﻿28 
  

  

  C. 
  K. 
  Wead 
  — 
  Intensity 
  of 
  Sound. 
  

  

  If 
  a 
  smooth 
  exponential 
  curve 
  be 
  drawn 
  with 
  these 
  ratios 
  for 
  

   the 
  experiments 
  of 
  table 
  I, 
  where 
  eight 
  keys 
  were 
  depressed 
  

   at 
  once, 
  it 
  will 
  be 
  found 
  to 
  fall 
  below 
  the 
  experimental 
  curve 
  

   in 
  the 
  first 
  and 
  third 
  octaves, 
  and 
  above 
  it 
  in 
  the 
  second 
  and 
  

   fourth 
  octaves 
  in 
  every 
  case 
  examined 
  ; 
  the 
  magnitude 
  of 
  the 
  

   difference 
  is 
  shown 
  in 
  column 
  9 
  ; 
  but 
  this 
  alternating 
  deviation 
  

   is 
  not 
  great, 
  and 
  is 
  probably 
  not 
  of 
  importance 
  ; 
  it 
  does 
  not 
  

   appear 
  in 
  table 
  III, 
  where 
  several 
  stops 
  are 
  combined. 
  

  

  Table 
  IV. 
  

  

  Table. 
  

  

  Stop 
  drawn. 
  

  

  No. 
  keys. 
  

  

  Range, 
  

   Octaves. 
  

  

  Number 
  

   equations. 
  

  

  Eatio. 
  

  

  Av. 
  Difference 
  

   per 
  cent. 
  

  

  I 
  

   I 
  

   I 
  

   I 
  

  

  III 
  

   III 
  

   III 
  

   II 
  

  

  Trumpet. 
  

  

  Open 
  Diapason. 
  

  

  Fifteenth. 
  

  

  Bourdon. 
  

  

  Three 
  stops. 
  

   Nine 
  stops. 
  

  

  8 
  

   8 
  

   8 
  

   8 
  

  

  1 
  

   1 
  

   1 
  

   1 
  

  

  4 
  

   4 
  

   3 
  

  

  2 
  

  

  3 
  

   4 
  

   4 
  

  

  1 
  

  

  4 
  

   4 
  

   2 
  

  

  4 
  

  

  5 
  

  

  4 
  

  

  13 
  

  

  •701 
  

   •620 
  

   •654 
  

   •71 
  

  

  •595 
  

   •595 
  

   •591 
  

   •5975 
  

  

  ± 
  T 
  

   ± 
  3-2 
  

  

  ±10 
  

  

  ±1-4 
  

   ±2-4 
  

   ±7 
  

   ±3-4 
  

  

  The 
  latter 
  half 
  of 
  table 
  IY 
  shows 
  that 
  when 
  the 
  stops 
  were 
  

   combined 
  as 
  in 
  ordinary 
  playing, 
  but 
  a 
  single 
  key 
  being 
  pressed, 
  

   there 
  is 
  a 
  remarkable 
  constancy 
  in 
  the 
  value 
  of 
  the 
  ratio 
  for 
  

   the 
  octave 
  however 
  it 
  is 
  determined, 
  and 
  its 
  value 
  for 
  the 
  Open 
  

   Diapason 
  differs 
  little 
  from 
  these 
  latter 
  values. 
  This 
  constancy 
  

   demands 
  an 
  explanation. 
  According 
  to 
  Topfer's 
  law 
  we 
  should 
  

   have 
  -50 
  = 
  V|; 
  we 
  do 
  have 
  very 
  nearly 
  VT 
  = 
  .5946 
  = 
  1 
  ^ 
  

   1/682. 
  This 
  I 
  believe 
  to 
  be 
  an 
  excellent 
  illustration 
  of 
  the 
  un- 
  

   conscious 
  recognition 
  by 
  the 
  artist 
  of 
  the 
  physical 
  or 
  mathe- 
  

   matical 
  laws 
  underlying 
  his 
  art. 
  At 
  present 
  we 
  cannot 
  explain 
  

   the 
  law, 
  any 
  more 
  than 
  the 
  laws 
  of 
  the 
  scale 
  could 
  be 
  explained 
  

   before 
  the 
  subject 
  of 
  harmonic 
  overtones 
  was 
  understood 
  ; 
  we 
  

   can 
  only 
  correlate 
  this 
  with 
  the 
  following 
  fact 
  relating 
  to 
  

   organ-pipes 
  —to 
  their 
  diameter, 
  or 
  "scale" 
  as 
  organ-builders 
  

   call 
  it. 
  It 
  is 
  a 
  matter 
  of 
  experience 
  that 
  to 
  produce 
  the 
  proper 
  

   loudness 
  of 
  sound 
  it 
  is 
  necessary 
  to 
  increase 
  the 
  ratio 
  of 
  the 
  

   diameter 
  to 
  the 
  length 
  as 
  the 
  pipe 
  becomes 
  shorter, 
  so 
  when 
  

   the 
  pitch 
  rises 
  an 
  octave 
  and 
  the 
  theoretical 
  length 
  becomes 
  

   one-half 
  that 
  of 
  the 
  fundamental 
  the 
  diameter 
  is 
  greater 
  than 
  

   half 
  that 
  of 
  the 
  fundamental 
  ; 
  usually 
  we 
  must 
  go 
  to 
  the 
  seven- 
  

   teenth 
  pipe, 
  as 
  from 
  C 
  to 
  e, 
  to 
  find 
  the 
  one 
  of 
  half 
  the 
  diam- 
  

   eter. 
  This 
  is 
  equivalent 
  to 
  saying 
  that 
  in 
  rising 
  4 
  octaves 
  the 
  

   theoretical 
  length 
  becomes 
  (|-) 
  4 
  , 
  but 
  the 
  diameter 
  (-J-) 
  3 
  ; 
  if 
  we 
  

   assume 
  an 
  exponential 
  series 
  all 
  the 
  way 
  up 
  the 
  ratio 
  of 
  diam_ 
  

   eters 
  of 
  pipes 
  an 
  octave 
  apart 
  is 
  therefore 
  (|-)* 
  or 
  -/-I-,— 
  the 
  

   ratio 
  already 
  found 
  ; 
  the 
  corresponding 
  ratio 
  for 
  the 
  semitone 
  

  

  