﻿140 
  Prof. 
  E. 
  H. 
  Barton 
  on 
  Range 
  and 
  

  

  measure 
  the 
  damping 
  and 
  frequency 
  ; 
  the 
  latter 
  is 
  known, 
  

   hut 
  to 
  the 
  former 
  we 
  can 
  only 
  approximate. 
  Thus, 
  staccato 
  

   notes 
  of 
  about 
  5 
  or 
  6 
  per 
  second 
  may 
  be 
  sounded 
  on 
  the 
  

   comet, 
  or 
  a 
  shake 
  of 
  10 
  or 
  12 
  notes 
  per 
  second 
  in 
  all. 
  It 
  

   would 
  thus 
  appear 
  that 
  the 
  decay 
  of 
  the 
  vibrations 
  expressed 
  

   by 
  e~ 
  kt 
  would 
  reduce 
  the 
  energy 
  to 
  one-tenth 
  in 
  something 
  

   like 
  1/14 
  of 
  a 
  second, 
  if 
  not 
  in 
  less. 
  This 
  gives 
  as 
  an 
  inferior 
  

   limit 
  for 
  k 
  the 
  value 
  107T 
  say. 
  A 
  similarly 
  rough 
  estimate 
  

   of 
  the 
  diminution 
  of 
  response 
  when 
  the 
  pedal 
  note 
  is 
  blown 
  

   out 
  of 
  tune 
  a 
  minor 
  third 
  (w//? 
  = 
  6/5, 
  M 
  = 
  l/3 
  nearly, 
  N 
  /N 
  = 
  6 
  

   say) 
  makes 
  k 
  of 
  the 
  order 
  30-rr. 
  Thus, 
  taking 
  the 
  mean, 
  we 
  

   may 
  regard 
  k 
  as 
  being 
  of 
  the 
  order 
  20-7T. 
  This 
  gives 
  for 
  the 
  

   sharpness 
  of 
  resonance 
  values 
  of 
  the 
  following 
  order 
  : 
  — 
  

  

  Notes 
  of 
  Cornet 
  Pedal, 
  2, 
  3, 
  4, 
  5, 
  6, 
  &c. 
  

  

  Sharpness 
  of 
  Resonance 
  ... 
  10, 
  20, 
  30, 
  40, 
  50, 
  60, 
  &c. 
  

  

  As 
  these 
  are 
  only 
  approximate, 
  no 
  attempt 
  has 
  been 
  made 
  

   to 
  allow 
  for 
  (i.) 
  a 
  possible 
  variation 
  of 
  k 
  with 
  pitch, 
  or 
  (ii.) 
  a 
  

   departure 
  of 
  the 
  pitch 
  of 
  the 
  pedal 
  from 
  the 
  relative 
  fre- 
  

   quency 
  denoted 
  by 
  unity. 
  The 
  above 
  values 
  for 
  the 
  sharp- 
  

   ness 
  of 
  resonance 
  were 
  adopted 
  in 
  drawing 
  the 
  resonance 
  

   curves 
  in 
  figs. 
  1 
  and 
  3. 
  

  

  Referring 
  now 
  to 
  either 
  of 
  these 
  figures, 
  we 
  see 
  that 
  (on 
  a 
  

   small 
  and 
  manageable 
  instrument) 
  the 
  pitch 
  of 
  a 
  low 
  note 
  in 
  

   this 
  harmonic 
  series 
  could 
  easily 
  be 
  forced 
  considerably 
  out 
  

   of 
  tune, 
  since 
  the 
  range 
  of 
  resonance 
  is 
  there 
  considerable. 
  

   But 
  if 
  this 
  is 
  done, 
  the 
  diagrams 
  also 
  show 
  that 
  there 
  would 
  

   be 
  very 
  slight 
  response 
  on 
  the 
  part 
  of 
  the 
  higher 
  tones 
  which 
  

   are 
  needed 
  as 
  the 
  upper 
  partials 
  of 
  the 
  lower. 
  Thus, 
  the 
  

   forcing 
  out 
  of 
  tune 
  by 
  a 
  semitone 
  of 
  the 
  pedal 
  note 
  on 
  the 
  

   cornet 
  (about 
  120 
  per 
  second) 
  would 
  involve 
  only 
  a 
  very 
  

   moderate 
  diminution 
  of 
  response 
  of 
  the 
  tone 
  of 
  this 
  pitch. 
  

   But 
  this 
  mistuning 
  would 
  involve 
  relatively 
  large 
  reductions 
  

   in 
  the 
  responses 
  of 
  the 
  higher 
  possible 
  tones 
  which 
  are 
  needed 
  

   as 
  its 
  upper 
  partials. 
  This 
  is 
  seen 
  by 
  the 
  single 
  ordinate 
  

   PQR 
  on 
  tig. 
  1 
  and 
  the 
  corresponding 
  three 
  related 
  ordinates 
  

   P, 
  Q, 
  and 
  R 
  on 
  fig. 
  3. 
  These 
  ordinates 
  are 
  all 
  a 
  semitone 
  

   out 
  of 
  tune 
  with 
  the 
  pitch 
  of 
  the 
  instrument. 
  This 
  progres- 
  

   sive 
  falling 
  off 
  in 
  response 
  of 
  the 
  notes 
  as 
  we 
  proceed 
  up 
  the 
  

   series 
  renders 
  the 
  quality 
  of 
  this 
  mistuned 
  note 
  very 
  dull 
  and 
  

   poor, 
  if 
  produced 
  by 
  a 
  novice. 
  Probably 
  an 
  expert 
  can 
  blow 
  

   a 
  note 
  slightly 
  out 
  of 
  tune 
  and 
  yet 
  so 
  favour 
  the 
  upper 
  par- 
  

   tials 
  by 
  his 
  special 
  skill 
  as 
  to 
  produce 
  a 
  tone 
  of 
  good 
  musical 
  

   quality 
  in 
  spite 
  of 
  the 
  inherent 
  difficulty. 
  

  

  This 
  leads 
  to 
  another 
  point 
  concerning 
  brass 
  instruments 
  

   which 
  is 
  elucidated 
  by 
  the 
  present 
  theory. 
  In 
  the 
  ordinary 
  

  

  