﻿4 
  

  

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

  

  the 
  first 
  two 
  being 
  written 
  and 
  read 
  after, 
  and 
  the 
  last 
  four 
  being 
  written 
  

   and 
  read 
  before, 
  the 
  letter 
  which 
  marks 
  the 
  note. 
  Thus 
  : 
  — 
  

  

  tAb=highAflat 
  = 
  g.|.^.e=|c, 
  

  

  +GJf 
  = 
  low 
  G 
  sharp 
  = 
  |?.|. 
  C 
  = 
  ^C. 
  

  

  The 
  precise 
  pitch 
  of 
  every 
  tone 
  is 
  therefore 
  indicated 
  by 
  its 
  symbol, 
  when 
  

   C 
  is 
  known. 
  

  

  "When 
  Italic 
  letters 
  are 
  used, 
  C 
  l9 
  C, 
  c, 
  c', 
  c", 
  &c. 
  indicate 
  Octaves, 
  and 
  

   is 
  the 
  pitch 
  of 
  the 
  lowest 
  note 
  of 
  the 
  violoncello. 
  When 
  Eoman 
  

   letters 
  are 
  used, 
  no 
  such 
  relative 
  values 
  are 
  attributed 
  to 
  large 
  and 
  small 
  

   letters. 
  

  

  B. 
  Temperament. 
  

  

  All 
  intervals 
  here 
  considered 
  can 
  be 
  made 
  up 
  of 
  Fifths, 
  major 
  Thirds, 
  

   and 
  Octaves, 
  taken 
  up 
  or 
  down. 
  In 
  other 
  words, 
  the 
  ratio 
  of 
  the 
  vibra- 
  

   tion-numbers 
  of 
  any 
  two 
  tones 
  can 
  be 
  represented 
  by 
  (^j™ 
  • 
  (^j 
  • 
  2 
  P 
  , 
  

  

  where 
  m, 
  n, 
  and 
  p 
  are 
  zero, 
  or 
  some 
  positive 
  or 
  negative 
  integers. 
  In 
  

   Table 
  I. 
  the 
  ratios 
  of 
  the 
  vibrational 
  numbers 
  of 
  all 
  the 
  tones 
  in 
  the 
  same 
  

  

  horizontal 
  line 
  is 
  (^j 
  * 
  • 
  2 
  P 
  , 
  and 
  in 
  the 
  same 
  vertical 
  column 
  is 
  (^j 
  m 
  • 
  2? 
  9 
  

  

  the 
  p 
  or 
  Octave 
  being 
  left 
  indeterminate. 
  If, 
  then, 
  we 
  proceed 
  from 
  any 
  

   note 
  supposed 
  to 
  be 
  the 
  low-er, 
  first 
  horizontally 
  to 
  the 
  column 
  containing 
  

   a 
  second 
  note, 
  supposed 
  to 
  be 
  higher 
  and 
  in 
  the 
  same 
  Octave, 
  and 
  then 
  

  

  vertically 
  to 
  the 
  second 
  note 
  itself, 
  and 
  multiply 
  or 
  divide 
  by 
  ^ 
  for 
  each 
  

  

  horizontal 
  step, 
  according 
  as 
  it 
  is 
  to 
  the 
  right 
  or 
  left, 
  and 
  multiply 
  or 
  

  

  3 
  

  

  divide 
  by 
  n 
  for 
  each 
  vertical 
  step, 
  according 
  as 
  it 
  is 
  upwards 
  or 
  down- 
  

   wards, 
  and 
  finally 
  multiply 
  or 
  divide 
  the 
  result 
  by 
  2, 
  until 
  the 
  final 
  result 
  

   lies 
  between 
  1 
  and 
  2, 
  that 
  fraction 
  will 
  be 
  the 
  ratio 
  of 
  the 
  higher 
  vibra- 
  

  

  tional 
  number 
  to 
  the 
  lower. 
  Thus 
  C 
  to 
  F# 
  gives 
  C 
  to 
  E, 
  ^ 
  ; 
  E 
  to 
  B, 
  

  

  3 
  3 
  5 
  /3\ 
  2 
  45 
  

  

  2 
  ; 
  B 
  to 
  F#, 
  ^ 
  ; 
  and 
  4 
  • 
  ( 
  9 
  j 
  = 
  vg, 
  which, 
  being 
  greater 
  than 
  2, 
  on 
  being 
  

  

  45 
  

  

  divided 
  by 
  2 
  gives 
  the 
  correct 
  ratio 
  of 
  the 
  Tritone 
  C 
  to 
  E# 
  as 
  ^ 
  An 
  

  

  inspection 
  of 
  the 
  Table 
  then 
  shows 
  that 
  to 
  jGr#, 
  f 
  Ob 
  to 
  fC, 
  &c. 
  are 
  

   the 
  same 
  intervals 
  as 
  C 
  to 
  E#, 
  because 
  they 
  have 
  the 
  same 
  relative 
  posi- 
  

   tion, 
  and 
  this 
  is 
  sometimes 
  very 
  convenient. 
  

  

  But 
  as 
  2, 
  3, 
  5 
  are 
  primes, 
  no 
  one 
  tone 
  in 
  Table 
  I., 
  supposing 
  it 
  to 
  be 
  

   indefinitely 
  extended, 
  will 
  have 
  the 
  same 
  pitch 
  as 
  any 
  other 
  tone. 
  The 
  

   object 
  of 
  Temperament 
  generally 
  is 
  to 
  obviate 
  this 
  inconvenience, 
  by 
  

   slightly 
  altering 
  the 
  ratios 
  of 
  the 
  Fifth 
  or 
  major 
  Third, 
  or 
  both. 
  Eroin 
  

   doing 
  so, 
  as 
  Helmholtz 
  has 
  proved, 
  more 
  or 
  less 
  dissonance 
  must 
  result. 
  

  

  