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Message: 4700

Date: Mon, 01 Oct 2001 23:14:27

Subject: 46 (was Re: Pauls fingerboard kit)

From: Paul Erlich

--- In tuning-math@y..., graham@m... wrote:
> Bob Valentine wrote:
> 
> > What does 46 have that everyone seems gaga about? Is 46
> > better at 13

Yes, but I'm not planning on using it for 13-limit. More important is 
that 46 is better than 41 in 5-limit (which is the main interval 
flavor for Indian music). Though 34 is better still in the 5-limit, 
46 allows me to access 7- and 11-flavors with much higher accuracy.


top of page bottom of page up down Message: 4701 Date: Mon, 01 Oct 2001 04:37:22 Subject: catching up From: genewardsmith@xxxx.xxx --- In tuning-math@y..., jon wild <wild@f...> wrote: > I understand from the rest of what you > wrote that the requirement is stronger still than connected: that the span > of the set is also limited. Thanks --Jon It is at least for classic blocks, which is what I just defined. The question is what to do about what I was calling "semiblocks"; simply requiring them to come from a convex region with a span of less than 1 on the first coordinate (which I think is in effect what Paul suggests in his Gentle Introduction) is pretty weak, and it allows for more extreme examples than he gave, but my definition may still be too restrictive--it doesn't include his example of the Indian diatonic, with sixth degree raised a comma, for instance. That makes for a distance of 8/7 to 4/3, which is a little far. Possibly a semiblock should just be convex and the diameter would serve as a measure of how extreme it is, so the above example would be an 8/7- semiblock.
top of page bottom of page up down Message: 4702 Date: Tue, 2 Oct 2001 16:08 +01 Subject: Re: 41, 46 and 58 From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9pbgq1+86qa@xxxxxxx.xxx> Gene wrote: > Here are relativized n-consistent goodness measures for odd n to 25, > for 41, 46 and 58: How are these calculated? It looks like lower numbers are better, so it's really a badness measure. Graham
top of page bottom of page up down Message: 4703 Date: Tue, 02 Oct 2001 16:19:03 Subject: Re: 41, 46 and 58 From: genewardsmith@xxxx.xxx --- In tuning-math@y..., graham@m... wrote: > In-Reply-To: <9pbgq1+86qa@e...> > Gene wrote: > > Here are relativized n-consistent goodness measures for odd n to 25, > > for 41, 46 and 58: > How are these calculated? It looks like lower numbers are better, so it's > really a badness measure. This is the same w-consistent measure cons(w, n) introduced in Yahoo groups: /tuning-math/message/860 * And yes, it is a badness measure, but you could always take the reciprocal. :)
top of page bottom of page up down Message: 4704 Date: Tue, 02 Oct 2001 16:22:07 Subject: How consistent are cents? From: genewardsmith@xxxx.xxx If you want an idea of what a randomly chosen consistent badness measure looks like, chew on this: Cons(w, 1200) for w from 2 to 49 3 53.998800 5 12.42619668 7 5.183728199 9 5.183728199 11 2.895954014 13 3.263002903 15 3.263002903 17 2.576191158 19 2.216354953 21 2.216354953 23 1.952782773 25 2.703605969 27 2.703605969 29 2.450061526 31 2.264455891 33 2.264455891 35 2.264455891 37 2.123104197 39 2.180931116 41 2.066877882 43 1.975042642 45 1.975042642 47 1.899581355 49 1.899581355
top of page bottom of page up down Message: 4705 Date: Tue, 2 Oct 2001 17:45 +01 Subject: Re: 41, 46 and 58 From: graham@xxxxxxxxxx.xx.xx In-Reply-To: <9pcpdn+pinr@xxxxxxx.xxx> > Gene wrote: > This is the same w-consistent measure cons(w, n) introduced in > Yahoo groups: /tuning-math/message/860 * > And yes, it is a badness measure, but you could always take the > reciprocal. :) And "less than or equal to l" should be "less than or equal to w". I was wondering how a prime could be less than 1. It looks like h(q_i) is the number of steps in the ET for the ratio q_i. So h(2) is the number of steps to the octave. So cons(w, n) assumes consistency which -- warning! -- doesn't hold for 46-equal in the 15-limit, because there's one ambiguous interval. This formula: n^(1/d) * max(abs(n*log_2(q_i) - h(q_i)) has a parenthesis unclosed. From the one lower down, I think it should be n^(1/d) * max(abs(n*log_2(q_i)) - h(q_i)) The n^(1/d) means you're scaling according to the size of the octave and the number of prime dimensions. Then you're taking the largest deviation for any interval within the limit, right? In which case, the parenthesising should be n^(1/d) * max(abs(n*log_2(q_i) - h(q_i))) and the other formula must be wrong. That could be re-written n^(1/d) * n * max(|tempered_pitch - just_pitch|) where pitches are in octaves. Which can be simplified to n^(1+1/d) * max(|tempered_pitch - just_pitch|) Is that right? I think I almost understand it! So why the 1+1/d? Graham
top of page bottom of page up down Message: 4706 Date: Tue, 02 Oct 2001 17:40:39 Subject: Re: Paul blocks From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote: > I see I was misreading Paul's construction, which actually only > allows for an extra comma which is either the sum or difference of > the other commas. This is a very nice idea, and can be formalized in > the same way as the Fokker block, via a norm. > > If we have a Fokker-type norm in the 5-limit, which is > 1/n [hn, v1, v2], where hn is an n-et and the other are defined on > octave equivalence classes, then instead of taking the maximum of the > absolute values of these valuations, we can take instead the maximum > of the absolute value of |hn(q)| together with the median of > {|v1(q)|, |v2(q)|, |v1(q)-v2(q)|}, or else > {|v1(q), |v2(q)|, |v1(q)+v2(q)|}--in other words, we sort three > absolute values, and take the one in the middle. This also gives us a > norm, and we can then define Paul blocks in the 5-limit in the same > way as Fokker blocks. To generalize to higher dimensions, it seems we > would need to take combinations of n vals k at a time, for k from 1 > to n, giving us the verticies of the n-measure polytope (=hypercube.) > We then would take a maximum over the n smallest. At least, that > seems right, but I haven't really thought it through carefully. I wish I could understand this. Can I ask you, what were you misreading, and what led you to a correct reading?
top of page bottom of page up down Message: 4707 Date: Tue, 02 Oct 2001 17:43:25 Subject: Re: 41, 46 and 58 From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote: > 41 is clearly an > excellent system for 7, 9 or 11, and deserves some respect! Oh yes . . . but it's mentioned far more than 46 in the literature, which is why I suspect Robert Valentine was puzzled about the 46 hoopla.
top of page bottom of page up down Message: 4708 Date: Tue, 02 Oct 2001 17:45:55 Subject: Re: 41, 46 and 58 From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote: > --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > > --- In tuning-math@y..., graham@m... wrote: > > > Bob Valentine wrote: > > > > > What does 46 have that everyone seems gaga about? Is 46 > > > > better at 13 > > > Yes, but I'm not planning on using it for 13-limit. More important > is > > that 46 is better than 41 in 5-limit (which is the main interval > > flavor for Indian music). Though 34 is better still in the 5- limit, > > 46 allows me to access 7- and 11-flavors with much higher accuracy. > > Here are relativized n-consistent goodness measures for odd n to 25, > for 41, 46 and 58: > > 41: > > 3 .67803586 > 5 1.380445520 > 7 .7433989824 > 9 .8004237371 > 11 .9169187332 > 13 1.010934526 > 15 1.010934526 > 17 1.138442704 > 19 1.042105199 > 21 1.042105199 > 23 1.399948567 > 25 1.399948567 > > 46: > > 3 4.21934770 > 5 1.297511950 > 7 1.181094581 > 9 1.181094581 > 11 .8584628865 > 13 .8850299838 > 15 1.082290028 > 17 .9526171496 > 19 1.188323665 > 21 1.188323665 > 23 1.109794688 > 25 1.270499309 > > 58: > > 3 4.18614652 > 5 2.499272068 > 7 1.270307523 > 9 1.270307523 > 11 .9740696116 > 13 .8435935409 > 15 .9018214272 > 17 .9309801788 > 19 1.393450457 > 21 1.393450457 > 23 1.295990287 > 25 1.721256452 > > We see that 41 has values less than 1 in the 3, 7, 9 and 11 limits; > 46 in 11, 13 and 17; and 58 in 11, 13, 15, and 17. 41 is clearly an > excellent system for 7, 9 or 11, and deserves some respect! Gene, I don't know what you mean by n-consistent here. 46 is only consistent through the 13-limit, so it's inconsistent in the 17- limit. So how can 46 have a finite value for "17-consistent goodness"?
top of page bottom of page up down Message: 4709 Date: Tue, 02 Oct 2001 17:47:26 Subject: Re: How consistent are cents? From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote: > If you want an idea of what a randomly chosen consistent badness > measure looks like, chew on this: > > Cons(w, 1200) for w from 2 to 49 > > 3 53.998800 > 5 12.42619668 > 7 5.183728199 > 9 5.183728199 > 11 2.895954014 > 13 3.263002903 > 15 3.263002903 > 17 2.576191158 > 19 2.216354953 > 21 2.216354953 > 23 1.952782773 > 25 2.703605969 > 27 2.703605969 > 29 2.450061526 > 31 2.264455891 > 33 2.264455891 > 35 2.264455891 > 37 2.123104197 > 39 2.180931116 > 41 2.066877882 > 43 1.975042642 > 45 1.975042642 > 47 1.899581355 > 49 1.899581355 Again, I don't know what you mean by consistent here. 1200-tET is consistent through the 9-limit, but not the 11-limit.
top of page bottom of page up down Message: 4710 Date: Tue, 02 Oct 2001 17:49:10 Subject: Re: Digest Number 124 From: Paul Erlich Gene -- do you have any response here? Have you been misunderstanding something all along? --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > --- In tuning-math@y..., genewardsmith@j... wrote: > > > The first system has jargon wherein 81/80 and 25/24 are "commatic > > unison vectors" and 16/15 is a "chromatic unison vector" in a > > situation where we are seeking a 7-note "periodiity block" scale; > > Something's wrong here . . . in a 7-tone PB, specificially the > diatonic scale, 81:80 is the commatic unison vector, 25:24 is a > chromatic unison vector, and 16:15 is not a unison vector at all, but > a "step vector". > > > No; to say the scale is "defined" by the fact that 16/15 is > > a "chromatic unison vector" (something of a misnomer, I fear) and > > 81/80 and 25/24 are "commatic unison vectors" > > Misnomer because it's incorrect!
top of page bottom of page up down Message: 4711 Date: Tue, 02 Oct 2001 03:32:53 Subject: Paul blocks From: genewardsmith@xxxx.xxx I see I was misreading Paul's construction, which actually only allows for an extra comma which is either the sum or difference of the other commas. This is a very nice idea, and can be formalized in the same way as the Fokker block, via a norm. If we have a Fokker-type norm in the 5-limit, which is 1/n [hn, v1, v2], where hn is an n-et and the other are defined on octave equivalence classes, then instead of taking the maximum of the absolute values of these valuations, we can take instead the maximum of the absolute value of |hn(q)| together with the median of {|v1(q)|, |v2(q)|, |v1(q)-v2(q)|}, or else {|v1(q), |v2(q)|, |v1(q)+v2(q)|}--in other words, we sort three absolute values, and take the one in the middle. This also gives us a norm, and we can then define Paul blocks in the 5-limit in the same way as Fokker blocks. To generalize to higher dimensions, it seems we would need to take combinations of n vals k at a time, for k from 1 to n, giving us the verticies of the n-measure polytope (=hypercube.) We then would take a maximum over the n smallest. At least, that seems right, but I haven't really thought it through carefully.
top of page bottom of page up down Message: 4712 Date: Tue, 02 Oct 2001 18:10:24 Subject: Re: Paul blocks From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > I wish I could understand this. Can I ask you, what were you > misreading, and what led you to a correct reading? It needs fixing anyway, so I will try again later. What I had thought you were saying would have been equivalent to saying that after a linear transformation of the parallepiped into a hypercube of measure 1, we allow ourselves to chop up the hypercube and reassemble it into a convex body, also of measure 1 (no Banach-Tarski, please!) This tiles the n-space, but it allows us too much latitude. Instead, you were putting restrictions on how the square could be chopped up and reassembled, and we should presumably have some restrictions also if we generalize this. I'll try to work the mess I posted out in a way which makes more sense, but I need to define the norm by first creating a region, and then defining the norm of a point by scaling the region and finding what scale factor makes the point lie on the boundry. We could try transforming back to a regular hexagon instead, but then I still want to know how to generalize it. Would the 3-D version be the bee-type honeycomb of rhombic dodecahedra?
top of page bottom of page up down Message: 4713 Date: Tue, 02 Oct 2001 03:54:46 Subject: Re: Some "ABC good" intervals From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > --- In tuning-math@y..., genewardsmith@j... wrote: > > I found a web page, ABC-Ratios >1.4 *, > with > > a list of all 148 known "good" ABCs according to the definition you > > will find there of "good" (there are others.) > I don't see a definition there of "good", and most of these are not > superparticular ratios at all . . . so what, really, is the ABC > conjecture? If we have three relatively prime positive integers A, B, C such that A + B = C, and if we define a "radical" function rad(N) as the product of the primes dividing N, we can look at C/rad(ABC); it turns out that this can be arbitarily large. The ABC conjecture says that C/rad(ABC)^e, for any e>1, cannot become arbitrarily large. It's a conjecture in elementary number theory with a large number of very powerful and not always elementary consequences, and since it is the latest fad I thought of it when I saw that the list of m/n in the p- limit, with |m-n|<d, should be finite. It seems it is also an easy consequence of Baker's theorem, and hence is true--a useful thing to know. Any ABC triple such that ln(C)/ln(rad(C)) > 1.4 is rather arbitarily termed "good"; they are pretty rare and some of them turn up in music theory, it seems.
top of page bottom of page up down Message: 4714 Date: Tue, 02 Oct 2001 18:27:33 Subject: Re: Paul blocks From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote: > --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > > > I wish I could understand this. Can I ask you, what were you > > misreading, and what led you to a correct reading? > > It needs fixing anyway, so I will try again later. What I had thought > you were saying would have been equivalent to saying that after a > linear transformation of the parallepiped into a hypercube of measure > 1, we allow ourselves to chop up the hypercube and reassemble it into > a convex body, also of measure 1 (no Banach-Tarski, please!) This > tiles the n-space, but it allows us too much latitude. Too much latitude? Why? > > Instead, you were putting restrictions on how the square could be > chopped up and reassembled, I was? > We could try transforming back to a regular hexagon instead, but then > I still want to know how to generalize it. Why a regular hexagon? And in what version of the lattice? No, I see a desirable class of 5-limit periodicity blocks defined as a hexagon of any shape, as long as opposite sides are parallel and congruent. > Would the 3-D version be > the bee-type honeycomb of rhombic dodecahedra? Yes, but they certainly don't have to be regular. Also, they can be "degenerate", with certain faces vanishing or combining with other faces, so that one ends up with hexagonal prisms or parallelepipeds.
top of page bottom of page up down Message: 4715 Date: Tue, 02 Oct 2001 04:06:18 Subject: Re: Some "ABC good" intervals From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > I don't see a definition there of "good", and most of these are not > superparticular ratios at all . . . so what, really, is the ABC > conjecture? If we have three relatively prime positive integers A, B, C such that A + B = C, and if we define a "radical" function rad(N) as the product of the primes dividing N, we can look at C/rad(ABC); it turns out that this can be arbitarily large. The ABC conjecture says that C/rad(ABC)^e, for any e>1, cannot become arbitrarily large. It's a conjecture in elementary number theory with a large number of very powerful and not always elementary consequences, and since it is the latest fad I thought of it when I saw that the list of m/n in the p-limit, with |m-n|<d, should be finite. It seems it is also an easy consequence of Baker's theorem, and hence is true--a useful thing to know. Any ABC triple such that ln(C)/ln(rad(ABC)) > 1.4 is rather arbitarily termed "good"; they are pretty rare and some of them turn up in music theory, it seems. For superparticular ratios this measure becomes ln(C)/ln(rad(BC)), and looking at the "goodness" of our favorite commas might be interesting, I suppose.
top of page bottom of page up down Message: 4716 Date: Tue, 02 Oct 2001 19:18:18 Subject: Re: Digest Number 124 From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > Gene -- do you have any response here? Have you been misunderstanding > something all along? It's the same thing which made it so hard for me to get the terminology in the first place--in the tempered situation, we have *two* scale-step vals. For instance, we could have h12 and h7, and then we would have h12(16/15) = h12(25/24) = 1, but h7(16/15) = 1 and h7(25/24) = 0. So 25/24 is a "unison" according to h7 but it is also a "step" on the piano keyboard--according to h12. We have (16/15, 25/24, 81/80)^(-1) = [h7, h5, h3]. For the purpose of constructing a JI scale approximating to h7, 16/15 is a step and 25/24 is a comma. If we temper out 81/80, we can pick an et of the form n h7 + m h5, and then in that et 16/15 will be n intervals and 25/24 will be m intervals, but in the scale 16/15 is still one step and 25/24 not an allowed step; it is still a unison so far as h7 is concerned.
top of page bottom of page up down Message: 4717 Date: Tue, 02 Oct 2001 04:45:53 Subject: 41, 46 and 58 From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > --- In tuning-math@y..., graham@m... wrote: > > Bob Valentine wrote: > > > What does 46 have that everyone seems gaga about? Is 46 > > > better at 13 > Yes, but I'm not planning on using it for 13-limit. More important is > that 46 is better than 41 in 5-limit (which is the main interval > flavor for Indian music). Though 34 is better still in the 5-limit, > 46 allows me to access 7- and 11-flavors with much higher accuracy. Here are relativized n-consistent goodness measures for odd n to 25, for 41, 46 and 58: 41: 3 .67803586 5 1.380445520 7 .7433989824 9 .8004237371 11 .9169187332 13 1.010934526 15 1.010934526 17 1.138442704 19 1.042105199 21 1.042105199 23 1.399948567 25 1.399948567 46: 3 4.21934770 5 1.297511950 7 1.181094581 9 1.181094581 11 .8584628865 13 .8850299838 15 1.082290028 17 .9526171496 19 1.188323665 21 1.188323665 23 1.109794688 25 1.270499309 58: 3 4.18614652 5 2.499272068 7 1.270307523 9 1.270307523 11 .9740696116 13 .8435935409 15 .9018214272 17 .9309801788 19 1.393450457 21 1.393450457 23 1.295990287 25 1.721256452 We see that 41 has values less than 1 in the 3, 7, 9 and 11 limits; 46 in 11, 13 and 17; and 58 in 11, 13, 15, and 17. 41 is clearly an excellent system for 7, 9 or 11, and deserves some respect!
top of page bottom of page up down Message: 4718 Date: Tue, 02 Oct 2001 19:35:36 Subject: Re: Digest Number 124 From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote: > --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > > > Gene -- do you have any response here? Have you been > misunderstanding > > something all along? > > It's the same thing which made it so hard for me to get the > terminology in the first place--in the tempered situation, we have > *two* scale-step vals. For instance, we could have h12 and h7, and > then we would have h12(16/15) = h12(25/24) = 1, but h7(16/15) = 1 and > h7(25/24) = 0. So 25/24 is a "unison" according to h7 but it is also > a "step" on the piano keyboard--according to h12. What if we substituted the word "second" instead of "step"? Better yet, can we avoid bringing 12 into this at all? I see no reason it should be brought in. > We have (16/15, 25/24, 81/80)^(-1) = [h7, h5, h3]. For the purpose of > constructing a JI scale approximating to h7, 16/15 is a step and > 25/24 is a comma. If we temper out 81/80, we can pick an et of the > form n h7 + m h5, and then in that et 16/15 will be n intervals and > 25/24 will be m intervals, but in the scale 16/15 is still one step > and 25/24 not an allowed step; it is still a unison so far as h7 is > concerned. Right . . .
top of page bottom of page up down Message: 4719 Date: Tue, 2 Oct 2001 21:48:50 Subject: Miracle web page From: Graham Breed I've added a page to my website on Miracle temperament. <Miracle temperaments *>. It's not connected up yet, I'll try and sort that out tomorrow. Graham "I toss therefore I am" -- Sartre
top of page bottom of page up down Message: 4720 Date: Tue, 02 Oct 2001 20:54:07 Subject: Re: Miracle web page From: Paul Erlich --- In tuning-math@y..., Graham Breed <graham@m...> wrote: > I've added a page to my website on Miracle temperament. Coincidentally, I just posted (to the tuning list) the design of a MIRACLE guitar fingerboard. It's actually the Canasta scale in 72- tET, centered on A, with the open strings tuned conventionally in 12- tET. Feel free to reproduce this on your website . . .
top of page bottom of page up down Message: 4721 Date: Tue, 02 Oct 2001 21:29:04 Subject: Re: Miracle web page From: genewardsmith@xxxx.xxx --- In tuning-math@y..., Graham Breed <graham@m...> wrote: > "I toss therefore I am" -- Sartre Are you sure that isn't "I am therefore I toss"? My favorite is "Slime is the agony of water", from Being and Nothingness.
top of page bottom of page up down Message: 4722 Date: Tue, 02 Oct 2001 21:26:55 Subject: Re: Digest Number 124 From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > Better yet, can we avoid bringing 12 into this at all? I see no > reason it should be brought in. Since we aren't tempering, leaving it out is the thing to do.
top of page bottom of page up down Message: 4723 Date: Tue, 02 Oct 2001 21:41:47 Subject: Re: Digest Number 124 From: Paul Erlich --- In tuning-math@y..., genewardsmith@j... wrote: > --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > > > Better yet, can we avoid bringing 12 into this at all? I see no > > reason it should be brought in. > > Since we aren't tempering, leaving it out is the thing to do. Well . . . 81:80 being a commatic unison vector sort of implies tempering . . . but in no way selects 12 over other meantones like 19, 31, 43, 55, or 74.
top of page bottom of page up down Message: 4724 Date: Tue, 02 Oct 2001 21:57:52 Subject: Re: Digest Number 124 From: genewardsmith@xxxx.xxx --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > Well . . . 81:80 being a commatic unison vector sort of implies > tempering . . . but in no way selects 12 over other meantones like > 19, 31, 43, 55, or 74. Those are all of the form n h7 + m h5, which is what I was talking about.
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