﻿Theory 
  of 
  the 
  Division 
  of 
  the 
  Octave. 
  

  

  401 
  

  

  Such 
  passages 
  as 
  this 
  can 
  be 
  played 
  on 
  the 
  harmonium 
  hereafter 
  

   described. 
  

  

  Principle 
  of 
  Symmetrical 
  Arrangement 
  in 
  Regular 
  Systems. 
  

  

  I£ 
  we 
  place 
  the 
  E. 
  T. 
  notes 
  in 
  the 
  order 
  of 
  the 
  scale, 
  and 
  set 
  off 
  the 
  

   departures 
  of 
  the 
  notes 
  of 
  any 
  regular 
  system 
  at 
  right 
  angles 
  to 
  the 
  E. 
  T. 
  

   line, 
  sharp 
  departures 
  up 
  and 
  flat 
  departures 
  down, 
  we 
  obtain 
  the 
  posi- 
  

   tions 
  of 
  what 
  may 
  be 
  called 
  a 
  symmetrical 
  arrangement. 
  

  

  The 
  distances 
  of 
  the 
  E. 
  T. 
  notes 
  from 
  the 
  starting-point 
  are 
  abscissae, 
  

   and 
  the 
  departures 
  ordinates. 
  

  

  Positive 
  Systems. 
  

  

  The 
  subjoined 
  is 
  a 
  symmetrical 
  arrangement 
  of 
  the 
  notes 
  of 
  General 
  

   Thompson's 
  enharmonic 
  organ 
  (p. 
  402). 
  It 
  is 
  selected 
  as 
  not 
  being 
  too 
  

   extensive 
  for 
  reproduction 
  as 
  being 
  of 
  historical 
  interest, 
  and 
  as 
  illustra- 
  

   ting 
  the 
  nature 
  of 
  the 
  difficulty 
  caused 
  by 
  the 
  distribution 
  of 
  such 
  systems 
  

   into 
  separate 
  key-boards. 
  Each 
  of 
  the 
  single 
  vertical 
  steps 
  represents 
  the 
  

   departure 
  of 
  one 
  fifth. 
  

  

  The 
  property 
  of 
  symmetrical 
  arrangements, 
  from 
  which 
  they 
  derive 
  

   ' 
  their 
  principal 
  importance, 
  is 
  that, 
  position 
  being 
  determined 
  only 
  by 
  

   relations 
  of 
  interval, 
  the 
  notes 
  of 
  a 
  combination 
  forming 
  given 
  intervals 
  

   present 
  always 
  the 
  same 
  form, 
  whatever 
  be 
  the 
  key 
  or 
  the 
  actual 
  notes 
  

   employed. 
  

  

  Let 
  us 
  express, 
  as 
  before, 
  the 
  number 
  of 
  E. 
  T. 
  semitones, 
  which 
  is 
  now 
  

   our 
  abscissa, 
  by 
  simple 
  integers, 
  and 
  the 
  number 
  of 
  departures 
  of 
  fifths, 
  

   . 
  which 
  is 
  our 
  ordinate, 
  by 
  a 
  coefficient 
  attached 
  to 
  $. 
  Then 
  we 
  have 
  only 
  

   to 
  note 
  tho 
  values 
  of 
  the 
  different 
  intervals 
  to 
  obtain 
  their 
  coordinates 
  

   with 
  respect 
  to 
  any 
  note 
  taken 
  as 
  origin. 
  • 
  . 
  

  

  Thus 
  the 
  third 
  is 
  4— 
  8#, 
  or 
  four 
  steps 
  to 
  the 
  right 
  and 
  eight 
  down 
  

   (c-\e) 
  ; 
  the 
  fifth 
  is 
  7+£, 
  seven 
  steps 
  to 
  the' 
  right 
  and 
  one 
  up 
  (c-g) 
  ; 
  

   the 
  minor 
  third 
  is 
  three 
  to 
  the 
  right 
  and 
  nine 
  up 
  (\e-g) 
  ; 
  and 
  so 
  on. 
  

  

  Two 
  notes 
  are 
  omitted 
  from 
  the 
  otherwise 
  complete 
  series, 
  5 
  anclwc?; 
  

   and 
  we 
  notice 
  the 
  number 
  of 
  otherwise 
  complete 
  chords 
  which 
  their 
  

   absence 
  destroys. 
  . 
  

  

  Distribution 
  over 
  three 
  Key-hoards. 
  — 
  As 
  an 
  example 
  of 
  the 
  effect 
  of 
  this, 
  

   we 
  note 
  that 
  the 
  notes 
  of 
  the 
  chord 
  of 
  a 
  minor 
  are 
  all 
  present 
  ; 
  but 
  they 
  

   are 
  ^.a-/^-^, 
  so 
  that 
  the 
  third 
  and 
  fifth 
  are 
  on 
  different 
  key-boards. 
  

  

  Negative 
  Systems, 
  

  

  According 
  to 
  the 
  enunciation 
  of 
  the 
  principle 
  of 
  symmetrical 
  arrauge- 
  

   . 
  ment, 
  the 
  positions 
  should 
  be 
  taken 
  lower 
  for 
  negative 
  systems 
  as 
  we 
  

   ascend 
  in 
  the 
  series 
  of 
  fifths 
  ; 
  but 
  it 
  is 
  practically 
  more 
  convenient 
  to 
  

   use 
  the 
  positive 
  form 
  in 
  negative 
  systems 
  as 
  well. 
  The 
  coordinates 
  of 
  

   • 
  some 
  intervals 
  become 
  different— 
  the 
  third 
  is 
  4 
  + 
  45, 
  the 
  minor 
  third 
  

   3-35, 
  &c. 
  

  

  