﻿Theory 
  of 
  the 
  Division 
  of 
  the 
  Octave. 
  

  

  397 
  

  

  Continuing 
  the 
  series 
  to 
  the 
  right, 
  each 
  note 
  of 
  the 
  next 
  12 
  fifths 
  is 
  

   affected 
  with 
  the 
  mark 
  / 
  (mark 
  of 
  elevation), 
  drawn 
  upwards 
  in 
  the 
  

   direction 
  of 
  writing. 
  These 
  notes 
  join 
  on 
  to 
  the 
  . 
  unmarked 
  duodene 
  as 
  

   follows 
  

  

  and 
  so 
  on. 
  

  

  Thus 
  /c 
  is 
  12 
  fifths 
  to 
  the 
  right 
  of 
  c, 
  and 
  the 
  interval 
  /c-cis 
  the 
  depar- 
  

   ture 
  of 
  12 
  fifths. 
  

  

  The 
  next 
  duodene 
  to 
  the 
  right 
  is 
  affected 
  with 
  the 
  mark 
  //, 
  which 
  

   joins 
  on 
  to 
  the 
  last 
  as 
  before 
  : 
  — 
  

  

  /e-/h-//f$ 
  

  

  and 
  so 
  on. 
  

  

  Proceeding 
  in 
  the 
  same 
  way, 
  we 
  have 
  notes 
  affected 
  with 
  such 
  marks 
  

   as 
  ///, 
  ////. 
  

  

  Return 
  to 
  the 
  unmarked 
  duodene, 
  and 
  let 
  it 
  be 
  continued 
  to 
  the 
  left 
  ; 
  

   the 
  notes 
  in 
  the 
  next 
  duodene 
  on 
  the 
  left 
  are 
  affected 
  with 
  the 
  mark 
  \, 
  

   (mark 
  of 
  depression), 
  drawn 
  downwards 
  in 
  the 
  direction 
  of 
  writing. 
  The 
  

   junction 
  with 
  the 
  unmarked 
  duodene 
  will 
  be 
  

  

  \c~\g-\d-\Or-\e-\b-f#-c# 
  

  

  The 
  next 
  junction 
  on 
  the 
  left 
  will 
  be 
  

  

  ; 
  

  

  and, 
  proceeding 
  in 
  the 
  same 
  way, 
  we 
  have 
  such 
  marks 
  as 
  \\\, 
  

  

  Thus 
  c-\e 
  is 
  a 
  major 
  third 
  determined 
  by 
  eight 
  fifths 
  down 
  in 
  the 
  whole 
  

   series; 
  and 
  \e 
  will 
  have 
  the 
  departure 
  (—8$) 
  from 
  the 
  E. 
  T. 
  note 
  e 
  

   derived 
  from 
  c. 
  

  

  Notation 
  applicable 
  to 
  all 
  Regular 
  Systems, 
  Negative 
  as 
  well 
  as 
  

   Positive. 
  

  

  As 
  this 
  notation 
  simply 
  consists 
  of 
  a 
  determination 
  of 
  position 
  in 
  a 
  

   continuous 
  series 
  of 
  fifths, 
  it 
  may 
  be 
  applied 
  to 
  all 
  regular 
  systems, 
  posi- 
  

   tive 
  or 
  negative 
  ; 
  but, 
  as 
  it 
  is 
  not 
  commonly 
  needed 
  for 
  negative 
  systems, 
  

   it 
  is 
  not 
  generally 
  applied 
  to 
  them. 
  

  

  Formation 
  of 
  Harmonic 
  Sevenths 
  in 
  Positive 
  and 
  Negative 
  Systems. 
  

  

  The 
  harmonic 
  seventh 
  is 
  the 
  interval 
  whose 
  ratio 
  is 
  7:4. 
  It 
  affords 
  a 
  

   smooth 
  combination, 
  free 
  from 
  beats. 
  

  

  The 
  departure 
  of 
  the 
  harmonic 
  seventh 
  from 
  the 
  note 
  which 
  gives 
  the 
  

   E. 
  T. 
  minor 
  seventh 
  is 
  — 
  -31174 
  (Eule 
  I.). 
  

  

  Helmholtz 
  observes 
  that 
  his 
  system 
  of 
  just 
  intonation 
  affords 
  an 
  approxi- 
  

  

  