﻿Theory 
  of 
  the 
  Division 
  of 
  the 
  Octave 
  . 
  

  

  399 
  

  

  An 
  illustration 
  may 
  be 
  made 
  as 
  follows, 
  which 
  shows 
  on 
  inspection 
  all 
  

   the 
  data 
  involved 
  in 
  the 
  above 
  Table, 
  and 
  the 
  properties 
  of 
  any 
  other 
  

   system 
  introduced 
  into 
  it. 
  

  

  Take 
  axes 
  of 
  abscissae 
  and 
  ordinates, 
  and 
  set 
  off 
  on 
  both 
  distances 
  

   representing 
  tenths 
  of 
  E. 
  T. 
  semitones 
  — 
  for 
  ordinary 
  purposes 
  10 
  inches 
  

   to 
  the 
  E. 
  T. 
  semitone 
  answers 
  best 
  ; 
  for 
  Lecture 
  scale, 
  1 
  metre 
  to 
  the 
  E. 
  T. 
  

   semitone. 
  

  

  On 
  the 
  axis 
  of 
  ordinates 
  set 
  off 
  points 
  representing 
  the 
  values 
  in 
  

   column 
  A 
  of 
  the 
  Table, 
  and 
  corresponding 
  values 
  for 
  any 
  other 
  system 
  

   required. 
  Through 
  each 
  of 
  these 
  points 
  rule 
  a 
  straight 
  line 
  parallel 
  to 
  

   the 
  axis 
  of 
  abscissae. 
  

  

  On 
  the 
  axis 
  of 
  abscissae 
  set 
  off 
  points 
  representing 
  the 
  values 
  — 
  -13686 
  

   and 
  —'31174. 
  Rule 
  lines 
  through 
  these 
  parallel 
  to 
  the 
  axis 
  of 
  ordi- 
  

   nates. 
  These 
  abscissae 
  represent 
  respectively 
  perfect 
  thirds 
  and 
  perfect 
  

   sevenths. 
  

  

  Draw 
  lines 
  inclined 
  to 
  the 
  axis 
  of 
  abscissae 
  at 
  angles 
  tan 
  -1 
  t> 
  and 
  

  

  tan 
  -1 
  ^. 
  These 
  give, 
  by 
  their 
  intersections 
  with 
  the 
  lines 
  of 
  the 
  different 
  

  

  positive 
  systems, 
  the 
  thirds 
  and 
  sevenths 
  respectively. 
  

  

  Draw 
  lines 
  inclined 
  to 
  the 
  axis 
  of 
  abscissae 
  at 
  angles 
  tan 
  -1 
  — 
  3 
  and 
  

  

  tan 
  -1 
  — 
  g. 
  These 
  give, 
  by 
  their 
  intersections 
  with 
  the 
  lines 
  of 
  the 
  different 
  

  

  negative 
  systems, 
  the 
  thirds 
  and 
  sevenths 
  respectively. 
  

  

  The 
  errors 
  of 
  the 
  thirds 
  and 
  sevenths 
  are 
  the 
  perpendicular 
  distances 
  

   of 
  the 
  intersections 
  which 
  determine 
  them 
  from 
  the 
  ordinates 
  of 
  perfect 
  

   thirds 
  and 
  sevenths 
  already 
  constructed. 
  

  

  Ia 
  Mer/alar 
  Cyclical 
  Si/stems, 
  to 
  find 
  the 
  number 
  of 
  Units 
  in 
  any 
  Interval 
  

  

  in 
  the 
  Scale. 
  

  

  Let 
  x 
  be 
  the 
  number 
  of 
  units 
  in 
  the 
  seven-fifths 
  semitone, 
  then 
  

  

  12 
  r 
  

   ^.— 
  = 
  1 
  + 
  7^=1 
  + 
  7- 
  

  

  n 
  n' 
  

  

  or 
  

  

  n 
  -f 
  7r 
  

  

  It 
  is 
  easy 
  to 
  see 
  that 
  x 
  will 
  always 
  be 
  integral 
  if 
  the 
  order 
  condition 
  is 
  

   satisfied 
  (Th. 
  iii.), 
  viz. 
  if 
  7n 
  + 
  r 
  is 
  a 
  multiple 
  of 
  12. 
  

  

  For 
  then 
  7(7n+?-)=49n 
  + 
  7r 
  ; 
  whence, 
  casting 
  out 
  48n, 
  n-\~7r 
  is 
  a 
  

   multiple 
  of 
  12. 
  

  

  We 
  can 
  now 
  determine 
  the 
  remaining 
  intervals 
  in 
  terms 
  of 
  x 
  

   and 
  r 
  : 
  — 
  

  

  