﻿16 
  

  

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

  

  this 
  paper, 
  to 
  explain 
  them. 
  A 
  means 
  of 
  putting 
  the 
  strange-looking 
  

   chords 
  1 
  , 
  £+a 
  ... 
  b 
  + 
  £d#, 
  and 
  f+a— 
  c 
  with 
  £d#, 
  or 
  f 
  + 
  a 
  with 
  £d#, 
  which 
  

   they 
  contain, 
  under 
  the 
  hands 
  of 
  the 
  performer 
  on 
  justly 
  intoned 
  instru- 
  

   ments, 
  is 
  absolutely 
  necessary. 
  

  

  The 
  Harmonic 
  Duodene 
  or 
  Element 
  of 
  Modulation, 
  as 
  distinguished 
  from 
  

   the 
  heptadecad 
  or 
  unit 
  of 
  modulation, 
  contains 
  the 
  12 
  tones 
  inclosed 
  

   within 
  an 
  oblong 
  in 
  the 
  figure 
  of 
  the 
  heptadecad. 
  It 
  is 
  seen 
  to 
  contain 
  

   a 
  complete 
  decad 
  of 
  C 
  and 
  two 
  additional 
  tones, 
  f 
  J 
  and 
  d[>, 
  which 
  I 
  term 
  

   mutators, 
  as 
  each 
  of 
  them 
  is 
  part 
  of 
  two 
  cells, 
  and 
  hence 
  lead 
  the 
  old 
  

   decad 
  to 
  change 
  into 
  the 
  new 
  decads 
  containing 
  them. 
  Thus 
  f# 
  is 
  

   part 
  of 
  

  

  the 
  vertical 
  cell 
  f 
  f 
  ta 
  and 
  of 
  the 
  lateral 
  cell 
  D 
  f# 
  

  

  Dfjf 
  B 
  £d#, 
  

  

  and 
  hence 
  leads 
  both 
  to 
  the 
  dominant 
  decad 
  and 
  to 
  the 
  right 
  correlative 
  

   decad. 
  Again, 
  d\) 
  is 
  a 
  part 
  of 
  

  

  the 
  vertical 
  cell 
  df? 
  F 
  and 
  of 
  the 
  lateral 
  cell 
  tf|> 
  f 
  A[> 
  

  

  bb 
  id 
  dt> 
  F, 
  

  

  and 
  hence 
  leads 
  to 
  the 
  subdomiuant 
  decad 
  and 
  to 
  the 
  left 
  correlative 
  

   decad. 
  But 
  these 
  mutators, 
  fjf 
  and 
  d\), 
  also 
  complete 
  two 
  scales 
  left 
  

   incomplete 
  in 
  the 
  decad 
  because 
  they 
  required 
  vertical 
  modulation 
  (or 
  

   decadation), 
  namely, 
  

  

  fA\) 
  majpdma. 
  ... 
  dp 
  + 
  f 
  — 
  tajj+c— 
  felp+g— 
  fb\}, 
  and 
  

   E 
  miphni 
  .... 
  a 
  — 
  c+ 
  e 
  — 
  g+ 
  b 
  — 
  d+ 
  f#, 
  

  

  and 
  also 
  complete 
  the 
  peculiar 
  chords, 
  d[? 
  + 
  F 
  . 
  .. 
  Gr 
  + 
  B, 
  tA|?-fC 
  . 
  .. 
  

   D+f#, 
  Dt>+E-tAt> 
  with 
  B, 
  tAb 
  + 
  C-tEb 
  with 
  f#, 
  which 
  occur 
  in 
  the 
  

   tetrachordals 
  of 
  minor 
  scales 
  already 
  mentioned. 
  

  

  A 
  duodene, 
  then, 
  consists 
  of 
  12 
  tones, 
  forming 
  four 
  trines 
  of 
  major 
  

   Thirds 
  arranged 
  in 
  three 
  quaternions 
  of 
  Fifths. 
  Hence 
  the 
  duodene 
  con- 
  

   structed 
  on 
  the 
  second 
  tone 
  C 
  of 
  any 
  trine, 
  fA^ 
  + 
  C 
  + 
  E, 
  contains 
  the 
  

   mapama 
  or 
  major 
  of 
  the 
  first 
  tone 
  tAJ?, 
  the 
  complete 
  decad 
  of 
  the 
  second 
  

   tone 
  C, 
  and 
  the 
  mvpimi 
  or 
  common 
  descending 
  minor 
  of 
  the 
  third 
  tone 
  E. 
  

   It 
  has 
  therefore 
  three 
  tonics, 
  tA|?, 
  C, 
  and 
  E 
  ; 
  but 
  the 
  tonic 
  of 
  the 
  decad 
  

   being 
  most 
  characteristic, 
  this 
  is 
  called 
  the 
  root 
  of 
  the 
  duodene, 
  and 
  the 
  

   duodene 
  is 
  named 
  after 
  it. 
  

  

  Any 
  duodene 
  is 
  clearly 
  and 
  sharply 
  separated 
  from 
  its 
  adjacent 
  trines 
  

   and 
  quaternions, 
  as 
  shown 
  in 
  Table 
  I., 
  where 
  the 
  small 
  innermost 
  oblong 
  

   marks 
  off 
  the 
  duodene 
  of 
  0. 
  For 
  in 
  the 
  duodene 
  the 
  smallest 
  intervals 
  

  

  between 
  two 
  adjacent 
  tones 
  are 
  the 
  Semitone, 
  j|, 
  (f 
  # 
  to 
  Gr, 
  B 
  to 
  0, 
  E 
  to 
  F, 
  

  

  1 
  The 
  justification 
  of 
  these 
  chords 
  is 
  that 
  the 
  interval 
  /to 
  \d'% 
  = 
  2 
  . 
  J£ 
  . 
  § 
  . 
  

   it 
  = 
  i 
  • 
  Mi' 
  an( 
  ^ 
  ^ 
  s 
  hence 
  very 
  nearly 
  the 
  interval 
  of 
  the 
  natural 
  Seventh 
  = 
  

  

  