﻿Theory 
  of 
  the 
  Division 
  of 
  the 
  Octave. 
  

  

  405 
  

  

  of 
  53 
  is 
  adopted. 
  It 
  is 
  required 
  to 
  find 
  rules 
  of 
  identification 
  for 
  passing 
  

   from 
  one 
  principal 
  division 
  of 
  the 
  octave 
  to 
  another. 
  

  

  Rule. 
  — 
  In 
  the 
  system 
  of 
  53 
  the 
  notation 
  of 
  positive 
  systems 
  becomes 
  

   subject 
  to 
  the 
  following 
  identifications 
  : 
  — 
  

  

  If 
  two 
  notes 
  in 
  adjoining 
  principal 
  divisions 
  (e. 
  g. 
  c 
  and 
  cjf) 
  be 
  so 
  

   situated 
  as 
  to 
  admit 
  of 
  identification 
  (e. 
  g. 
  a 
  high 
  c 
  and 
  a 
  low 
  cj), 
  they 
  

   will 
  be 
  the 
  same 
  if 
  the 
  sum 
  of 
  the 
  elevation- 
  and 
  depression-marks 
  =4 
  ; 
  

   unless 
  the 
  lower 
  of 
  the 
  two 
  divisions 
  is 
  black 
  (accidental), 
  then 
  the 
  sum 
  

   of 
  the 
  marks 
  of 
  identical 
  notes 
  = 
  5. 
  

  

  This 
  can 
  only 
  be 
  proved 
  by 
  enumeration 
  of 
  a 
  case 
  in 
  each 
  pair 
  of 
  divi- 
  

   sions. 
  This 
  enumeration 
  is 
  made 
  in 
  the 
  writer's 
  original 
  paper. 
  It 
  is 
  

   founded 
  on 
  the 
  followiug 
  principles 
  : 
  — 
  

  

  Noting 
  that 
  the 
  5-fifths 
  semitone 
  is 
  4 
  units 
  (scheme 
  following 
  Th. 
  L), 
  

   we 
  see 
  that 
  c-c#is 
  4 
  units, 
  whence 
  ////o-cjf, 
  ///c-Xcjf, 
  //c-Wcf 
  .... 
  

   are 
  identities 
  ; 
  or, 
  again, 
  c$-\d 
  is 
  4 
  units, 
  and 
  ////c#-\d, 
  ///c-\\d 
  

   .... 
  are 
  identities. 
  

  

  Application 
  of 
  the 
  System 
  of 
  53 
  to 
  the 
  " 
  Generalized 
  Key 
  -board." 
  

  

  An 
  harmonium 
  has 
  been 
  constructed 
  which 
  is 
  arranged 
  as 
  follows 
  : 
  — 
  

   The 
  note 
  \\\c 
  is 
  taken 
  as 
  the 
  first 
  note 
  of 
  the 
  series, 
  and 
  receives 
  the 
  

   characteristic 
  number 
  1. 
  Then 
  c 
  is 
  4, 
  and 
  the 
  remaining 
  numbers 
  can 
  

   be 
  assigned 
  by 
  the 
  rules 
  for 
  the 
  identifications 
  in 
  the 
  system 
  of 
  53 
  given 
  

   above. 
  

  

  A 
  number 
  of 
  notes 
  at 
  the 
  top 
  of 
  the 
  key-board 
  are 
  thus 
  identical 
  with 
  

   corresponding 
  notes 
  in 
  the 
  adjacent 
  principal 
  divisions 
  on 
  the 
  right 
  at 
  

   the 
  bottom, 
  e.g. 
  //c=J=\\cf. 
  These 
  permit 
  the 
  infinite 
  freedom 
  of 
  

   modulation 
  which 
  is 
  the 
  characteristic 
  of 
  cyclical 
  systems 
  ; 
  for 
  in 
  moving 
  

   upwards 
  on 
  the 
  key-board 
  we 
  can, 
  on 
  arriving 
  near 
  the 
  top, 
  change 
  the 
  

   hands 
  on 
  to 
  identical 
  notes 
  near 
  the 
  bottom, 
  and 
  so 
  proceed 
  further 
  in 
  

   the 
  same 
  direction, 
  and 
  vice 
  versa. 
  

  

  It 
  is 
  to 
  be 
  noted 
  that, 
  in 
  positive 
  systems, 
  displacement 
  upwards 
  or 
  

   downwards 
  on 
  the 
  key-board 
  takes 
  place 
  most 
  readily 
  by 
  modulation 
  

   between 
  related 
  major 
  and 
  minor 
  keys 
  — 
  not, 
  as 
  has 
  been 
  commonly 
  

   assumed, 
  only 
  by 
  modulation 
  round 
  the 
  circles 
  of 
  fifths. 
  In 
  negative 
  

   systems, 
  on 
  the 
  contrary, 
  displacements 
  take 
  place 
  only 
  by 
  modulations 
  

   of 
  the 
  latter 
  type. 
  

  

  Application 
  of 
  the 
  System 
  of 
  118 
  to 
  the 
  " 
  Generalized 
  Key-board." 
  

  

  The 
  5-fifths 
  semitone 
  is 
  here 
  9 
  units, 
  and 
  the 
  7-fifths 
  semitone 
  is 
  11 
  

   units. 
  The 
  major 
  tone 
  (2-fifths 
  tone) 
  is 
  consequently 
  20, 
  and 
  the 
  minor 
  

   tone 
  (10-fifths 
  tone) 
  is 
  18. 
  Hence 
  the 
  notes 
  in 
  the 
  successive 
  principal 
  

   divisions 
  are 
  alternately 
  odd 
  and 
  even, 
  and 
  the 
  identifications 
  lie 
  in 
  alter- 
  

   nate 
  columns. 
  These 
  are 
  not 
  here 
  further 
  investigated, 
  as 
  no 
  practical 
  

   use 
  has 
  been 
  made 
  of 
  the 
  system. 
  

  

  2 
  i 
  2 
  

  

  