﻿396 
  

  

  Mr. 
  R. 
  H. 
  M. 
  Bosanquet 
  on 
  the 
  

  

  repeated 
  over 
  and 
  oyer 
  again. 
  To 
  obtain 
  the 
  rest 
  of 
  the 
  notes 
  we 
  should 
  

   have 
  to 
  change 
  the 
  starting-point. 
  

  

  On 
  the 
  whole, 
  we 
  may 
  regard 
  the 
  system 
  hi 
  as 
  consisting 
  of 
  Jc 
  dif- 
  

   ferent 
  systems 
  n, 
  Having 
  starting-points 
  distant 
  from 
  each 
  other 
  by 
  7 
  

  

  of 
  the 
  unit 
  of 
  the 
  system 
  n. 
  

  

  It 
  follows 
  immediately 
  that 
  the 
  system 
  Jen 
  is 
  of 
  the 
  &rth 
  order 
  ; 
  for 
  in 
  

   every 
  unit 
  of 
  the 
  system 
  n 
  there 
  are 
  Jc 
  units 
  of 
  system 
  hi 
  ; 
  and 
  so 
  in 
  r 
  

   units 
  of 
  system 
  n 
  there 
  are 
  Jcr 
  units 
  of 
  system 
  Jen. 
  

  

  Any 
  system, 
  when 
  n 
  is 
  not 
  a 
  prime, 
  can 
  be 
  regarded 
  as 
  a 
  multiple 
  

   system. 
  

  

  Thus 
  the 
  system 
  of 
  59 
  is 
  of 
  the 
  7th 
  order 
  ; 
  118 
  consequently 
  a 
  multiple 
  

   system 
  of 
  the 
  14th 
  order, 
  in 
  which 
  point 
  of 
  view 
  it 
  is 
  of 
  no 
  interest 
  ; 
  but, 
  

   casting 
  out 
  the 
  12 
  from 
  the 
  order, 
  it 
  may 
  be 
  also 
  regarded 
  as 
  an 
  inde- 
  

   pendent 
  system 
  of 
  the 
  2nd 
  order, 
  in 
  which 
  point 
  of 
  view 
  it 
  is 
  of 
  con- 
  

   siderable 
  interest. 
  

  

  Formation 
  of 
  Major 
  Thirds 
  in 
  Positive 
  and 
  Negative 
  Systems, 
  

  

  The 
  departure 
  of 
  the 
  perfect 
  third 
  is 
  — 
  *13686. 
  Hence 
  negative 
  

   systems 
  (where 
  the 
  fifth 
  is 
  7 
  — 
  8) 
  form 
  their 
  thirds 
  in 
  accordance 
  with 
  

   the 
  ordinary 
  notation 
  of 
  music. 
  For 
  if 
  we 
  take 
  4 
  negative 
  fifths 
  up, 
  we 
  

   have 
  a 
  third 
  with 
  negative 
  departure 
  (— 
  4c>) 
  which 
  can 
  approximately 
  

   represent 
  the 
  departure 
  of 
  the 
  perfect 
  third. 
  Thus 
  cjf 
  is 
  either 
  the 
  

   third 
  to 
  a, 
  or 
  four 
  fifths 
  up 
  from 
  «, 
  in 
  accordance 
  with 
  the 
  usage 
  of 
  

   musicians. 
  

  

  Positive 
  systems 
  form 
  their 
  thirds 
  by 
  8 
  fifths 
  down 
  ; 
  for 
  their 
  fifths 
  

   are 
  of 
  the 
  form 
  (7 
  + 
  £), 
  and 
  8 
  fifths 
  down 
  give 
  the 
  negative 
  departure 
  

   ( 
  — 
  8$). 
  Thus 
  the 
  third 
  of 
  a 
  should 
  be 
  d\), 
  which 
  is 
  inconsistent 
  with 
  

   musical 
  usage. 
  Hence 
  positive 
  systems 
  require 
  a 
  separate 
  notation. 
  

   Helmholtz 
  proposed 
  a 
  notation 
  for 
  this 
  purpose, 
  which, 
  however, 
  is 
  

   unsuitable 
  for 
  use 
  with 
  written 
  music. 
  The 
  following 
  notation 
  is 
  here 
  

   adopted 
  for 
  positive 
  systems 
  in 
  general; 
  it 
  is 
  not 
  intended 
  to 
  'be 
  limited 
  

   to 
  any 
  one 
  system, 
  like 
  Helmholtz's. 
  In 
  fact 
  it 
  may, 
  on 
  occasions, 
  be 
  

   used 
  even 
  for 
  negative 
  systems. 
  

  

  Notation 
  for 
  Positive 
  Regular 
  Systems. 
  

  

  The 
  notes 
  are 
  arranged 
  in 
  series, 
  each 
  containing 
  12 
  fifths, 
  from 
  /# 
  up 
  

   to 
  5. 
  These 
  may 
  be 
  called 
  duodenes, 
  adopting 
  a 
  term 
  introduced 
  by 
  Mr. 
  

   Ellis. 
  The 
  duodene 
  

  

  f%-c%-g%-d%-a%-f~c-g-d-a-e~l, 
  

  

  which 
  contains 
  the 
  standard 
  c, 
  is 
  called 
  the 
  unmarked 
  duodene. 
  ISTo 
  

   distinction 
  is 
  made 
  in 
  these 
  series 
  between 
  such 
  notes 
  as 
  c% 
  and 
  djp. 
  These 
  

   signs 
  refer 
  only 
  to 
  the 
  E. 
  T. 
  note 
  from 
  which 
  the 
  note 
  in 
  question 
  is 
  

   derived 
  ; 
  the 
  place 
  in 
  the 
  series 
  of 
  fifths 
  is 
  determined 
  by 
  the 
  notation. 
  

  

  