﻿1874.] 
  

  

  Mr. 
  A. 
  J. 
  Ellis 
  on 
  Musical 
  Duodenes. 
  

  

  17 
  

  

  1 
  3t 
  

  

  D 
  to 
  tE|?, 
  Gr 
  to 
  tA|?, 
  C 
  to 
  dt>) 
  ; 
  the 
  Sharp, 
  ~ 
  (E 
  to 
  £#, 
  d[> 
  to 
  D) 
  ; 
  and 
  

  

  the 
  low 
  Sharp, 
  || 
  (tBp 
  to 
  B, 
  fE|? 
  to 
  E, 
  fA\) 
  to 
  A). 
  But 
  if 
  we 
  take 
  the 
  

  

  trine 
  above, 
  tf 
  + 
  fa 
  + 
  cjf, 
  we 
  have 
  two 
  intervals 
  of 
  a 
  comma, 
  t, 
  (E 
  to 
  tf 
  , 
  

   A 
  to 
  ta), 
  and 
  one 
  of 
  a 
  diaskhisma, 
  tjjp, 
  (c# 
  to 
  d[>). 
  If 
  we 
  take 
  the 
  trine 
  

   below, 
  g|? 
  + 
  b[? 
  + 
  +d, 
  we 
  have 
  the 
  same 
  intervals 
  of 
  a 
  comma 
  (b\) 
  to 
  tB|?, 
  

   Jd 
  to 
  D), 
  and 
  a 
  diaskhisma 
  (f£ 
  to 
  g\)). 
  If 
  we 
  take 
  the 
  quaternion 
  to 
  the 
  

   right, 
  as 
  ic# 
  x 
  +g# 
  x 
  +d£ 
  x 
  ajf, 
  we 
  have 
  three 
  intervals 
  of 
  a 
  diesis, 
  ttj^, 
  

   (a# 
  to 
  f 
  B[>, 
  JdJ 
  to 
  tE|?, 
  tg# 
  to 
  f 
  A\}, 
  and 
  tcjf 
  to 
  d|?) 
  ; 
  and 
  similarly 
  if 
  

   we 
  proceed 
  to 
  the 
  left. 
  Hence 
  the 
  intervals 
  introduced 
  by 
  adjacent 
  

   trines 
  and 
  quaternions 
  are 
  all 
  less 
  than 
  two 
  commas. 
  In 
  equal 
  tempera- 
  

   ment 
  no 
  new 
  intervals 
  would 
  be 
  thus 
  introduced 
  ; 
  for 
  all 
  the 
  Eifths 
  are 
  

   there 
  so 
  altered 
  that 
  the 
  new 
  upper 
  trine, 
  tempered 
  tf 
  + 
  ta 
  + 
  c$, 
  would 
  

   become 
  identical 
  with 
  the 
  original 
  bottom 
  trine, 
  tempered 
  d[}-}-E 
  + 
  A, 
  

   except 
  in 
  order 
  of 
  terms 
  ; 
  and 
  the 
  new 
  quaternion 
  to 
  the 
  right, 
  tempered 
  

   +cjf 
  x 
  Zg% 
  X 
  +djf 
  x 
  ajf, 
  would 
  be 
  identical 
  both 
  in 
  value 
  and 
  order 
  of 
  terms 
  

   with 
  the 
  old 
  quaternion 
  to 
  the 
  left, 
  tempered 
  d\} 
  x 
  t 
  A\> 
  x 
  f 
  E|? 
  x 
  f 
  B|>. 
  The 
  

   consequence 
  is 
  that 
  only 
  one 
  duodene 
  exists 
  for 
  equal 
  temperament, 
  and 
  

   the 
  real 
  nature 
  of 
  modulation 
  is 
  thoroughly 
  disguised. 
  In 
  tertian 
  tem- 
  

   perament 
  this 
  would 
  not 
  be 
  the 
  case 
  ; 
  the 
  quaternions 
  would 
  be 
  distin- 
  

   guished, 
  but 
  the 
  trines 
  would 
  partly 
  coincide, 
  and 
  hence 
  some, 
  but 
  not 
  all, 
  

   of 
  the 
  meaning 
  of 
  modulation 
  would 
  be 
  lost 
  1 
  . 
  

  

  1 
  If 
  in 
  Table 
  I. 
  the 
  signs 
  t 
  \ 
  be 
  omitted, 
  and 
  the 
  letters 
  and 
  the 
  signs 
  % 
  b 
  be 
  taken 
  

   to 
  have 
  their 
  values 
  in 
  Tertian 
  or 
  any 
  uniform 
  commatic 
  temperament 
  (except 
  the 
  

   Equal, 
  which 
  is 
  also 
  skhismatic), 
  the 
  Table 
  will 
  represent 
  the 
  corresponding 
  duodenes. 
  

   But 
  if 
  the 
  letters 
  and 
  signs 
  % 
  b 
  are 
  taken 
  to 
  have 
  their 
  value 
  in 
  the 
  Equal 
  tempera- 
  

   ment, 
  so 
  that 
  

  

  C 
  D 
  E 
  F 
  G 
  A 
  B 
  

   =Dbb 
  Ebb 
  Fb 
  Gbb 
  Abb 
  Bbb 
  Cb 
  

   and 
  =B# 
  Ctfjf 
  Dfltf 
  E* 
  Ftftf 
  G-Jptf 
  AJf* 
  

  

  and 
  

  

  C# 
  DS 
  E« 
  Ftf 
  G* 
  Atf 
  BJt 
  

  

  =Db 
  Eb 
  F 
  Gb 
  Ab 
  Bb 
  C 
  

  

  (showing 
  the 
  utterly 
  absurd 
  relations 
  between 
  symbolization 
  and 
  signification), 
  then 
  the 
  

   same 
  Table 
  will 
  reduce 
  to 
  the 
  one 
  central 
  duodene 
  with 
  its 
  tones 
  differently 
  distributed. 
  

   This 
  will 
  be 
  still 
  better 
  shown 
  by 
  using 
  

  

  C 
  cd 
  D 
  de 
  E 
  F 
  fg 
  G 
  ga 
  A 
  ab 
  B 
  

  

  for 
  the 
  12 
  digitals 
  on 
  a 
  piano, 
  so 
  that 
  the 
  central 
  duodene 
  and 
  its 
  adjacent 
  trines 
  and 
  

   quaternions 
  reduce 
  to 
  

  

  cd 
  

  

  F 
  

  

  A 
  

  

  cd 
  

  

  F 
  

  

  fg 
  

  

  ab 
  

  

  D 
  

  

  fg 
  

  

  ab 
  

  

  B 
  

  

  de 
  

  

  G 
  

  

  B 
  

  

  de 
  

  

  E 
  

  

  g 
  a 
  

  

  C 
  

  

  E 
  

  

  g 
  a 
  

  

  A 
  

  

  cd 
  

  

  F 
  

  

  A 
  

  

  cd 
  

  

  D 
  

  

  fg 
  

  

  ab 
  

  

  D 
  

  

  fg 
  

  

  VOL. 
  XXIII. 
  

  

  