﻿Theory 
  of 
  the 
  Division 
  of 
  the 
  Octave. 
  

  

  391 
  

  

  commonly 
  employed 
  by 
  others. 
  Great 
  advantages 
  have 
  been 
  found, 
  

   however, 
  to 
  result 
  from 
  the 
  adoption 
  of 
  the 
  equal 
  temperament 
  (E. 
  T.) 
  

   semitone, 
  which 
  is 
  j 
  1 
  ^- 
  of 
  an 
  octave, 
  as 
  the 
  unit 
  of 
  interval. 
  It 
  is 
  the 
  unit 
  

   most 
  familiar 
  to 
  musicians, 
  and 
  has 
  been 
  found 
  to 
  admit 
  of 
  the 
  expres- 
  

   sion 
  of 
  the 
  theory 
  of 
  cyclical 
  systems 
  by 
  means 
  of 
  formula) 
  of 
  the 
  simplest 
  

   character. 
  The 
  writer 
  therefore 
  devised 
  the 
  following 
  rules 
  for 
  the 
  

   transformation 
  of 
  ratios 
  into 
  E. 
  T. 
  semitones 
  and 
  vice 
  versa, 
  and 
  subse- 
  

   quently 
  found 
  that 
  De 
  Morgan 
  had 
  given 
  rules 
  for 
  the 
  same 
  purpose 
  

   which 
  are 
  substantially 
  the 
  same 
  (Camb. 
  Phil. 
  Trans, 
  vol. 
  x. 
  p. 
  129). 
  

   The 
  rules 
  obviously 
  depend 
  on 
  the 
  form 
  of 
  log 
  2. 
  The 
  form 
  of 
  the 
  first 
  

   rule 
  affords 
  a 
  little 
  more 
  accuracy 
  than 
  De 
  Morgan's. 
  

  

  Eule 
  I. 
  To 
  find 
  the 
  equivalent 
  of 
  a 
  given 
  vibrations-ratio 
  in 
  E. 
  T. 
  

   semitones. 
  

  

  Take 
  log 
  (ratio), 
  subtract 
  and 
  call 
  this 
  the 
  first 
  improved 
  value. 
  

  

  From 
  log 
  (ratio) 
  subtract 
  ^ 
  of 
  the 
  first 
  improved 
  value 
  and 
  10 
  qqq 
  

  

  of 
  the 
  first 
  improved 
  value. 
  Multiply 
  the 
  remainder 
  by 
  40. 
  We 
  can 
  

   rely 
  on 
  five 
  places 
  in 
  the 
  result. 
  

  

  The 
  following 
  data 
  are 
  introduced 
  here 
  ; 
  they 
  can 
  be 
  verified 
  by 
  

   numbers 
  given 
  in 
  Woolhouse's 
  tract 
  : 
  — 
  

  

  Eifth= 
  7*019,550,008,654. 
  

   Third=4- 
  -136,862,861,351. 
  

  

  Five 
  places 
  are 
  ordinarily 
  sufficient. 
  

  

  Eule 
  II. 
  To 
  find 
  the 
  vibrations-ratio 
  of 
  an 
  interval 
  given 
  in 
  E. 
  T. 
  

   semitones. 
  

  

  To 
  the 
  given 
  number 
  add 
  and 
  jq^qq 
  of 
  itself. 
  Divide 
  by 
  40. 
  The 
  

  

  result 
  is 
  the 
  logarithm 
  of 
  the 
  ratio 
  required. 
  We 
  can 
  rely 
  on 
  five 
  places 
  

   in 
  the 
  result, 
  or 
  on 
  six, 
  if 
  six 
  are 
  taken. 
  

  

  Ex. 
  The 
  E. 
  T. 
  third 
  is 
  4 
  semitones. 
  The 
  vibrations-ratio 
  found 
  as 
  

   above 
  is 
  1*259921. 
  

  

  Hence 
  the 
  vibrations-ratio 
  of 
  the 
  E. 
  T. 
  third 
  to 
  the 
  perfect 
  third 
  

   is 
  very 
  nearly 
  126 
  : 
  125. 
  

  

  Definitions. 
  

  

  Regular 
  systems 
  are 
  such 
  that 
  all 
  their 
  notes 
  can 
  be 
  arranged 
  in 
  a 
  con- 
  

   tinuous 
  series 
  of 
  equal 
  fifths. 
  

  

  Regular 
  cyclical 
  systems 
  are 
  not 
  only 
  regular, 
  but 
  return 
  into 
  the 
  same 
  

   pitch 
  after 
  a 
  certain 
  number 
  of 
  fifths. 
  Every 
  such 
  system 
  divides 
  the 
  

   octave 
  into 
  a 
  certain 
  number 
  of 
  equal 
  intervals. 
  

  

  Error 
  is 
  deviation 
  from 
  a 
  perfect 
  interval. 
  

  

  Departure 
  is 
  deviation 
  from 
  an 
  E. 
  T. 
  interval. 
  

  

  