﻿Theory 
  of 
  the 
  Division 
  of 
  the 
  Octave. 
  

  

  393 
  

  

  Regular 
  Cyclical 
  Systems, 
  

  

  The 
  importance 
  of 
  regular 
  cyclical 
  systems 
  arises 
  from 
  the 
  infinite 
  

   freedom 
  of 
  modulation 
  in 
  every 
  direction 
  which 
  is 
  possible 
  in 
  such 
  

   systems 
  when 
  properly 
  arranged 
  ; 
  whereas 
  in 
  non-cyclical 
  systems 
  

   required 
  modulations 
  are 
  liable 
  to 
  be 
  impossible, 
  owing 
  to 
  the 
  demand 
  for 
  

   notes 
  lying 
  outside 
  the 
  material 
  provided. 
  

  

  Theorem 
  i. 
  In 
  a 
  regular 
  cyclical 
  system 
  of 
  the 
  + 
  rth 
  order 
  the 
  difference 
  

   between 
  the 
  seven-fifths 
  semitone 
  and 
  five-fifths 
  semitone 
  is 
  + 
  r 
  units 
  of 
  

   the 
  system, 
  or 
  s—f= 
  j-r 
  units. 
  

  

  Eecalling 
  the 
  definition 
  of 
  rth 
  order 
  (12£= 
  +>• 
  units), 
  the 
  proposition 
  

   follows 
  from 
  Th. 
  /3. 
  

  

  Cor. 
  This 
  proposition, 
  taken 
  with 
  Th. 
  a, 
  enables 
  us 
  to 
  ascertain 
  the 
  

   number 
  of 
  divisions 
  in 
  the 
  octave 
  in 
  systems 
  of 
  any 
  order, 
  by 
  introducing 
  

   the 
  consideration 
  that 
  each 
  semitone 
  must 
  consist 
  of 
  an 
  integral 
  number 
  

   of 
  units. 
  The 
  principal 
  known 
  systems 
  are 
  here 
  enumerated 
  : 
  — 
  • 
  

  

  Primary 
  (1st 
  order) 
  Positive. 
  

  

  7-fifths 
  semitone 
  5-fifths 
  semitone 
  Number 
  of 
  units 
  in 
  octave 
  

  

  =x 
  units. 
  =y 
  units. 
  (Th. 
  a) 
  bx+7y 
  = 
  n. 
  

  

  2 
  ' 
  1 
  17 
  

  

  3 
  2 
  29 
  

  

  4 
  3 
  41 
  

  

  5 
  4 
  53 
  

  

  6 
  5 
  65 
  

  

  Secondary 
  (2nd 
  order) 
  Positive. 
  

  

  11 
  9 
  118 
  

  

  Primary 
  Negative. 
  

  

  1 
  2 
  . 
  19 
  

  

  2 
  .m'-sim-mj 
  3 
  ^1 
  

  

  Secondary 
  Negative. 
  

  

  3 
  ........... 
  5 
  50 
  

  

  Theorem 
  ii. 
  In 
  any 
  regular 
  cyclical 
  system, 
  if 
  the 
  octave 
  be 
  divided 
  

   into 
  n 
  equal 
  intervals, 
  and 
  r 
  be 
  "the 
  order 
  of 
  the 
  system, 
  the 
  departure 
  of 
  

  

  T 
  

  

  each 
  fifth 
  of 
  the 
  system 
  is 
  - 
  E. 
  T. 
  semitones. 
  

   i 
  n 
  

  

  Eor 
  departure 
  of 
  12 
  fifths 
  = 
  128 
  = 
  r 
  units 
  by 
  definition 
  and 
  the 
  unit= 
  

  

  12 
  

  

  — 
  E. 
  T. 
  semitones 
  ; 
  

  

  n 
  

  

  Theorem 
  iii. 
  If, 
  in 
  a 
  system 
  of 
  the 
  rth 
  order, 
  the 
  octave 
  be 
  divided 
  into 
  

  

  