﻿32 
  C. 
  K. 
  Wead 
  — 
  Intensity 
  of 
  Sound. 
  

  

  For 
  all 
  the 
  pipes 
  of 
  any 
  given 
  stop 
  K' 
  and 
  K" 
  should 
  remain 
  

   constant 
  (pp. 
  99, 
  112). 
  This 
  assumption 
  underlies 
  his 
  elabo- 
  

   rate 
  table 
  of 
  " 
  normal 
  scales." 
  But 
  his 
  experiments 
  do 
  not 
  

   seem 
  to 
  establish 
  this 
  constancy. 
  Thus, 
  for 
  the 
  16 
  ft. 
  Prin- 
  

   cipal 
  : 
  

  

  

  Length. 
  

  

  Q. 
  

  

  K'. 
  

  

  K". 
  

  

  e' 
  

  

  20" 
  

  

  61-5 
  

  

  91-7 
  

  

  496 
  

  

  e° 
  

  

  40"3'" 
  

  

  99 
  

  

  72-5 
  

  

  408 
  

  

  E° 
  

  

  82-6 
  

  

  177 
  

  

  58 
  

  

  352 
  

  

  E' 
  

  

  100-4 
  

  

  162 
  

  

  25*6 
  

  

  165 
  tone 
  dull. 
  

  

  In 
  other 
  cases 
  the 
  values 
  of 
  K' 
  and 
  K" 
  vary 
  considerably 
  

   without 
  showing 
  any 
  regular 
  increase. 
  

  

  The 
  second 
  constant, 
  K", 
  appears 
  to 
  be 
  in 
  some 
  sense 
  a 
  

   measure 
  of 
  the 
  quality 
  of 
  the 
  note, 
  the 
  note 
  being 
  duller 
  as 
  K" 
  

   is 
  smaller. 
  For 
  pipes 
  of 
  the 
  same 
  length 
  obviously 
  K' 
  is 
  pro- 
  

   portioned 
  to 
  the 
  mass 
  of 
  air 
  used 
  per 
  unit 
  section 
  of 
  the 
  pipe, 
  

   and 
  so 
  to 
  the 
  energy 
  of 
  vibration 
  at 
  any 
  point 
  within 
  the 
  pipe, 
  

   if 
  we 
  make 
  the 
  violent 
  assumption 
  of 
  equal 
  efficiency 
  for 
  pipes 
  

   of 
  all 
  diameters. 
  In 
  the 
  same 
  way 
  K" 
  is 
  proportioned 
  to 
  the 
  

   energy 
  of 
  vibration 
  at 
  the 
  mouth. 
  But 
  we 
  are 
  not 
  concerned 
  

   with 
  the 
  intensity 
  of 
  vibration 
  in 
  the 
  pipe 
  ; 
  we 
  want 
  the 
  ex- 
  

   ternal 
  effect 
  due 
  to 
  the 
  total 
  cross-section. 
  

  

  The 
  introduction 
  of 
  the 
  square 
  root 
  of 
  the 
  length 
  has 
  no 
  

   physical 
  meaning 
  or 
  justification 
  that 
  I 
  can 
  discover 
  ; 
  but 
  it 
  is 
  

   needed 
  to 
  make 
  all 
  parts 
  of 
  Topfer's 
  theory 
  hang 
  together. 
  

   This 
  may 
  be 
  shown 
  as 
  follows 
  : 
  Assuming 
  equal 
  temperament 
  

   and 
  that 
  diameters 
  double 
  at 
  the 
  17th 
  pipe, 
  and 
  putting 
  a 
  for 
  

   the 
  diameter 
  of 
  any 
  pipe, 
  the 
  diameter 
  of 
  the 
  nth. 
  pipe 
  above 
  

   becomes 
  : 
  

  

  n 
  

  

  D 
  = 
  a 
  (|) 
  T6 
  

   Similarly 
  for 
  length 
  

  

  L 
  = 
  b 
  (1)t* 
  

   And 
  for 
  quantities 
  of 
  wind 
  on 
  Topfer's 
  assumption 
  

  

  Q 
  = 
  c 
  (i)A 
  

  

  .-. 
  Q 
  -- 
  D 
  2 
  = 
  (|)"A 
  c/a\ 
  QL-J 
  -4- 
  D 
  a 
  = 
  c 
  b^/a* 
  = 
  K'. 
  

  

  Evidently 
  it 
  is 
  necessary 
  to 
  introduce 
  L 
  2 
  to 
  obtain 
  a 
  constant 
  

   factor. 
  

  

  Topfer 
  then 
  goes 
  on 
  to 
  establish 
  a 
  "scale" 
  or 
  series 
  of 
  diam- 
  

   eters 
  for 
  a 
  set 
  of 
  pipes. 
  He 
  has 
  found 
  in 
  tables 
  published 
  by 
  

   Dom 
  Bedos 
  in 
  1766, 
  on 
  whose 
  work 
  his 
  own 
  treatise 
  is 
  largely 
  

   based, 
  that 
  the 
  ratio 
  of 
  sections 
  of 
  pipes 
  differing 
  an 
  8ve 
  in 
  

   pitch 
  ranges 
  between 
  1 
  : 
  4 
  and 
  1:2; 
  experience 
  shows 
  that 
  

   these 
  are 
  extreme 
  ; 
  so 
  it 
  is 
  safe 
  to 
  take 
  their 
  mean 
  1 
  : 
  v/8. 
  

  

  