﻿1874.] 
  Mr. 
  A. 
  J. 
  Ellis 
  on 
  Musical 
  Duodenes. 
  

  

  II. 
  Semidiurnal 
  Tide. 
  

   Mean 
  Lunitidal 
  Interval 
  (observed), 
  

  

  H. 
  W. 
  L. 
  W. 
  

  

  -0 
  h 
  26 
  m 
  -7. 
  6 
  h 
  l 
  m 
  -l. 
  

  

  III. 
  " 
  On 
  Musical 
  Duodenes, 
  or 
  the 
  Theory 
  of 
  Constructing 
  

   Instruments 
  with 
  Fixed 
  Tones 
  in 
  Just 
  or 
  Practically 
  Just 
  

   Intonation." 
  By 
  Alexander 
  J. 
  Ellis, 
  F.R.S., 
  F.S.A., 
  

   F.C.P.S., 
  F.C.P. 
  Received 
  October 
  28, 
  1874. 
  

  

  This 
  paper 
  is 
  intended 
  to 
  complete 
  and 
  supplement 
  three 
  papers 
  on 
  

   Music 
  which 
  I 
  have 
  already 
  read 
  before 
  the 
  Royal 
  Society 
  1 
  . 
  It 
  contains 
  

   a 
  more 
  complete 
  theory 
  of 
  temperament, 
  embracing 
  that 
  indicated 
  by 
  

   Helmholtz 
  2 
  , 
  but 
  not 
  worked 
  out 
  by 
  him, 
  and 
  its 
  application 
  to 
  the 
  theory 
  

   of 
  constructing 
  musical 
  instruments 
  with 
  an 
  intonation 
  practically 
  just, 
  

   without 
  change 
  of 
  fingering, 
  and, 
  if 
  there 
  are 
  three 
  or 
  four 
  performers, 
  

   without 
  change 
  of 
  mechanism. 
  The 
  name 
  Duodene 
  refers 
  to 
  that 
  collec- 
  

   tion 
  of 
  twelve 
  notes, 
  suitable 
  to 
  the 
  present 
  manuals, 
  which 
  is 
  made 
  the 
  

   unit 
  of 
  construction. 
  To 
  obtain 
  its 
  precise 
  form, 
  and 
  determine 
  the 
  

   number 
  and 
  value 
  of 
  all 
  such 
  duodenes 
  as 
  it 
  is 
  necessary 
  to 
  tune, 
  I 
  have 
  

   been 
  obliged 
  to 
  indicate 
  a 
  theory 
  of 
  harmonic 
  scales 
  and 
  modulation, 
  

   which 
  I 
  believe 
  to 
  be 
  entirely 
  new, 
  and 
  which 
  has 
  of 
  course 
  other 
  uses. 
  

   The 
  great 
  extent 
  of 
  the 
  subject 
  obliges 
  me 
  to 
  confine 
  this 
  part 
  of 
  my 
  

   paper 
  to 
  a 
  mere 
  indication. 
  

  

  A. 
  Notation 
  of 
  Pitch. 
  

  

  The 
  letters 
  C, 
  D, 
  E, 
  F, 
  Gr, 
  A, 
  B 
  indicate 
  both 
  musical 
  tones 
  and 
  the 
  

   number 
  of 
  vibrations 
  made 
  by 
  the 
  prime 
  or 
  lowest 
  partial 
  tone 
  of 
  each 
  in 
  

   a 
  second 
  ; 
  so 
  that, 
  C 
  being 
  known, 
  

  

  D 
  = 
  f 
  C, 
  E=| 
  C, 
  F=f 
  C, 
  Gr=f 
  C, 
  A 
  = 
  f 
  C, 
  B 
  = 
  lf 
  C. 
  

  

  The 
  marks 
  £ 
  t> 
  f 
  i 
  1[ 
  are 
  used 
  for 
  fractional 
  multipliers, 
  having 
  the 
  

   following 
  names 
  and 
  values 
  : 
  — 
  

  

  , 
  135 
  r, 
  i. 
  h 
  128 
  

  

  sharp, 
  ft 
  = 
  m 
  ; 
  flat, 
  9 
  = 
  m 
  > 
  

  

  i 
  • 
  -i 
  «i, 
  81 
  i 
  80 
  

  

  high, 
  i 
  = 
  go; 
  low, 
  t 
  = 
  gr' 
  

  

  , 
  , 
  . 
  . 
  m 
  32805 
  , 
  , 
  , 
  . 
  . 
  32768 
  

   skhismic, 
  t 
  = 
  g2768 
  hyposkhismic, 
  = 
  g^og, 
  

  

  1 
  " 
  On 
  the 
  Conditions, 
  Extent, 
  and 
  Eealization 
  of 
  a 
  Perfect 
  Musical 
  Scale 
  on 
  Instru- 
  

   ments 
  with 
  Fixed 
  Tones," 
  read 
  Jan. 
  21, 
  1864, 
  printed 
  at 
  length 
  in 
  Proceedings, 
  

   vol. 
  xiii. 
  p. 
  93 
  ; 
  " 
  On 
  the 
  Physical 
  Constitution 
  and 
  Eelations 
  of 
  Musical 
  Chords," 
  and, 
  

   lastly, 
  " 
  On 
  the 
  Temperament 
  of 
  Musical 
  Instruments 
  with 
  Fixed 
  Tones," 
  both 
  read 
  

   on 
  June 
  16, 
  1864, 
  and 
  printed 
  at 
  length 
  in 
  the 
  Proceedings, 
  vol. 
  xiii. 
  p. 
  392 
  and 
  

   p. 
  404. 
  

  

  2 
  Tonempfindungen, 
  3rd 
  ed. 
  p. 
  495. 
  

  

  B 
  2 
  

  

  