﻿Theory 
  of 
  the 
  Division 
  of 
  the 
  Octave. 
  

  

  407 
  

  

  Now, 
  when 
  r=2 
  we 
  have 
  the 
  system 
  of 
  118, 
  which 
  affords 
  the 
  closest 
  

   approximation 
  to 
  what 
  is 
  required 
  of 
  any 
  cyclical 
  system 
  known 
  hitherto, 
  

   the 
  error 
  of 
  its 
  third 
  being 
  -00127. 
  

  

  Referring 
  to 
  Th. 
  v., 
  it 
  is 
  easy 
  to 
  see 
  that 
  no 
  other 
  even 
  system 
  of 
  an 
  

   order 
  much 
  below 
  the 
  24th 
  can 
  afford 
  a 
  better 
  approximation 
  ; 
  for 
  the 
  

   number 
  118 
  differs 
  from 
  the 
  value 
  given 
  by 
  the 
  above 
  condition 
  by 
  little 
  

   more 
  than 
  unity. 
  Its 
  multiple 
  is 
  always 
  of 
  the 
  right 
  order 
  (Th. 
  v.); 
  

   there 
  can 
  therefore 
  be 
  no 
  other 
  system 
  of 
  the 
  right 
  order 
  within 
  12 
  digits 
  

   of 
  the 
  multiple 
  either 
  way, 
  and 
  the 
  deviation 
  of 
  the 
  value 
  given 
  by 
  the 
  

   condition 
  cannot 
  amount 
  to 
  12 
  digits 
  till 
  near 
  the 
  24th 
  order 
  ; 
  we 
  there- 
  

   fore 
  confine 
  ourselves 
  to 
  systems 
  of 
  uneven 
  orders. 
  

  

  Casting 
  out 
  12's 
  from 
  58*4526, 
  we 
  can 
  take 
  the 
  remainder 
  as 
  10*45 
  

   for 
  the 
  purposes 
  of 
  the 
  search 
  :— 
  

  

  3 
  

   5 
  

   7 
  

   9 
  

   11 
  

  

  r 
  . 
  10 
  . 
  45. 
  

  

  31*35 
  

   52-25 
  

   73-15 
  

   94-05 
  

   114*95 
  

  

  Remainder, 
  

   casting 
  out 
  12's. 
  

  

  7-35 
  

   4-25 
  

   1-15 
  

   10-05 
  

   6*95 
  

  

  Remainder 
  re- 
  

   quired 
  for 
  order 
  

   (Th. 
  iii.). 
  

  

  3 
  

   1 
  

   11 
  

  

  The 
  coincidence 
  at 
  the 
  11th 
  order 
  is 
  the 
  closest 
  so 
  far; 
  and 
  it 
  is 
  

   easy 
  to 
  see, 
  by 
  considerations 
  analogous 
  to 
  those 
  above, 
  that 
  no 
  subse- 
  

   quent 
  system 
  can 
  afford 
  another 
  till 
  a 
  much 
  higher 
  order 
  is 
  reached. 
  

  

  Eor 
  the 
  11th 
  order, 
  then, 
  we 
  have 
  

  

  11x58-4526 
  = 
  642*9786; 
  

  

  and 
  643 
  is 
  a 
  system 
  of 
  the 
  11th 
  order, 
  as 
  shown 
  by 
  its 
  giving 
  remainder 
  

   7 
  on 
  dividing 
  by 
  12 
  (Th. 
  iii.). 
  

  

  Calculating 
  the 
  third 
  of 
  this 
  system 
  ^^ 
  = 
  dep.^, 
  and 
  taking 
  seven 
  

  

  places, 
  we 
  have 
  : 
  — 
  

  

  Departure 
  of 
  perfect 
  third 
  = 
  - 
  -1368629 
  

   Departure 
  of 
  third 
  of 
  643 
  = 
  - 
  -1368585 
  

  

  Error 
  = 
  -0000044 
  sharp. 
  

  

  To 
  five 
  places 
  both 
  thirds 
  are 
  represented 
  by 
  — 
  -13686. 
  

  

  The 
  intervals 
  of 
  this 
  system 
  will 
  furnish 
  us 
  with 
  simple 
  numerical 
  

   ratios, 
  which 
  represent 
  with 
  great 
  accuracy 
  the 
  intervals 
  of 
  the 
  perfect 
  

   system. 
  

  

  We 
  have 
  (see 
  the 
  section 
  on 
  the 
  number 
  of 
  units 
  in 
  any 
  interval) 
  — 
  

  

  7-fifths 
  semitone 
  = 
  60 
  units, 
  

   5-fifths 
  semitone 
  = 
  49 
  units 
  ; 
  

  

  