﻿6 
  Mr. 
  A. 
  J. 
  Ellis 
  on 
  Musical 
  Duodenes. 
  [Nov. 
  19, 
  

  

  whence 
  by 
  (7), 
  

  

  log 
  V=log 
  ?-! 
  log 
  t. 
  

  

  In 
  my 
  previous 
  paper 
  I 
  have 
  shown 
  this 
  to 
  be 
  the 
  best 
  suited 
  for 
  

   harmony 
  of 
  all 
  in 
  this 
  system 
  ; 
  but 
  it 
  requires 
  27 
  tones 
  to 
  the 
  Octave. 
  

  

  3. 
  Hemitonia 
  or 
  Equal 
  Temperament. 
  Assume 
  Jc 
  = 
  l, 
  and 
  also 
  s 
  = 
  l, 
  

   then 
  by 
  (7), 
  

  

  logV=log 
  !_l 
  .(logt4-log 
  f), 
  

  

  by 
  (8), 
  

  

  log 
  T 
  = 
  log 
  1 
  . 
  (2 
  log 
  t-log 
  

  

  Since 
  log 
  f 
  =0-005 
  3950 
  319 
  (whence 
  ^ 
  log 
  t= 
  0-000 
  4904 
  574) 
  and 
  

   log 
  1[ 
  = 
  0-000 
  4901 
  071, 
  we 
  may, 
  for 
  all 
  acoustical 
  purposes, 
  assume 
  

   log 
  1F=tx 
  ^' 
  anc 
  ^ 
  nence 
  ^ 
  na 
  ^ 
  

  

  log 
  V=log 
  • 
  log 
  t=log 
  |-log 
  % 
  

  

  log 
  T=log 
  I 
  + 
  ^ 
  • 
  log 
  t=log 
  | 
  + 
  7 
  log 
  f. 
  

  

  This 
  is 
  the 
  only 
  uniform 
  temperament 
  which 
  requires 
  no 
  more 
  than 
  

   12 
  tones, 
  and 
  hence, 
  although 
  very 
  ill 
  suited 
  to 
  harmony, 
  it 
  has 
  been 
  

   theoretically 
  adopted 
  by 
  almost 
  all 
  musicians. 
  

  

  Skhismatic 
  System 
  (for 
  which 
  s 
  = 
  l). 
  

  

  4. 
  SJchismic 
  or 
  Arabic, 
  according 
  to 
  Helmholtz's 
  indication 
  (op. 
  cit. 
  

   p. 
  441). 
  Assume 
  s=l 
  and 
  V=o5 
  then 
  by 
  (7), 
  

  

  log 
  fc=log 
  t+log 
  f, 
  

  

  and 
  by 
  (8), 
  

  

  log 
  T=log 
  |-log 
  If. 
  

  

  The 
  Arabs 
  tune 
  17 
  tones 
  to 
  form 
  16 
  perfect 
  Eifths, 
  and 
  thus 
  obtain 
  

   8 
  major 
  Thirds 
  which 
  are 
  only 
  one 
  skhisma 
  too 
  flat. 
  But 
  they 
  do 
  not 
  

   ise 
  harmony 
  in 
  our 
  sense 
  of 
  the 
  word, 
  although 
  this 
  would 
  give 
  almost 
  

   3'ist 
  intonation 
  in 
  certain 
  keys. 
  

  

  5. 
  SMiistic, 
  or 
  Helmholtzian, 
  as 
  it 
  may 
  be 
  called, 
  because 
  suggested 
  by 
  

  

  5 
  

  

  Helmholtz 
  (op. 
  cit. 
  p. 
  495). 
  Assume 
  s 
  = 
  l 
  and 
  T 
  = 
  ^ 
  ; 
  then 
  by 
  (8), 
  

   2 
  log 
  h 
  =2 
  log 
  t-log 
  IF, 
  

  

  and 
  by 
  (7), 
  

  

  logV= 
  log|-glog1F. 
  

  

  The 
  name 
  slchistie 
  is 
  derived 
  from 
  sJchist, 
  or 
  by 
  abbreviation 
  cr, 
  which 
  is 
  

  

  