﻿C. 
  K. 
  Wead 
  — 
  Intensity 
  of 
  Sound. 
  29 
  

  

  is 
  of 
  course 
  (-J) 
  1 
  ** 
  = 
  *9576. 
  This 
  ratio 
  for 
  the 
  diameters 
  is 
  only 
  

   a 
  mathematical 
  expression 
  of 
  a 
  mechanical 
  fact, 
  there 
  is 
  no 
  

   theory 
  about 
  it. 
  Such 
  a 
  "scale" 
  gives 
  convenient 
  rules 
  in 
  

   practice 
  for 
  laying 
  out 
  the 
  pipes, 
  and 
  satisfies 
  the 
  ear, 
  or 
  it 
  

   would 
  not 
  have 
  found 
  such 
  general 
  adoption. 
  In 
  this 
  organ 
  

   Open 
  Diapason 
  c' 
  has 
  an 
  internal 
  diameter 
  of 
  57 
  min., 
  the 
  e" 
  of 
  

   29^; 
  the 
  Dulciana 
  & 
  of 
  31, 
  the 
  e" 
  of 
  15J*. 
  It 
  is 
  not 
  for 
  a 
  

   moment 
  to 
  be 
  assumed 
  that 
  the 
  amount 
  of 
  wind 
  required 
  is 
  

   directly 
  determined 
  by 
  the 
  diameter 
  of 
  the 
  pipe 
  ; 
  for 
  the 
  

   organ 
  builder 
  would 
  point 
  out 
  that 
  the 
  shape 
  of 
  the 
  mouth 
  

   is 
  an 
  important 
  factor, 
  and 
  that 
  the 
  voicer 
  or 
  finisher 
  varies 
  

   the 
  amount 
  of 
  wind 
  by 
  plugging 
  the 
  holes 
  through 
  the 
  feet 
  

   of 
  wooa 
  pipes, 
  cutting 
  out 
  or 
  closing 
  the 
  feet 
  of 
  metal 
  pipes, 
  

   varying 
  the 
  width 
  of 
  the 
  slit 
  for 
  the 
  wind, 
  etc., 
  till 
  his 
  ear 
  is 
  

   satisfied 
  with 
  the 
  loudness 
  and 
  quality 
  of 
  the 
  sound. 
  But 
  in 
  

   the 
  light 
  of 
  these 
  experiments 
  we 
  must 
  conclude 
  that 
  for 
  

   similar 
  pipes 
  the 
  volume 
  of 
  air 
  used 
  per 
  second, 
  and 
  therefore 
  

   the 
  energy 
  expended 
  per 
  second, 
  varies 
  as 
  the 
  f-power 
  of 
  the 
  

   wave-length 
  of 
  the 
  note, 
  or 
  inversely 
  as 
  the 
  f-power 
  of 
  the 
  vi- 
  

   bration-ratio 
  ; 
  and 
  further 
  conclude 
  that 
  the 
  voicer 
  uncon- 
  

   sciously 
  strives 
  to 
  secure 
  this 
  ratio 
  just 
  as 
  the 
  tuner 
  uncon- 
  

   sciously 
  strives 
  to 
  get 
  the 
  familiar 
  vibration-ratios 
  in 
  the 
  tuning 
  

   of 
  any 
  instrument. 
  It 
  is 
  to 
  be 
  remembered 
  that 
  we 
  cannot 
  

   recognize 
  small 
  differences 
  of 
  intensity 
  with 
  much 
  accuracy. 
  

   Volkman 
  could 
  always 
  detect 
  a 
  difference 
  of 
  25 
  per 
  cent 
  ; 
  

   Renz 
  & 
  Wolff 
  one 
  of 
  28 
  per 
  cent 
  ; 
  the 
  latter 
  experimenters 
  

  

  *In 
  Clarke's 
  little 
  book 
  on 
  '"The 
  Pipe 
  Organ" 
  a 
  simple 
  construction 
  is 
  given 
  

   for 
  finding 
  the 
  diameters 
  of 
  intermediate 
  pipes 
  when 
  the 
  diameters 
  are 
  given 
  for 
  

   two 
  pipes 
  16, 
  8, 
  4. 
  &c. 
  semitones 
  apart. 
  At 
  the 
  ends 
  of 
  any 
  convenient 
  base 
  line 
  

   AB 
  erect 
  perpendiculars 
  AC, 
  BD 
  proportional 
  to 
  the 
  given 
  diameters 
  and 
  join 
  

   the 
  ends 
  C, 
  D 
  : 
  Draw 
  the 
  two 
  diagonals 
  of 
  the 
  trapezoid 
  thus 
  formed 
  and 
  erect 
  

   through 
  their 
  point 
  of 
  intersection 
  a 
  perpendicular 
  to 
  the 
  base 
  line. 
  The 
  part 
  of 
  

   this 
  perpendicular 
  between 
  AB 
  and 
  CD 
  is 
  proportional 
  to 
  the 
  diameter 
  of 
  the 
  pipe 
  

   midway 
  between 
  the 
  given 
  extremes. 
  By 
  continuing 
  the 
  construction 
  the 
  diame- 
  

   ters 
  of 
  the 
  other 
  pipes 
  will 
  be 
  obtained, 
  

  

  A 
  little 
  calculation 
  shows 
  that 
  this 
  gives 
  a 
  harmonic 
  series, 
  and 
  if 
  the 
  first 
  

   diameter 
  be 
  2. 
  and 
  the 
  seventeenth 
  1, 
  the 
  series 
  is 
  32 
  -t- 
  16, 
  17, 
  18 
  . 
  . 
  . 
  31, 
  32. 
  

   All 
  of 
  the 
  intermediate 
  quotients 
  are 
  slightly 
  less 
  than 
  the 
  numbers 
  derived 
  from 
  

   the 
  exponential 
  series 
  whose 
  ratio 
  is 
  the 
  16th 
  root 
  of 
  £. 
  the 
  value 
  for 
  the 
  8ve 
  

   being 
  £§ 
  = 
  571 
  instead 
  of 
  "59-16. 
  The 
  maximum 
  difference 
  is 
  about 
  5 
  per 
  cent 
  — 
  a 
  

   quantity 
  entirely 
  negligible 
  to 
  ordinary 
  ears. 
  

  

  If 
  a 
  series 
  of 
  pipes 
  were 
  made 
  on 
  this 
  harmonic 
  scale 
  and 
  the 
  quantities 
  of 
  

   wind 
  could 
  be 
  accurately 
  adjusted 
  in 
  the 
  ratio 
  of 
  the 
  diameters, 
  an 
  exponential 
  

   curve 
  deduced 
  from 
  experiments 
  on 
  them 
  would 
  show 
  an 
  •• 
  alternating 
  deviation 
  " 
  

   similar 
  to 
  that 
  referred 
  to 
  above. 
  The 
  sign 
  of 
  the 
  deviation 
  in 
  a 
  given 
  8ve 
  would 
  

   depend 
  on 
  where 
  the 
  starting 
  point 
  of 
  the 
  harmonic 
  scale 
  was 
  taken. 
  

  

  The 
  sum 
  of 
  8 
  terms 
  of 
  the 
  harmonic 
  series 
  corresponding 
  to 
  the 
  key 
  of 
  C, 
  the 
  

   lowest 
  term 
  being 
  1, 
  is 
  5 
  95 
  : 
  of 
  the 
  same 
  terms 
  of 
  the 
  exponential 
  series 
  6'19; 
  

   of 
  13 
  terms 
  in 
  the 
  exponential 
  series 
  9 
  - 
  4 
  Therefore 
  to 
  find 
  the 
  amount 
  of 
  wind 
  

   (or 
  of 
  energy) 
  used 
  by 
  the 
  lowest 
  pipe 
  of 
  any 
  group 
  of 
  eight 
  in 
  the 
  tables 
  divide 
  

   by 
  6 
  the 
  amount 
  given 
  for 
  the 
  group. 
  

  

  f 
  Pogg. 
  Ann., 
  xcviii, 
  595, 
  1856. 
  

  

  