﻿400 
  

  

  Mr. 
  R. 
  H. 
  M. 
  Bosanquet 
  on 
  the 
  

  

  "No. 
  of 
  units. 
  

  

  Interval. 
  Positive 
  systems. 
  Negative 
  systems. 
  

  

  5 
  -fifths 
  semitone 
  so 
  — 
  r 
  ...... 
  x— 
  r 
  

  

  Minor 
  tone 
  \ 
  

  

  10-fifths 
  tone 
  j2x-2r 
  

  

  Major 
  tone 
  ■. 
  . 
  . 
  "I 
  _ 
  

  

  2-fifthstone 
  J 
  2x 
  ~ 
  r 
  2x 
  ~ 
  r 
  

  

  Minor 
  third 
  .. 
  , 
  3x- 
  r 
  . 
  3x—2r 
  

  

  Major 
  third 
  4^ 
  — 
  3?- 
  * 
  4x— 
  2r 
  

  

  Fourth 
  bx 
  — 
  3r 
  5x—3r 
  

  

  Fifth 
  7a? 
  — 
  4r 
  7x 
  — 
  4r 
  

  

  Sixth 
  9x-6r 
  . 
  9x-5r 
  

  

  Harmonic 
  seventh 
  lOx 
  — 
  7r 
  . 
  lOx 
  — 
  or 
  

  

  Major 
  seventh 
  lias— 
  7 
  r 
  llx 
  — 
  6r 
  

  

  Octave 
  12x 
  — 
  7r 
  12x 
  — 
  7r=n. 
  

  

  The 
  — 
  r's 
  in 
  negative 
  systems 
  are, 
  of 
  course, 
  positive 
  quantities. 
  

  

  Employment 
  of 
  Positive 
  Systems 
  in 
  Music. 
  

  

  Rule 
  for 
  thirds. 
  — 
  If 
  we 
  write 
  down 
  one 
  of 
  the 
  duodenes 
  of 
  the 
  notation, 
  

  

  f%-c%-cyt.-d%~a 
  %-f-c-g-d-a-e-b 
  , 
  

  

  and 
  remember 
  that 
  positive 
  systems 
  form 
  their 
  thirds 
  by 
  8 
  fifths 
  down, 
  

   we 
  have 
  the 
  rule 
  :— 
  

  

  The 
  four 
  accidentals 
  on 
  the 
  left 
  in 
  any 
  duodene 
  of 
  the 
  notation 
  form 
  

   major 
  thirds 
  to 
  the 
  four 
  notes 
  on 
  the 
  extreme 
  right 
  in 
  the 
  same 
  duodene. 
  

   All 
  other 
  notes 
  have 
  their 
  major 
  thirds 
  in 
  the 
  next 
  duodene 
  below. 
  Thus 
  

   d-f%, 
  c-\e 
  are 
  major 
  thirds.] 
  

  

  Use 
  of 
  the 
  Notation 
  with 
  Musical 
  Symhols. 
  

  

  It 
  is 
  an 
  essential 
  point 
  in 
  this 
  notation 
  that 
  it 
  can 
  be 
  used 
  with 
  musical 
  

   symbols. 
  The 
  following 
  example 
  shows 
  the 
  major 
  and 
  minor 
  chords 
  and 
  

   the 
  interval 
  used 
  for 
  the 
  harmonic 
  seventh 
  

  

  The 
  first 
  chord 
  is 
  the 
  major 
  triad 
  ; 
  the 
  second 
  involves 
  g-\f 
  the 
  har- 
  

   monic 
  seventh 
  ; 
  the 
  fourth 
  crotchet 
  gives 
  the 
  minor 
  common 
  chord 
  ; 
  and 
  

   the 
  first 
  chord 
  of 
  the 
  second 
  bar 
  is 
  the 
  sharp 
  sixth, 
  rendered 
  peculiarly 
  

   smooth 
  by 
  employment 
  of 
  the 
  approximate 
  harmonic 
  seventh 
  for 
  the 
  

   interval 
  /a\y-f$. 
  

  

  The 
  employment 
  of 
  positive 
  systems 
  is 
  presupposed 
  with 
  this 
  notation, 
  

   unless 
  the 
  contrary 
  is 
  expressly 
  stated, 
  

  

  