﻿18 
  

  

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

  

  In 
  skhistic 
  intonation, 
  the 
  modification 
  of 
  the 
  Fifth 
  leads 
  to 
  a 
  modi- 
  

   fication 
  of 
  the 
  comma 
  and 
  obliteration 
  of 
  the 
  skhisma 
  ; 
  so 
  that 
  the 
  two 
  

   first 
  tones, 
  skhistic 
  ff, 
  fa, 
  of 
  the 
  new 
  upper 
  trine, 
  tf 
  + 
  ta 
  + 
  c#, 
  are 
  one 
  

   skhistic 
  comma 
  higher, 
  and 
  the 
  third, 
  skhistic 
  cjf, 
  is 
  one 
  skhistic 
  comma 
  

   lower 
  than 
  the 
  two 
  last 
  tones, 
  F, 
  A, 
  and 
  the 
  first 
  tone 
  D|? 
  of 
  the 
  old 
  

   trine, 
  D|? 
  + 
  F 
  + 
  A. 
  And 
  the 
  tones 
  of 
  the 
  new 
  right 
  quaternion 
  will 
  be 
  

   in 
  the 
  same 
  order, 
  exactly 
  tivo 
  skhistic 
  commas 
  flatter 
  than 
  the 
  old 
  left- 
  

   hand 
  quaternion 
  1 
  . 
  

  

  The 
  consequence 
  is 
  that 
  if 
  we 
  took 
  4 
  independent 
  duodenes 
  (that 
  is, 
  

   such 
  that 
  no 
  tone 
  of 
  one 
  is 
  common 
  to 
  any 
  tone 
  of 
  the 
  other) 
  as 
  the 
  duo- 
  

   denes 
  of 
  f 
  B|?, 
  A#, 
  G\} 
  } 
  and 
  +F#, 
  the 
  tones 
  of 
  which 
  are 
  contained 
  within 
  

   the 
  dotted 
  lines 
  and 
  right 
  side 
  of 
  the 
  inner 
  oblong 
  of 
  Table 
  L, 
  the 
  tones 
  

   of 
  the 
  duodenes 
  A# 
  and 
  t~F$ 
  will 
  be 
  two 
  commas 
  flatter 
  than 
  those 
  of 
  

   f~B\) 
  and 
  Gr[> 
  ; 
  and 
  the 
  tones 
  of 
  the 
  two 
  first 
  quaternions 
  of 
  the 
  f 
  B\> 
  and 
  

   A# 
  duodenes 
  will 
  be 
  one 
  comma 
  sharper 
  than 
  those 
  of 
  the 
  two 
  last 
  

   quaternions 
  of 
  Gj? 
  and 
  JFJ, 
  while 
  the 
  tones 
  of 
  the 
  third 
  quaternions 
  of 
  

   fB[> 
  and 
  Ajf 
  will 
  be 
  one 
  comma 
  flatter 
  than 
  those 
  of 
  the 
  duodenes 
  of 
  Gf> 
  

   and 
  +F# 
  respectively. 
  

  

  The 
  result, 
  then, 
  is 
  that 
  the 
  48 
  tones 
  will 
  consist 
  of 
  four 
  corresponding 
  

   sets 
  of 
  12 
  tones 
  each 
  appearing 
  in 
  4 
  forms, 
  differing 
  in 
  pitch 
  by 
  one 
  

   skhistic 
  comma. 
  This 
  will 
  appear 
  more 
  clearly 
  by 
  the 
  following 
  Table, 
  

   in 
  which 
  the 
  value 
  in 
  octs 
  of 
  the 
  cycle 
  of 
  301 
  is 
  given 
  for 
  73 
  tones, 
  being 
  

   those 
  in 
  cols. 
  I. 
  to 
  vi. 
  of 
  Table 
  I., 
  less 
  those 
  in 
  col. 
  i., 
  lines 
  I, 
  m, 
  n, 
  and 
  

   col. 
  vi., 
  lines 
  y, 
  z. 
  The 
  48 
  of 
  those 
  tones 
  contained 
  in 
  4 
  independent 
  duo- 
  

   denes 
  are 
  in 
  Roman 
  capitals, 
  the 
  other 
  of 
  the 
  73 
  tones, 
  which 
  are 
  some 
  

   of 
  their 
  skhistic 
  synonyms, 
  are 
  in 
  Eoman 
  small 
  letters, 
  and 
  other 
  syno- 
  

   nyms 
  are 
  added 
  in 
  Italics 
  ; 
  the 
  whole 
  are 
  divided 
  into 
  groups 
  of 
  4, 
  the 
  

   constituents 
  of 
  which 
  differ 
  from 
  one 
  another 
  by 
  5 
  octs, 
  or 
  one 
  skhistic 
  

   comma. 
  

  

  1 
  This 
  is 
  readily 
  seen 
  by 
  expressing 
  the 
  tones 
  in 
  terms 
  of 
  the 
  octs 
  of 
  the 
  cycle 
  of 
  301 
  , 
  

   by 
  continually 
  adding 
  and 
  subtracting 
  176 
  for 
  the 
  Fifths 
  and 
  97 
  for 
  the 
  major 
  Thirds, 
  

   adding 
  or 
  subtracting 
  301 
  as 
  often 
  as 
  is 
  necessary 
  to 
  reduce 
  to 
  the 
  same 
  Octave. 
  A 
  

   skhistic 
  comma 
  is 
  represented 
  by 
  5 
  octs. 
  This 
  gives 
  

  

  Tones. 
  Octs. 
  

  

  tdb 
  

  

  tf 
  

  

  ta 
  

  

  c# 
  

  

  e# 
  

  

  33 
  

  

  130 
  

  

  227 
  

  

  23 
  

  

  120 
  

  

  tgb 
  

  

  tBb 
  

  

  D 
  

  

  Ffl 
  

  

  a# 
  

  

  158 
  

  

  255 
  

  

  51 
  

  

  148 
  

  

  245 
  

  

  tcb 
  

  

  tEb 
  

  

  Gt 
  

  

  B 
  

  

  Jd* 
  

  

  283 
  

  

  79 
  

  

  176 
  

  

  273 
  

  

  69 
  

  

  tfb 
  

  

  tAb 
  

  

  G 
  

  

  E 
  

  

  tgfl 
  

  

  107 
  

  

  204 
  

  

  

  

  97 
  

  

  194 
  

  

  tbbb 
  

  

  Db 
  

  

  F 
  

  

  A 
  

  

  

  232 
  

  

  28 
  

  

  125 
  

  

  222 
  

  

  18 
  

  

  tebb 
  

  

  gb 
  

  

  bb 
  

  

  

  itt 
  

  

  56 
  

  

  153 
  

  

  250 
  

  

  46 
  

  

  143 
  

  

  