﻿Theory 
  of 
  the 
  Division 
  of 
  the 
  Octave. 
  

  

  403 
  

  

  Application 
  of 
  Principle 
  of 
  Symmetrical 
  Arrangement 
  to 
  a 
  " 
  Generalized 
  

   Key-board 
  " 
  for 
  Regular 
  Systems. 
  

  

  A 
  key-board 
  has 
  been 
  constructed, 
  on 
  the 
  principle 
  of 
  " 
  symmetrical 
  

   arrangement," 
  in 
  the 
  following 
  manner 
  : 
  — 
  

  

  The 
  octave 
  is 
  taken 
  = 
  6 
  inches 
  horizontally 
  (in 
  ordinary 
  key-boards 
  the 
  

   octave 
  is 
  6| 
  inches). 
  This 
  is 
  divided 
  into 
  12 
  spaces, 
  each 
  \ 
  inch 
  broad. 
  

   These 
  are 
  called 
  the 
  12 
  principal 
  divisions 
  of 
  the 
  octave. 
  A 
  horizontal 
  

   line 
  gives 
  the 
  positions 
  of 
  an 
  E. 
  T. 
  series 
  where 
  it 
  crosses 
  them 
  all. 
  

  

  The 
  keys 
  are 
  then 
  placed 
  at 
  vertical 
  and 
  horizontal 
  distances 
  from 
  the 
  

   E. 
  T. 
  line 
  corresponding 
  to 
  their 
  departures, 
  on 
  the 
  supposition 
  that 
  the 
  

   arrangement 
  is 
  positive. 
  

  

  The 
  departure 
  of 
  12 
  fifths 
  up 
  corresponds 
  to 
  a 
  horizontal 
  displacement 
  

   of 
  3 
  inches 
  from 
  the 
  player, 
  and 
  a 
  vertical 
  displacement 
  of 
  1 
  inch 
  up. 
  

  

  These 
  displacements 
  are 
  divided 
  equally 
  among 
  the 
  fifths 
  to 
  which 
  they 
  

   may 
  be 
  regarded 
  as 
  due, 
  i. 
  e. 
  the 
  displacement 
  of 
  g 
  with 
  respect 
  to 
  c 
  is 
  

   | 
  inch 
  back 
  and 
  T 
  ^ 
  inch 
  up 
  ; 
  so 
  of 
  cl 
  with 
  respect 
  to 
  g, 
  of 
  a 
  with 
  respect 
  

   to 
  d, 
  and 
  so 
  on. 
  

  

  Although 
  only 
  3 
  inches 
  of 
  each 
  key 
  are 
  thus 
  exposed 
  on 
  a 
  plan, 
  yet 
  

   the 
  keys 
  are 
  all 
  made 
  to 
  overhang 
  \ 
  inch, 
  and 
  thus 
  the 
  tangible 
  length 
  of 
  

   each 
  key 
  is 
  3| 
  inches. 
  

  

  The 
  accompanying 
  figure 
  (p. 
  404) 
  shows 
  a 
  small 
  portion 
  of 
  the 
  key- 
  

   board, 
  on 
  a 
  scale 
  of 
  half 
  the 
  real 
  size. 
  

  

  The 
  keys 
  are 
  each 
  § 
  inch 
  broad, 
  and 
  their 
  centres 
  are 
  \ 
  inch 
  apart. 
  

   There 
  is 
  thus 
  \ 
  inch 
  free 
  between 
  the 
  adjacent 
  surfaces 
  of 
  each 
  pair 
  of 
  

   keys, 
  and 
  -| 
  inch 
  altogether 
  between 
  the 
  two 
  keys 
  which 
  rise 
  on 
  each 
  side 
  

   of 
  any 
  given 
  key. 
  This 
  is 
  of 
  importance 
  ; 
  e.g., 
  in 
  the 
  chord 
  c-\e—g-c, 
  

   taken 
  with 
  the 
  right 
  hand, 
  the 
  first 
  finger 
  has 
  to 
  reach 
  \e 
  between 
  e\)-f 
  

   and 
  under 
  the 
  overhanging 
  e. 
  

  

  The 
  keys 
  in 
  the 
  five 
  principal 
  divisions 
  which 
  have 
  "accidental" 
  names 
  

   (e. 
  g. 
  c# 
  or 
  d\)) 
  are 
  black, 
  the 
  rest 
  white. 
  

  

  There 
  are 
  seven 
  keys 
  in 
  each 
  principal 
  division; 
  the 
  seven 
  c's 
  are 
  

   marked 
  from 
  \\\c 
  to 
  ///c, 
  the 
  unmarked 
  c 
  being 
  in 
  the 
  middle. 
  Thus 
  

   there 
  are 
  84 
  keys 
  in 
  each 
  octave. 
  The 
  key-board 
  controls 
  an 
  harmo- 
  

   nium 
  which 
  contains 
  the 
  system 
  of 
  53. 
  

  

  Application 
  of 
  the 
  Positive 
  System 
  of 
  Perfect 
  Thirds 
  to 
  the 
  " 
  Generalized 
  

   Key-board 
  " 
  (Helmholtz's 
  system, 
  just 
  intonation}. 
  

  

  If 
  the 
  thirds, 
  such 
  as 
  c-\e, 
  are 
  made 
  perfect, 
  and 
  the 
  fifths 
  flat 
  by 
  

   •00244, 
  a 
  quantity 
  which 
  escapes 
  the 
  ear, 
  we 
  have 
  the 
  system 
  here 
  men- 
  

   tioned. 
  Helmholtz 
  makes 
  a 
  mistake 
  in 
  describing 
  it 
  (' 
  Die 
  Lehre 
  von 
  

   den 
  Tonempfindungen,' 
  ed. 
  3, 
  p. 
  495) 
  ; 
  he 
  supposes 
  that 
  the 
  fifths 
  are 
  

   sharp 
  instead 
  of 
  flat 
  by 
  the 
  above 
  interval 
  ; 
  it 
  is 
  easy 
  to 
  see 
  from 
  the 
  

   context 
  that 
  this 
  is 
  a 
  mistake. 
  

  

  The 
  notation 
  of 
  positive 
  systems 
  is 
  applicable 
  without 
  specialization. 
  

  

  VOL. 
  XXIII. 
  2 
  I 
  

  

  