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Message: 4526 - Contents - Hide Contents Date: Sun, 7 Apr 2002 03:05:00 Subject: Re: Blocks and convexity From: Pierre Lamothe I wrote: Don't forget to press "enter" or use "play" to start the animation. It would have been useful with attached image but it's useless with http since the plug-in start the animation and don't show a menu. Pierre [This message contained attachments]
Message: 4529 - Contents - Hide Contents Date: Sun, 07 Apr 2002 03:59:23 Subject: Re: A common notation for JI and ETs From: David C Keenan George, Here's another pass at a full set of 31-limit symbols, taken simply as one symbol per prime from 5 to 31. Whadya think? [If you're reading this on the yahoogroups website you will need to choose Message Index, Expand Messages, to see the following symbols rendered correctly.] 5-comma 80:81 /| / | | \ / | | 7-comma 63:64 _ | \ | | | L P | | 11-comma 32:33 /|\ / | \ | v ^ | | 13-comma 1024:1053 _ /| \ / | | | { } flags based on vanishing of schisma 4095:4096 | | 17-comma 2176:2187 | _/| | j f | | 19-comma 512:513 _ (_) | | o * | | 23-comma 729:736 | |\_ | w m | | 29-comma 256:261 _ / | | | | q d flag based on vanishing of schisma 20735:20736 | | 31-comma 243:248 _ (_) | \ | y h flags based on vanishing of schisma 253935:253952 | | We also have optional symbols for larger 11, 13 and 23 commas. 11'-comma 704:729 _ _ / | \ | | | | [ ] flags based on vanishing of schisma 5103:5104 | | 13'-comma 26:27 _ / |\ | | \ | ; | flags based on vanishing of schisma 20735:20736 | | 23'-comma 16384:16767 |\ _/| \ | W M flags based on vanishing of schisma 3519:3520 | | -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 4532 - Contents - Hide Contents Date: Sun, 07 Apr 2002 22:16:37 Subject: Re: question for my more learned friends From: genewardsmith --- In tuning-math@y..., Robert C Valentine <BVAL@I...> wrote:> > Probably more of a set theory question, I'll try to phrase what > I was looking for such that a mathematician might be able to come > up with an answer.It's a combinatorics question, and has a vague relationship to the theory of cyclic difference sets. Mod 7, {1,2,4} will give you all the non-zero elements exactly once: 2-1=1, 4-2=2, 4-1=3, and then the negatives of those. This makes {1,2,4} a (7,3,1)-difference set. That's not what you were looking for, of course, but it seems to have the same theme.
Message: 4533 - Contents - Hide Contents Date: Sun, 07 Apr 2002 22:49:24 Subject: Re: A common notation for JI and ETs From: dkeenanuqnetau Or perhaps the 19 and 31 commas should be: 19-comma 512:513 _ (_) | | | | and 31-comma 243:248 _ (_)\ | \ | | | or 31-comma 243:248 _ (_) |\ | \ | | The circle was always intended to be filled, and is now a kind of left flag rather than central. This eliminates a lot of possible redundant combinations, and the attendant lateral confusability, by making it only combinable with right flags. It is also nice that the 17 and 19 flags look a little like the digits 7 and 9 respectively.
Message: 4535 - Contents - Hide Contents Date: Mon, 08 Apr 2002 07:18:50 Subject: Re: question for my more learned friends From: genewardsmith --- In tuning-math@y..., Robert C Valentine <BVAL@I...> wrote:> > Probably more of a set theory question, I'll try to phrase what > I was looking for such that a mathematician might be able to come > up with an answer.By the way, I find Math World has a page for perfect difference sets: Perfect Difference Set -- from MathWorld * [with cont.] If q is a prime power, then there is such a beast for q^2+q+1, corresponding to the finite projective plane over Fq with q^2+q+1 points and lines. Now that I've actually read your question, I see that in fact this whole business *is* closely connected to cyclic difference sets and modular Golomb rulers. The 7-et solution you found is from q=2, and the 13-et is from q=3. The projective plane corresponding to q=4 gives a 21-et solution, and q=5 a 31-et solution, which is what you were looking for.
Message: 4536 - Contents - Hide Contents Date: Mon, 08 Apr 2002 07:35:06 Subject: Program to construct Golomb rulers from projective planes From: genewardsmith cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c Program to construct Golomb rulers from projective planes c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c IBM SOFTWARE DISCLAIMER c c conpp.f (version 1.1) c Copyright (1998,1986) c International Business Machines Corporation c c Permission to use, copy, modify and distribute this software for c any purpose and without fee is hereby granted, provided that this c copyright and permission notice appear on all copies of the software. c The name of the IBM corporation may not be used in any advertising or c publicity pertaining to the use of the software. IBM makes no c warranty or representations about the suitability of the software c for any purpose. It is provided "AS IS" without any express or c implied warranty, including the implied warranties of merchantability, c fitness for a particular purpose and non-infringement. IBM shall not c be liable for any direct, indirect, special or consequential damages c resulting from the loss of use, data or projects, whether in action c of contract or tort, arising out or in the connection with the use or c performance of this software. c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c Author: James B. Shearer c email: jbs@xxxxxx.xxx.xxx c website: James B. Shearer's home page * [with cont.] (Wayb.) c date: 1998 (based on code written in 1986) c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c Version 1.1 (12/11/98) - Renamed variables to conform with c exhaustive search routines. c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c Requires: ESSL library or portable versions of ESSL routines c durand, isort (see essl.f) c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c This program constructs good but not necessarily optimal c Golomb rulers from finte projective planes. c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c Theory c c Suppose p is a prime power. Consider the Galois field GF(p) and c the extension field GF(p**3). Let x be a generator of the cyclic c multiplicative group of GF(p**3). Then the elements GF(p**3) can c be represented in the form a+b*x+c*x**2 where a,b,c are elements of c GF(p). (Note in particular x**3 can be so written, so we may take c x to be the root of a cubic polynomial over GF(p).) Let two non- c zero elements, {y,z}, of GF(p**3) be equivalent if one is a scalar c of the other (ie y/z is an element of the base field GF(p)). This c partitions the p**3-1 nonzero elements of GF(p**3) into p**2+p+1 c classes (of size p-1). As is well known these classes can be c thought of as the points of a finite projective plane. Consider c such a point consisting of the class {y1, y2 ...}. Let y1=x**n1, c y2=x**n2 ... . We claim n1=n2 mod (p**2+p+1). (Because the c elements of the base field are generated by x**(p**2+p+1).) Hence c it is easy to see that we can associate each point of the plane c with an unique residue mod p**2+p+1. Consider the residues c associated with the p+1 points on a line in the projective plane. c We claim these p+1 residues form a distinct difference set mod c p**2+p+1. Consider for example the points (a+b*x+c*x**2) with c third coordinate (c) zero. There are p+1 such points which we c can take to be a+x (a in GF(p)) and 1. Suppose the associated c residues are not a modular distinct difference set. Then we c would have for example (a+x)/(b+x)=e*(c+x)/(d+x) (a,b,c,d,e in c GF(p)). But then x**2+(a+d)*x+a*d=e*(x**2+(b+c)*x+b*c). Or c equating powers of x, e=1, a+d=b+c, a*d=b*c. So {a,d}={b,c} c (since they are roots of the same quadratic polynomial). The c claim follows by contradiction. The other cases involving the c point 1 are similar. c Modular distinct difference sets can be unwound and truncated c to form Golomb rulers. Note we may multiply a modular distinct c difference set by anything prime to the modulus to obtain another c modular distinct difference set. The program below tests all c possibilities to obtain the shortest Golomb rulers. c The modular difference set construction is due to Singer [2], c the application to Golomb rulers to Robinson and Bernstein [1]. c The program below finds the best Golomb rulers using this c construction for prime powers up to maxn-1. It will start to c fail as maxn**4 overflows integer*4 arithmetic (loop 230). A c program which just handled primes and not prime powers would be c simpler. cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c References c c 1. J. P. Robinson and A. J. Bernstein, "A class of binary recurrent c codes with limited error propagation", IEEE Transactions on c Information Theory, IT-13(1967), p. 106-113. c 2. J. Singer, "A theorem in finite projective geometry and some c applications to number theory", Transactions American c Mathematical Society, 43(1938), p.377-385. c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc Parameter (maxn=160,maxpow=10) integer*4 len(maxn),nval(maxn),mrec(maxn,maxn) integer*4 ids(maxn) integer*4 mw(2*maxn) integer*4 ipc(3*maxpow),iit(3*maxpow),itemp(3*maxpow) real*8 buf(3*maxpow) integer*4 ibase(3*maxpow,2*maxpow),iperp(3*maxpow,maxpow) integer*4 irep(maxn),ipiv(maxn) c initialize best rulers so far do 5 j=2,maxn len(j)=maxn*maxn nval(j)=0 5 continue c loop over n do 10 n=2,maxn-1 c check if n is a prime power c first find smallest prime divisor do 20 j=2,n if(mod(n,j).eq.0)go to 30 20 continue stop "error 20" 30 np=j npow=1 nprod=j c next check if n is a power of the smallest prime divisor 40 if(nprod.eq.n)go to 50 nprod=nprod*np npow=npow+1 if(nprod.le.n)go to 40 c n is not a prime power, go to next n go to 10 c n is a prime power, construct GF(n**3)=GF(np**ndeg) 50 ndeg=3*npow c generate random coefficients for monic polynomial P, c with degree ndeg over GF(np) 60 call irand(ipc,np,ndeg,buf) c check if constant term is 0, if so generate another polynomial if(ipc(1).eq.0)go to 60 c check if x is a multiplicative generator mod P c initialize iit (x**0) to the unit vector do 70 j=1,ndeg iit(j)=0 70 continue iit(1)=1 c generate powers of x do 80 j=1,n*n*n-2 c multiply iit by x itemp(1)=ipc(1)*iit(ndeg) do 90 i=2,ndeg itemp(i)=iit(i-1)+ipc(i)*iit(ndeg) 90 continue do 100 i=1,ndeg iit(i)=mod(itemp(i),np) 100 continue c check if power of x is 1 prematurely if(iit(1).ne.1)go to 80 do 110 i=2,ndeg if(iit(i).ne.0)go to 80 110 continue c this polynomial no good, go generate another go to 60 80 continue c All powers of x ok. We now have a representation of GF(n**3) as c arithmetic mod a polynomial over GF(np) c The field GF(n**3) is an extension of the field GF(n). We need c a way of identifying elements of the subspace generated by GF(n) c and x. This is easy when n is a prime rather than a prime power, c these are the elements with x**2 term 0. The prime power case is c more complicated, we find a linear basis for the space and then c for the perpendicular space. We can then test by looking at c inner products with the basis of the perpendicular space. c First npow powers of x**(n**2+n+1) span GF(n) c Set first element of GF(n) basis to 1 do 85 j=1,ndeg ibase(j,1)=0 85 continue ibase(1,1)=1 c find remaining elements of basis ibp=n**2+n+1 c initialize iit (x**0) to the unit vector do 120 j=1,ndeg iit(j)=0 120 continue iit(1)=1 c generate powers of x do 125 ja=2,npow do 130 j=1,ibp c multiply iit by x itemp(1)=ipc(1)*iit(ndeg) do 140 i=2,ndeg itemp(i)=iit(i-1)+ipc(i)*iit(ndeg) 140 continue do 150 i=1,ndeg iit(i)=mod(itemp(i),np) 150 continue 130 continue c add to basis vectors do 160 j=1,ndeg ibase(j,ja)=iit(j) 160 continue 125 continue c now find basis of space generated by GF(n) and x. c generate n additional basis vectors by multiplying first n vectors by x do 170 ja=1,npow itemp(1)=ipc(1)*ibase(ndeg,ja) do 180 i=2,ndeg itemp(i)=ibase(i-1,ja)+ipc(i)*ibase(ndeg,ja) 180 continue do 190 i=1,ndeg ibase(i,npow+ja)=mod(itemp(i),np) 190 continue 170 continue c we now generate a table of inverses to help us do arithmetic in GF(np) do 200 j=1,np-1 do 210 i=1,np-1 if(mod(i*j,np).ne.1)go to 210 irep(j)=i go to 200 210 continue stop "error 210" 200 continue c put basis in a normal form. do 220 j=1,2*npow c find first non-zero in column j do 230 i=1,ndeg if(ibase(i,j).ne.0)go to 240 230 continue stop "error 230" c record position of first non-zero 240 ipiv(j)=i c multiply column so non-zero becomes 1 imult=irep(ibase(i,j)) do 250 i=1,ndeg ibase(i,j)=mod(ibase(i,j)*imult,np) 250 continue c zero remaining elements in row do 260 ja=1,2*npow if(ja.eq.j)go to 260 imult=ibase(ipiv(j),ja) do 270 i=1,ndeg ibase(i,ja)=mod(ibase(i,ja)-imult*ibase(i,j),np) if(ibase(i,ja).lt.0)ibase(i,ja)=ibase(i,ja)+np 270 continue 260 continue 220 continue c now construct perpendicular basis jp=0 do 280 j=1,ndeg c look for row not among the 2*npow recorded in ipiv do 290 i=1,2*npow if(ipiv(i).eq.j)go to 280 290 continue jp=jp+1 c zero jp'th vector in perpendicular basis do 300 i=1,ndeg iperp(i,jp)=0 300 continue c fill in elements in ipiv rows so as to make inner products 0 do 310 ja=1,2*npow iperp(ipiv(ja),jp)=ibase(j,ja) 310 continue c note -1 = np - 1 mod np iperp(j,jp)=np-1 280 continue c construct perfect difference set mod n**2+n+1 c look for powers of x in space spanned by GF(n), x c put 0 in set ids(1)=0 nc=1 c initialize iit do 320 j=1,ndeg iit(j)=0 320 continue iit(1)=1 c look for powers of x in space spanned by GF(n), x do 330 j=1,n*n+n c multiply by x itemp(1)=ipc(1)*iit(ndeg) do 340 i=2,ndeg itemp(i)=iit(i-1)+ipc(i)*iit(ndeg) 340 continue do 350 i=1,ndeg iit(i)=mod(itemp(i),np) 350 continue c check inner products do 360 ja=1,npow iip=0 do 370 i=1,ndeg iip=iip+iperp(i,ja)*iit(i) 370 continue iip=mod(iip,np) if(iip.ne.0)go to 330 360 continue nc=nc+1 ids(nc)=j 330 continue c check that difference set is right size if(nc.ne.n+1)stop "error 330" c output difference set c write(6,1000)n,(ids(j),j=1,n+1) c1000 format(1x,i5,5x,(10i5)) c check for better rulers c cycle over multipliers, don't need to try j and -j mod (n*n+n+1) c do 420 j=1,n*n+n do 420 j=1,(n*n+n+1)/2 c check if multiplier prime to modulus if(igcd(j,n*n+n+1).ne.1)go to 420 c multiply difference set do 430 i=1,n+1 mw(i)=mod(ids(i)*j,n*n+n+1) 430 continue c sort new difference set call isort(mw,1,n+1) c unwrap difference set do 440 i=1,n+1 mw(i+n+1)=mw(i)+n*n+n+1 440 continue c check for new records do 450 ia=1,n+1 do 460 ib=1,n if(mw(ia+ib)-mw(ia).ge.len(ib+1))go to 460 c new record ruler len(ib+1)=mw(ia+ib)-mw(ia) nval(ib+1)=n do 470 ja=1,ib+1 mrec(ja,ib+1)=mw(ia+ja-1)-mw(ia) 470 continue 460 continue 450 continue 420 continue 10 continue c output maximum rulers do 500 j=2,maxn if(nval(j).eq.0)go to 500 c put ruler in standard form (flip if needed) if(mrec((j+1)/2,j)+mrec((j+2)/2,j).lt.len(j))go to 520 c flip ruler do 510 i=1,(j+1)/2 mtemp=mrec(i,j) mrec(i,j)=len(j)-mrec(j+1-i,j) mrec(j+1-i,j)=len(j)-mtemp 510 continue 520 continue c c write results for j marks c c j,length and prime power to unit 6 (terminal) c j,length and prime power to unit 1 (disk) c ruler to unit 1 (disk) c write(6,1010)j,len(j),nval(j) write(1,1020)j,len(j),nval(j) write(1,1030)(mrec(i,j),i=1,j) 1010 format(1x,3i10) 1020 format(3i10) 1030 format(10i6) 500 continue c mark end of disk file write(1,1020)0,0,0 stop end c generate random vector of n integers 0,...,np-1 subroutine irand(l,np,n,x) integer*4 l(*) real*8 x(*) real*8 dseed/1.d0/ save dseed c generate random 0-1 real*8 vector call durand(dseed,n,x) c convert to integer 0,...,np-1 do 10 j=1,n l(j)=np*x(j) 10 continue return end c find gcd of ia,ib using Euler's method function igcd(ia,ib) ja=ia jb=ib 1 jc=mod(ja,jb) ja=jb jb=jc if(jb.ne.0)go to 1 igcd=ja return end
Message: 4537 - Contents - Hide Contents Date: Mon, 08 Apr 2002 08:05:07 Subject: Re: Program to construct Golomb rulers from projective planes From: genewardsmith Here's output from the program. There are alternating lines; first the number of marks, length and prime power are given, then the modular ruler. So we have 3 3 2, meaning 3 notes making up 3 scale steps, using p=2 (which implies n=2^2+2+1=7, the 7-et.) Before that is the truncated ruler of 2 marks on a ruler of length 1 obtained from it. We have a {0,1,4,6} ruler with p=3 and n=13, a {0,3,4,9,11} ruler with p=4 and n=21, a {0,1,4,10,12,17} ruler with p=5 and n=31, and so forth. 2 1 2 0 1 3 3 2 0 1 3 4 6 3 0 1 4 6 5 11 4 0 3 4 9 11 6 17 5 0 1 4 10 12 17 7 28 7 0 3 9 11 23 24 28 8 35 7 0 7 10 16 18 30 31 35 9 45 8 0 3 9 16 20 21 35 43 45 10 55 9 0 1 6 10 23 26 34 41 53 55 11 72 11 0 1 9 19 24 31 52 56 58 69 72 12 85 11 0 2 6 24 29 40 43 55 68 75 76 85 13 114 13 0 3 7 18 20 39 51 61 77 85 86 91 114 14 127 13 0 5 28 38 41 49 50 68 75 92 107 121 123 127 15 155 17 0 7 13 16 30 38 50 77 96 98 122 140 150 151 155 16 179 17 0 9 21 43 47 61 66 67 96 103 135 151 166 168 176 179 17 201 16 0 5 15 34 35 42 73 75 86 89 98 134 151 155 177 183 201 18 216 17 0 2 10 22 53 56 82 83 89 98 130 148 153 167 188 192 205 216 19 246 19 0 4 13 15 42 56 59 77 93 116 126 138 146 174 214 221 240 245 246 20 283 19 0 24 30 43 55 71 75 89 104 125 127 162 167 189 206 215 272 275 282 283 21 333 23 0 4 23 37 40 48 68 78 138 147 154 189 204 238 250 251 256 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Message: 4538 - Contents - Hide Contents Date: Mon, 08 Apr 2002 09:35:14 Subject: Ets with good Golomb rulers From: genewardsmith Modular Golomb rulers * [with cont.] (Wayb.) This mentions three constructions for modular Golomb rulers--the projective plane one, for the q^2+q+1 ets, an affine plane one for q^2-1 ets, and one of size q^2-q, constructed I know not how. Putting together all of these for primes and prime powers with the result less than 1000, I got the following list: 2, 3, 6, 7, 8, 12, 13, 15, 20, 21, 24, 31, 42, 48, 56, 57, 63, 72, 73, 80, 91, 110, 120, 133, 156, 168, 183, 240, 255, 272, 273, 288, 307, 342, 360, 381, 506, 528, 553, 600, 624, 651, 702, 728, 757, 812, 840, 871, 930, 960, 992, 993 We see the 7-et, of course, from 2^2+2+1, we have a Golomb ruler for the 12-et coming from 4^2-4, one for the 15-et from 4^2-1, the 31-et from 5^2+5+1, and the 72-et from 9^2-9, the 80-et from 9^2-1, and even the 342 et from 19^2-19. (I never knew about this rational point on the elliptic curve y^2-y=x^3-1 before; cute.) Now I want to know what Robert wants these for...it seems to be they are, musically speaking, at opposite poles from what we usually contruct as scales--they are anti-scales of a sort.
Message: 4544 - Contents - Hide Contents Date: Tue, 09 Apr 2002 19:44:09 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:>> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:>>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:>>>> ... So the question now becomes: Are we left with any good reason for >>>> basing the JI notation on 311 instead of 217? >>>>>> From your point of view, I would say that you are better off with 217-ET. >>>> This amounts, then, to a 19-limit-unique-&-consistent, polyphonic- >> readable sagittal notation with non-unique capability up to the 35- >> odd limit. That sounds like something that fulfills (and in some >> ways exceeds) our original objective (as I understood it). >> Sure. But I don't understand what 217-ET or 311-ET have to do with it. > 217-ET just happens to be the highest ET that you can notate with it. > The definitions of the symbols must be based on the commas, not the > degrees of 217-ET. > > What do you mean by "polyphonic-readable"? As opposed to what?As opposed to polyphonic-confusible or polyphonic-difficult-or-slow- to-read. This was just my way of putting in another plea for single- symbol modifications to notes -- my obsession, as you call it.>>> However I do not wish to base a JI (rational) notation on _any_ temperament >>> that has errors larger than 0.5 c. For me, 217-ET and 311-ETwere merely a>>> way of looking for schismas that might be notationally usable (less than >>> 0.5 c), and of checking that things were working sensibly, andit was nice>>> to actually be able to notate those ETs themselves. But I'm taking Johnny >>> Reinhard at his word when he says (or implies) that nothing less than >>> 1200-ET is good enough as an ET-based JI notation.So (as I see it) Johnny's obsession has become yours as well. As I said before, I really don't think that an underlying ET needs to have that much accuracy -- it's going to take a great deal of skill and concentration to hold a sustained pitch that steady on an instrument of flexible pitch, and if it's of short duration, then it would be pretty difficult to perceive an error of, say, 3 cents, except on laboratory equipment (which I wouldn't expect anyone to bring to a concert). This is why I feel that 217-ET is adequate: it puts you close enough for most purposes, and if that is not close enough (meaning that can still hear that you're not close enough), then you can make a super-fine correction in intonation by ear. I should emphasize that those intervals in which you are most likely to be able to hear 2-cent errors are the 5-limit consonances, none of which have an error greater than 1 cent in 217-ET. Anyway, I expect that we can allow for each other's obsessions and can continue to work on this together to achieve both of our objectives.>> There is a question that needs to be asked: are we notating JI or are >> we notating 217-ET? I understood that we were notating JI (mapped >> onto 217 for convenience in understanding some of the size >> relationships among the various ratios), which makes discussion about >> 3-cent errors a bit irrelevant. >> OK. Good. So I wish you'd stop talking about it being "based on" or > "going with" 217-ET, or any other ET with larger than 0.5 cent errors.How about a compromise in which we "go with" both 217 and 1600-ET (37- limit), with a specific set of symbols for 217 and a superset for 1600? (This might also make it possible to notate 311-ET using the full set of symbols.) I am suggesting this in light of your observation:> Yes, so 217-ET is just one ET that could be used in this way. The > notation is not based on it. It just happens to be the highest one > that is fully notatable with single symbols.This is one point that has become all too apparent, as you have proceeded (in your subsequent messages) to suggest changes in the symbols that: 1) Go beyond the three types of flags (straight, convex, & concave) that work so elegantly for 217 (remember that I said that something that could be regarded as "overkill" was immune to criticism as long as the additional complexity didn't make it more difficult to do the simpler things; this introduces more complexity for 217-ET); 2) Introduce new symbols that I have no idea how to incorporate into a single-symbol notation (this makes it difficult to do something that I was previously able to do with 217-ET); and 3) Employ semantics inconsistent with 217-ET, as in the following: --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:>>> 3) It conflates all three of my schismas: 4095:4096, 3519:3520, and >>> 20735:20736 (but not the 31-schisma that I tried, 59024:59049, which >>> was also unusable in 311); >>>> 59024:59049 (2^4*7*17*31:3^10) doesn't pass the Reinhard test anyway, >> being 0.73 c, however it might tempt me if it could be combined with >> other suitable schismas, as per my challenge. >> We can forget about that 31-schisma. What's wrong with 253935:253952 > (3^5*5*11*19 : 2^13*31) 0.12 cents. Consistent with 311-ET 388-ET > 1600-ET, but not 217-ET. > > 31 comma = (11 comma - 5 comma) + 19 comma > > Since (11 comma - 5 comma) is a single flag and 19 comma is a single > flag (or blob) then this 31 comma can be represented by a pair of > flags. The fact that it doesn't work in 217-ET doesn't matter because > the notation is not "based on" 217-ET and the 31 comma is not needed > in order to notate 217-ET.I made it a point to think very carefully before replying to your subsequent messages, because I know you spent a lot of time and effort on the content and have come up with some very good things, such as the 31-schisma (above). During the two weeks or so that I spent leading up to my 17-limit (183-tone) and 23-limit (217-tone) approaches, I also spent a lot of time trying various things, and I don't consider the time wasted that I spent on ideas that I subsequently discarded. In the process of developing a notation such as this, you want to try as many things as you can possibly think of, because that best enables you to see why the method that is finally chosen is the best one. I wanted to find a way to resolve this that would satisfy both of our requirements. Here is the compromise that I am proposing: Let's keep the 217-ET- based symbols as they are, defining 2176:2187 as xL and 512:513 as xR, with their combination allowed to represent either 4096:4131 or 729:736 as required (in 217-ET or another ET, where consistent, but incapable of being combined with anything else). Then, for the 1600- based notation, let's expand on that with a combination of the following methods: 1) Allow two flags to appear on the same side, as was suggested for 6400:6561, the 25 comma. This would then allow us to use sR+vR (with the concave flag at the top of an upward-pointing arrow) to notate the 31-comma 243:248, using the schisma 353935:253952. Also, the alternate 37-comma 999:1024 could be notated with xL+vL, using the schisma 570236193:570425344. (We would have to experiment to see how this would be done. With the convex flag at the end, the two would form a sort of loop; or they might be made to interlock.) 2) Define one or more additional types of flags to notate new primes, beginning with a new left one for the 23-comma, 729:736. This would then allow us to use newL+xR+vR to notate the 37-comma 36:37, using the schisma 6992:6993. (Thus, the symbols for the two 37-commas both contain a combination of a convex and concave flag on the same side, which is most appropriate!) --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:> This is probably all pretty silly, catering for 37, and we should probably > just forget it and keep the large 23 comma symbol, but here's apass at a> full set of 37-limit symbols anyway.Silly or not, I think we should keep whatever capability we can, as long as it is consistent. And I would prefer to keep *both* the large 23 comma symbol and a full set of 37-limit symbols, as with this "compromise." Overkill? Maybe, but it keeps the simpler things simple, while serving those who want a lot of capability. And it does follow the no-more-than-one-new-comma-per prime guideline throughout. Also, there are some divisions between 100 and 217 that the 217-notation won't handle (such as 140), for which I would expect that the extended set of symbols could be used. So how does that grab you? --George
Message: 4545 - Contents - Hide Contents Date: Wed, 10 Apr 2002 02:48:39 Subject: The (19,9,4) difference set scale From: Gene W Smith David Bowen wrote on tuning-math: <<By coincidence, the April 2002 issue of the Mathematics Magazine arrived at my house yesterday and the lead article discusses the many applications of the 7-et set. One of the first theorems inthe article is that if p is a prime of the form 4n+3, then the squares mod p will give you a set of 2n+1 elements where each difference occurs n times. So for 19 we have the set {1, 4, 5, 6, 7, 9, 11, 16, 17} where each difference occurs 4 times and for 31 we have the set {1, 2, 4, 5, 7, 8, 9, 10, 14, 16, 18, 19, 20, 25, 28} where each difference occurs 7 times.>> The scale consisting of the 9 quadradic residues mod 19 seemed worth investigating. This is [1, 4, 5, 6, 7, 9, 11, 16, 17]; the characteristic polynomials for the odd limits to 11 are given below; the x^7 term gives the number of edges, and the x^6 twice the number of triads. If this experiment works, you should be able to see graphs of the scale in the various limits in the attachments. p03 x^9-4*x^7+4*x^5-x^3 p05 x^9-12*x^7-6*x^6+40*x^5+30*x^4-38*x^3-32*x^2+7*x+6 p07 x^9-20*x^7-28*x^6+53*x^5+100*x^4-6*x^3-66*x^2-24*x p09 x^9-28*x^7-74*x^6-35*x^5+54*x^4+42*x^3 p11 x^9-32*x^7-116*x^6-160*x^5-70*x^4+39*x^3+44*x^2+10*x [This message contained attachments]
Message: 4546 - Contents - Hide Contents Date: Wed, 10 Apr 2002 20:43:31 Subject: Re: A common notation for JI and ETs From: David C Keenan Hi George, ----------- The 19 flag ----------- I don't require that the new type of flag be small irrespective of what it is used for. I only want the flag for the 3.3 cent 19 comma to be smaller than all the others, because it is less than half the size of any other flag comma and less than 1/6th of the size of all but the 17 comma. If this is allowed, then it follows that it must be a new kind of comma, not convex, striaght or concave. It seems, from an RT point of view, that the 19 comma flag could equally well be a left flag or a right flag, I have no great attachment to either. However in notating 217-ET you need to use 19 flag + 17 flag to notate 3 steps and so the 19 flag would be best on the opposite side from the 17 flag. And if we want the large 23 comma not to have flags on the same side, then the 17 comma must be on the opposite side from the 11-5 flag, which means that the 17 flag must be a left flag and the 19 flag a right flag (I mistakenly had 19 as a left flag in my previous message). ---------- Priorities ---------- It seems that there is a significant difference of priorities between an approach (a) that seeks to notate a particular large ET, which is 19-odd-limit-unique, using single symbols spanning from double-flat to double-sharp (or even just from flat to sharp), and use subsets of it to notate lower ETs, and extend it to uniquely notate 19-or-higher-prime-limit RTs (rational tunings), and an approach (b) that seeks to notate 19-or-higher-prime-limit RTs and use subsets of it to notate low enough ETs, and extend it to allow those ETs to be notated using single symbols spanning from double-flat to double-sharp (or even just from flat to sharp). I believe I've understood your points, but I don't have any suggestions yet that might satisfy us both, so I'm just going to put it in the too hard basket for a while, or let it churn away in my subconscious. ----------------- The new flag type ----------------- In the meantime, let's try to agree on what the new type of flag should look like, irrespective of what it is used for. I realise now that my earlier suggestions of blobs or circles failed to take account of the need to work with multiple shafts and X shafts. I believe the following proposal does. It resulted from asking myself the question "What could be more concave than concave and yet still indicate a direction, and work with multiple shafts?". Of course I also wanted it to look smaller (just in case it might get used for the 19 comma :-), but I figured that since straight looks smaller than convex and concave looks smaller than straight, then "more concave than concave" is bound to look smaller than concave. I settled on a _right-angle_ flag. It indicates direction simply by being close to one end of the shaft. Since none of our arrows have "tail-feathers" there can be no confusion about which direction is meant, and in any case I find that it invites the eye to complete a small 45 degree right triangle. But I don't want this triangle completed literally, since it would then look too large, and would no longer be "more concave than concave". In addition to its angularity (not straight, not curved), its smallness is part of what distinguishes it, at a glance, from a concave flag. Here's my best attempt at showing, in ASCII-graphics, all the possible combinations for up arrows (with no more than one flag to a side). I haven't bothered to show combinations which are merely left/right reversals of those shown, and I've given no consideration to possible meanings of flags or which combinations may be irrelevant. _ / | | | | | | /| / | | | | | _/| | | | | _| | | | | ___ / | \ | | | | | | /|\ / | \ | | | | _/|\_ | | | _|_ | | | | _ / |\ | | \ | | | /| / |\_ | | | _ | \ _/| | | | | |_ _/| | | | _ _| \ | | | | | _|\ | \ | | | ___ / | | | | | | | | | | | /| / | /| | | | | | | /| _/ | | | | | | __| | | | | | | _____ / | | \ | | | | | | | | | | / \ /| |\ | | | | | | | _/ \_ | | | | | | _|_ | | | | | | | | ___ / | |\ | | | \ | | | | | | /| / | /| |\_ | | | | ___ | | \ _/| | | | | | | | | |_ /| _/ | | | | | ___ _| | \ | | | | | | | | | _|\ | \ | |\ | | | | ___ / ||| | ||| ||| ||| ||| /| /|| /||| ||| ||| | /| _/|| ||| ||| | __| ||| ||| ||| _____ / ||| \ | ||| | ||| ||| ||| /|\ /|||\ ||| ||| ||| | _/|\_ ||| ||| ||| _|_ ||| ||| ||| ||| ___ / |||\ | ||| \ ||| ||| ||| /| /|| /|||\_ ||| ||| ___ ||| \ _/||| | ||| ||| ||| |_ /| _/|| ||| ||| ___ _||| \ ||| | ||| ||| ||| _|\ ||\ |||\ ||| ||| ___ / | | | | | \ / X / \ /| / | /\ / X / \ | /| _/ / X / \ | __| \ / X / \ _____ / | | \ | | | | \ / X / \ / \ /| |\ \ / X / \ | _/ \_ \ / X / \ _|_ | | \ / X / \ ____ / | |\ | | | \ \ / X / \ /| / | /\ /\_ X / \ ___ | | \ _/| | | \ / X / \ |_ /| _/ / X / \ ___ _| | \ | | | \ / X / \ _|\ | \ \ /\ X / \ I think the fact that they can be made distinct using the extremely limited resolution of the above ASCII-graphics, bodes well for the real, high resolution symbols. Notice how a lot of problems are eliminated by bending the lines of the X shafts so they become parallel near the head of the arrow. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 4547 - Contents - Hide Contents Date: Wed, 10 Apr 2002 03:36:52 Subject: Re: A common notation for JI and ETs From: David C Keenan --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:>> Sure. But I don't understand what 217-ET or 311-ET have to do with > it.>> 217-ET just happens to be the highest ET that you can notate with > it.>> The definitions of the symbols must be based on the commas, not the >> degrees of 217-ET. >> >> What do you mean by "polyphonic-readable"? As opposed to what? >> As opposed to polyphonic-confusible or polyphonic-difficult-or-slow- > to-read. This was just my way of putting in another plea for single- > symbol modifications to notes -- my obsession, as you call it.And a very fine obsession it is. I do not want to deflect you from it in the slightest. I would merely like it recognised that it is not the most general use of the notation. The most general is: by using more than one symbol at a time one can uniquely notate any rational pitch up to a 37 prime limit. So in this way of using it, it is not even "based on" 1600-ET. 1600-ET was merely used in determining the symbols for the prime commas, after which the symbols are considered atomic.> So (as I see it) Johnny's obsession has become yours as well.Not personally, but I think it wise to recognise that his opinions are widely respected in microtonal circles, and so if we hope for this notation to achieve wide acceptance we might as well eliminate the possible objection that it does not allow one to uniquely notate 19-prime-limit rational pitches which may be _more_ than 3 cents apart. e.g. 19/14 and 34/25 are the same in 217-ET, but differ by 475:476 or 3.6 cents. Here's an extreme example, which I admit is unlikely to be encountered in real life. The comma 29229255:29360128 (3^12*5*11 : 7*2^22 vanishes in 217-ET but is 7.7 cents in rational tuning.> As I > said before, I really don't think that an underlying ET needs to have > that much accuracy -- it's going to take a great deal of skill and > concentration to hold a sustained pitch that steady on an instrument > of flexible pitch, and if it's of short duration, then it would be > pretty difficult to perceive an error of, say, 3 cents, except on > laboratory equipment (which I wouldn't expect anyone to bring to a > concert). This is why I feel that 217-ET is adequate: it puts you > close enough for most purposes, and if that is not close enough > (meaning that can still hear that you're not close enough), then you > can make a super-fine correction in intonation by ear. I should > emphasize that those intervals in which you are most likely to be > able to hear 2-cent errors are the 5-limit consonances, none of which > have an error greater than 1 cent in 217-ET.Except for the last sentence, I have posted similar opinions to the tuning list myself many times over the years. It's curious that I chose 2.8 cent maximum error as my (fairly arbitrary) cutoff for what I consider a "microtemperament", without ever considering it as a half-step of 217-ET. For example, I consider 72-ET to be a 7-limit microtemperament, but not a 9-limit or higher one. 217-ET is therefore the smallest ET that is a 21-limit microtemperament, and if that cutoff were bumped to 2.9 cents it would be a 35-limit microtemperament (max 37-limit error is 4.6 cents). 311-ET is a 45-limit microtemperament and has no error greater than 1.9 cents in the 41-limit. So 311-ET is way more than we need from this point of view, and 217-ET is just right. By the way, 1600-ET gets us to the 45-limit without exceeding 0.5 cents error, but there is no way to get its 41 or 43 commas by combining existing flag commas, not even 3 or more of them with multiple flags allowed per side. Thank goodness! 37 is already more than we need.> Anyway, I expect that we can allow for each other's obsessions and > can continue to work on this together to achieve both of our > objectives.Absolutely. I am immensely enjoying working with you on this.>> OK. Good. So I wish you'd stop talking about it being "based on" or >> "going with" 217-ET, or any other ET with larger than 0.5 cent > errors. >> How about a compromise in which we "go with" both 217 and 1600-ET (37- > limit), with a specific set of symbols for 217 and a superset for > 1600? (This might also make it possible to notate 311-ET using the > full set of symbols.)OK. Except I'd probably prefer to put it this way: The notation is based on pythagorean A-G,#,b, with the addition of a pair of arrow symbols (up and down) for each prime number from 5 to 37. Each pair of arrow symbols corresponds to a comma that is smaller than a half-apotome (56.8 cents) and relates the prime number to a chain of between -4 and 7 fifths, ignoring octaves. That's from Ab to C# relative to C. This requires 10 new pairs of symbols, which might be hard to learn and might result in some notes having a ridiculous number of accidentals before them, except that the symbols are not atomic. They are themselves made up of a vertical shaft with only 4 kinds of half-arrowhead or flag. Most of these flags come in left and right varieties for a total of 7 kinds of flag (ignoring up and down varieties). These 7 flags correspond to the commas for the primes 5, 7, 11*, 17, 19, 23, 29. The symbols for the commas for 13, 31 and 37 and some optional additional commas, are obtained by combining flags on the same shaft according to an arithmetic which corresponds to simple addition of the nearest 1/1600ths of an octave. * The 11 comma is symbolised, not by a single flag but by a new flag combined with the 5 flag, and so we refer to this new flag as the 11-5 flag. Because we use 1600-ET for this flag arithmetic, if we choose to combine multiple symbols into a single symbol we can do so without introducing any error greater than about half a cent. The system is designed so that at each prime limit lower than 37, it is as simple as possible. No higher prime has been allowed to complicate the system for those who don't need it. Here are the numbers of different flags that must be learnt at each prime limit 5 1 7 2 11 3 13 3 17 4 19 5 23 6 29 7 31 7 37 7 Although we've so far described this as a notation for purely rational scales, it works beautifully for equal temperaments too. [explain how - choose your fifth etc.] In the case of equal temperaments we use only the symbols for the lowest primes, or combinations thereof, that are necessary to notate each step. It turns out that one doesn't need to go past 19-limit to notate most ETs of interest. 217-ET is the largest ET that can be notated by this method, using only one symbol per note (in addition to a possible sharp or flat symbol). 217-ET has no error greater than 2.9 cents in the 35-limit, and so provided that such errors are acceptable, we can use it to notate up to 35-limit rational scales using only one symbol per note. So far we have assumed that the arrow symbols will be used in conjunction with conventional sharp and flat symbols, but this is not necessary either. The system includes additional arrow symbols, which use the same flags (half arrowheads) but have multiple shafts to the arrow. These can cover the range from a double-flat to a double-sharp using single symbols.> I am suggesting this in light of your > observation: >>> Yes, so 217-ET is just one ET that could be used in this way. The >> notation is not based on it. It just happens to be the highest one >> that is fully notatable with single symbols. >> This is one point that has become all too apparent, as you have > proceeded (in your subsequent messages) to suggest changes in the > symbols that: > > 1) Go beyond the three types of flags (straight, convex, & concave) > that work so elegantly for 217 (remember that I said that something > that could be regarded as "overkill" was immune to criticism as long > as the additional complexity didn't make it more difficult to do the > simpler things; this introduces more complexity for 217-ET);217-ET only needs 19-limit, correct? I don't understand why you consider that changing the 19-flag to something other than a concave flag is an increase in complexity. The 5 limit uses only a straight left flag. We didn't require that the 7 limit use the straight right flag but went to a convex flag and didn't use the straight right until 11-limit. This would be similar; delaying the use of the other convex flag until 23 limit; and could be justified on exactly the same grounds, namely eliminating lateral confusability from the 19-limit (and thereby greatly reducing it in 217-ET).> 2) Introduce new symbols that I have no idea how to incorporate into > a single-symbol notation (this makes it difficult to do something > that I was previously able to do with 217-ET); andI think this is the big one, but I have a proposed solution. Later.> 3) Employ semantics inconsistent with 217-ET, ...I don't see this as a problem because I don't think that anything employing those semantics is required in order to notate 217-ET> I made it a point to think very carefully before replying to your > subsequent messages, because I know you spent a lot of time and > effort on the content and have come up with some very good things, > such as the 31-schisma (above). During the two weeks or so that I > spent leading up to my 17-limit (183-tone) and 23-limit (217-tone) > approaches, I also spent a lot of time trying various things, and I > don't consider the time wasted that I spent on ideas that I > subsequently discarded. In the process of developing a notation such > as this, you want to try as many things as you can possibly think of, > because that best enables you to see why the method that is finally > chosen is the best one. I wanted to find a way to resolve this that > would satisfy both of our requirements.I totally agree.> Here is the compromise that I am proposing: Let's keep the 217-ET- > based symbols as they are, defining 2176:2187 as xL and 512:513 as > xR, with their combination allowed to represent either 4096:4131 or > 729:736 as required (in 217-ET or another ET, where consistent, but > incapable of being combined with anything else). Then, for the 1600- > based notation, let's expand on that with a combination of the > following methods: > > 1) Allow two flags to appear on the same side, as was suggested for > 6400:6561, the 25 comma. This would then allow us to use sR+vR (with > the concave flag at the top of an upward-pointing arrow) to notate > the 31-comma 243:248, using the schisma 353935:253952. Also, the > alternate 37-comma 999:1024 could be notated with xL+vL, using the > schisma 570236193:570425344. (We would have to experiment to see how > this would be done. With the convex flag at the end, the two would > form a sort of loop; or they might be made to interlock.)I have no objection to using multiple flags on the same side, to notate primes beyond 29. However I consider 999:1024 to be the standard 37 comma because it is smaller than 36:37, also because it only requires 2 lower-prime flags instead of 3. Can you explain why you want 36:37 to be the standard 37 comma?> 2) Define one or more additional types of flags to notate new primes, > beginning with a new left one for the 23-comma, 729:736.Beginning and ending with a new 23-flag. 7 flags is enough.> This would > then allow us to use newL+xR+vR to notate the 37-comma 36:37, using > the schisma 6992:6993. (Thus, the symbols for the two 37-commas both > contain a combination of a convex and concave flag on the same side, > which is most appropriate!)Other combinations might have other kinds of appropriateness, such as one containing the other flipped horizontally.> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:>> This is probably all pretty silly, catering for 37, and we should > probably>> just forget it and keep the large 23 comma symbol, but here's a> pass at a>> full set of 37-limit symbols anyway. >> Silly or not, I think we should keep whatever capability we can, as > long as it is consistent. And I would prefer to keep *both* the > large 23 comma symbol and a full set of 37-limit symbols, as with > this "compromise."OK. But I'd prefer a slightly different compromise where the 19 flag is the one that is other than straight, convex or concave and gives the impression of being smaller than any of them. So the following has the 19 and 23 flags swapped relative to your suggestion. 17 vL 19 smallL 23 vR 23' vL + sR 31 smallL + sR 37 xL + vL (999:1024) 37' smallL + vR + xR (36:37) Now to the problems that occur when you try to make this work for 217-ET with the full sagittal treatment, i.e. no # or b. Here's what you wrote earlier about the notation of apotome complements:>By the way, something else I figured out over the weekend is how to >notate 13 through 20 degrees of 217 with single symbols, i.e., how to >subtract the 1 through 8-degree symbols from the sagittal apotome >(/||\). The symbol subtraction for notation of apotome complements >works like this: > >For a symbol consisting of: >1) a left flag (or blank) >2) a single (or triple) stem, and >3) a right flag (or blank): >4) convert the single stem to a double (or triple to an X); >5) replace the left and right flags with their opposites according to >the following: > a) a straight flag is the opposite of a blank (and vice versa); > b) a convex flag is the opposite of a concave flag (and vice versa). > >This produces a reasonable and orderly progression of symbols >(assuming that 63:64 is a curved convex flag; it does not work as >well with 63:64 as a straight flag) that is consistent with the >manner in which I previously employed the original sagittal symbols >for various ET's.The problem I have with this (even assuming _your_ suggested compromise) is that, while the opposite of sL and sR must certainly be blanks if the apotome is to be a double-shafted sL+sR, the other opposites are entirely arbitrary. What I dislike about the result of your choice is that, having learnt that xL is larger than sL, I now find that when they have a double shaft under them, the order of these two is reversed, while all the others remain the same. Why can't we simply give a fixed comma value to the second shaft (and so on for subsequent shafts), so the ordering of flag combinations learnt for the first half-apotome is simply repeated in the second half-apotome (and all other half-apotomes). To do this, the second shaft need only be declared equal in value to xL+xR. Another advantage of this is that one does not need to use flags that properly belong to higher limits in the second and subsequent half-apotomes of lower limit rational notations, or of ET notations based on lower limits. e.g. There will be no concave flags (or small flag) in 72-ET. And there will be no need for xL or vR in 217-ET. This also solves your problem number 2 above. Objections? -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 4548 - Contents - Hide Contents Date: Wed, 10 Apr 2002 21:33:46 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >>>> How about a compromise in which we "go with" both 217 and 1600-ET >> (37-limit), with a specific set of symbols for 217 and a superset for >> 1600? (This might also make it possible to notate 311-ET using the >> full set of symbols.) >> OK. Except I'd probably prefer to put it this way: > > The notation is based on pythagorean A-G,#,b, with the addition of a pair > of arrow symbols (up and down) for each prime number from 5 to 37. Each > pair of arrow symbols corresponds to a comma that is smaller than a > half-apotome (56.8 cents) and relates the prime number to a chain of > between -4 and 7 fifths, ignoring octaves. That's from Ab to C#relative to C.> > This requires 10 new pairs of symbols, which might be hard to learn and > might result in some notes having a ridiculous number of accidentals before > them, except that the symbols are not atomic. They are themselves made up > of a vertical shaft with only 4 kinds of half-arrowhead or flag. Most of > these flags come in left and right varieties for a total of 7 kinds of flag > (ignoring up and down varieties). > > These 7 flags correspond to the commas for the primes 5, 7, 11*, 17, 19, > 23, 29. The symbols for the commas for 13, 31 and 37 and some optional > additional commas, are obtained by combining flags on the same shaft > according to an arithmetic which corresponds to simple addition of the > nearest 1/1600ths of an octave. > > * The 11 comma is symbolised, not by a single flag but by a new flag > combined with the 5 flag, and so we refer to this new flag as the 11-5 flag. > > Because we use 1600-ET for this flag arithmetic, if we choose to combine > multiple symbols into a single symbol we can do so without introducing any > error greater than about half a cent. > > The system is designed so that at each prime limit lower than 37,it is as> simple as possible. No higher prime has been allowed to complicate the > system for those who don't need it. Here are the numbers of different flags > that must be learnt at each prime limit > > 5 1 > 7 2 > 11 3 > 13 3 > 17 4 > 19 5 > 23 6 > 29 7 > 31 7 > 37 7 > > Although we've so far described this as a notation for purely rational > scales, it works beautifully for equal temperaments too. [explain how - > choose your fifth etc.] > > In the case of equal temperaments we use only the symbols for the lowest > primes, or combinations thereof, that are necessary to notate each step. It > turns out that one doesn't need to go past 19-limit to notate most ETs of > interest. > > 217-ET is the largest ET that can be notated by this method, using only one > symbol per note (in addition to a possible sharp or flat symbol). 217-ET > has no error greater than 2.9 cents in the 35-limit, and so provided that > such errors are acceptable, we can use it to notate up to 35-limit rational > scales using only one symbol per note. > > So far we have assumed that the arrow symbols will be used in conjunction > with conventional sharp and flat symbols, but this is not necessary either. > The system includes additional arrow symbols, which use the same flags > (half arrowheads) but have multiple shafts to the arrow. These can cover > the range from a double-flat to a double-sharp using single symbols.Okay, that sounds like a good description of what we are are very close to achieving. I might prefer to call the 11-comma a diesis (although it is plain that you are using the term "comma" in a broader sense here), which would further justify the introduction of the 11-5 comma that is used in achieving it, just as the 13-diesis is also the (approximate) sum of two commas.>> I am suggesting this in light of your >> observation: >>>>> Yes, so 217-ET is just one ET that could be used in this way. The >>> notation is not based on it. It just happens to be the highest one >>> that is fully notatable with single symbols. >>>> This is one point that has become all too apparent, as you have >> proceeded (in your subsequent messages) to suggest changes in the >> symbols that: >> >> 1) Go beyond the three types of flags (straight, convex, & concave) >> that work so elegantly for 217 (remember that I said that something >> that could be regarded as "overkill" was immune to criticism as long >> as the additional complexity didn't make it more difficult to do the >> simpler things; this introduces more complexity for 217-ET); >> 217-ET only needs 19-limit, correct? I don't understand why you consider > that changing the 19-flag to something other than a concave flag is an > increase in complexity. The 5 limit uses only a straight left flag. We > didn't require that the 7 limit use the straight right flag butwent to a> convex flag and didn't use the straight right until 11-limit. This would be > similar; delaying the use of the other convex flag until 23 limit; and > could be justified on exactly the same grounds, namely eliminating lateral > confusability from the 19-limit (and thereby greatly reducing it in 217-ET).It was getting more complicated inasmuch as I was leading up to my next point:>> 2) Introduce new symbols that I have no idea how to incorporate into >> a single-symbol notation (this makes it difficult to do something >> that I was previously able to do with 217-ET); and >> I think this is the big one, but I have a proposed solution. Later.It doesn't work (see my reply below).>> 3) Employ semantics inconsistent with 217-ET, ... >> I don't see this as a problem because I don't think that anything employing > those semantics is required in order to notate 217-ETI had the impression that the 23-flag used in combination with something else defined another prime inconsistenly in 217, but that one (for the 37-comma 36:37) requires 3 flags, so it wouldn't be used anyway.>> I made it a point to think very carefully before replying to your >> subsequent messages, because I know you spent a lot of time and >> effort on the content and have come up with some very good things, >> such as the 31-schisma (above). During the two weeks or so that I >> spent leading up to my 17-limit (183-tone) and 23-limit (217- tone) >> approaches, I also spent a lot of time trying various things, and I >> don't consider the time wasted that I spent on ideas that I >> subsequently discarded. In the process of developing a notation such >> as this, you want to try as many things as you can possibly think of, >> because that best enables you to see why the method that is finally >> chosen is the best one. I wanted to find a way to resolve this that >> would satisfy both of our requirements. >> I totally agree. >>> Here is the compromise that I am proposing: Let's keep the 217- ET- >> based symbols as they are, defining 2176:2187 as xL and 512:513 as >> xR, with their combination allowed to represent either 4096:4131 or >> 729:736 as required (in 217-ET or another ET, where consistent, but >> incapable of being combined with anything else).In the preceding sentence it should be obvious to you that I meant to say "defining 2176:2187 as *vL* and 512:513 as *vR*", but just so no one else misunderstands, I am correcting this here.>> Then, for the 1600- >> based notation, let's expand on that with a combination of the >> following methods: >> >> 1) Allow two flags to appear on the same side, as was suggested for >> 6400:6561, the 25 comma. This would then allow us to use sR+vR (with >> the concave flag at the top of an upward-pointing arrow) to notate >> the 31-comma 243:248, using the schisma 353935:253952. Also, the >> alternate 37-comma 999:1024 could be notated with xL+vL, using the >> schisma 570236193:570425344. (We would have to experiment to see how >> this would be done. With the convex flag at the end, the two would >> form a sort of loop; or they might be made to interlock.) >> I have no objection to using multiple flags on the same side, to notate > primes beyond 29. However I consider 999:1024 to be the standard 37 comma > because it is smaller than 36:37, also because it only requires 2 > lower-prime flags instead of 3. Can you explain why you want 36:37 to be > the standard 37 comma?Using primes this high has more legitimacy, in my opinion, in otonal chords than in utonal chords. If C is 1/1, then 37/32 would be D (9/8) raised by 37:36. With 1024:999 the 37 factor is in the smaller number of the ratio, which is not where I need it. For a similar reason I regard 26:27 as the principal 13-diesis. Taking C as 1/1, to get 13/8 I want to lower A (27/16) by a semiflat (26:27) instead of raising A-flat by a semisharp (1053:1024), even if 1053:1024 is the smaller diesis. But considering that 26:27 is more than half an apotome (and that we are adequately representing both of these in the notation anyway), I have no problem that you prefer to state it the other way. While we are on the subject of higher primes, I have one more schisma, just for the record. This is one that you probably won't be interested in, inasmuch as it is inconsistent in both 311 and 1600, but consistent and therefore usable in 217. It is 6560:6561 (2^5*5*41:3^8, ~0.264 cents), the difference between 80:81 and 81:82, the latter being the 41-comma, which can be represented by the sL flag. I don't think I ever found a use for any ratios of 37, but Erv Wilson and I both found different practical applications for ratios involving the 41st harmonic back in the 1970's, so I find it rather nice to be able to notate this in 217.>> 2) Define one or more additional types of flags to notate new primes, >> beginning with a new left one for the 23-comma, 729:736. >> Beginning and ending with a new 23-flag. 7 flags is enough.Yes, in light of the additional schismas that you have found.>> This would >> then allow us to use newL+xR+vR to notate the 37-comma 36:37, using >> the schisma 6992:6993. (Thus, the symbols for the two 37-commas both >> contain a combination of a convex and concave flag on the same side, >> which is most appropriate!) >> Other combinations might have other kinds of appropriateness, such as one > containing the other flipped horizontally. >>> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:>>> This is probably all pretty silly, catering for 37, and we should probably >>> just forget it and keep the large 23 comma symbol, but here's apass at a>>> full set of 37-limit symbols anyway. >>>> Silly or not, I think we should keep whatever capability we can, as >> long as it is consistent. And I would prefer to keep *both* the >> large 23 comma symbol and a full set of 37-limit symbols, as with >> this "compromise." >> OK. But I'd prefer a slightly different compromise where the 19flag is the> one that is other than straight, convex or concave and gives the impression > of being smaller than any of them. So the following has the 19 and 23 flags > swapped relative to your suggestion. > > 17 vL > 19 smallL > 23 vR > 23' vL + sR > 31 smallL + sR > 37 xL + vL (999:1024) > 37' smallL + vR + xR (36:37)Why are you requiring that the new type of flag (whether for 19 or 23) be smaller in size? I would have the new flag represent 23 on the basis that it is a *higher prime* than 19. Then with 217-ET (which is unique only through 19 and completely consistent only through 21) we need only the three types of flags that are used for the 19-limit notation, with a *newL* (different-looking *left*) flag for the 23 comma being foreign to all three: the 19-limit, 217-ET, and the single-symbol notation. Otherwise, I would need to have a way to incorporate the new flag into the single-symbol notation, which will be discussed next.> Now to the problems that occur when you try to make this work for 217-ET > with the full sagittal treatment, i.e. no # or b. > > Here's what you wrote earlier about the notation of apotome complements: >>> By the way, something else I figured out over the weekend is how to >> notate 13 through 20 degrees of 217 with single symbols, i.e., how to >> subtract the 1 through 8-degree symbols from the sagittal apotome >> (/||\). The symbol subtraction for notation of apotome complements >> works like this: >> >> For a symbol consisting of: >> 1) a left flag (or blank) >> 2) a single (or triple) stem, and >> 3) a right flag (or blank): >> 4) convert the single stem to a double (or triple to an X); >> 5) replace the left and right flags with their opposites accordingto the following:>> a) a straight flag is the opposite of a blank (and vice versa); >> b) a convex flag is the opposite of a concave flag (and vice versa). >> >> This produces a reasonable and orderly progression of symbols >> (assuming that 63:64 is a curved convex flag; it does not work as >> well with 63:64 as a straight flag) that is consistent with the >> manner in which I previously employed the original sagittal symbols >> for various ET's. >> The problem I have with this (even assuming _your_ suggested compromise) is > that, while the opposite of sL and sR must certainly be blanks if the > apotome is to be a double-shafted sL+sR, the other opposites are entirely > arbitrary. What I dislike about the result of your choice is that, having > learnt that xL is larger than sL, I now find that when they have a double > shaft under them, the order of these two is reversed, while all the others > remain the same.The heart of the problem is that, in order to have a completely consistent order of symbols, sL and xL should be swapped, so that straight flags are *always* larger than curved flags. However, this would make both the 5-comma and 7-comma flags convex, which re- introduces the problem of lateral confusibility, not only between ratios of 5 and 7, but also for the two 11-dieses, which I think is a more serious issue. (In addition, a curved 5-flag would not have a constant slope, thereby obscuring the comma-up meaning.) Another inconsistency is that vL||sR is a smaller interval than ||sR (in effect making vL alter by -2 degrees when used with || ), but this one is fortunately avoided in 217: vL||sR does not have to be used, inasmuch as it is the same number of degrees as sL||. (And vL||xR can also be avoided, being almost the same size as xL||vR.) All of these problems are easily avoided in lesser divisions by a judicious selection of symbols. So I would consider this an example of a situation that is (to quote a joke I once heard) "hopeless but not serious."> Why can't we simply give a fixed comma value to the second shaft(and so on> for subsequent shafts), so the ordering of flag combinations learnt for the > first half-apotome is simply repeated in the second half-apotome (and all > other half-apotomes). To do this, the second shaft need only be declared > equal in value to xL+xR.That's the way I did it way back (about 3 months ago) when life was much simpler: I was using only straight flags and 72-ET was the most complicated system I had to deal with. The problem in doing that now is that the ratios that we're trying to represent don't ascend in the same order from a half-apotome (now what ratio is that anyway?) as they do from a unison; instead they occur in reverse order from the apotome downward. So scratch that idea.> Another advantage of this is that one does not need to use flags that > properly belong to higher limits in the second and subsequent half- apotomes > of lower limit rational notations, or of ET notations based on lower > limits. e.g. There will be no concave flags (or small flag) in 72- ET. And > there will be no need for xL or vR in 217-ET.I would want xL in 217 anyway, since it does handle ratios of 29. After all, this is supposed to allow 35-limit (nonunique) notation, and it would be better not to have a new flag appearing out of the blue, just for 29. Now regarding 72-ET, you will recall that I said this earlier: << Using curved flags in the 72-ET native notation to alleviate lateral confusibility complicates this a little when we wish to notate the apotome's complement (4deg72) of 64/63 (2deg72), a single *convex right* flag. I was doing it with two stems plus a *convex left* flag, but the above rules dictate two stems with *straight left* and *concave right* flags. As it turns out, the symbol having a single stem with *concave left* and *straight right* flags is also 2deg72, and its apotome complement is two stems plus a *convex left* flag (4deg72), which gives me what I was using before for 4 degrees. So with a little bit of creativity I can still get what I had (and really want) in 72; the same thing can be done in 43-ET. This is the only bit of trickery that I have found any need for in divisions below 100. >> By using a "faux complement," I can avoid using any concave flags for both 72-ET and 43-ET. In fact, the only ET's under 100 that need concave flags (that I have tried so far) are 50, 58, 94, and 96, and none of the more important ones do. I still need to prepare a diagram that illustrates the sequence of symbols in various ET's, and I'd like to do a full-octave diagram for 217 as well, just so we have a better idea of how everything comes out. --George
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