﻿1874.] 
  Mr. 
  A. 
  J. 
  Ellis 
  on 
  Musical 
  Duodenes. 
  5 
  

  

  Another 
  object, 
  therefore, 
  is 
  to 
  make 
  that 
  dissonance 
  as 
  little 
  annoying 
  

   as 
  possible. 
  

  

  "We 
  find 
  immediately 
  by 
  actual 
  multiplication, 
  

  

  /3\ 
  12 
  /1\ 
  7 
  531441 
  n-n^i. 
  / 
  t 
  \ 
  

  

  1 
  2) 
  ' 
  [2) 
  — 
  594288 
  = 
  comma 
  or 
  Pythagoras, 
  (1) 
  

  

  (t) 
  3 
  ' 
  2 
  = 
  m 
  =tt^ 
  the 
  diesis 
  (2) 
  

  

  Multiplying 
  these 
  equations 
  together, 
  and 
  extracting 
  the 
  cube 
  root, 
  

  

  (1) 
  ' 
  \ 
  ' 
  (1) 
  = 
  8T) 
  = 
  ^' 
  ^ 
  e 
  comma 
  Didymus) 
  (3) 
  

  

  Dividing 
  (1) 
  by 
  (3), 
  

  

  (2) 
  ' 
  4 
  " 
  (2) 
  =3i76l 
  = 
  ^' 
  the 
  skhisma 
  - 
  ■ 
  ( 
  4 
  ) 
  

  

  On 
  these 
  equations 
  depend 
  all 
  uniform 
  temperaments 
  in 
  which 
  every 
  

   Fifth 
  and 
  major 
  Third 
  preserves 
  the 
  same 
  ratio 
  throughout. 
  

  

  Let 
  V, 
  T, 
  k, 
  s 
  be 
  any 
  four 
  fractions 
  having 
  relations 
  similar 
  to 
  J, 
  J, 
  

  

  t 
  and 
  % 
  respectively 
  in 
  (1) 
  and 
  (2) 
  ; 
  then 
  

  

  V 
  l2 
  +2 
  7 
  =Jcs, 
  and 
  2-^T 
  3 
  = 
  F-rs 
  (5,6) 
  

  

  Subtracting 
  the 
  logarithms 
  of 
  (1) 
  and 
  (2) 
  from 
  the 
  logarithms 
  of 
  (5) 
  

   and 
  (6) 
  respectively, 
  we 
  find 
  

   3 
  1 
  

  

  log 
  V=log 
  2_j2 
  • 
  ( 
  lo 
  & 
  t— 
  log 
  7^ 
  + 
  log 
  If 
  -log 
  s), 
  (7) 
  

  

  log 
  T=log 
  | 
  + 
  ? 
  • 
  (log 
  f-log 
  k)-l 
  (log 
  f 
  -log 
  s), 
  (8) 
  

  

  which 
  are 
  the 
  fundamental 
  equations 
  of 
  all 
  temperament, 
  and 
  are 
  identi- 
  

   ties, 
  of 
  course, 
  for 
  just 
  intonation. 
  As 
  they 
  contain 
  4 
  unknown 
  expres- 
  

   sions, 
  two 
  may 
  be 
  assumed 
  and 
  the 
  rest 
  found, 
  giving 
  rise 
  to 
  an 
  endless 
  

   variety 
  of 
  temperaments. 
  Without 
  discussing 
  these 
  generally, 
  the 
  fol- 
  

   lowing 
  cases 
  should 
  be 
  mentioned 
  : 
  — 
  

  

  Commotio 
  System 
  (for 
  which 
  Tc=l). 
  — 
  This 
  is 
  the 
  only 
  system 
  discussed 
  

   in 
  my 
  previous 
  paper, 
  where 
  50 
  cases 
  were 
  considered. 
  

  

  1. 
  Quintal 
  or 
  Pythagorean 
  Temperament. 
  Assume 
  ^ 
  = 
  1, 
  and 
  V=^ 
  

   then 
  by 
  (7), 
  

  

  log 
  f 
  4- 
  log 
  f 
  =log 
  s; 
  

  

  whence 
  by 
  (8), 
  

  

  logT=log| 
  + 
  logt. 
  

   This 
  temperament 
  proves 
  to 
  be 
  thoroughly 
  unsuitable 
  for 
  harmony. 
  

  

  2. 
  Tertian, 
  Mesotonic 
  or 
  Mean 
  Temperament. 
  Assume 
  1=1 
  and 
  

   T=|, 
  then 
  by 
  (8), 
  

  

  2 
  log 
  t 
  = 
  log 
  H— 
  log 
  s: 
  

  

  