﻿408 
  On 
  the 
  Theory 
  of 
  the 
  Division 
  of 
  the 
  Octave. 
  

  

  whence 
  we 
  can 
  deduce 
  the 
  remaining 
  intervals. 
  These 
  values 
  o£ 
  the 
  

   semitones 
  suggest 
  the 
  following 
  curious 
  derivation 
  of 
  this 
  system 
  : 
  — 
  

  

  Referring 
  to 
  the 
  Table 
  of 
  characteristic 
  numbers, 
  we 
  notice 
  that 
  the 
  

   errors 
  of 
  the 
  thirds 
  of 
  the 
  systems 
  of 
  53 
  and 
  65 
  are 
  nearly 
  equal 
  and 
  

   opposite. 
  

  

  The 
  system 
  of 
  53 
  is 
  derived 
  on 
  the 
  assumption 
  that 
  the 
  interval 
  ratio 
  

   4 
  

  

  of 
  the 
  semitones 
  is 
  g 
  (Th. 
  i. 
  Cor.), 
  and 
  that 
  of 
  65 
  on 
  the 
  assumption 
  

  

  5 
  ... 
  9 
  

  

  g 
  for 
  the 
  same 
  ratio 
  ; 
  taking, 
  then, 
  an 
  intermediate 
  ratio, 
  yp 
  we 
  get 
  the 
  

  

  system 
  of 
  118, 
  which 
  has 
  very 
  good 
  thirds. 
  

  

  But 
  if 
  we 
  take 
  an 
  intermediate 
  ratio 
  in 
  the 
  following 
  manner, 
  we 
  get 
  

   the 
  new 
  system 
  of 
  643 
  : 
  — 
  

  

  4 
  5 
  

  

  Reducing 
  the 
  fractions 
  ^, 
  g 
  to 
  a 
  common 
  denominator, 
  we 
  have 
  

  

  or 
  doubling, 
  ^ 
  ; 
  and 
  if 
  we 
  take 
  the 
  intermediate 
  ratio 
  ~, 
  we 
  

  

  get 
  the 
  system 
  of 
  643, 
  by 
  the 
  formula 
  5x-\-7y=n, 
  derived 
  from 
  Th. 
  a 
  of 
  

   Eegular 
  Systems. 
  

  

  The 
  systems 
  of 
  the 
  fifth 
  order 
  are 
  not 
  particularly 
  good 
  ; 
  the 
  best 
  is 
  

   289, 
  then 
  301. 
  They 
  derive 
  their 
  interest 
  from 
  the 
  logarithmic 
  properties 
  

   of 
  301. 
  

  

  Negative 
  Si/stems. 
  — 
  The 
  condition 
  for 
  the 
  excellence 
  of 
  the 
  thirds 
  of 
  

   negative 
  systems 
  is 
  that 
  

  

  4^=- 
  -13686 
  nearly, 
  

  

  or 
  

  

  -=-29-2263 
  nearly. 
  

  

  Searching 
  as 
  before, 
  we 
  find 
  for 
  order 
  — 
  7, 
  

  

  7x29-2263=204-5841; 
  

  

  and 
  205 
  is 
  a 
  system 
  of 
  order 
  —7. 
  

   Comparing 
  thirds, 
  

  

  Departure 
  of 
  perfect 
  third 
  = 
  — 
  -1368629 
  

   Departure 
  of 
  third 
  of 
  205 
  = 
  - 
  -1369002 
  

  

  Error 
  = 
  -0000373 
  flat. 
  

   The 
  following 
  is 
  a 
  resume 
  of 
  the 
  properties 
  of 
  these 
  higher 
  systems 
  :- 
  

   System. 
  Order. 
  A 
  = 
  12tf. 
  Error 
  of 
  fifth. 
  Error 
  of 
  third 
  

  

  289 
  5 
  -20761 
  - 
  -00225 
  - 
  -00155 
  

  

  643 
  11 
  -20529 
  - 
  -00244 
  + 
  -0000044 
  

  

  301 
  5 
  -19934 
  - 
  -00294 
  + 
  -00397 
  

  

  205 
  -7 
  - 
  -41070 
  - 
  -05377 
  - 
  -000037 
  

  

  