﻿134 
  Prof. 
  E. 
  H. 
  Barton 
  on 
  Range 
  and 
  

  

  But 
  these 
  changes 
  somewhat 
  vitiate 
  the 
  harmonic 
  series 
  of 
  

   tones. 
  To 
  obviate 
  this, 
  as 
  far 
  as 
  may 
  be, 
  the 
  makers 
  adopt 
  

   a 
  quasi-hyperbolic 
  outline, 
  and 
  aim 
  at 
  so 
  modelling 
  the 
  tube 
  

   as 
  to 
  preserve 
  in 
  tune 
  all 
  the 
  intervals 
  in 
  musical 
  use. 
  This 
  

   usually 
  leaves 
  the 
  lowest 
  interval 
  out 
  of 
  tune. 
  But 
  this 
  is 
  

   the 
  interval 
  (nominally 
  an 
  octave) 
  between 
  tone 
  number 
  two 
  

   and 
  the 
  fundamental. 
  This 
  last 
  is 
  called 
  the 
  pedal, 
  and 
  is 
  

   seldom 
  employed 
  musically. 
  To 
  set 
  the 
  upper 
  notes 
  of 
  the 
  

   series 
  right 
  the 
  pedal 
  of 
  some 
  instruments 
  may 
  be 
  lowered 
  in 
  

   pitch, 
  say 
  in 
  the 
  ratio 
  6 
  :5. 
  There 
  seem 
  to 
  be 
  other 
  instru- 
  

   ments, 
  of 
  a 
  different 
  form 
  of 
  tubing, 
  in 
  which, 
  for 
  the 
  same 
  

   reason, 
  the 
  pedal 
  is 
  similarly 
  raised, 
  say, 
  in 
  the 
  ratio 
  6:7. 
  

   The 
  two 
  cases 
  might 
  then 
  be 
  represented 
  by 
  the 
  following 
  

   schemes 
  of 
  relative 
  frequencies 
  : 
  — 
  

  

  5/6, 
  2, 
  3, 
  4, 
  5, 
  6, 
  &c. 
  

   7/6, 
  2, 
  3, 
  4, 
  5, 
  6, 
  &c. 
  

  

  In 
  each 
  of 
  these 
  schemes 
  the 
  first 
  figure 
  would 
  have 
  been 
  

   unity, 
  if 
  not 
  purposely 
  disturbed 
  to 
  set 
  the 
  others 
  in 
  their 
  

   right 
  relation. 
  Now, 
  if 
  the 
  pedal 
  of 
  frequency 
  represented 
  

   by 
  unity 
  is 
  ever 
  required, 
  it 
  may 
  be 
  expected, 
  on 
  the 
  theory 
  

   of 
  this 
  paper, 
  that 
  the 
  range 
  of 
  resonance 
  at 
  this 
  low 
  pitch 
  

   would 
  be 
  great 
  enough 
  to 
  allow 
  the 
  player 
  to 
  produce 
  the 
  

   note 
  by 
  forcing 
  it 
  to 
  the 
  desired 
  pitch. 
  But 
  if 
  the 
  resonance 
  

   can 
  be 
  spread 
  in 
  one 
  direction 
  (say 
  down, 
  from 
  7/6 
  to 
  1), 
  it 
  

   seems 
  only 
  natural 
  to 
  suppose 
  that 
  it 
  could 
  also 
  be 
  spread 
  in 
  

   the 
  opposite 
  direction 
  (say 
  up, 
  from 
  7/6 
  to 
  4/3). 
  

  

  In 
  this 
  way 
  it 
  seems 
  possible 
  to 
  account 
  for 
  the 
  objective 
  

   reality 
  of 
  a 
  certain 
  low 
  note 
  whose 
  existence 
  on 
  the 
  euphonion 
  

   and 
  bombardon 
  is 
  traditionally 
  asserted 
  among 
  musicians. 
  

  

  Indeed, 
  the 
  present 
  paper 
  is 
  the 
  outcome 
  of 
  an 
  endeavour 
  

   to 
  answer 
  the 
  question 
  as 
  to 
  how 
  this 
  note 
  arises 
  (asked 
  me 
  

   by 
  Mr. 
  F. 
  C. 
  Edwards 
  of 
  Messrs. 
  Besson 
  & 
  Co., 
  Ltd.). 
  

  

  The 
  note 
  to 
  which 
  reference 
  is 
  made 
  lies 
  below 
  note 
  No. 
  2, 
  

   but 
  above 
  the 
  pedal 
  (if 
  of 
  its 
  true 
  pitch 
  unity). 
  If 
  the 
  notes 
  

   are 
  represented 
  in 
  relative 
  frequency 
  by 
  the 
  natural 
  numbers, 
  

   as 
  in 
  the 
  previous 
  schemes, 
  the 
  low 
  note 
  in 
  question 
  would 
  

   be 
  represented 
  by 
  the 
  fraction 
  4/3. 
  Or, 
  when 
  the 
  notes 
  are 
  

   represented 
  by 
  letters 
  as 
  though 
  the 
  instrument 
  were 
  in 
  the 
  

   key 
  of 
  C 
  (which 
  is 
  the 
  practice 
  in 
  writing 
  music 
  for 
  valved 
  

   instruments) 
  then 
  the 
  note 
  is 
  called 
  a 
  low 
  F. 
  The 
  actual 
  

   pitch 
  of 
  the 
  note 
  varies 
  with 
  that 
  of 
  the 
  instrument. 
  This 
  

   is 
  illustrated 
  in 
  the 
  following 
  schemes 
  of 
  notes 
  w 
  7 
  hich 
  can 
  be 
  

   sounded 
  on 
  these 
  instruments 
  as 
  open 
  notes, 
  i. 
  e. 
  without 
  

   using 
  the 
  valves. 
  

  

  The 
  anomalous 
  note 
  is 
  inclosed 
  in 
  brackets, 
  and 
  the 
  pedal 
  

  

  