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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
154 189 204 238 250 251 256 277 309 331
333
22 358 23
0 2 25 32 58 69 121 130 140 148
152 176 216 233 254 293 307 308 313 342
355 358
23 372 23
0 6 22 24 43 56 95 126 137 146
172 173 201 213 258 273 281 306 311 355
365 369 372
24 425 23
0 9 33 37 38 97 122 129 140 142
152 191 205 208 252 278 286 326 332 353
368 384 403 425
25 480 25
0 12 29 39 72 91 146 157 160 161
166 191 207 214 258 290 316 354 372 394
396 431 459 467 480
26 492 25
0 5 17 28 36 52 62 106 136 149
174 178 234 241 243 289 292 307 329 368
382 388 409 459 491 492
27 553 27
0 3 15 41 66 95 97 106 142 152
220 221 225 242 295 330 338 354 382 388
402 415 486 504 523 546 553
28 585 27
0 3 15 41 66 95 97 106 142 152
220 221 225 242 295 330 338 354 382 388
402 415 486 504 523 546 553 585
29 623 29
0 7 11 31 43 53 100 121 144 150
202 220 229 268 284 285 356 371 390 416
430 465 467 528 582 590 595 620 623
30 680 29
0 12 32 39 49 82 85 100 147 166
206 207 211 286 302 310 316 344 388 399
462 475 500 529 531 552 623 645 671 680
31 747 31
0 17 22 46 72 78 146 176 186 187
245 273 281 288 308 361 365 384 398 436
521 542 555 586 602 604 668 693 735 738
747
32 784 31
0 7 15 26 28 57 112 118 136 176
177 181 211 214 258 309 318 341 389 403
456 476 512 528 582 628 671 696 745 762
772 784
33 859 32
0 22 38 47 91 108 123 136 141 229
256 263 293 319 329 358 360 400 406 505
516 524 559 573 633 684 685 705 708 763
767 847 859
34 938 37
0 4 19 35 77 99 125 148 162 236
249 282 290 366 387 404 431 459 470 506
540 585 679 685 703 746 771 849 869 878
879 881 931 938
35 997 37
0 46 58 64 98 109 114 139 147 221
276 293 364 387 429 431 466 490 509 556
563 625 673 712 733 743 765 769 864 884
893 969 970 984 997
36 1032 37
0 19 54 67 101 112 140 234 290 322
332 362 368 382 419 437 452 481 533 576
607 656 678 739 763 816 839 842 880 905
907 1011 1015 1016 1023 1032
37 1099 37
0 14 63 87 113 169 220 286 289 328
361 363 381 439 464 475 507 519 529 535
566 700 717 746 798 832 839 862 877 962
981 1002 1029 1086 1090 1091 1099
38 1146 37
0 7 57 59 60 69 89 167 192 235
253 259 353 398 432 468 479 507 534 551
572 648 656 689 702 776 790 813 839 861
903 919 934 938 1050 1090 1141 1146
39 1252 41
0 36 51 94 121 125 126 147 186 257
339 350 391 394 441 509 521 539 585 587
593 622 699 762 802 819 918 941 960 974
1041 1069 1079 1103 1128 1148 1236 1245 1252
40 1288 41
0 4 26 38 65 146 169 174 182 206
261 308 333 411 427 446 517 523 532 599
606 639 690 736 739 837 891 893 922 936
966 986 1054 1111 1177 1225 1235 1246 1287 1288
41 1305 41
0 4 26 38 65 146 169 174 182 206
261 308 333 411 427 446 517 523 532 599
606 639 690 736 739 837 891 893 922 936
966 986 1054 1111 1177 1225 1235 1246 1287 1288
1305
42 1397 41
0 34 49 75 88 91 143 160 272 312
318 402 410 420 431 440 517 582 610 635
642 705 706 756 838 861 883 897 941 1014
1047 1090 1095 1114 1151 1230 1234 1261 1296 1308
1395 1397
43 1513 43
0 15 20 84 128 187 287 288 316 397
443 462 485 494 496 518 581 620 732 738
742 745 799 815 836 881 931 967 1081 1107
1121 1159 1251 1292 1322 1340 1371 1398 1433 1445
1488 1505 1513
44 1596 43
0 58 72 133 190 193 214 319 344 351
353 382 438 553 554 590 608 636 655 678
698 728 795 805 821 874 978 986 1027 1127
1133 1207 1211 1222 1255 1321 1389 1434 1485 1498
1520 1525 1537 1596
45 1687 47
0 42 47 54 83 139 148 165 167 279
319 365 426 500 501 521 531 581 599 650
689 714 741 776 898 920 987 1030 1036 1044
1089 1206 1279 1283 1294 1327 1457 1480 1512 1550
1584 1608 1621 1684 1687
46 1703 47
0 2 10 23 54 115 153 237 255 295
311 338 341 457 519 544 551 617 668 685
705 738 780 877 927 936 1005 1050 1054 1131
1143 1157 1179 1198 1288 1348 1456 1503 1527 1532
1538 1566 1623 1638 1702 1703
47 1830 47
0 11 12 116 157 191 224 231 253 290
326 391 396 419 445 513 566 623 702 715
750 813 822 873 900 931 961 976 986 1117
1138 1194 1278 1347 1390 1429 1437 1461 1475 1479
1481 1605 1730 1747 1811 1827 1830
48 1887 49
0 23 36 93 110 161 177 227 252 409
412 433 473 488 536 580 619 651 688 716
770 817 860 905 913 919 954 965 1118 1127
1239 1301 1319 1320 1433 1459 1489 1531 1589 1609
1754 1764 1776 1849 1853 1880 1882 1887
49 1958 49
0 17 20 86 119 140 166 227 240 255
353 430 520 559 564 565 602 675 724 781
817 833 905 929 961 970 980 1131 1162 1189
1212 1319 1403 1433 1437 1451 1462 1497 1504 1589
1601 1680 1763 1785 1825 1880 1888 1956 1958
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1329 1391 1525 1536 1540 1543 1577 1593 1601 1671
1672 1710 1809 1829 1834 1973 2027 2046 2063 2094
51 2190 53
0 1 8 26 39 149 223 247 355 384
419 439 485 506 508 548 585 644 650 761
900 936 950 1018 1035 1040 1110 1168 1217 1244
1289 1322 1337 1480 1489 1492 1508 1670 1732 1785
1815 1826 1858 1912 1983 2043 2095 2099 2146 2156
2190
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0 80 81 88 106 119 229 303 327 435
464 499 519 565 586 588 628 665 724 730
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1324 1369 1402 1417 1560 1569 1572 1588 1750 1812
1865 1895 1906 1938 1992 2063 2123 2175 2179 2226
2236 2270
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0 1 9 30 107 211 248 273 297 330
372 386 452 528 572 587 600 655 708 778
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1211 1249 1358 1423 1469 1473 1619 1636 1707 1788
1804 1807 1852 1937 1947 2081 2120 2154 2245 2257
2280 2341 2347
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0 1 9 30 107 211 248 273 297 330
372 386 452 528 572 587 600 655 708 778
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1211 1249 1358 1423 1469 1473 1619 1636 1707 1788
1804 1807 1852 1937 1947 2081 2120 2154 2245 2257
2280 2341 2347 2373
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0 20 86 112 126 141 215 242 284 355
361 525 561 606 703 734 782 842 936 953
957 992 1016 1035 1088 1139 1140 1149 1224 1379
1387 1425 1489 1547 1642 1759 1827 1849 1903 1936
1947 2054 2127 2215 2265 2272 2306 2399 2431 2531
2568 2580 2593 2596 2598
56 2725 59
0 45 52 110 134 169 190 247 298 316
434 489 661 727 757 759 800 838 884 901
1000 1034 1054 1067 1094 1213 1235 1303 1306 1356
1398 1554 1563 1582 1686 1730 1734 1837 2012 2048
2087 2095 2118 2124 2209 2273 2408 2433 2494 2495
2505 2510 2625 2699 2713 2725
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0 16 137 227 231 245 301 373 405 439
534 545 631 686 699 817 829 866 908 929
962 1010 1163 1194 1199 1216 1277 1350 1365 1412
1472 1570 1629 1669 1808 1832 1852 1890 1909 1917
1919 2022 2246 2281 2405 2428 2431 2456 2520 2604
2647 2656 2697 2727 2766 2772 2773
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0 24 87 106 158 270 343 381 446 459
464 471 481 620 636 711 725 755 907 953
968 1015 1052 1094 1122 1128 1148 1314 1357 1386
1443 1528 1536 1567 1659 1668 1852 1856 1892 1966
2019 2060 2083 2128 2178 2395 2397 2446 2548 2645
2646 2678 2705 2726 2782 2793 2848 2851
59 2911 59
0 4 14 133 162 218 400 415 435 440
473 517 576 671 684 805 879 930 980 1056
1073 1122 1154 1200 1242 1261 1272 1313 1347 1412
1577 1599 1608 1688 1691 1840 1897 1936 2006 2100
2162 2163 2169 2190 2216 2300 2397 2542 2550 2587
2610 2674 2710 2722 2765 2789 2893 2895 2911
60 3019 59
0 15 123 168 171 203 208 227 284 386
446 546 584 595 623 656 774 790 840 852
904 910 1016 1143 1157 1241 1250 1315 1332 1378
1407 1461 1502 1592 1732 1734 1757 1843 1851 1939
2006 2042 2131 2141 2175 2228 2246 2372 2504 2511
2654 2658 2684 2685 2705 2727 2806 2951 3006 3019
61 3184 61
0 2 22 71 103 153 187 282 375 408
415 488 555 567 584 625 644 689 744 862
983 1092 1107 1115 1145 1253 1315 1339 1340 1383
1497 1511 1694 1704 1760 1819 1858 1909 1946 1994
2000 2003 2021 2144 2287 2292 2318 2334 2370 2538
2637 2729 2757 2792 2803 2864 2868 2970 3067 3080
3184
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0 16 21 38 63 71 74 81 200 251
291 310 418 462 497 569 593 608 683 729
752 766 872 967 997 1029 1095 1097 1179 1273
1302 1351 1382 1436 1500 1506 1702 1722 1815 1819
1867 1876 2014 2147 2175 2262 2263 2339 2366 2471
2505 2627 2640 2729 2821 2877 2922 2963 2989 3062
3203 3215
63 3391 64
0 22 126 143 172 187 272 278 395 429
487 517 598 654 687 699 736 738 846 933
976 1001 1117 1177 1244 1339 1438 1451 1470 1510
1517 1541 1615 1619 1624 1835 2009 2020 2072 2095
2145 2148 2308 2336 2377 2378 2432 2497 2631 2728
2744 2764 2821 3019 3045 3083 3163 3177 3198 3225
3373 3381 3391
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0 8 44 210 231 247 318 319 510 524
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1234 1251 1279 1398 1422 1468 1483 1533 1536 1617
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2278 2318 2404 2464 2543 2582 2664 2676 2699 2733
2789 3000 3027 3075 3104 3124 3167 3197 3256 3297
3303 3465 3517 3527
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0 8 44 210 231 247 318 319 510 524
626 651 693 706 725 867 869 878 998 1156
1234 1251 1279 1398 1422 1468 1483 1533 1536 1617
1643 1648 1681 1965 1969 1987 2070 2124 2220 2227
2278 2318 2404 2464 2543 2582 2664 2676 2699 2733
2789 3000 3027 3075 3104 3124 3167 3197 3256 3297
3303 3465 3517 3527 3593
66 3757 67
0 17 25 44 135 180 223 276 345 391
426 440 505 609 675 709 732 894 899 905
909 925 972 1002 1026 1247 1337 1429 1462 1484
1513 1545 1573 1615 1671 1739 1859 1868 2007 2020
2106 2240 2278 2281 2462 2483 2590 2602 2654 2661
2791 2933 2934 3039 3079 3291 3293 3341 3378 3450
3532 3606 3645 3681 3739 3757
67 3819 67
0 17 25 44 135 180 223 276 345 391
426 440 505 609 675 709 732 894 899 905
909 925 972 1002 1026 1247 1337 1429 1462 1484
1513 1545 1573 1615 1671 1739 1859 1868 2007 2020
2106 2240 2278 2281 2462 2483 2590 2602 2654 2661
2791 2933 2934 3039 3079 3291 3293 3341 3378 3450
3532 3606 3645 3681 3739 3757 3819
68 3956 67
0 23 70 137 142 178 241 332 342 350
372 564 629 654 693 697 713 820 853 881
964 1112 1131 1197 1229 1310 1354 1425 1446 1598
1693 1694 1720 1749 1751 1799 1879 1896 1931 2105
2156 2339 2345 2457 2526 2600 2603 2615 2703 2741
2765 2819 2994 3031 3040 3133 3294 3369 3376 3418
3429 3463 3539 3736 3749 3822 3942 3956
69 4145 71
0 41 55 111 171 191 276 322 442 450
495 517 697 751 763 769 853 939 996 1013
1046 1117 1128 1155 1324 1356 1420 1462 1478 1554
1555 1758 1872 1990 2027 2219 2244 2280 2327 2332
2342 2371 2405 2504 2517 2552 2662 2801 2810 2950
3015 3019 3172 3198 3301 3395 3398 3438 3539 3562
3851 3902 3930 3932 3951 4019 4114 4138 4145
70 4217 71
0 18 24 47 99 103 143 176 252 369
431 489 519 591 625 718 757 779 798 810
869 1043 1052 1112 1150 1233 1234 1254 1328 1443
1454 1551 1629 1684 1880 1923 2041 2092 2127 2141
2289 2376 2430 2433 2446 2601 2647 2692 2729 2757
2793 2916 2924 3029 3260 3286 3349 3397 3463 3513
3647 3771 3910 3977 4045 4175 4185 4190 4192 4217
71 4330 71
0 13 14 31 132 212 218 316 376 526
628 655 667 670 696 705 781 847 891 921
953 1147 1367 1374 1421 1458 1651 1655 1745 1764
1785 1833 1884 1920 1987 1992 2152 2233 2267 2340
2395 2478 2543 2571 2668 2739 2876 2928 2939 3003
3028 3061 3085 3177 3298 3300 3320 3343 3467 3650
3729 3745 3841 3890 3900 3946 4063 4199 4269 4277
4330
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0 51 118 140 211 264 283 425 502 618
635 655 666 762 843 945 960 1021 1176 1225
1238 1419 1453 1507 1553 1576 1658 1728 1865 1957
1964 2014 2047 2089 2092 2133 2201 2281 2387 2482
2620 2730 2734 2735 2793 2996 3048 3056 3080 3219
3298 3364 3465 3477 3493 3495 3520 3661 3667 3765
3852 3873 3887 4144 4217 4314 4343 4353 4379 4417
4464 4473
73 4513 73
0 40 91 158 180 251 304 323 465 542
658 675 695 706 802 883 985 1000 1061 1216
1265 1278 1459 1493 1547 1593 1616 1698 1768 1905
1997 2004 2054 2087 2129 2132 2173 2241 2321 2427
2522 2660 2770 2774 2775 2833 3036 3088 3096 3120
3259 3338 3404 3505 3517 3533 3535 3560 3701 3707
3805 3892 3913 3927 4184 4257 4354 4383 4393 4419
4457 4504 4513
74 4753 73
0 40 91 158 180 251 304 323 465 542
658 675 695 706 802 883 985 1000 1061 1216
1265 1278 1459 1493 1547 1593 1616 1698 1768 1905
1997 2004 2054 2087 2129 2132 2173 2241 2321 2427
2522 2660 2770 2774 2775 2833 3036 3088 3096 3120
3259 3338 3404 3505 3517 3533 3535 3560 3701 3707
3805 3892 3913 3927 4184 4257 4354 4383 4393 4419
4457 4504 4513 4753
75 4982 79
0 12 79 82 166 171 209 348 359 423
485 491 629 735 883 1006 1053 1086 1125 1243
1271 1312 1454 1467 1475 1558 1931 1949 1953 1979
2034 2100 2151 2165 2174 2209 2262 2462 2472 2535
2640 2762 2764 2796 2898 3201 3294 3313 3353 3390
3414 3598 3652 3669 3698 3747 3797 3833 3962 4038
4069 4163 4179 4355 4370 4412 4468 4528 4707 4708
4822 4930 4955 4975 4982
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0 21 68 74 82 104 265 382 409 593
655 657 781 851 1044 1125 1134 1213 1279 1355
1384 1422 1455 1468 1686 1807 1818 1819 1905 1946
2045 2052 2212 2313 2361 2364 2420 2483 2538 2697
2800 2860 2895 2929 3052 3101 3146 3198 3276 3308
3348 3601 3618 3805 3907 3987 4018 4044 4062 4072
4230 4267 4272 4420 4445 4554 4774 4790 4794 4867
4886 4910 4925 4975 5066 5089
77 5204 79
0 21 68 74 82 104 265 382 409 593
655 657 781 851 1044 1125 1134 1213 1279 1355
1384 1422 1455 1468 1686 1807 1818 1819 1905 1946
2045 2052 2212 2313 2361 2364 2420 2483 2538 2697
2800 2860 2895 2929 3052 3101 3146 3198 3276 3308
3348 3601 3618 3805 3907 3987 4018 4044 4062 4072
4230 4267 4272 4420 4445 4554 4774 4790 4794 4867
4886 4910 4925 4975 5066 5089 5204
78 5299 79
0 9 33 160 205 331 395 443 523 608
817 818 875 890 937 951 979 1004 1020 1177
1199 1211 1352 1457 1477 1568 1665 1749 1815 1915
1918 1953 1970 2154 2204 2214 2260 2445 2515 2541
2580 2673 2678 2825 2856 3024 3098 3246 3328 3341
3347 3368 3565 3653 3696 3745 3747 3777 3921 4080
4229 4247 4337 4458 4526 4601 4612 4866 4965 5036
5043 5072 5080 5159 5182 5236 5295 5299
79 5408 79
0 109 118 142 269 314 440 504 552 632
717 926 927 984 999 1046 1060 1088 1113 1129
1286 1308 1320 1461 1566 1586 1677 1774 1858 1924
2024 2027 2062 2079 2263 2313 2323 2369 2554 2624
2650 2689 2782 2787 2934 2965 3133 3207 3355 3437
3450 3456 3477 3674 3762 3805 3854 3856 3886 4030
4189 4338 4356 4446 4567 4635 4710 4721 4975 5074
5145 5152 5181 5189 5268 5291 5345 5404 5408
80 5563 79
0 9 23 48 131 157 190 212 335 425
449 585 621 695 770 814 857 935 975 1067
1114 1163 1166 1193 1290 1309 1322 1506 1541 1613
1641 1782 1839 1843 1897 1912 1983 2025 2294 2300
2494 2582 2662 2861 2877 2911 2923 2928 3048 3086
3150 3333 3364 3417 3424 3571 3641 3744 3861 3924
3965 4157 4225 4246 4323 4331 4761 4790 4855 4856
4866 4967 5100 5118 5120 5270 5425 5481 5518 5563
81 5717 83
0 19 27 30 95 111 157 265 433 492
561 593 732 867 907 1017 1083 1090 1114 1139
1308 1391 1427 1512 1693 1721 1763 1907 2000 2063
2113 2152 2165 2200 2212 2318 2332 2355 2396 2652
2658 2748 2882 2936 3011 3015 3097 3115 3336 3462
3523 3659 3805 3850 3879 4027 4136 4169 4174 4189
4191 4314 4417 4568 4612 4692 4726 4797 4885 4928
5145 5239 5346 5355 5356 5413 5528 5554 5605 5626
5717
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0 24 52 66 106 142 192 235 396 428
616 667 754 803 924 969 1054 1113 1202 1260
1276 1329 1339 1409 1679 1740 1778 1834 1859 1898
2012 2043 2072 2077 2089 2335 2406 2718 2719 2741
2842 2850 2917 2985 3098 3170 3232 3258 3262 3273
3444 3463 3541 3703 3810 3932 3967 4164 4212 4219
4310 4414 4461 4586 4698 4782 4800 4803 4809 4959
5003 5098 5263 5265 5380 5417 5490 5590 5698 5718
5731 5814
83 6020 83
0 59 148 159 231 324 334 351 525 740
885 947 952 1029 1049 1157 1186 1317 1325 1483
1506 1541 1615 1639 1646 1725 1793 1836 1874 1920
1989 1991 2036 2245 2549 2563 2564 2582 2603 2833
2849 2945 2967 3066 3096 3190 3378 3412 3468 3582
3614 3618 3705 3843 3949 3962 4012 4037 4092 4285
4350 4398 4697 4757 4823 4849 5036 5140 5177 5218
5262 5313 5319 5480 5533 5718 5760 5788 5895 5959
5968 5971 6020
84 6159 83
0 42 77 124 288 339 368 407 434 494
495 644 682 765 784 788 936 1005 1209 1254
1257 1300 1414 1477 1518 1531 1713 1763 1878 1904
1998 2256 2309 2340 2441 2448 2485 2503 2537 2559
2675 2878 2888 2899 3077 3205 3286 3310 3385 3444
3461 3551 3553 3686 3833 3986 4140 4165 4170 4237
4243 4301 4366 4503 4539 4625 4725 4823 4837 4894
5004 5037 5053 5443 5528 5536 5556 5568 5698 5872
5881 5951 6144 6159
85 6477 89
0 45 75 208 219 228 392 398 449 453
465 511 547 596 610 809 852 1164 1204 1228
1282 1497 1550 1651 1741 1767 1827 1877 1929 1947
2176 2310 2341 2354 2369 2425 2462 2491 2621 2798
2930 3037 3102 3240 3287 3288 3368 3391 3560 3601
3757 3796 3892 3925 3997 4016 4251 4276 4483 4552
4594 4677 4694 4783 4817 4958 4965 5104 5125 5399
5487 5525 5599 5601 5604 5696 5862 5939 5949 5971
6187 6255 6442 6469 6477
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0 22 83 129 211 248 295 300 316 348
650 683 868 930 933 1075 1109 1132 1160 1295
1324 1479 1570 1577 1636 1647 1667 1837 1875 2011
2066 2183 2322 2335 2365 2380 2444 2453 2695 2766
2853 2961 3001 3075 3076 3111 3235 3289 3415 3487
3526 3620 3823 3916 4063 4112 4179 4356 4383 4469
4486 4592 4642 4754 4835 4960 5293 5349 5425 5444
5524 5565 5625 5786 5849 6018 6043 6087 6297 6438
6440 6446 6450 6464 6542 6584
87 6708 89
0 33 36 81 91 109 156 186 497 593
639 784 821 835 852 923 945 1120 1252 1260
1326 1367 1618 1629 1679 1746 1805 1844 1867 1923
1957 2198 2200 2333 2368 2525 2729 2735 2843 2872
2929 3051 3103 3163 3252 3301 3386 3533 3751 3772
3804 3876 3920 4020 4039 4150 4166 4193 4371 4383
4564 4590 4628 4731 4830 5013 5033 5164 5242 5322
5326 5506 5521 5546 5758 5782 6068 6122 6131 6223
6228 6387 6394 6481 6523 6695 6708
88 6745 89
0 3 48 58 76 123 153 464 560 606
751 788 802 819 890 912 1087 1219 1227 1293
1334 1585 1596 1646 1713 1772 1811 1834 1890 1924
2165 2167 2300 2335 2492 2696 2702 2810 2839 2896
3018 3070 3130 3219 3268 3353 3500 3718 3739 3771
3843 3887 3987 4006 4117 4133 4160 4338 4350 4531
4557 4595 4698 4797 4980 5000 5131 5209 5289 5293
5473 5488 5513 5725 5749 6035 6089 6098 6190 6195
6354 6361 6448 6490 6662 6675 6744 6745
89 6778 89
0 33 36 81 91 109 156 186 497 593
639 784 821 835 852 923 945 1120 1252 1260
1326 1367 1618 1629 1679 1746 1805 1844 1867 1923
1957 2198 2200 2333 2368 2525 2729 2735 2843 2872
2929 3051 3103 3163 3252 3301 3386 3533 3751 3772
3804 3876 3920 4020 4039 4150 4166 4193 4371 4383
4564 4590 4628 4731 4830 5013 5033 5164 5242 5322
5326 5506 5521 5546 5758 5782 6068 6122 6131 6223
6228 6387 6394 6481 6523 6695 6708 6777 6778
90 6967 89
0 189 222 225 270 280 298 345 375 686
782 828 973 1010 1024 1041 1112 1134 1309 1441
1449 1515 1556 1807 1818 1868 1935 1994 2033 2056
2112 2146 2387 2389 2522 2557 2714 2918 2924 3032
3061 3118 3240 3292 3352 3441 3490 3575 3722 3940
3961 3993 4065 4109 4209 4228 4339 4355 4382 4560
4572 4753 4779 4817 4920 5019 5202 5222 5353 5431
5511 5515 5695 5710 5735 5947 5971 6257 6311 6320
6412 6417 6576 6583 6670 6712 6884 6897 6966 6967
91 7570 97
0 36 52 79 105 142 165 187 356 452
599 648 677 683 702 856 977 1221 1232 1294
1422 1517 1616 1626 1628 1674 1823 1884 2051 2065
2310 2344 2437 2528 2556 2622 2626 2788 2925 2964
3123 3136 3352 3353 3420 3475 3834 3885 4026 4033
4204 4306 4314 4356 4439 4444 4519 4578 4619 4663
4693 4780 5018 5021 5161 5287 5319 5451 5469 5637
5977 5997 6054 6069 6078 6224 6340 6380 6444 6500
6533 6733 6864 7101 7208 7225 7246 7322 7434 7499
7570
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0 36 52 79 105 142 165 187 356 452
599 648 677 683 702 856 977 1221 1232 1294
1422 1517 1616 1626 1628 1674 1823 1884 2051 2065
2310 2344 2437 2528 2556 2622 2626 2788 2925 2964
3123 3136 3352 3353 3420 3475 3834 3885 4026 4033
4204 4306 4314 4356 4439 4444 4519 4578 4619 4663
4693 4780 5018 5021 5161 5287 5319 5451 5469 5637
5977 5997 6054 6069 6078 6224 6340 6380 6444 6500
6533 6733 6864 7101 7208 7225 7246 7322 7434 7499
7570 7617
93 7726 97
0 17 97 139 192 211 315 379 550 692
699 704 762 840 950 1183 1288 1342 1353 1535
1587 1621 1656 1740 1811 1905 1978 2301 2319 2328
2404 2445 2453 2588 2590 2746 2750 2779 2959 2987
3209 3212 3308 3318 3324 3503 3528 3585 3758 3813
3839 4039 4254 4255 4278 4322 4405 4452 4683 4758
4860 5140 5160 5280 5572 5604 5634 5670 5836 5936
5982 5997 6224 6245 6337 6468 6596 6743 6924 7015
7123 7280 7319 7359 7396 7409 7452 7483 7497 7615
7666 7704 7726
94 7884 97
0 36 52 79 105 142 165 187 356 452
599 648 677 683 702 856 977 1221 1232 1294
1422 1517 1616 1626 1628 1674 1823 1884 2051 2065
2310 2344 2437 2528 2556 2622 2626 2788 2925 2964
3123 3136 3352 3353 3420 3475 3834 3885 4026 4033
4204 4306 4314 4356 4439 4444 4519 4578 4619 4663
4693 4780 5018 5021 5161 5287 5319 5451 5469 5637
5977 5997 6054 6069 6078 6224 6340 6380 6444 6500
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7570 7617 7853 7884
95 7967 97
0 22 28 270 280 295 329 433 574 610
626 666 864 926 1047 1066 1286 1319 1414 1415
1432 1548 1574 1637 1867 1943 1954 2051 2153 2246
2317 2377 2424 2428 2492 2497 2766 2875 2889 3093
3174 3198 3343 3389 3556 3657 3700 3847 3868 3876
3921 3959 3982 4047 4383 4571 4610 4737 4827 4854
5035 5070 5090 5242 5284 5296 5445 5454 5646 5732
5883 6037 6040 6159 6226 6558 6571 6602 6643 6701
6767 6799 7088 7322 7401 7458 7495 7632 7735 7742
7817 7819 7867 7897 7967
96 8150 97
0 40 57 205 295 361 579 581 612 789
886 961 1026 1087 1146 1267 1329 1377 1435 1445
1482 1689 1887 2011 2034 2112 2225 2246 2257 2342
2513 2612 2656 2682 2694 2695 2745 2831 2884 3209
3227 3318 3385 3571 3644 3814 3848 3936 3960 4036
4064 4260 4275 4522 4571 4673 4788 4823 4852 4868
4955 4975 4982 5117 5607 5667 5736 5761 5880 5935
6046 6138 6144 6365 6529 6538 6610 6656 6816 6979
7136 7179 7320 7372 7376 7633 7647 7652 7655 7726
7881 7989 8019 8073 8114 8150
97 8357 97
0 61 134 184 233 274 312 340 544 628
744 747 767 974 983 1020 1036 1103 1245 1349
1367 1499 1504 1525 1818 1852 1915 1985 2060 2100
2102 2156 2367 2439 2443 2491 2521 2622 2711 2816
2885 3028 3257 3263 3349 3691 3789 3802 3821 3889
3948 3991 4036 4117 4188 4479 4486 4812 4836 4837
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5721 5831 5896 6097 6133 6141 6419 6650 6910 6927
6984 7048 7075 7087 7228 7372 7484 7577 7702 7940
8025 8054 8076 8131 8189 8249 8357
98 8462 101
0 47 60 115 161 194 231 290 403 524
711 796 951 978 996 1050 1082 1131 1581 1634
1656 1676 1822 1833 1895 2091 2093 2143 2223 2329
2393 2487 2609 2709 2882 2916 3041 3097 3179 3286
3455 3552 3578 3592 3617 3661 3829 3983 4086 4107
4197 4330 4333 4551 4690 4741 4830 4868 5100 5163
5192 5304 5409 5467 5497 5722 5732 5809 5850 6058
6124 6406 6407 6422 6571 6747 6855 6867 6890 6898
7046 7050 7274 7279 7476 7661 7732 7874 8077 8084
8101 8175 8194 8203 8251 8395 8456 8462
99 8540 101
0 47 60 115 161 194 231 290 403 524
711 796 951 978 996 1050 1082 1131 1581 1634
1656 1676 1822 1833 1895 2091 2093 2143 2223 2329
2393 2487 2609 2709 2882 2916 3041 3097 3179 3286
3455 3552 3578 3592 3617 3661 3829 3983 4086 4107
4197 4330 4333 4551 4690 4741 4830 4868 5100 5163
5192 5304 5409 5467 5497 5722 5732 5809 5850 6058
6124 6406 6407 6422 6571 6747 6855 6867 6890 6898
7046 7050 7274 7279 7476 7661 7732 7874 8077 8084
8101 8175 8194 8203 8251 8395 8456 8462 8540
100 8831 101
0 291 338 351 406 452 485 522 581 694
815 1002 1087 1242 1269 1287 1341 1373 1422 1872
1925 1947 1967 2113 2124 2186 2382 2384 2434 2514
2620 2684 2778 2900 3000 3173 3207 3332 3388 3470
3577 3746 3843 3869 3883 3908 3952 4120 4274 4377
4398 4488 4621 4624 4842 4981 5032 5121 5159 5391
5454 5483 5595 5700 5758 5788 6013 6023 6100 6141
6349 6415 6697 6698 6713 6862 7038 7146 7158 7181
7189 7337 7341 7565 7570 7767 7952 8023 8165 8368
8375 8392 8466 8485 8494 8542 8686 8747 8753 8831
101 8897 101
0 20 43 44 99 106 231 244 302 433
562 592 883 1066 1103 1213 1251 1302 1311 1392
1497 1563 1597 1609 1624 1720 1777 2122 2191 2219
2250 2462 2527 2696 2826 2894 3067 3243 3264 3293
3346 3483 3596 3602 3618 3650 3690 3883 4053 4098
4234 4308 4319 4327 4576 4581 4896 4991 5005 5040
5131 5428 5432 5505 5508 5592 5670 5819 5886 5993
6032 6171 6223 6241 6365 6375 6547 6731 6814 6847
6968 7143 7207 7309 7413 7460 7688 7729 7843 7951
7976 7993 8068 8094 8286 8463 8578 8739 8741 8861
8897
102 9218 101
0 104 120 373 412 453 551 624 646 688
861 889 1027 1088 1221 1258 1371 1385 1610 1648
1769 1894 1964 1975 2288 2315 2323 2455 2505 2573
2665 2777 2940 2960 2984 2986 3071 3228 3285 3295
3364 3487 3642 3671 3678 3787 3850 4110 4116 4206
4284 4406 4436 4453 4742 4824 4968 5044 5110 5220
5225 5229 5285 5474 5677 5692 5695 5778 6272 6273
6372 6421 6469 6528 6634 7172 7265 7298 7323 7400
7412 7452 7484 7503 7546 7782 7837 7912 7925 8053
8327 8358 8381 8645 8753 8798 8915 8986 9020 9144
9197 9218
103 9408 103
0 111 246 266 373 453 455 534 585 807
871 912 1009 1013 1187 1418 1454 1508 1516 1668
1708 1854 2115 2180 2342 2508 2540 2593 2712 2737
2804 2972 3152 3166 3208 3280 3329 3445 3629 3690
3717 3785 3932 3960 3961 4352 4359 4510 4540 4555
4639 4644 4663 4896 4922 5130 5232 5506 5615 5670
5701 5841 5880 5917 5990 6000 6023 6034 6523 6545
6728 6744 6929 6967 7025 7042 7274 7280 7326 7419
7493 7543 7556 7643 7713 7784 7861 8109 8156 8433
8490 8499 8511 8559 8602 8925 8960 9019 9150 9272
9275 9390 9408
104 9581 103
0 111 246 266 373 453 455 534 585 807
871 912 1009 1013 1187 1418 1454 1508 1516 1668
1708 1854 2115 2180 2342 2508 2540 2593 2712 2737
2804 2972 3152 3166 3208 3280 3329 3445 3629 3690
3717 3785 3932 3960 3961 4352 4359 4510 4540 4555
4639 4644 4663 4896 4922 5130 5232 5506 5615 5670
5701 5841 5880 5917 5990 6000 6023 6034 6523 6545
6728 6744 6929 6967 7025 7042 7274 7280 7326 7419
7493 7543 7556 7643 7713 7784 7861 8109 8156 8433
8490 8499 8511 8559 8602 8925 8960 9019 9150 9272
9275 9390 9408 9581
105 9893 107
0 5 85 182 241 263 276 283 354 386
470 614 660 811 1198 1335 1404 1422 1437 1494
1554 1647 1702 1739 1747 2026 2162 2173 2189 2293
2435 2466 2658 2896 2930 2949 3133 3150 3359 3483
3731 3733 3780 3794 4046 4155 4176 4288 4338 4437
4686 4726 4727 4756 4794 4800 5247 5303 5368 5464
5589 5695 5784 5870 5965 5989 6100 6154 6177 6311
6409 6555 6567 6643 6694 6769 6907 6969 7074 7140
7314 7443 7469 7790 7942 8025 8189 8193 8241 8359
8369 8555 8677 8892 9007 9032 9035 9071 9114 9371
9525 9666 9724 9884 9893
106 10135 107
0 178 252 339 349 554 674 703 733 966
1002 1066 1123 1126 1138 1192 1340 1347 1439 1608
1625 1766 1818 1995 2170 2362 2364 2506 2584 2600
2646 2686 2719 2975 2983 3063 3166 3229 3478 3563
3673 3696 4139 4171 4197 4248 4293 4298 4382 4654
4821 4929 4950 4968 4977 5180 5248 5318 5379 5422
5568 5684 5847 6087 6255 6331 6353 6378 6459 6496
6515 6696 6749 6763 6791 7036 7049 7237 7342 7558
7599 7690 7797 7909 8020 8085 8091 8255 8400 8525
8655 8842 8932 8981 9015 9240 9446 9753 9757 9777
9788 9832 9870 9984 9985 10135
107 10241 109
0 44 123 224 343 372 479 593 620 657
765 845 955 1218 1230 1376 1430 1647 1721 1725
1853 1889 1920 1978 2271 2295 2506 2842 2944 2962
3240 3256 3288 3369 3422 3516 3521 3562 3577 3760
3780 3902 3957 3964 3965 4014 4369 4465 4661 4680
4752 4927 4995 5106 5149 5236 5319 5336 5452 5800
5842 5918 5963 6080 6132 6450 6606 6617 6631 6889
6936 7236 7320 7360 7425 7446 7455 7552 7775 7788
7814 7865 8015 8153 8188 8222 8297 8379 8382 8649
8894 8900 9043 9065 9131 9202 9204 9581 9783 9842
9934 9957 10110 10143 10203 10213 10241
108 10415 109
0 28 146 148 208 281 293 374 394 446
772 859 1022 1134 1181 1288 1377 1725 1728 1736
1754 1880 1959 2273 2286 2290 2389 2479 2547 2563
2618 2677 2788 2815 3153 3158 3322 3368 3483 3493
3525 3589 3765 3780 3843 3947 3953 4337 4480 4499
4536 4550 4616 4851 4970 4991 4992 5067 5120 5262
5303 5312 5606 5790 6001 6039 6263 6303 6386 6440
6463 6507 6682 6863 6912 6945 7231 7362 7584 7620
7678 7925 8158 8201 8249 8481 8538 8583 8842 8867
8991 9079 9230 9269 9299 9330 9364 9486 9713 9799
9907 9931 10109 10116 10214 10288 10323 10415
109 10583 109
0 28 146 148 208 281 293 374 394 446
772 859 1022 1134 1181 1288 1377 1725 1728 1736
1754 1880 1959 2273 2286 2290 2389 2479 2547 2563
2618 2677 2788 2815 3153 3158 3322 3368 3483 3493
3525 3589 3765 3780 3843 3947 3953 4337 4480 4499
4536 4550 4616 4851 4970 4991 4992 5067 5120 5262
5303 5312 5606 5790 6001 6039 6263 6303 6386 6440
6463 6507 6682 6863 6912 6945 7231 7362 7584 7620
7678 7925 8158 8201 8249 8481 8538 8583 8842 8867
8991 9079 9230 9269 9299 9330 9364 9486 9713 9799
9907 9931 10109 10116 10214 10288 10323 10415 10583
110 10767 109
0 13 47 97 183 257 299 411 517 619
673 712 812 897 906 941 1023 1257 1358 1515
1530 1859 1861 1884 1921 2109 2135 2249 2628 2632
2783 2926 2932 3024 3185 3197 3214 3215 3304 3360
3540 3567 3663 3746 4102 4211 4260 4341 4393 4398
4412 4579 4856 4966 5079 5087 5103 5279 5343 5384
5546 5597 5645 6006 6086 6156 6233 6401 6479 6566
6606 6627 6638 6836 7035 7057 7102 7177 7374 7540
7872 7967 7987 8025 8091 8156 8199 8202 8209 8334
8624 8655 8683 8746 8962 8995 9031 9337 9416 9471
9612 9740 9816 9904 10124 10197 10341 10596 10699 10767
111 11108 113
0 77 109 297 371 560 603 700 704 790
817 975 1023 1032 1044 1603 1616 1632 1773 1783
1856 1919 2114 2168 2201 2384 2419 2466 2468 2557
2601 2667 2767 2974 3177 3288 3289 3436 3544 3744
3866 3871 3921 4267 4395 4545 4569 4633 4792 4829
4907 4982 5013 5021 5150 5184 5509 5526 5571 5629
5860 6041 6246 6287 6367 6439 6491 6531 6703 6821
6881 6900 7340 7358 7549 7577 7653 7675 7818 7829
7888 7914 8122 8267 8282 8320 8548 8690 8821 8824
8926 9112 9362 9487 9709 9755 9848 9871 9878 10027
10157 10252 10351 10562 10882 10976 11001 11037 11043 11057
11108
112 11292 113
0 76 79 145 181 206 451 462 483 495
542 663 998 1003 1153 1195 1226 1235 1393 1508
1660 1860 1887 2065 2128 2181 2236 2446 2563 2577
2625 2724 2725 2860 2878 3048 3152 3171 3217 3391
3481 3756 3785 3866 3998 4000 4004 4194 4424 4565
4635 4648 4685 4749 4772 4794 4824 4957 5015 5323
5495 5733 5793 5906 6009 6093 6191 6211 6297 6462
6604 6628 7016 7088 7105 7173 7489 7698 7752 7809
7847 8042 8268 8278 8375 8624 8632 8720 8849 8884
8923 9371 9564 9683 9690 9724 9739 9970 9986 10098
10126 10169 10220 10312 10598 10675 10852 10930 11074 11199
11225 11292
113 11423 113
0 57 81 209 228 396 438 490 550 576
578 809 821 867 888 942 1065 1088 1251 1477
1814 1864 1988 2079 2114 2122 2132 2236 2425 2458
2676 2785 3077 3143 3170 3192 3206 3309 3417 3476
3586 3611 3899 3960 4060 4138 4233 4346 4384 4416
4586 4601 4691 4929 4976 5082 5126 5166 5171 5258
5605 5703 6023 6062 6091 6211 6275 6416 6423 6657
6677 6739 6876 7019 7374 7492 7566 7575 7789 7862
7865 7896 7992 8368 8398 8453 8469 8470 8628 8634
8897 9093 9192 9248 9317 9328 9600 9731 9779 9876
9995 10032 10036 10330 10537 10550 10741 10947 11119 11196
11261 11312 11423
114 11764 113
0 77 109 297 371 560 603 700 704 790
817 975 1023 1032 1044 1603 1616 1632 1773 1783
1856 1919 2114 2168 2201 2384 2419 2466 2468 2557
2601 2667 2767 2974 3177 3288 3289 3436 3544 3744
3866 3871 3921 4267 4395 4545 4569 4633 4792 4829
4907 4982 5013 5021 5150 5184 5509 5526 5571 5629
5860 6041 6246 6287 6367 6439 6491 6531 6703 6821
6881 6900 7340 7358 7549 7577 7653 7675 7818 7829
7888 7914 8122 8267 8282 8320 8548 8690 8821 8824
8926 9112 9362 9487 9709 9755 9848 9871 9878 10027
10157 10252 10351 10562 10882 10976 11001 11037 11043 11057
11108 11400 11468 11764
115 12212 121
0 118 132 189 209 443 467 527 724 880
1028 1287 1534 1608 1650 1656 1771 1896 2006 2022
2031 2081 2168 2176 2189 2379 2382 2383 2803 2835
2931 3023 3239 3413 3439 3468 3515 3649 4111 4177
4280 4451 4482 4523 4635 4779 4852 5260 5282 5345
5396 5442 5569 5597 5710 5737 5861 5876 5965 6143
6188 6250 6308 6521 6817 6827 7066 7287 7327 7420
7506 7714 7744 7823 8027 8203 8297 8302 8493 8529
8688 8741 8852 9004 9027 9245 9346 9475 9518 9740
9807 9809 9863 9897 10040 10339 10404 10422 10439 10868
11060 11112 11190 11337 11443 11763 11827 12013 12025 12032
12093 12130 12163 12174 12212
116 12412 121
0 180 261 291 305 418 451 553 607 741
788 932 1179 1230 1326 1541 1715 1734 1771 1797
1956 1974 2023 2274 2291 2327 2555 2713 2737 3056
3467 3551 3566 3582 3593 3902 3954 4162 4171 4226
4231 4266 4367 4444 4519 4611 4650 4845 4917 4951
5175 5329 5639 5722 5862 5864 5971 6056 6064 6076
6285 6531 6569 6572 6692 6928 6935 6978 7394 7404
7426 7523 7754 8029 8057 8406 8557 8578 8603 8651
8721 8727 8835 8925 9327 9393 9464 9493 9538 9603
9626 9758 9845 9995 10008 10125 10469 10581 10585 10642
10764 10912 10980 11151 11509 11571 11674 11772 11850 11971
12057 12146 12225 12226 12284 12412
117 12517 121
0 102 145 361 397 509 681 703 719 769
928 1321 1501 1513 1541 1555 1637 1638 1745 1752
1912 1983 2101 2247 2286 2413 2662 2792 2867 2890
2931 3000 3060 3336 3340 3493 3625 3655 3756 3933
3952 4009 4087 4460 4508 4563 4598 4989 5000 5219
5256 5369 5390 5414 5513 5542 5569 5716 5879 5894
6112 6177 6254 6516 6565 6830 7044 7053 7193 7344
7362 7436 7527 7579 7649 7930 8014 8034 8382 8714
8795 8912 8915 8959 9021 9205 9236 9427 9478 9654
9726 9912 9925 9998 10093 10204 10640 10642 10648 10727
11199 11232 11299 11487 11699 11716 11953 12069 12094 12128
12174 12330 12335 12398 12424 12456 12517
118 12741 125
0 123 257 271 330 353 382 406 485 757
776 840 866 1025 1065 1268 1367 1698 1758 1826
1836 1842 1996 2005 2275 2324 2710 2756 2939 3025
3037 3386 3442 3503 3609 3679 3783 3974 4088 4089
4092 4127 4234 4354 4556 4618 4749 4796 4807 4968
5081 5132 5221 5488 5887 5912 5987 6109 6204 6443
6524 6654 6661 6797 6862 7013 7018 7033 7195 7208
7449 7649 7743 7815 7836 8067 8111 8276 8343 8664
9040 9067 9141 9159 9351 9353 9385 9537 9663 9897
9945 10002 10030 10318 10553 10608 10641 10814 10856 11114
11157 11377 11413 11454 11504 11534 11956 11978 12136 12202
12233 12304 12341 12349 12537 12591 12678 12741
119 12911 121
0 53 394 496 539 755 791 903 1075 1097
1113 1163 1322 1715 1895 1907 1935 1949 2031 2032
2139 2146 2306 2377 2495 2641 2680 2807 3056 3186
3261 3284 3325 3394 3454 3730 3734 3887 4019 4049
4150 4327 4346 4403 4481 4854 4902 4957 4992 5383
5394 5613 5650 5763 5784 5808 5907 5936 5963 6110
6273 6288 6506 6571 6648 6910 6959 7224 7438 7447
7587 7738 7756 7830 7921 7973 8043 8324 8408 8428
8776 9108 9189 9306 9309 9353 9415 9599 9630 9821
9872 10048 10120 10306 10319 10392 10487 10598 11034 11036
11042 11121 11593 11626 11693 11881 12093 12110 12347 12463
12488 12522 12568 12724 12729 12792 12818 12850 12911
120 13089 121
0 6 44 86 191 241 297 379 425 621
625 648 1075 1116 1169 1311 1429 1733 1985 2022
2093 2209 2216 2249 2307 2308 2359 2448 2705 2768
2782 2844 3145 3153 3313 3362 3489 3506 3515 3679
3783 3794 3814 3986 4112 4281 4355 4745 4857 4966
4994 5173 5256 5334 5377 5445 5597 5783 6028 6241
6276 6295 6606 6696 6781 6783 6999 7011 7332 7489
7554 7590 7651 7773 7902 7932 7998 8027 8161 8176
8208 8416 8708 8730 8775 8854 8875 8878 8888 9008
9125 9290 9617 9642 9690 10033 10378 10447 10583 10653
10734 11106 11181 11199 11238 11464 11548 11828 11930 11994
12101 12232 12433 12488 12493 12607 12819 13001 13017 13089
121 13280 121
0 6 46 97 252 256 297 333 555 608
619 992 999 1024 1025 1187 1211 1504 1634 1776
1969 2070 2174 2176 2315 2401 2404 2499 2538 2557
2665 2674 2708 2758 2889 3034 3086 3232 3352 3375
3746 3763 3896 3995 4107 4173 4369 4452 4494 4613
4693 4748 4919 4995 5022 5174 5619 5649 5721 5759
5796 5888 5961 6079 6360 6445 6864 6879 6923 7036
7180 7194 7318 7374 7688 7770 7877 7955 7973 7986
8190 8404 8527 8790 8811 8819 8880 9158 9252 9272
9326 9426 9674 9823 10133 10212 10561 10626 10790 10795
10916 11003 11406 11441 11453 11469 11706 11854 11924 11986
12046 12114 12162 12219 12229 12450 12538 13098 13209 13258
13280
122 13521 125
0 1 225 301 418 423 612 629 975 1070
1252 1317 1339 1496 1543 1554 1651 1923 1939 1975
2057 2204 2306 2316 2445 2482 2508 2533 2601 2619
2829 3189 3317 3330 3478 3563 3730 3931 4249 4276
4310 4322 4376 4414 4481 4485 4529 4630 4940 4949
4979 5019 5293 5476 5740 5768 5912 6047 6217 6277
6516 6547 6678 6680 6737 7003 7072 7092 7113 7312
7520 7817 7895 7898 8001 8133 8188 8391 8406 8549
8673 8825 8921 9367 9412 9562 9581 9762 9856 10033
10113 10229 10264 10293 10355 10515 10703 10871 10946 11132
11438 11470 11661 11924 11948 11998 12047 12281 12643 12785
12828 12905 12958 13041 13083 13155 13367 13400 13423 13507
13513 13521
123 13802 127
0 65 109 122 154 213 309 450 502 527
621 857 880 950 1048 1081 1122 1468 1505 1526
1568 1650 2106 2185 2310 2321 2548 2587 2614 2753
2893 3006 3025 3028 3277 3279 3327 3500 3535 3730
3916 3963 4065 4329 4412 4568 4597 4993 5219 5365
5382 5499 5559 5579 5628 5666 5709 5804 5916 6258
6573 6626 6911 6912 7110 7114 7140 7427 7434 7489
7505 7701 7737 7920 7984 8089 8095 8246 8254 8698
8765 8816 8850 8862 8976 9347 9375 9474 9528 9616
9708 9836 9851 10248 10272 10464 10570 10579 10654 10740
10946 11429 11485 11490 11669 11745 11785 12104 12135 12478
12568 12640 12829 12937 13038 13111 13255 13269 13392 13402
13512 13580 13802
124 13991 125
0 23 193 323 347 458 465 537 548 1045
1050 1058 1064 1210 1338 1429 1477 1829 1922 2038
2039 2071 2324 2475 2542 2699 2753 2757 2882 2898
2994 3082 3230 3232 3565 3602 3678 3833 4008 4144
4229 4350 4620 4640 5192 5265 5345 5387 5444 5519
5578 5808 5825 6068 6083 6265 6429 6606 6618 6627
6721 6882 6923 7005 7015 7249 7312 7571 7948 8134
8168 8194 8221 8239 8341 8397 8537 8680 8709 8865
8896 8931 9245 9291 9343 9346 9453 9514 9719 9763
9998 10020 10082 10129 10179 10248 10487 10593 10895 11349
11387 11575 11664 11788 11831 11856 11896 11926 12000 12318
12357 12605 12656 12742 12959 12987 13296 13377 13557 13751
13828 13877 13955 13991
125 14055 125
0 23 193 323 347 458 465 537 548 1045
1050 1058 1064 1210 1338 1429 1477 1829 1922 2038
2039 2071 2324 2475 2542 2699 2753 2757 2882 2898
2994 3082 3230 3232 3565 3602 3678 3833 4008 4144
4229 4350 4620 4640 5192 5265 5345 5387 5444 5519
5578 5808 5825 6068 6083 6265 6429 6606 6618 6627
6721 6882 6923 7005 7015 7249 7312 7571 7948 8134
8168 8194 8221 8239 8341 8397 8537 8680 8709 8865
8896 8931 9245 9291 9343 9346 9453 9514 9719 9763
9998 10020 10082 10129 10179 10248 10487 10593 10895 11349
11387 11575 11664 11788 11831 11856 11896 11926 12000 12318
12357 12605 12656 12742 12959 12987 13296 13377 13557 13751
13828 13877 13955 13991 14055
126 14348 127
0 25 113 243 438 492 564 577 578 582
588 638 891 989 1188 1367 1559 1726 1902 2018
2254 2256 2285 2300 2323 2391 2544 2574 2772 2998
3098 3110 3207 3327 3330 3408 3478 3956 4138 4239
4302 4343 4350 4416 4499 4606 4658 4779 4922 4939
5275 5311 5413 5453 5660 5719 5777 5822 5932 6056
6431 6513 6546 6679 6916 7107 7162 7201 7281 7309
7346 7577 7793 7846 7921 7968 8086 8148 8536 8794
8837 8858 9051 9144 9379 9399 9456 9585 9872 9959
10043 10373 10405 10504 10638 10795 10922 10949 10998 11090
11222 11430 11588 11850 12031 12192 12281 12417 12433 12468
12678 12687 12850 12872 13149 13191 13360 13379 13549 13575
13583 13654 13906 14031 14253 14348
127 14460 128
0 92 108 250 347 380 531 542 563 847
1016 1041 1107 1370 1390 1409 1431 1473 1772 1774
1808 1982 2012 2041 2244 2379 2429 2518 2561 3087
3127 3165 3209 3293 3338 3568 3868 3891 3895 3963
4122 4311 4320 4371 4429 4760 4835 4948 5058 5105
5205 5348 5383 5469 5698 5920 6060 6306 6404 6521
6601 6677 6734 6762 6765 6933 6986 7010 7023 7320
7483 8083 8129 8137 8208 8273 8354 8591 8768 8778
8969 8970 8987 9074 9089 9205 9342 9752 9867 10017
10144 10243 10463 10557 10612 10757 10932 11043 11113 11236
11285 11450 11464 11576 11588 11743 12187 12260 12467 12472
12568 12620 12754 12959 13155 13389 13549 13701 13808 13815
13871 13877 14213 14319 14367 14393 14460
128 14821 128
0 136 312 427 446 526 698 699 810 1051
1060 1076 1178 1281 1319 1394 1479 1823 1825 2047
2498 2529 2547 2579 2783 2852 2936 2989 3065 3215
3320 3385 3528 3808 3842 3886 4073 4077 4521 4542
4638 4690 4760 4998 5130 5288 5423 5485 5521 5585
5906 5920 5966 5993 6131 6299 6405 6460 6557 6650
6708 6852 6857 7067 7108 7114 7197 7483 8043 8111
8199 8266 8306 8392 8448 8573 8581 8632 8740 9292
9469 9511 9703 9807 9947 9964 9986 10314 10362 10388
10423 10542 10608 10687 10996 11003 11026 11273 11368 11397
11469 11597 11600 12176 12196 12239 12267 12359 12413 12689
13025 13036 13200 13224 13237 13314 13347 13541 13556 13727
13843 14131 14141 14561 14643 14655 14700 14821
129 15075 128
0 42 110 186 202 267 308 351 487 639
671 801 883 954 969 1112 1228 1241 1545 1609
1620 1869 1984 2033 2092 2175 2205 2402 2491 2576
2654 2723 2749 2756 2996 3357 3485 3507 3928 4005
4206 4235 4333 4424 4451 4572 4617 4677 4910 5034
5071 5249 5284 5419 5424 5575 5713 5765 5779 5964
5982 5984 6078 6259 6588 6878 6981 7140 7237 7281
7348 7354 7404 8123 8144 8178 8216 8224 8277 8600
8679 8737 8933 9115 9169 9324 9411 9450 9501 9621
10087 10226 10335 10409 10578 10640 10765 10835 11105 11114
11115 11162 11417 11562 11693 11797 11800 11825 12195 12212
12231 12235 12423 12785 12848 13236 13412 13443 13680 13704
13904 14023 14229 14547 14741 14829 14963 14975 15075
130 15275 131
0 142 211 227 332 430 501 654 663 845
1045 1096 1282 1527 1645 1838 2012 2056 2095 2365
2571 2637 2650 2652 2686 2987 3226 3342 3359 3620
3624 3652 3754 3878 3883 3943 4029 4124 4284 4371
4956 4977 5059 5120 5121 5445 5567 5796 5816 5819
5864 5936 5991 6044 6145 6321 6339 6398 6605 6697
6705 6901 7084 7467 7473 7525 7551 7767 7863 7890
7999 8196 8355 8380 8494 8557 8707 9204 9316 9353
9629 9636 9726 9755 9767 9802 10265 10375 10479 10503
10533 10743 10783 11086 11100 11330 11566 11727 11884 11894
12019 12353 12459 12552 12622 12774 12930 12997 13138 13232
13270 13343 13417 13632 13688 13731 13777 13948 14169 14347
14633 14652 14683 14740 14820 14831 14853 14895 15087 15275
131 15548 131
0 67 93 115 221 506 571 977 1161 1212
1216 1237 1274 1531 1561 1747 1817 2016 2164 2180
2325 2372 2378 2447 2452 2480 2652 2729 2862 2881
2992 3174 3406 3445 3458 3704 3947 3958 4082 4211
4343 4441 4730 4761 4773 4829 4970 5084 5445 5481
5491 5581 5598 5630 5718 5809 6057 6183 6288 6478
6505 6780 6781 6783 6974 7118 7152 7175 7633 7657
7974 8056 8224 8400 8516 8589 8603 8760 8841 8979
9278 9447 9536 9621 9630 9659 9713 9777 9872 10068
10193 10522 10643 11160 11231 11456 11464 11565 11606 11651
11896 12063 12175 12238 12485 12564 12636 12798 13013 13156
13217 13314 13321 13380 13424 13694 13975 14145 14619 14703
14738 14753 14884 14926 15086 15106 15146 15312 15452 15530
15548
132 15893 131
0 80 195 477 705 775 798 1032 1207 1235
1399 1420 1578 1785 1892 2144 2192 2294 2316 2556
2668 2800 3000 3036 3094 3278 3317 3323 3335 3391
3497 3584 3626 3651 3698 3773 3786 3850 4361 4391
4445 4446 4662 4950 4974 5039 5312 5350 5358 5449
5604 5919 6215 6265 6318 6548 6688 6885 6946 7067
7099 7133 7140 7150 7331 7371 7621 7767 7786 7884
8060 8202 8311 8373 8616 8742 8893 9050 9081 9287
9407 9597 9824 9928 10044 10046 10213 10424 10771 10857
10976 11066 11137 11229 11262 11278 11305 11406 11488 11848
12168 12276 12424 12551 12555 12560 12694 12909 13315 13352
13396 13448 13809 13888 13914 14238 14307 14456 14485 14939
14959 15125 15184 15281 15284 15295 15625 15685 15748 15763
15858 15893
133 16192 137
0 50 56 147 185 222 224 323 579 762
769 953 1154 1195 1205 1206 1375 1610 1618 1703
1712 1736 1928 2000 2428 2440 2673 2752 2826 3352
3509 3592 3628 3717 3748 3770 3790 3963 4055 4070
4252 4269 4627 4736 4802 4847 4963 5084 5247 5586
5654 5672 5716 5903 5985 6275 6471 6501 6549 6767
6848 6851 6912 7015 7156 7227 7361 7629 7808 7954
8216 8263 8303 8457 8597 8622 8734 8799 8951 9074
9090 9385 9483 9737 9804 10082 10292 10473 10624 10883
10988 10993 11007 11083 11115 11143 11243 11286 11544 11573
11887 12352 12411 12415 12559 12900 12954 12980 13086 13113
13324 13675 13721 13744 13978 13991 14048 14197 14311 14426
14677 14698 14794 14907 15065 15269 15357 15552 15683 15741
15969 16157 16192
134 16296 137
0 9 16 38 62 114 178 267 463 744
775 955 1027 1225 1379 1390 1651 1782 1937 2023
2126 2175 2256 2351 2352 2536 2642 2781 2974 3029
3037 3329 3611 3655 3798 4077 4319 4331 4615 4685
4807 4948 4982 5105 5138 5165 5183 5213 5256 5312
5427 5677 5774 5980 5983 6034 6117 6351 6453 6545
6672 6838 6918 7194 7417 7504 7529 7617 7726 7728
7854 7874 8033 8169 8360 8437 8933 8952 8956 9188
9227 9346 9447 9659 9724 9977 10141 10518 10791 10857
10898 10977 11433 11438 11571 11779 11840 11872 11940 11982
12022 12324 12382 12418 12542 12705 12969 13348 13358 13415
13636 13662 13806 13875 14010 14237 14250 14418 14465 14590
15075 15090 15149 15473 15487 15508 15558 15675 15703 15807
15824 16206 16212 16296
135 16622 139
0 35 43 163 301 354 370 453 525 584
650 870 904 922 1013 1139 1387 1489 1814 1860
1914 2178 2295 2370 2373 2449 2507 2891 3089 3410
3461 3491 3635 3750 4048 4132 4189 4408 4493 4512
4532 4777 4806 5056 5103 5192 5217 5531 5932 6024
6239 6240 6262 6276 6440 6444 6472 6629 6725 6906
6966 7073 7205 7407 7837 7854 8030 8178 8205 8220
8624 8753 8786 9015 9046 9056 9392 9441 9489 9552
9576 9692 9698 9864 10272 10581 10679 10686 10750 10844
11281 11348 11434 11507 11690 11751 11839 11901 11945 11957
12144 12687 12757 12797 12936 13018 13039 13044 13252 13475
13617 13712 13750 13863 13990 14221 14366 14368 14377 14489
14554 14845 14913 15140 15241 15647 15702 15875 15920 15994
16230 16452 16532 16545 16622
136 16766 137
0 50 56 147 185 222 224 323 579 762
769 953 1154 1195 1205 1206 1375 1610 1618 1703
1712 1736 1928 2000 2428 2440 2673 2752 2826 3352
3509 3592 3628 3717 3748 3770 3790 3963 4055 4070
4252 4269 4627 4736 4802 4847 4963 5084 5247 5586
5654 5672 5716 5903 5985 6275 6471 6501 6549 6767
6848 6851 6912 7015 7156 7227 7361 7629 7808 7954
8216 8263 8303 8457 8597 8622 8734 8799 8951 9074
9090 9385 9483 9737 9804 10082 10292 10473 10624 10883
10988 10993 11007 11083 11115 11143 11243 11286 11544 11573
11887 12352 12411 12415 12559 12900 12954 12980 13086 13113
13324 13675 13721 13744 13978 13991 14048 14197 14311 14426
14677 14698 14794 14907 15065 15269 15357 15552 15683 15741
15969 16157 16192 16662 16717 16766
137 17031 139
0 48 69 81 303 323 387 570 687 691
725 955 1091 1153 1250 1291 1292 1365 1432 1603
1885 1908 1917 2035 2632 2828 2889 2929 3051 3320
3337 3344 3359 3525 3764 3827 4178 4188 4458 4533
4718 4864 5067 5117 5273 5510 5599 5644 5655 5820
5875 5927 5933 6101 6374 6444 6907 7023 7094 7172
7198 7264 7520 7588 7779 7932 8025 8069 8112 8148
8177 8302 8321 9072 9167 9204 9287 9381 9550 9585
9743 9869 9971 10323 10442 10533 10700 10730 10757 11097
11226 11244 11275 11329 11439 11600 11724 12025 12185 12261
12275 12357 12403 12720 12733 12832 13040 13093 13224 13226
13443 13451 13691 14038 14063 14066 14143 14502 14616 14704
14764 14815 14915 15058 15144 15250 15577 16135 16151 16314
16386 16630 16635 16765 16824 16922 17031
138 17124 139
0 35 43 163 301 354 370 453 525 584
650 870 904 922 1013 1139 1387 1489 1814 1860
1914 2178 2295 2370 2373 2449 2507 2891 3089 3410
3461 3491 3635 3750 4048 4132 4189 4408 4493 4512
4532 4777 4806 5056 5103 5192 5217 5531 5932 6024
6239 6240 6262 6276 6440 6444 6472 6629 6725 6906
6966 7073 7205 7407 7837 7854 8030 8178 8205 8220
8624 8753 8786 9015 9046 9056 9392 9441 9489 9552
9576 9692 9698 9864 10272 10581 10679 10686 10750 10844
11281 11348 11434 11507 11690 11751 11839 11901 11945 11957
12144 12687 12757 12797 12936 13018 13039 13044 13252 13475
13617 13712 13750 13863 13990 14221 14366 14368 14377 14489
14554 14845 14913 15140 15241 15647 15702 15875 15920 15994
16230 16452 16532 16545 16622 16944 17074 17124
139 17587 139
0 27 32 216 269 346 371 633 774 778
809 843 1039 1058 1192 1427 1715 1730 1738 1880
1998 2257 2340 2705 2751 2808 2995 3109 3206 3432
3938 4024 4233 4247 4331 4581 4609 4687 4778 4808
4814 5032 5169 5445 5448 5474 5544 5605 6017 6083
6093 6130 6188 6349 6436 6524 6706 6788 6897 6991
7193 7234 7283 7383 7434 7523 7556 7608 8502 8576
8631 8843 8958 8978 9026 9106 9278 9328 9656 9812
9964 10390 10463 10534 10667 10847 10860 10979 11054 11146
11450 11512 11576 11620 11744 12111 12123 12324 12340 12444
12483 12621 12664 12675 12903 13065 13175 13177 13383 13782
13791 13898 13970 13977 14037 14191 14215 14308 14466 14506
14928 14970 14987 15355 15436 15602 16224 16347 16458 16583
16604 16605 16768 17097 17233 17486 17531 17549 17587
140 17938 139
0 80 83 209 332 404 509 561 671 877
933 991 1115 1302 1751 1867 1915 1934 2042 2198
2253 2288 2351 2561 2740 2887 2931 3083 3182 3271
3382 3735 3759 3900 3961 3993 4006 4063 4249 4353
4421 4485 4634 4735 4762 4928 4970 5185 5193 5194
5210 5490 5849 5924 6054 6151 6227 6239 6562 6647
6766 6817 7110 7130 7329 7467 7510 7726 8048 8215
8226 8504 8834 8903 8980 9003 9062 9374 9460 9572
9605 10275 10289 10410 10519 10663 10670 10716 10744 10884
10933 10938 11300 11431 11718 11797 11833 12004 12045 12051
12066 12158 12184 12188 12649 12699 12841 12991 13309 13396
13803 13834 14132 14335 14369 14406 14408 14958 15334 15412
15430 15946 15956 16104 16188 16349 16621 16755 16795 16931
16996 17034 17056 17151 17341 17435 17501 17818 17847 17938
141 18601 149
0 77 205 208 281 327 422 555 586 587
600 684 774 983 1030 1312 1415 1606 1613 1782
1809 1870 1934 2040 2052 2532 2551 2963 3242 3404
3414 3606 3643 3878 3957 3979 4390 4494 4553 4645
4720 4806 5016 5126 5292 5465 5530 5820 5907 6021
6057 6151 6466 6510 6526 6801 7105 7123 7318 7607
7775 7790 7917 7969 8026 8278 8306 8464 8546 8813
8912 8951 8960 9097 9514 9595 9667 9921 9975 10207
10309 10532 10667 10707 11125 11180 11249 11251 11467 11497
11651 11674 11785 11790 12087 12128 12394 12543 12549 12742
12942 13207 13560 13675 13708 13775 13831 13920 14260 14480
14484 14509 14558 14592 14616 15346 15354 15389 15439 15459
15579 15722 15813 15879 16157 16583 16825 16878 16946 17262
17547 17573 17794 17938 18055 18097 18118 18135 18324 18505
18601
142 18751 149
0 3 174 223 309 489 549 623 650 779
781 871 982 1258 1367 1449 1465 1829 1939 2221
2282 2344 2465 2861 2862 3126 3172 3313 3565 3629
3801 3991 4156 4261 4293 4331 4431 4794 4815 5093
5122 5322 5339 5367 5378 5418 5521 5580 5928 6060
6219 6274 6331 6337 6856 6890 6962 7003 7129 7262
7338 7356 7726 7774 7876 8311 8341 8466 8489 8509
8608 8701 8715 8987 9053 9582 9604 9613 9777 9999
10014 10024 10293 10343 10626 10709 10886 11080 11194 11907
11943 12070 12103 12200 12288 12357 12404 12612 12828 12863
13014 13518 13940 13964 14031 14387 14472 14710 14714 14785
15139 15279 15332 15560 15644 15709 15813 15855 15928 15933
16117 16143 16464 16532 16586 17184 17197 17363 17515 17818
17898 18006 18237 18249 18256 18373 18517 18554 18598 18606
18693 18751
143 18971 151
0 28 70 85 138 199 234 333 343 509
758 805 869 891 1122 1152 1421 1744 2181 2308
2531 2535 2647 2778 2953 2992 3075 3481 3533 3581
3607 3621 3784 3846 3851 3857 3901 4182 4439 4528
4608 4713 4826 5005 5098 5353 5550 5653 5748 6149
6167 6203 6236 6427 6614 6616 6718 6930 7087 7108
7592 7599 7608 7894 8115 8206 8238 8303 8669 8692
8954 9138 9383 9402 9530 9637 9818 9974 9986 10148
10156 10427 10586 10790 11083 11165 11323 11394 11466 11542
11605 11755 11798 11801 12311 12497 12557 12595 12948 13040
13099 13282 13307 13327 13428 13892 13948 14306 14390 14515
14544 14709 14925 15234 15426 15556 15633 15646 15973 16369
16522 16559 16637 16695 16774 17116 17197 17224 17248 17342
17552 17829 18070 18237 18437 18454 18503 18578 18609 18810
18811 18930 18971
144 19123 151
0 152 180 222 237 290 351 386 485 495
661 910 957 1021 1043 1274 1304 1573 1896 2333
2460 2683 2687 2799 2930 3105 3144 3227 3633 3685
3733 3759 3773 3936 3998 4003 4009 4053 4334 4591
4680 4760 4865 4978 5157 5250 5505 5702 5805 5900
6301 6319 6355 6388 6579 6766 6768 6870 7082 7239
7260 7744 7751 7760 8046 8267 8358 8390 8455 8821
8844 9106 9290 9535 9554 9682 9789 9970 10126 10138
10300 10308 10579 10738 10942 11235 11317 11475 11546 11618
11694 11757 11907 11950 11953 12463 12649 12709 12747 13100
13192 13251 13434 13459 13479 13580 14044 14100 14458 14542
14667 14696 14861 15077 15386 15578 15708 15785 15798 16125
16521 16674 16711 16789 16847 16926 17268 17349 17376 17400
17494 17704 17981 18222 18389 18589 18606 18655 18730 18761
18962 18963 19082 19123
145 19325 149
0 5 148 152 160 233 321 785 798 926
1042 1133 1326 1393 1432 1567 1729 1969 1971 2101
2177 2796 2874 2921 3100 3101 3266 3488 3932 4250
4514 4548 4559 4591 4594 4611 4967 4989 5300 5327
5667 5696 5850 5944 6174 6236 6524 6642 6716 6809
6839 6858 7071 7210 7300 7318 7576 7646 7830 8028
8064 8301 8334 8554 8579 8755 8847 8854 8908 8976
9097 9211 9569 9695 10015 10128 10152 10178 10193 10460
10563 10709 10749 10995 11140 11405 11992 12036 12206 12415
12502 12706 12870 12891 12929 12987 13001 13087 13192 13641
13792 13798 13840 13893 14205 14309 14648 14872 14961 15197
15399 15408 15463 15523 15718 15825 15900 15937 16081 16112
16284 16312 16395 16461 16563 16983 17142 17422 17479 17753
17862 17872 17941 18012 18365 18485 18501 18672 18695 18828
18910 18966 19100 19227 19325
146 19628 149
0 122 231 315 567 870 911 913 924 950
1212 1399 1565 1634 1664 1791 1956 2023 2035 2343
2572 2648 2682 2856 3061 3260 3449 3612 3690 4076
4086 4095 4180 4405 4541 4658 4680 4822 4871 5043
5707 5918 5924 5980 6020 6025 6048 6204 6296 6568
6676 6737 6807 6823 6974 7118 7276 7627 7671 7878
7983 8094 8118 8121 8192 8247 8289 8304 8613 8728
8890 9157 9244 9393 9444 9482 9748 9907 10130 10176
10380 10473 10607 10747 10799 10944 11122 11648 11721 11927
12155 12335 12487 12815 12836 13034 13048 13209 13297 13410
13491 13556 13894 14112 14373 14618 14643 14718 14738 15045
15078 15128 15136 15590 15653 15875 15907 16023 16030 16089
16319 16323 16437 16565 16613 16896 17111 17695 17772 17897
18029 18030 18429 18697 18803 18838 18867 18885 18957 18988
19235 19372 19408 19425 19568 19628
147 19757 149
0 72 227 249 377 508 540 544 872 892
1032 1171 1184 1248 1485 1531 1641 1730 1732 1739
1820 1990 2014 2126 2414 2580 2815 2979 3028 3448
3477 3709 3813 3971 4176 4239 4245 4256 4290 4293
4583 4978 5126 5317 5329 5348 5671 6049 6288 6394
6499 6555 6671 6880 6965 6980 7051 7106 7251 7567
7585 7760 8503 8504 8672 8768 8906 8941 9172 9182
9239 9369 9428 9624 9647 10224 10232 10282 10414 10439
10532 10643 10861 10953 10996 11035 11249 11393 11552 11599
11782 11822 11968 12255 12323 12398 12472 12596 12634 12729
12844 13009 13146 13187 13249 13371 13897 14073 14149 14215
14535 14642 14656 15024 15054 15132 15153 15205 15266 15779
16042 16169 16234 16357 16600 16694 16991 17018 17051 17138
17358 17553 18031 18084 18185 18264 18269 18703 19062 19159
19383 19485 19604 19687 19713 19729 19757
148 20037 149
0 138 176 353 368 595 599 898 1158 1290
1570 1642 1656 1831 1895 2044 2200 2210 2336 2417
2451 2527 2615 2660 2789 2871 3043 3316 3467 3495
3620 3813 3975 4037 4222 4448 4503 4587 4981 5046
5152 5165 5620 5628 5727 5982 6076 6130 6137 6239
6241 6507 6576 6690 6730 6790 6848 6870 7493 7522
7572 7605 8206 8227 8284 8631 8871 9168 9219 9468
9488 9535 9610 9656 9894 9906 10007 10096 10302 10341
10407 10677 10991 11203 11334 11407 11515 11567 11684 11714
11841 11900 12044 12070 12889 12890 12895 12914 12982 13271
13288 13750 13792 13803 13840 13970 14013 14040 14111 14235
14298 14769 14904 15109 15205 15632 15752 16061 16222 16419
16450 16553 16644 16840 16863 16937 16986 17022 17138 17613
17763 17891 18177 18193 18411 18429 18506 18750 18893 19215
19360 19497 19705 19761 19993 19996 20028 20037
149 20265 151
0 8 66 123 248 312 323 384 473 555
723 992 1063 1211 1650 1785 1838 2117 2122 2298
2343 2766 2864 2974 3007 3152 3404 3494 3613 3941
4087 4154 4175 4227 4307 4347 4521 4757 4874 4878
5090 5431 5446 5767 5826 5851 5860 5895 6128 6142
6439 6471 6513 6896 6902 7014 7295 7344 7524 7699
7746 8123 8140 8536 8742 9027 9040 9302 9340 9526
9692 9718 9834 9852 9888 10161 10253 10358 10359 10444
10731 10750 11242 11346 11373 11460 11601 11695 11791 11839
11868 12110 12213 12237 12320 12420 12625 12905 13108 13159
13403 14066 14078 14179 14317 14356 14773 14796 14833 14895
14990 15350 15428 15504 15554 15595 15651 15753 16396 16526
16533 16685 16695 16850 16880 16896 16959 16961 17029 17231
17487 17835 17855 18193 18317 18456 18620 18776 19007 19029
19158 19266 19321 19742 20007 20035 20038 20222 20265
150 20521 149
0 5 148 152 160 233 321 785 798 926
1042 1133 1326 1393 1432 1567 1729 1969 1971 2101
2177 2796 2874 2921 3100 3101 3266 3488 3932 4250
4514 4548 4559 4591 4594 4611 4967 4989 5300 5327
5667 5696 5850 5944 6174 6236 6524 6642 6716 6809
6839 6858 7071 7210 7300 7318 7576 7646 7830 8028
8064 8301 8334 8554 8579 8755 8847 8854 8908 8976
9097 9211 9569 9695 10015 10128 10152 10178 10193 10460
10563 10709 10749 10995 11140 11405 11992 12036 12206 12415
12502 12706 12870 12891 12929 12987 13001 13087 13192 13641
13792 13798 13840 13893 14205 14309 14648 14872 14961 15197
15399 15408 15463 15523 15718 15825 15900 15937 16081 16112
16284 16312 16395 16461 16563 16983 17142 17422 17479 17753
17862 17872 17941 18012 18365 18485 18501 18672 18695 18828
18910 18966 19100 19227 19325 19929 20147 20198 20437 20521
151 20841 151
0 185 234 642 660 665 698 1001 1130 1151
1154 1183 1252 1386 1416 1721 1993 2027 2139 2269
2407 2724 2784 2820 3154 3382 3651 3697 3824 4346
4363 4389 4576 4601 4673 4708 4762 5001 5020 5200
5421 5437 5530 5609 5720 6167 6224 6326 6337 6535
6663 6675 6702 6730 6780 6927 7364 7434 7623 8012
8219 8223 8358 8473 8514 8691 8755 8845 9094 9272
9492 9643 9663 9844 9886 10164 10268 10327 10460 10521
10767 10888 10950 11092 11340 11355 11362 11443 11649 12045
12128 12168 12591 12732 12808 12907 12951 13075 13230 13570
13578 13580 13944 14120 14489 14541 14651 14842 14924 15117
15319 15435 15503 15651 15949 16029 16237 16303 16477 16595
16682 16774 16819 16939 17045 17349 17751 17760 17825 17896
17909 17982 18040 18209 18284 18369 18565 18690 18691 19124
19243 19334 19624 20480 20527 20575 20727 20741 20804 20835
20841
152 20892 151
0 51 236 285 693 711 716 749 1052 1181
1202 1205 1234 1303 1437 1467 1772 2044 2078 2190
2320 2458 2775 2835 2871 3205 3433 3702 3748 3875
4397 4414 4440 4627 4652 4724 4759 4813 5052 5071
5251 5472 5488 5581 5660 5771 6218 6275 6377 6388
6586 6714 6726 6753 6781 6831 6978 7415 7485 7674
8063 8270 8274 8409 8524 8565 8742 8806 8896 9145
9323 9543 9694 9714 9895 9937 10215 10319 10378 10511
10572 10818 10939 11001 11143 11391 11406 11413 11494 11700
12096 12179 12219 12642 12783 12859 12958 13002 13126 13281
13621 13629 13631 13995 14171 14540 14592 14702 14893 14975
15168 15370 15486 15554 15702 16000 16080 16288 16354 16528
16646 16733 16825 16870 16990 17096 17400 17802 17811 17876
17947 17960 18033 18091 18260 18335 18420 18616 18741 18742
19175 19294 19385 19675 20531 20578 20626 20778 20792 20855
20886 20892
153 21715 157
0 3 123 144 169 201 347 424 492 892
1435 1668 1720 1937 2069 2200 2259 2322 2677 2771
2793 2895 2966 3231 3391 3479 3681 3818 4001 4048
4104 4318 4354 4597 4648 4664 4747 4765 5105 5106
5145 5175 5220 5965 6191 6217 6380 6505 6520 6654
6812 6836 7058 7123 7187 7300 7568 7595 7686 7851
8038 8177 8486 8541 8628 8694 8747 8873 8875 9072
9578 9678 9687 9878 9954 9983 10140 10304 10311 10353
10459 10832 10918 11216 11444 11606 11629 11702 11762 11974
12439 12566 12627 12655 12926 13018 13238 13449 13553 13762
13969 14366 14570 14620 14731 15134 15167 15288 15418 15431
15569 15650 16070 16111 16128 16263 16669 17164 17340 17576
17648 17683 17914 18288 18331 18368 18710 18920 19104 19276
19386 19470 19481 19578 19774 19805 19853 20299 20559 20695
20729 20733 20739 20954 20968 20973 21066 21458 21548 21633
21695 21707 21715
154 21833 157
0 20 49 249 628 681 704 736 880 1182
1196 1294 1537 1643 1646 2071 2140 2226 2296 2374
2865 3017 3048 3255 3337 3430 3437 3555 3643 3821
3840 3960 4008 4079 4623 4979 5015 5096 5210 5340
5341 5383 5442 5659 5940 6075 6379 6929 7016 7067
7072 7089 7217 7300 7309 7366 7884 7930 8088 8099
8114 8193 8370 8374 8624 8870 9206 9236 9398 9422
9569 9695 9762 9824 9865 9904 10163 10471 10562 10685
11033 11078 11286 11452 11637 11790 11906 11927 11981 12184
12516 12541 12554 12606 12695 13062 13367 13594 13855 14170
14594 14804 14806 14967 15027 15071 15134 15402 15479 15512
15576 15774 15993 16021 16117 16157 16322 16779 17005 17194
17241 17435 17548 17575 17670 17707 17858 18172 18244 18581
18642 18692 18727 19019 19343 19477 19701 20054 20070 20088
20430 20488 20587 20899 20909 20983 21173 21179 21294 21306
21585 21653 21825 21833
155 22035 157
0 202 222 251 451 830 883 906 938 1082
1384 1398 1496 1739 1845 1848 2273 2342 2428 2498
2576 3067 3219 3250 3457 3539 3632 3639 3757 3845
4023 4042 4162 4210 4281 4825 5181 5217 5298 5412
5542 5543 5585 5644 5861 6142 6277 6581 7131 7218
7269 7274 7291 7419 7502 7511 7568 8086 8132 8290
8301 8316 8395 8572 8576 8826 9072 9408 9438 9600
9624 9771 9897 9964 10026 10067 10106 10365 10673 10764
10887 11235 11280 11488 11654 11839 11992 12108 12129 12183
12386 12718 12743 12756 12808 12897 13264 13569 13796 14057
14372 14796 15006 15008 15169 15229 15273 15336 15604 15681
15714 15778 15976 16195 16223 16319 16359 16524 16981 17207
17396 17443 17637 17750 17777 17872 17909 18060 18374 18446
18783 18844 18894 18929 19221 19545 19679 19903 20256 20272
20290 20632 20690 20789 21101 21111 21185 21375 21381 21496
21508 21787 21855 22027 22035
156 22348 157
0 202 222 251 451 830 883 906 938 1082
1384 1398 1496 1739 1845 1848 2273 2342 2428 2498
2576 3067 3219 3250 3457 3539 3632 3639 3757 3845
4023 4042 4162 4210 4281 4825 5181 5217 5298 5412
5542 5543 5585 5644 5861 6142 6277 6581 7131 7218
7269 7274 7291 7419 7502 7511 7568 8086 8132 8290
8301 8316 8395 8572 8576 8826 9072 9408 9438 9600
9624 9771 9897 9964 10026 10067 10106 10365 10673 10764
10887 11235 11280 11488 11654 11839 11992 12108 12129 12183
12386 12718 12743 12756 12808 12897 13264 13569 13796 14057
14372 14796 15006 15008 15169 15229 15273 15336 15604 15681
15714 15778 15976 16195 16223 16319 16359 16524 16981 17207
17396 17443 17637 17750 17777 17872 17909 18060 18374 18446
18783 18844 18894 18929 19221 19545 19679 19903 20256 20272
20290 20632 20690 20789 21101 21111 21185 21375 21381 21496
21508 21787 21855 22027 22035 22348
157 22683 157
0 19 62 197 316 665 689 795 1100 1127
1163 1256 1548 1573 1646 1681 1697 1762 1801 1948
1961 1982 2259 2569 2908 2910 2930 3149 3154 3458
3620 3630 3690 3822 4047 4048 4264 4338 4434 4452
4663 4667 4746 5018 5321 5388 5487 5757 5893 6200
6252 6393 6667 6756 6877 6935 7038 7278 7412 7527
7596 7707 7735 8087 8290 8330 8344 8650 8837 9032
9073 9331 9423 9561 9673 9908 10062 10252 10275 10631
10939 10984 11062 11091 11106 11148 11679 11852 11855 11885
12247 12507 12644 12790 12840 12940 13103 13167 13174 13254
13262 13372 13707 14187 14234 14272 14331 14476 14553 14724
15550 15567 15598 15715 15843 15898 15974 16226 16368 16394
16400 16656 16740 17001 17565 17718 18005 18114 18196 18205
18271 18526 18652 18708 18720 18757 18810 19676 19803 20078
20247 20432 20533 20710 20756 20931 21269 21382 21507 21647
21708 21719 21926 22173 22446 22589 22683
158 22954 157
0 19 62 197 316 665 689 795 1100 1127
1163 1256 1548 1573 1646 1681 1697 1762 1801 1948
1961 1982 2259 2569 2908 2910 2930 3149 3154 3458
3620 3630 3690 3822 4047 4048 4264 4338 4434 4452
4663 4667 4746 5018 5321 5388 5487 5757 5893 6200
6252 6393 6667 6756 6877 6935 7038 7278 7412 7527
7596 7707 7735 8087 8290 8330 8344 8650 8837 9032
9073 9331 9423 9561 9673 9908 10062 10252 10275 10631
10939 10984 11062 11091 11106 11148 11679 11852 11855 11885
12247 12507 12644 12790 12840 12940 13103 13167 13174 13254
13262 13372 13707 14187 14234 14272 14331 14476 14553 14724
15550 15567 15598 15715 15843 15898 15974 16226 16368 16394
16400 16656 16740 17001 17565 17718 18005 18114 18196 18205
18271 18526 18652 18708 18720 18757 18810 19676 19803 20078
20247 20432 20533 20710 20756 20931 21269 21382 21507 21647
21708 21719 21926 22173 22446 22589 22683 22954
0 0 0
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.
_
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\ /\
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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