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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]


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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.)


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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.
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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.


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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.
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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


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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
        50      2094        49
     0    51    83   110   191   224   226   271   313   413
   468   514   591   619   640   683   767   790   796   865
   878   887   950   980   990  1046  1199  1211  1225  1285
  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
        52      2270        53
     0    80    81    88   106   119   229   303   327   435
   464   499   519   565   586   588   628   665   724   730
   841   980  1016  1030  1098  1115  1120  1190  1248  1297
  1324  1369  1402  1417  1560  1569  1572  1588  1750  1812
  1865  1895  1906  1938  1992  2063  2123  2175  2179  2226
  2236  2270
        53      2347        53
     0     1     9    30   107   211   248   273   297   330
   372   386   452   528   572   587   600   655   708   778
   809   940   960   962   967  1003  1014  1054  1072  1132
  1211  1249  1358  1423  1469  1473  1619  1636  1707  1788
  1804  1807  1852  1937  1947  2081  2120  2154  2245  2257
  2280  2341  2347
        54      2373        53
     0     1     9    30   107   211   248   273   297   330
   372   386   452   528   572   587   600   655   708   778
   809   940   960   962   967  1003  1014  1054  1072  1132
  1211  1249  1358  1423  1469  1473  1619  1636  1707  1788
  1804  1807  1852  1937  1947  2081  2120  2154  2245  2257
  2280  2341  2347  2373
        55      2598        59
     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
        57      2773        59
     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
        58      2851        59
     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
        62      3215        61
     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
        64      3527        64
     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
        65      3593        64
     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
        72      4473        73
     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
        76      5089        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
        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
        82      5814        83
     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
        86      6584        89
     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
        92      7617        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  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
  6533  6733  6864  7101  7208  7225  7246  7322  7434  7499
  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
  4847  4998  5029  5217  5250  5416  5431  5511  5525  5674
  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


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Message: 4538 - Contents - Hide Contents

Date: Mon, 08 Apr 2002 09:35:14

Subject: Ets with good Golomb rulers

From: genewardsmith

Modular Golomb rulers * [with cont.]  (Wayb.)

This mentions three constructions for modular Golomb rulers--the projective plane one, for the q^2+q+1 ets, an affine plane one for
q^2-1
ets, and one of size q^2-q, constructed I know not how. Putting
together all of these for primes and prime powers with the result less
than 1000, I got the following list:

2, 3, 6, 7, 8, 12, 13, 15, 20, 21, 24, 31, 42, 48, 56, 57, 63, 72, 
73, 80, 91, 110, 120, 133, 156, 168, 183, 240, 255, 272, 273, 288,
307, 342, 360, 381, 506, 528, 553, 600, 624, 651, 702, 728, 757, 812,
840, 871, 930, 960, 992, 993

We see the 7-et, of course, from 2^2+2+1, we have a Golomb ruler for
the 12-et coming from 4^2-4, one for the 15-et from 4^2-1, the 31-et
from 5^2+5+1, and the 72-et from 9^2-9, the 80-et from 9^2-1, and even
the 342 et from 19^2-19. (I never knew about this rational point on
the elliptic curve y^2-y=x^3-1 before; cute.)

Now I want to know what Robert wants these for...it seems to be they
are, musically speaking, at opposite poles from what we usually
contruct as scales--they are anti-scales of a sort.


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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-ET
were 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, and
it 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 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." 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
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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]


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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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 _____
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    |_
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 _|\
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/ | |
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  \ /
   X
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   /|
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 /\ /
   X
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    |
   /|
 _/ /
   X
  / \

    |
  __|
  \ /
   X
  / \
 _____
/ | | \
| | | |
  \ /
   X
  / \
 
  / \
 /| |\
  \ /
   X
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   |
 _/ \_
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  / \

  _|_
  | | 
  \ /
   X
  / \
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/ | |\
| | | \
  \ /
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   /|
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 /\ /\_
   X
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_/| | |
  \ /
   X
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    |_
   /|
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   X
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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.)


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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); and
I 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.)
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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 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).
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-ET
I 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 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)
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 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.
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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