﻿394 
  

  

  Mr. 
  R. 
  H. 
  M. 
  Bosanquet 
  on 
  the 
  

  

  n 
  equal 
  intervals, 
  r-\-7n 
  is 
  a 
  multiple 
  of 
  12, 
  and 
  ~^9~ 
  * 
  s 
  ^ 
  ne 
  number 
  of 
  

   units 
  in 
  the 
  fifth 
  of 
  the 
  system. 
  

  

  Let 
  cf> 
  be 
  the 
  number 
  of 
  units 
  in 
  the 
  fifth. 
  

  

  •'• 
  ^="12"' 
  

  

  and 
  is 
  an 
  integer 
  by 
  hypothesis 
  ; 
  whence 
  the 
  proposition. 
  

  

  Cor. 
  From 
  this 
  proposition 
  we 
  can 
  deduce 
  corresponding 
  values 
  of 
  n 
  

   and 
  r. 
  It 
  is 
  useful 
  in 
  the 
  investigation 
  of 
  systems 
  of 
  the 
  higher 
  orders. 
  

   Casting 
  out 
  multiples 
  of 
  12, 
  where 
  necessary, 
  from 
  n 
  and 
  r, 
  we 
  have 
  the 
  

   following 
  relations 
  between 
  the 
  remainders 
  : 
  — 
  

  

  Remainder 
  of 
  

  

  n 
  .... 
  1 
  2 
  3 
  4 
  5 
  6 
  7 
  8 
  9 
  10 
  11 
  

   r 
  .... 
  5 
  10 
  3 
  8 
  1 
  6 
  11 
  4 
  9 
  2 
  7 
  

  

  Theorem 
  iv. 
  If 
  a 
  system 
  divide 
  the 
  octave 
  into 
  n 
  equal 
  intervals, 
  the 
  

   total 
  departure 
  of 
  all 
  the 
  n 
  fifths 
  of 
  the 
  system 
  =r 
  E. 
  T. 
  semitones, 
  where 
  

   r 
  is 
  the 
  order 
  of 
  the 
  system. 
  

  

  Tor 
  by 
  Th. 
  ii. 
  £=- 
  ; 
  whence 
  

  

  n 
  

  

  n$ 
  = 
  r, 
  

  

  or 
  the 
  departure 
  of 
  n 
  fifths 
  = 
  r 
  semitones. 
  

  

  This 
  gives 
  rise 
  to 
  a 
  curious 
  mode 
  of 
  deriving 
  the 
  different 
  systems. 
  

  

  Suppose 
  the 
  notes 
  of 
  an 
  E. 
  T. 
  series 
  arranged 
  in 
  order 
  of 
  fifths, 
  and 
  

   proceeding 
  onwards 
  indefinitely, 
  thus 
  : 
  — 
  

  

  c 
  g 
  d 
  a 
  e 
  b 
  /# 
  c# 
  g$ 
  d$ 
  aft 
  f 
  c 
  g 
  . 
  . 
  ., 
  

  

  and 
  so 
  on. 
  Let 
  a 
  regular 
  system 
  of 
  fifths 
  start 
  from 
  c. 
  If 
  they 
  are 
  

   positive, 
  then 
  at 
  each 
  step 
  the 
  pitch 
  rises 
  further 
  from 
  E. 
  T. 
  It 
  can 
  only 
  

   return 
  to 
  c 
  by 
  sharpening 
  an 
  E. 
  T. 
  note. 
  

  

  Suppose 
  that 
  b 
  is 
  sharpened 
  one 
  E. 
  T. 
  semitone, 
  so 
  as 
  to 
  become 
  c 
  ; 
  

   then 
  the 
  return 
  may 
  be 
  effected 
  

  

  at 
  the 
  first 
  b 
  in 
  5 
  fifths, 
  

  

  at 
  the 
  second 
  b 
  in 
  17 
  fifths, 
  

  

  at 
  the 
  third 
  b 
  in 
  29 
  fifths 
  ; 
  and 
  so 
  on. 
  

  

  Thus 
  we 
  obtain 
  the 
  primary 
  positive 
  systems. 
  Secondary 
  positive 
  

   systems 
  may 
  be 
  got 
  by 
  sharpening 
  b\) 
  2 
  semitones 
  ; 
  and 
  so 
  on. 
  

  

  If 
  the 
  fifths 
  are 
  negative, 
  the 
  return 
  may 
  be 
  effected 
  by 
  depressing 
  eft 
  

   a 
  semitone 
  in 
  7, 
  19, 
  31 
  . 
  . 
  . 
  fifths 
  ; 
  we 
  thus 
  obtain 
  the 
  primary 
  nega- 
  

   tive 
  systems 
  ; 
  or 
  by 
  depressing 
  el 
  two 
  semitones, 
  by 
  which 
  we 
  get 
  the 
  

   secondary 
  negative 
  systems 
  ; 
  and 
  so 
  on. 
  

  

  