﻿Theory 
  of 
  the 
  Division 
  of 
  the 
  Octave. 
  395 
  

  

  An 
  instructive 
  illustration 
  may 
  be 
  made 
  as 
  follows; 
  it 
  requires 
  too 
  

   large 
  dimensions 
  for 
  convenient 
  reproduction 
  here 
  : 
  — 
  

  

  Set 
  off 
  on 
  the 
  axis 
  of 
  abscissae 
  the 
  equal 
  temperament 
  series 
  in 
  order 
  

   of 
  fifths, 
  as 
  above, 
  taking 
  about 
  10 
  complete 
  periods. 
  If 
  the 
  distances 
  

   of 
  the 
  single 
  terms 
  are 
  made 
  1 
  centimetre, 
  this 
  will 
  take 
  l 
  ni, 
  20 
  in 
  length, 
  

   starting 
  from 
  the 
  origin 
  on 
  the 
  left. 
  

  

  Select 
  a 
  unit 
  for 
  the 
  E. 
  T. 
  semitone 
  of 
  departure, 
  say 
  1 
  decimetre. 
  

  

  Eule 
  a 
  series 
  of 
  lines 
  parallel 
  to 
  the 
  axis 
  of 
  abscissae, 
  at 
  distances 
  repre- 
  

   senting 
  integral 
  numbers 
  of 
  E. 
  T. 
  semitones, 
  both 
  above 
  and 
  below. 
  

  

  Eule, 
  parallel 
  to 
  the 
  axis 
  of 
  ordinates, 
  straight 
  lines 
  through 
  the 
  points 
  

   representing 
  the 
  E. 
  T. 
  notes. 
  

  

  Enter 
  on 
  the 
  intersections 
  the 
  names 
  of 
  the 
  E. 
  T. 
  notes 
  they 
  represent. 
  

   Thus 
  the 
  notes 
  on 
  the 
  positive 
  ordinate 
  of 
  c 
  are 
  c-c#-d 
  . 
  . 
  ., 
  and 
  so 
  

   on, 
  each 
  pair 
  separated 
  by 
  1 
  decimetre, 
  and 
  the 
  notes 
  on 
  the 
  negative 
  

   ordinate 
  of 
  c 
  are 
  c-b-blp 
  .... 
  

  

  If 
  we 
  then 
  join 
  the 
  c 
  on 
  the 
  left 
  hand 
  of 
  the 
  axis 
  of 
  abscissae 
  to 
  all 
  the 
  

   other 
  c's 
  on 
  the 
  figure, 
  except, 
  of 
  course, 
  those 
  on 
  the 
  axis, 
  we 
  obtain 
  a 
  

   complete 
  graphic 
  representation 
  of 
  all 
  the 
  systems 
  whose 
  orders 
  are 
  

   included. 
  The 
  rth. 
  order 
  is 
  represented 
  by 
  lines 
  drawn 
  to 
  the 
  c's 
  in 
  the 
  

   rth 
  line 
  above, 
  the 
  — 
  Hh 
  by 
  the 
  lines 
  drawn 
  to 
  the 
  c's 
  in 
  the 
  rth 
  line 
  

   below. 
  

  

  This 
  illustration 
  brings 
  specially 
  into 
  prominence 
  the 
  singularity 
  of 
  

   multiple 
  systems, 
  as 
  all 
  the 
  multiples 
  of 
  any 
  system 
  lie 
  on 
  the 
  same 
  

   straight 
  line 
  with 
  it, 
  and 
  the 
  representation 
  fails 
  to 
  give 
  all 
  the 
  notes 
  of 
  

   such 
  systems. 
  

  

  Multiple 
  Systems. 
  

  

  Multiple 
  systems 
  are 
  such 
  that 
  the 
  number 
  of 
  divisions 
  in 
  the 
  octave 
  

   {hn) 
  in 
  any 
  such 
  system 
  is 
  a 
  multiple 
  (Jc) 
  of 
  the 
  number 
  of 
  divisions 
  (n) 
  

   of 
  some 
  other 
  system. 
  

  

  Multiple 
  systems 
  have 
  not 
  been 
  as 
  yet 
  practically 
  applied. 
  

  

  These 
  systems 
  are 
  not 
  strictly 
  regular 
  ; 
  for 
  though 
  their 
  fifths 
  are 
  all 
  

   equal, 
  yet 
  they 
  do 
  not 
  form 
  one 
  continuous 
  series, 
  but 
  several. 
  They 
  

   are 
  strictly 
  cyclical, 
  i. 
  e. 
  they 
  divide 
  the 
  octave 
  into 
  n 
  equal 
  intervals. 
  

  

  Theorem 
  v. 
  A 
  multiple 
  system, 
  Ten, 
  may 
  be 
  regarded 
  as 
  being 
  of 
  order 
  

   Jcr, 
  where 
  n 
  is 
  a 
  system 
  of 
  order 
  r. 
  

  

  Eor, 
  n 
  being 
  a 
  system 
  of 
  order 
  r, 
  r+7n 
  is 
  a 
  multiple 
  of 
  12 
  ; 
  .*. 
  also 
  

   &(>+7n) 
  is 
  a 
  multiple 
  of 
  12, 
  which 
  is 
  the 
  condition 
  that 
  the 
  system 
  hi 
  

   be 
  of 
  order 
  h\ 
  

  

  This 
  is 
  useful 
  in 
  the 
  investigation 
  of 
  systems 
  of 
  the 
  higher 
  orders. 
  

   If 
  n 
  is 
  a 
  multiple 
  of 
  12, 
  the 
  system 
  is 
  a 
  multiple 
  of 
  the 
  E. 
  T., 
  and 
  of 
  

   order 
  zero. 
  

  

  In 
  the 
  illustration 
  described 
  under 
  Th. 
  iv. 
  the 
  notes 
  of 
  a 
  multiple 
  

   system 
  (Icri) 
  are 
  the 
  same 
  as 
  those 
  of 
  system 
  n, 
  until 
  the 
  latter 
  is 
  com- 
  

   plete. 
  The 
  rest 
  of 
  the 
  representation 
  consists 
  simply 
  of 
  the 
  same 
  notes 
  

  

  