of New Music should be out,
according to their website <Perspectives of New Music Home Page * [with cont.] (Wayb.)>. Note
Mark Gould's article "Balzano and Zweifel: Another Look at Generalized
Diatonic Scales" (in <http://www.perspectivesofnewmusic.org/TOC382.pdf - Type Ok * [with cont.] (Wayb.)>).
He did e-mail me a while back mentioning this. Something about lattices
and whatever Balzano does being essentially the same thing. If anybody
has access to a copy, I'd be interested in a summary.
Graham
Message: 1865 - Contents - Hide Contents Date: Mon, 15 Oct 2001 22:04:21 Subject: Re: It's the 15th! From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> That means the new issue of Perspectives of New Music should be out, > according to their website <Perspectives of New Music Home Page * [with cont.] (Wayb.)>. Note > Mark Gould's article "Balzano and Zweifel: Another Look at Generalized > Diatonic Scales" (in <http://www.perspectivesofnewmusic.org/TOC382.pdf - Type Ok * [with cont.] (Wayb.)>). > He did e-mail me a while back mentioning this. Something about lattices > and whatever Balzano does being essentially the same thing. If anybody > has access to a copy, I'd be interested in a summary.Graham, have you seen Balzano's older papers? He does use square lattices of major thirds and minor thirds, but makes a mistake (I feel) in identifying the importance of major thirds and minor thirds in their ability to generate the tuning, i.e., C(3)*C(4)=C(12), rather than in their acoustical consonance. He ignores the fact that 19- and 31-tone systems were actually used by a few early musicians, sounded wonderful, but 12-tone simply proved more economical. When he moves on to 20-equal, and constructs triads from 4/20-oct. and 5/20- oct. intervals (since C(4)*C(5)=C(20)), all vestiges of acoustical foundation are lost.
Message: 1867 - Contents - Hide Contents Date: Tue, 16 Oct 2001 13:59 +0 Subject: Re: It's the 15th! From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9qfmh5+hos3@xxxxxxx.xxx> Paul wrote:> Graham, have you seen Balzano's older papers?No, I know very little about Balzano. And even less about Zweifel.> He does use square > lattices of major thirds and minor thirds, but makes a mistake (I > feel) in identifying the importance of major thirds and minor thirds > in their ability to generate the tuning, i.e., C(3)*C(4)=C(12), > rather than in their acoustical consonance. He ignores the fact that > 19- and 31-tone systems were actually used by a few early musicians, > sounded wonderful, but 12-tone simply proved more economical. When he > moves on to 20-equal, and constructs triads from 4/20-oct. and 5/20- > oct. intervals (since C(4)*C(5)=C(20)), all vestiges of acoustical > foundation are lost.That doesn't sound very interesting to me. Perhaps it's better to go on not knowing much about it. Graham
Message: 1868 - Contents - Hide Contents Date: Wed, 17 Oct 2001 11:10 +0 Subject: Re: It's the 15th! From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <001e01c156cb$216b1420$2258d63f@stearns> Dan Stearns wrote:> I haven't read any Balzano or any of the related generalized diatonic > scale articles, but do any of these even try to argue or hint at an > acoustical foundation for any of this... if not, then why bother > criticizing or expecting in a direction that doesn't really apply?The impression I got from the e-mails he sent me was that Mark Gould's article does cover JI lattices, similar to those on my website (which is how he found me). But I don't have access to a library that has this stuff, and I don't know if it's worth ordering a copy. If there's no (psycho)acoustical foundation, I'm much less likely to be interested in it. Graham
Message: 1869 - Contents - Hide Contents Date: Sat, 20 Oct 2001 02:02:54 Subject: Relative error theorems From: genewardsmith@xxxx.xxx Relative Error Theorem: Let l be an odd number, h a val and f a note of the p-limit, where p is the largest prime less than or equal to l. Define the error in relative cents of the val h for the note f to be e(h, f) = 100*(h(f) - h(2) log2(f)). Define the badness measures e_inf(h, l) = max |e(h, r)|, e_2(h, l) = sqrt(sum e(h, r)^2), and e_1(h, l) = sum |e(h, r)|, where in each case we take r to range over rational numbers p/q where p/q is reduced to its lowest terms, 1<p/q, and 1<=p,q<=l. Let j and k be two vals such that j+k=h; we then have e_inf(h, l) <= e_inf(j, l) + e_inf(k, l) e_2(h, l) <= e_2(j, l) + e_2(k, l) e_1(h, l) <= e_1(j, l) + e_1(k, l) Proof: It follows from its definition and the multiplicative linearity of the logarithm that relative error is linear, ie that e(j+k, l) = e(j, l) + e(k, l). We may define error vectors by defining a fixed ordering of the numbers r, so that if vj is the vector of errors for j, vk is the vector of errors for k, and vh for h=j+k, the linearity of relative error entails that vh = vj+vk. We may consider these vectors to reside in a normed vector space with norm L_inf (the maximum of the absolute values of the coordinates), L_2 (Euclidean norm), or L_1 (the sum of the absolute values.) These norms are respectively e_inf, e_2 and e_1; from this and the triangle inequality for each of these norms, the result follows. Definition: Associated generator Let j and k be valid vals, such that a=j(2) < b=k(2) are distinct, so that j and k are 2-distinct. We reduce to lowest terms by dividing through by the gcd, so that if d = gcd(a, b), q=a/d, s=b/d. We form the fraction s/q, and define r/p by the condition that p is the least denominator for which we have |rq - sp| = 1. Exchanging the names of j and k, q and s, p and r if necessary we define matters so that rq - sp = 1. We then define the associated generator val g as j(2) k - k(2) j, and the reduced generator val as g/d. Theorem: Let l, h, f, and p be as above. Define com_inf(h, l) = max |h(r)|, com_2(h, l) = sqrt(sum h(r)^2) and com_1(h, l) = sum |h(r)|, where r is defined as before. If j and k are 2-distinct valid vals, and if g is the associated generator val, then com_inf(g, l) <= (k(2) e_inf(j, l) + j(2) e_inf(k, l))/100 (Graham's complexity) com_2(g, l) <= (k(2) e_2(j, l) + j(2) e_2(k, l))/100 com_1(g, l) <= (k(2) e_1(j, l) + j(2) e_1(k, l))/100 Proof: Defining p and f as before, we define a linear temperament associated to j and k by A^j(f) B^k(f) for some fixed A and B. If we set G = A^p B^r and E = A^q B^s we may also express the linear temperament in terms of G and E, since the transformation matrix [[p r] [q s]]has determinant -1, and hence is invertible as an integral matrix. Inverting it, we find that in terms of G and E, the approximation to f is given by G^h(f) E^e(f), where h = q k - s j is the reduced generator val and e = r j - p k is the interval of equivalence val. (In the particular case of the j+k et, we have j(2)+k(2)= n, A = B = 2^(1/n), so that G= 2^((p+r)/n) is the generator within an interval of equivalence of E = 2^((q+s)/n) = 2^(1/d).) We have e(j, f) = 100(j(f) - j(2) log2(f)), e(k, f) = 100(k(f) - k(2) log2(f)). From this we get j(f) = j(2) log2(f) + e(j, f)/100, k(f) = k(2) log2(f) + e(k, f)/100 Then h(f) = j(2) k - k(2) j = j(2) (k(2) log2(f) + e(k, f)/100) - k(2) (j(2) log2(f) + e(j, f)1/00) = (e(j, f) + e(k, f))/100. Forming vectors as before and applying the triangle inequality, we get the theorem.
Message: 1870 - Contents - Hide Contents Date: Mon, 22 Oct 2001 11:30:42 Subject: question about quadratics From: monz To all the math-geeks: It's well-established that the Babylonians had highly developed algebraic methods, altho they never actually developed an algebraic notation. There's also concrete proof that they had very good sexagesimal approximations to both square and cube roots. I'm investigating the possible applications some of these methods may have had to tuning problems, but my extremely math-challenged self needs some help. Can someone tell me what application quadratic equations may have to determining string-lengths? I'm interested in possible applications for the purposes of determining both JIs and/or temperaments. For a reference to a modern explanation of Babylonian algebra, please see the following, p 30-50: Neugebauer, Otto. 1957. _The Exact Sciences in Antiquity_. Providence, Brown University Press, 2d ed L.O.C.#: QA22 .N36 1957 Thanks. love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 1871 - Contents - Hide Contents Date: Mon, 22 Oct 2001 11:35:51 Subject: Re: question about quadratics From: monz This may also be useful / helpful / interesting: "Pythagorean Triangles and Musical Proportions" by Martin Euser Pythagorean triangles and musical proportions ... * [with cont.] (Wayb.) ----- Original Message ----- From: monz <joemonz@xxxxx.xxx> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Monday, October 22, 2001 11:30 AM Subject: [tuning-math] question about quadratics> To all the math-geeks: > > It's well-established that the Babylonians had > highly developed algebraic methods, altho they > never actually developed an algebraic notation. > > There's also concrete proof that they had very good > sexagesimal approximations to both square and cube roots. > > I'm investigating the possible applications some > of these methods may have had to tuning problems, > but my extremely math-challenged self needs some help. > > Can someone tell me what application quadratic equations > may have to determining string-lengths? I'm interested > in possible applications for the purposes of determining > both JIs and/or temperaments. > > > For a reference to a modern explanation of Babylonian > algebra, please see the following, p 30-50: > > Neugebauer, Otto. 1957. > _The Exact Sciences in Antiquity_. > Providence, Brown University Press, 2d ed > L.O.C.#: QA22 .N36 1957 > > > Thanks.love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 1872 - Contents - Hide Contents Date: Sat, 27 Oct 2001 10:57 +0 Subject: Re: more 72-tET From: graham@xxxxxxxxxx.xx.xx Paul:>>>> What others can you find? Gene: >>> The>>> 31+22 line, through 53, 75, 84 and 97, is clearly visible, which >>> makes me happy as I'm still working away on my Orwell piece. Paul:>> And the 5-limit UV is . . . ? Gene: > 3^3*5^7/2^21The 11-limit UVs, if anybody wants them, are 385:384, 225:224 and 121:120. The answers to Paul's original questions are temper.getRatio( temper.Temperament( 25,12,temper.primes[:2]).getUnisonVectors()[0]) temper.getRatio( temper.Temperament( 22,31,temper.primes[:2]).getUnisonVectors()[0]) and <3 4 5 7 8 9 10 12 15 16 18 19 22 23 25 26 27 2... * [with cont.] (Wayb.)> Graham
Message: 1873 - Contents - Hide Contents Date: Sat, 27 Oct 2001 22:55:48 Subject: Osmium generators From: genewardsmith@xxxx.xxx Paul has taken up the challenge of finding the exact value of the Osmium Meantone, which is the limit of the sequence 4/7, 7/12, 7/12, 11/19, 14/24, 18/31, 25/43, 32/55 ..., and is about 1/5-comma meantone. If he solves that, this should be easy: 3/8, 4/11, 7/19, 7/19, 11/30, 14/38, 18/49, 25/68, 32/87, 43/117, 57/155... . If you multiply the denominators by 9, you get 72, 99, 171, 171, 270, 342, 441, 612, 783, 1053, 1395 ..., so this is the Osmium version of a generator which doesn't yet have a name, so far as I know, but which has been discussed several times. (Why Osmium? I'll explain that after Paul either solves the problem or asks for my solution.) I'd also like to know what to do with the sequence 9, 10, 12, 19, 22, 31, 41, 53, 72, 94, ... It doesn't seem like Miracle, or Orwell either, but the ets make sense, at any rate, even if the resultant commas don't. Apparently this Osmium business doesn't always give you a generator that makes much sense.
Message: 1874 - Contents - Hide Contents Date: Mon, 29 Oct 2001 19:09:56 Subject: Tribonacci From: Paul Erlich Hey Gene, any thoughts on this year-old tuning list post? I think Dan Stearns thinks he understands the phenomena, but I didn't follow his answers too closely . . . --- In tuning@y..., "Paul Erlich" <PERLICH@A...> wrote: For MOS scales we've seen the noble generators allow for a uniquely regular expansion of resources, in that each MOS will have L = s*phi and the steps in the old MOS change into the steps of the new MOS as follows: L(old) -> L(new) + s(new) s(old) -> L(new) so that the sizes (number of notes) of any three consecutive scales in the "evolution" obey a generalized Fibonacci recursion: siz(n-1) + siz(n) = siz(n+1); and the ratios of adjacent sizes approaches phi. The most famous example is the Kornerup system, which has the same scale sizes as Yasser's proposed evolution 2, 5, 7, 12, 19, 31, 50, 81 . . . but maintains the same (noble) generator throughout. What if we allow three step sizes, and posit the following evolution rules: siz(n-2) + siz(n-1) + siz(n) = siz(n+1)? Then the ratio of adjacent sizes approaches the solution of x^3 - x^2 - x - 1 = 0 = 1.839286755... One example is the so-called "Tribonacci Sequence": 1, 1, 2, 4, 7, 13, 24, 44, 81,... while another may be more musically relevant and may be familiar to followers of Kraehenbuehl & Schmidt: 2, 2, 3, 7, 12, 22, 41, . . . although the next term would be 75 instead of K&S's 78 -- since K&S started with 3-limit JI and forced "inflections" reflecting successively higher prime limits to "deform" the scale at each stage. Like the Fibonacci sequence, the Tribonacci sequence and its relatives can be derived from a continued fraction representation, but this time using generalized, "third-order" continued fractions (see 404 Object Not Found * [with cont.] (Wayb.)). Can anyone demonstrate a rule for how the steps of one scale transform into the steps of the next scale, and find the set of scales which are to Kraehenbuehl & Schmidt as Kornerup's set of scales is to Yasser? Bonus question: What is the object here analogous to the generator in MOS scales, and in what way is this generator built upon itself to create these "hyper-MOS" scales? Is there more than one solution? --- End forwarded message ---
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