﻿116 
  

  

  PfoE. 
  E. 
  H. 
  Barton 
  on 
  Mange 
  and 
  

  

  abscissse 
  and 
  ordinates 
  respectively, 
  have 
  the 
  same 
  maximum 
  

   central 
  ordinate. 
  

  

  Such 
  curves, 
  for 
  frequencies 
  proportional 
  to 
  the 
  first 
  six 
  

   natural 
  numbers, 
  are 
  plotted 
  in 
  fig. 
  1, 
  and 
  so 
  represent 
  the 
  

   responses 
  (by 
  these 
  tones 
  of 
  the 
  harmonic 
  series) 
  to 
  the 
  

   approximate 
  forcings. 
  

  

  Fig. 
  1. 
  — 
  Resonance 
  Curves 
  for 
  Commensurate 
  Frequencies. 
  

  

  Zoo 
  

  

  , 
  6/S 
  

   5oo 
  cents 
  J 
  

  

  The 
  diagram 
  also 
  shows 
  clearly 
  the 
  force 
  of 
  equation 
  (11). 
  

   Thus, 
  taking 
  in 
  it 
  a 
  horizontal 
  line 
  at 
  any 
  level, 
  we 
  find 
  that 
  

   the 
  values 
  of 
  M 
  for 
  the 
  intersections 
  of 
  the 
  different, 
  curves 
  

   are 
  inversely 
  as 
  the 
  frequencies 
  to 
  which 
  they 
  refer. 
  Indeed, 
  

   any 
  lower 
  curve 
  might 
  be 
  derived 
  from 
  any 
  upper 
  one 
  by 
  

   proportional 
  shrinkage 
  of 
  all 
  its 
  abscissa? 
  ; 
  or, 
  by 
  projection, 
  

   after 
  it 
  had 
  made 
  a 
  determinate 
  rotation 
  about 
  the 
  axis 
  of 
  

   ordinates. 
  Each 
  curve 
  has 
  two 
  points 
  of 
  inflexion, 
  which 
  all 
  

   occur 
  at 
  the 
  same 
  level, 
  viz. 
  at 
  the 
  height 
  of 
  three-fourths 
  

   of 
  the 
  maximum 
  ordinate. 
  

  

  The 
  apparently 
  strange 
  way 
  of 
  measuring 
  the 
  mistiming, 
  

   equation 
  (13), 
  which, 
  however, 
  the 
  analysis 
  almost 
  forced 
  

   upon 
  us, 
  is 
  now 
  seen 
  to 
  have 
  the 
  advantage 
  of 
  yielding 
  a 
  

   symmetrical 
  resonance 
  curve. 
  Further, 
  for 
  small 
  values 
  

   of 
  this 
  mistuning 
  we 
  have 
  the 
  relation 
  

  

  ■M- 
  2 
  log, 
  g), 
  nearly 
  (16) 
  

  

  And 
  we 
  know 
  that 
  to 
  measure 
  musical 
  intervals 
  by 
  the 
  

   logarithms 
  of 
  their 
  frequency-ratio 
  is 
  the 
  only 
  mode 
  that 
  makes 
  

  

  