﻿398 
  

  

  Mr. 
  It. 
  H. 
  M. 
  Bosanquet 
  on 
  the 
  

  

  mation 
  to 
  the 
  harmonic 
  seventh. 
  In 
  fact, 
  if 
  we 
  form 
  a 
  seventh 
  by 
  14 
  

   fifths 
  clown 
  in 
  positive 
  systems 
  (fifth 
  =7+ 
  £), 
  we 
  obtain 
  a 
  note 
  with 
  

   negative 
  departure 
  (— 
  14£), 
  which 
  can 
  approximately 
  represent 
  the 
  har- 
  

   monic 
  seventh 
  : 
  c-\b\) 
  represents 
  such 
  an 
  interval. 
  

  

  Mr. 
  Ellis 
  has 
  observed 
  (Boy. 
  Soc. 
  Proc. 
  1864) 
  that 
  the 
  mean-tone 
  

   system, 
  which 
  is 
  negative, 
  affords 
  a 
  good 
  approximation 
  to 
  the 
  harmonic 
  

   seventh. 
  In 
  fact, 
  if 
  we 
  form 
  a 
  seventh 
  by 
  10 
  fifths 
  up 
  in 
  negative 
  systems 
  

   (fifth 
  = 
  7— 
  £), 
  we 
  obtain 
  a 
  note 
  with 
  negative 
  departure 
  ( 
  — 
  10£), 
  which 
  

   can 
  approximately 
  represent 
  the 
  harmonic 
  seventh. 
  

  

  Concords 
  of 
  Regular 
  and 
  Regular 
  Cyclical 
  Systems. 
  

  

  These 
  considerations 
  permit 
  us 
  to 
  calculate 
  the 
  departures 
  and 
  errors 
  

   of 
  concords 
  in 
  the 
  various 
  regular 
  and 
  regular 
  cyclical 
  systems. 
  There 
  

   is, 
  however, 
  one 
  quantity 
  which 
  may 
  be 
  also 
  conveniently 
  taken 
  into 
  

   consideration 
  in 
  all 
  cases, 
  viz. 
  the 
  departure 
  of 
  12 
  fifths 
  of 
  the 
  system. 
  

   We 
  will 
  call 
  this 
  A, 
  putting 
  A=12S. 
  

  

  "We 
  have 
  then 
  the 
  following 
  Table 
  of 
  the 
  characteristic 
  quantities 
  for 
  

   the 
  more 
  important 
  systems 
  hitherto 
  known. 
  

  

  /81\ 
  

  

  The 
  value 
  of 
  the 
  ordinary 
  comma 
  (g^J 
  is 
  -21560. 
  It 
  is 
  comparable 
  

  

  with 
  the 
  values 
  of 
  A, 
  and 
  if 
  introduced 
  in 
  its 
  place 
  in 
  the 
  Table 
  would 
  

   give 
  rise 
  to 
  a 
  regular 
  non-cyclical 
  system, 
  lying 
  between 
  the 
  system 
  of 
  53 
  

   and 
  the 
  positive 
  system 
  of 
  perfect 
  thirds, 
  the 
  condition 
  of 
  which 
  would 
  

   be 
  that 
  the 
  departure 
  of 
  12 
  fifths 
  = 
  a 
  comma. 
  

  

  Error 
  of 
  har- 
  

   monic 
  seventh, 
  

   •31174-14^ 
  

  

  Name, 
  

  

  Order, 
  

  

  A=12d, 
  

  

  Error 
  of 
  fifth, 
  

  

  or 
  n. 
  

  

  r. 
  

  

  or 
  12-. 
  

  

  n 
  

  

  £-•01955. 
  

  

  

  

  

  17 
  

  

  1 
  

  

  •70588 
  

  

  •03927 
  

  

  29 
  

  

  1 
  

  

  •41379 
  

  

  •01493 
  

  

  41 
  

  

  1 
  

  

  •29268 
  

  

  •00484 
  

  

  Perfect 
  fifths. 
  

  

  

  •23460 
  

  

  

  53 
  

  

  1 
  

  

  •22642 
  

  

  -•00068 
  

  

  Positive 
  perfect 
  

   thirds. 
  

  

  }- 
  

  

  •20529 
  

  

  -•00244 
  

  

  118 
  

  

  2 
  

  

  •20339 
  

  

  -•00260 
  

  

  65 
  

  

  1 
  

  

  •18462 
  

  

  -•00417 
  

  

  Error 
  of 
  third, 
  

   •13686-81?. 
  

  

  -•33373 
  

   -•27586 
  

   -•19512 
  

   -•01954 
  

   -•01409 
  

  

  •00127 
  

   •01378 
  

  

  -•51178 
  

   -•17101 
  

   -•02970 
  

   •03804 
  

   •04758 
  

  

  •07223 
  

  

  •07445 
  

   •09635 
  

  

  (£= 
  - 
  is 
  here 
  negative.) 
  

  

  71 
  

  

  43 
  -1 
  - 
  -29707 
  

  

  31 
  _i 
  _ 
  -38710 
  

  

  Mesotonic. 
  Nega-1 
  — 
  -41058 
  

  

  tive 
  perfect 
  thirds. 
  J 
  " 
  ' 
  

  

  50 
  _ 
  2 
  - 
  -48000 
  

  

  19 
  _i 
  _ 
  -63158 
  

  

  13686+4^. 
  -311744-10^. 
  

  

  -•04431 
  

   -•05181 
  

  

  -•05376 
  

  

  -•05955 
  

   -•07218 
  

  

  •03784 
  

   •00783 
  

  

  •02314 
  

   •05367 
  

  

  •06418 
  

   -•01084 
  

  

  -•03041 
  

  

  -•08826 
  

   -•21458 
  

  

  A 
  few 
  systems 
  of 
  the 
  higher 
  orders, 
  which 
  possess 
  some 
  interest, 
  will 
  

   be 
  given 
  separately. 
  

  

  