﻿22 
  

  

  Mr. 
  A. 
  J. 
  Ellis 
  on 
  Musical 
  Duodenes. 
  [Nov. 
  19, 
  

  

  tion 
  with 
  great 
  ease, 
  as 
  shown 
  in 
  Table 
  II., 
  even 
  if 
  columns 
  1 
  and 
  2 
  in 
  

   Table 
  I. 
  were 
  omitted. 
  Hence 
  we 
  may 
  begin 
  by 
  cancelling 
  columns 
  1 
  and 
  

   2 
  of 
  Table 
  II. 
  In 
  view 
  of 
  the 
  greater 
  frequency 
  of 
  right 
  lateral 
  or 
  

   descending 
  duodenation, 
  we 
  need 
  only 
  reject 
  column 
  9 
  to 
  the 
  right 
  1 
  . 
  We 
  

   have 
  thus 
  reduced 
  the 
  just 
  tones 
  to 
  the 
  6 
  x 
  13 
  = 
  78 
  in 
  columns 
  i. 
  to 
  vi. 
  

   of 
  Table 
  I. 
  The 
  skhistic 
  identifications 
  reduce 
  these 
  further 
  to 
  the 
  

   6 
  x 
  8=48 
  tones 
  in 
  lines 
  p 
  to 
  x 
  of 
  the 
  same 
  columns, 
  together 
  with 
  the 
  

   three 
  tones 
  in 
  col. 
  I., 
  I, 
  m, 
  n, 
  and 
  the 
  two 
  in 
  col. 
  VI., 
  y, 
  z. 
  As 
  these 
  tones 
  

   may, 
  I 
  think, 
  be 
  always 
  avoided 
  by 
  properly 
  choosing 
  the 
  original 
  root, 
  

   motives 
  of 
  convenience 
  induce 
  me 
  to 
  reduce 
  the 
  number 
  of 
  skhistic 
  tones 
  

   necessary 
  to 
  the 
  48 
  included 
  by 
  the 
  dotted 
  lines 
  in 
  cols. 
  I. 
  to 
  vi. 
  of 
  

   Table 
  I. 
  

  

  In 
  my 
  former 
  paper 
  (' 
  Proceedings,' 
  vol. 
  xiii. 
  p. 
  98), 
  not 
  having 
  taken 
  a 
  

   sufficiently 
  comprehensive 
  view 
  of 
  the 
  nature 
  of 
  modulation, 
  I 
  fixed 
  the 
  

   number 
  of 
  just 
  tones 
  required 
  at 
  72 
  instead 
  of 
  117, 
  and 
  showed 
  that 
  they 
  

   would 
  reduce 
  by 
  skhismatic 
  substitution 
  (for 
  I 
  had 
  not 
  then 
  worked 
  out 
  

   the 
  theory 
  of 
  skhistic 
  temperament) 
  to 
  45 
  ; 
  and 
  on 
  examination 
  it 
  will 
  be 
  

   found 
  that 
  these 
  45 
  include 
  the 
  48 
  which 
  I 
  have 
  just 
  named, 
  with 
  the 
  

   exception 
  of 
  those 
  in 
  col. 
  i., 
  lines 
  p 
  and 
  q, 
  and 
  col. 
  vi., 
  line 
  x 
  of 
  Table 
  I. 
  

   The 
  tones 
  used 
  in 
  Mr. 
  Liston's 
  organ 
  (according 
  to 
  the 
  statement 
  I 
  was 
  

   able 
  to 
  give 
  in 
  ' 
  Proceedings,' 
  vol. 
  xiii. 
  p. 
  417, 
  note 
  §), 
  on 
  being 
  treated 
  

   skhistically, 
  include 
  44 
  of 
  these 
  48 
  tones, 
  omitting 
  the 
  4 
  tones 
  in 
  col. 
  i., 
  

   lines 
  w, 
  x, 
  col. 
  v., 
  line 
  x, 
  and 
  col. 
  vi., 
  line 
  x, 
  and 
  introduces 
  two 
  others 
  

   found 
  in 
  col. 
  2, 
  lines 
  p 
  and 
  x, 
  and 
  probably 
  only 
  due 
  to 
  his 
  system 
  of 
  

   tuning. 
  He 
  has 
  thus 
  46 
  tones 
  in 
  all. 
  Gren. 
  T. 
  Perronet 
  Thompson's 
  organ 
  

   (see 
  my 
  paper 
  in 
  ' 
  Proceedings,' 
  vol. 
  xiii. 
  p. 
  102), 
  when 
  similarly 
  reduced, 
  

   has 
  38 
  of 
  my 
  48 
  tones, 
  omitting 
  all 
  col. 
  i., 
  col. 
  n., 
  line 
  x, 
  and 
  col. 
  vi., 
  

   line 
  r 
  in 
  Table 
  I., 
  and 
  retaining 
  the 
  two 
  tones 
  of 
  col. 
  VI., 
  lines 
  y, 
  z, 
  which 
  

   I 
  do 
  not 
  find 
  necessary. 
  Mr. 
  Poole's 
  latest 
  organ 
  (Silliman's 
  Journal 
  

   for 
  1867, 
  pp. 
  1 
  to 
  45), 
  after 
  rejecting 
  his 
  39 
  natural 
  Sevenths, 
  which 
  I 
  

   expressly 
  exclude, 
  has 
  61 
  just 
  tones, 
  which 
  reduce 
  to 
  the 
  36 
  skhistic 
  

   tones 
  in 
  col. 
  n., 
  lines 
  q 
  to 
  x, 
  cols, 
  in., 
  iv., 
  and 
  v., 
  lines 
  p 
  to 
  x, 
  and 
  col. 
  vi., 
  

   lines 
  p 
  to 
  t. 
  These 
  are 
  the 
  principal 
  attempts 
  at 
  limiting 
  the 
  scale 
  actually 
  

   made 
  up 
  to 
  this 
  time 
  • 
  and 
  hence 
  I 
  conclude 
  that 
  my 
  reduction 
  to 
  48 
  

   skhistic 
  tones 
  (that 
  is, 
  practically, 
  unequally 
  just 
  tones) 
  would 
  embrace 
  

   almost 
  every 
  case, 
  though 
  it 
  is 
  conceivable 
  that 
  some 
  extraordinary 
  music 
  

   might 
  make 
  it 
  advisable 
  to 
  introduce 
  12 
  more, 
  namely, 
  col. 
  2, 
  lines 
  p 
  to 
  

   x, 
  and 
  col. 
  9, 
  lines 
  p 
  to 
  s, 
  making 
  60 
  tones 
  in 
  all, 
  for 
  adding 
  which 
  pro- 
  

   vision 
  should 
  be 
  made. 
  

  

  1 
  If 
  in 
  changes 
  of 
  key 
  in 
  the 
  movements 
  of 
  a 
  long 
  piece 
  modulation 
  took 
  place 
  first 
  

   much 
  to 
  the 
  left 
  and 
  then 
  much 
  to 
  the 
  right, 
  we 
  might 
  perhaps 
  make 
  commatic 
  changes 
  

   between 
  the 
  movements, 
  without 
  disturbing 
  the 
  connexion. 
  And 
  when 
  changes 
  are 
  

   introduced 
  by 
  successions 
  of 
  discords, 
  such 
  commatic 
  changes 
  could 
  not 
  be 
  observed 
  at 
  

   all 
  by 
  the 
  listener. 
  By 
  the 
  use 
  of 
  the 
  duodenal, 
  however, 
  they 
  will 
  be 
  rendered 
  per- 
  

   fectly 
  simple 
  to 
  the 
  performer. 
  

  

  