﻿392 
  

  

  Mr. 
  R. 
  H. 
  M. 
  Bosanquet 
  on 
  the 
  

  

  Intervals 
  taken 
  upwards 
  are 
  called 
  positive, 
  taken 
  downwards, 
  nega- 
  

   tive. 
  

  

  Systems 
  are 
  said 
  to 
  be 
  of 
  the 
  rth 
  order, 
  positive 
  or 
  negative, 
  when 
  the 
  

   departure 
  of 
  12 
  fifths 
  is 
  + 
  r 
  units 
  of 
  the 
  system, 
  

  

  Intervals 
  formed 
  hy 
  Fifths. 
  

  

  "When 
  successions 
  of 
  fifths 
  are 
  spoken 
  of, 
  it 
  is 
  intended 
  that 
  octaves 
  be 
  

   disregarded. 
  If 
  the 
  result 
  of 
  a 
  number 
  of 
  fifths 
  is 
  expressed 
  in 
  E. 
  T. 
  

   semitones, 
  any 
  multiples 
  of 
  12 
  (octaves) 
  are 
  cast 
  out. 
  [Representing 
  the 
  

   fifth 
  of 
  any 
  system 
  by 
  7+8, 
  where 
  8 
  is 
  the 
  departure 
  of 
  one 
  fifth 
  

   expressed 
  in 
  E. 
  T. 
  semitones, 
  we 
  form 
  the 
  following 
  intervals 
  amongst 
  

   others 
  : 
  — 
  

  

  Departure 
  of 
  12 
  fifths 
  = 
  128 
  

  

  (12 
  x 
  (7 
  +8) 
  = 
  84 
  +128, 
  and 
  84 
  is 
  cast 
  out). 
  

   Two-fifths 
  tone 
  = 
  2 
  + 
  28 
  

  

  (2 
  x 
  (7+ 
  ^) 
  ==14+ 
  2^, 
  and 
  12 
  is 
  cast 
  out). 
  

  

  Seven-fifths 
  semitone, 
  formed 
  by 
  seven 
  fifths 
  up, 
  = 
  1 
  + 
  73 
  

  

  (7x(7 
  + 
  8) 
  = 
  49 
  + 
  78, 
  and 
  48 
  is 
  cast 
  out). 
  

  

  Eive-fifths 
  semitone, 
  formed 
  by 
  five 
  fifths 
  down, 
  = 
  1 
  — 
  55 
  

  

  (5x 
  —(7 
  + 
  8)=— 
  (35 
  + 
  58), 
  and 
  36 
  is 
  added). 
  

  

  The 
  seven-fifths 
  semitone 
  will 
  be 
  denoted 
  by 
  s 
  (=1 
  + 
  78) 
  j 
  the 
  five- 
  

   fifths 
  semitone 
  by 
  / 
  ( 
  = 
  1 
  — 
  58). 
  

  

  Regular 
  Systems. 
  

  

  The 
  importance 
  of 
  regular 
  systems 
  arises 
  from 
  the 
  symmetry 
  of 
  the 
  

   scales 
  which 
  they 
  form, 
  

  

  Theorem 
  a. 
  In 
  any 
  regular 
  system 
  five 
  seven-fifths 
  semitones 
  + 
  seven 
  

   five-fifths 
  semitones 
  make 
  an 
  exact 
  octave, 
  or 
  5s 
  + 
  7/= 
  12. 
  

  

  Eor 
  the 
  departures 
  (from 
  E. 
  T.) 
  of 
  the 
  5 
  seven-fifths 
  semitones 
  are 
  

   due 
  to 
  35 
  fifths 
  up, 
  and 
  those 
  of 
  the 
  7 
  five-fifths 
  semitones 
  to 
  35 
  fifths 
  

   down, 
  leaving 
  12 
  E. 
  T. 
  semitones, 
  which 
  form 
  an 
  exact 
  octave 
  ; 
  or, 
  

  

  5(l 
  + 
  78) 
  + 
  7(l-5£)=12. 
  

  

  Theorem 
  /3. 
  In 
  any 
  regular 
  system 
  the 
  difference 
  between 
  the 
  seven- 
  

   fifths 
  semitone 
  and 
  the 
  five-fifths 
  semitone 
  is 
  the 
  departure 
  of 
  12 
  fifths, 
  

   having 
  regard 
  to 
  sign 
  ; 
  or, 
  

  

  s— 
  /= 
  departure 
  of 
  12 
  fifths. 
  

  

  Let 
  J 
  be 
  the 
  departure 
  of 
  each 
  fifth 
  of 
  the 
  system, 
  thens=l 
  + 
  7£, 
  

   f=l-53; 
  whence 
  s 
  -/== 
  12£, 
  

  

  