﻿1874.] 
  

  

  Mr. 
  A. 
  J. 
  Ellis 
  on 
  Musical 
  Duodenes. 
  

  

  21 
  

  

  harmonies 
  by 
  a 
  tone 
  of 
  not 
  more 
  than 
  two 
  commas 
  different, 
  which 
  must 
  

   be 
  considered 
  as 
  their 
  proper 
  representative 
  in 
  just 
  intonation. 
  

  

  E. 
  Number 
  of 
  Tones 
  required. 
  

  

  The 
  next 
  important 
  point 
  is 
  to 
  determine 
  how 
  many 
  duodenes 
  must 
  be 
  

   provided. 
  In 
  Table 
  I. 
  the 
  large 
  inner 
  oblong 
  contains 
  all 
  the 
  duodenes 
  

   which 
  have 
  at 
  least 
  one 
  tone 
  in 
  common 
  with 
  the 
  original 
  duodene 
  of 
  C. 
  

   Thus 
  the 
  duodenes 
  tD|? 
  and 
  $B 
  have 
  respectively 
  the 
  tones 
  f 
  B|? 
  and 
  A 
  

   in 
  common 
  with, 
  the 
  original 
  duodene, 
  and 
  no 
  others. 
  If 
  we 
  proceeded 
  

   further, 
  as 
  vertically 
  from 
  the 
  f 
  D[? 
  to 
  the 
  ttA[> 
  duodene, 
  we 
  should 
  no 
  

   longer 
  have 
  any 
  connexion 
  with 
  the 
  decad 
  of 
  the 
  original 
  tonic 
  C. 
  The 
  

   confusions 
  of 
  modern 
  equal 
  temperament 
  might 
  lead 
  much 
  further 
  ; 
  but 
  

   in 
  that 
  case 
  we 
  must 
  restore 
  the 
  commatic 
  changes 
  which 
  equal 
  tempera- 
  

   ment 
  ignored, 
  considering, 
  for 
  example, 
  that 
  when 
  the 
  composer 
  modu- 
  

   lated 
  into 
  tempered 
  A\> 
  from 
  tempered 
  D\) 
  he 
  really 
  meant 
  to 
  make 
  a 
  

   modulation 
  from 
  just 
  D|? 
  into 
  just 
  tAf>, 
  and 
  not 
  from 
  f 
  D[> 
  to 
  just 
  tfA[>. 
  

   It 
  would 
  probably 
  have 
  never 
  been 
  the 
  composer's 
  intention 
  to 
  proceed 
  to 
  

   such 
  unrelated 
  duodenes 
  as 
  these 
  two 
  last. 
  

  

  The 
  limits 
  of 
  the 
  original 
  roots 
  of 
  duodenes 
  may 
  be 
  taken 
  to 
  be 
  the 
  tones 
  

   of 
  the 
  duodene 
  of 
  C. 
  Practically, 
  composers 
  had 
  no 
  others 
  in 
  their 
  minds. 
  

   Any 
  smaller 
  changes 
  of 
  pitch 
  were 
  relegated 
  to 
  differences 
  in 
  the 
  pitch 
  

   of 
  C, 
  whence 
  all 
  the 
  others 
  were 
  derived. 
  If, 
  then, 
  we 
  construct 
  the 
  

   limiting 
  duodenes 
  to 
  the 
  extreme 
  tones 
  of 
  the 
  duodene 
  of 
  C 
  as 
  original 
  

   roots, 
  we 
  shall 
  obtain 
  all 
  the 
  tones 
  in 
  Table 
  I., 
  being 
  9x13=117 
  in 
  

   number. 
  This 
  is 
  the 
  number 
  of 
  tones 
  required, 
  therefore, 
  in 
  just 
  

   intonation. 
  

  

  Skhistic 
  intonation 
  would 
  introduce 
  identifications 
  which 
  would 
  reduce 
  

   this 
  number 
  to 
  the 
  9 
  x 
  8 
  = 
  72 
  tones 
  in 
  the 
  lines 
  _p 
  to 
  oo 
  in 
  Table 
  I., 
  toge- 
  

   ther 
  with 
  three 
  in 
  col. 
  1, 
  lines 
  I, 
  m, 
  n, 
  and 
  two 
  in 
  col. 
  9, 
  lines 
  y 
  and 
  z, 
  that 
  

   is 
  to 
  77 
  skhistic 
  tones 
  in 
  all. 
  The 
  last 
  5 
  are 
  so 
  extremely 
  unlikely 
  to 
  

   occur, 
  however, 
  that 
  we 
  may 
  consider 
  these 
  72 
  skhistic 
  tones 
  as 
  suffi- 
  

   ciently 
  representing 
  the 
  whole 
  117 
  of 
  the 
  Table. 
  These 
  72 
  tones 
  form 
  

   6 
  independent 
  duodenes, 
  those 
  of 
  f 
  tE[?t>, 
  D 
  and 
  iC##, 
  and 
  of 
  f 
  Qp\>, 
  B\) 
  

   and 
  +A#. 
  It 
  will 
  be 
  shown 
  that 
  there 
  is 
  really 
  no 
  difficulty 
  in 
  playing 
  

   them 
  all, 
  with 
  existing 
  means, 
  if 
  required; 
  but 
  they 
  would 
  not 
  be 
  

   required. 
  The 
  tendency 
  of 
  musicians 
  is 
  not 
  to 
  modulate 
  to 
  both 
  right 
  

   and 
  left 
  equally 
  in 
  the 
  same 
  piece. 
  It 
  has 
  been 
  already 
  noted 
  that 
  on 
  

   account 
  of 
  the 
  prevalence 
  of 
  major 
  scales 
  duodenation 
  is 
  generally 
  to 
  the 
  

   right. 
  The 
  fingering 
  for 
  the 
  duodenes 
  of 
  tB|? 
  and 
  AJf 
  would 
  be 
  the 
  same 
  

   on 
  manuals 
  constructed 
  on 
  the 
  duodenary 
  theory, 
  although 
  the 
  tones 
  in 
  

   skhistic 
  intonation 
  would 
  differ 
  by 
  two 
  skhistic 
  commas. 
  If, 
  then, 
  a 
  piece 
  

   in 
  fB[> 
  duodenated 
  much 
  to 
  the 
  left, 
  that 
  is 
  (for 
  skhistic 
  intonation), 
  

   ascended 
  vertically, 
  we 
  could 
  play 
  it 
  as 
  AJf. 
  It 
  would 
  simply 
  be 
  neces- 
  

   sary 
  to 
  write 
  +Ep 
  as 
  its 
  duodenal, 
  as 
  that 
  is 
  shown 
  to 
  be 
  identical 
  with 
  

   A# 
  in 
  the 
  last 
  Table. 
  We 
  should 
  then 
  be 
  able 
  to 
  use 
  ascending 
  duodena- 
  

  

  