﻿Sharpness 
  of 
  Resonance 
  under 
  Sustained 
  Forcing. 
  135 
  

  

  is 
  shown 
  as 
  if 
  of 
  the 
  true 
  nominal 
  pitch 
  to 
  form 
  the 
  base 
  of: 
  the 
  

   harmonic 
  series. 
  

  

  Relative 
  frequencies 
  ;— 
  1 
  (4/3) 
  2 
  3 
  4 
  5 
  6 
  &c. 
  

  

  Notes 
  as 
  written 
  for 
  all 
  ] 
  ^ 
  /T 
  ^ 
  X 
  n 
  ^ 
  n 
  i?' 
  N 
  r± 
  s 
  

  

  , 
  , 
  . 
  . 
  \ 
  C 
  (F) 
  C 
  G 
  C 
  E 
  G 
  &c. 
  

  

  valved 
  instruments 
  ... 
  J 
  v 
  ' 
  

  

  Actual 
  notes 
  for 
  B 
  p 
  

   instruments 
  

  

  1 
  B? 
  (EJT) 
  B> 
  F 
  B? 
  D 
  F 
  &c. 
  

  

  Actual 
  notes 
  for 
  E 
  p 
  

  

  instruments 
  

  

  E|? 
  (A?) 
  E? 
  B!7 
  E? 
  G 
  B?&c. 
  

  

  This 
  special 
  note 
  may 
  have 
  been 
  discovered 
  because 
  of 
  its 
  

   usefulness. 
  For 
  the 
  three-valve 
  mechanism 
  is 
  designed 
  to 
  

   supply 
  the 
  chromatic 
  scale 
  in 
  the 
  interval 
  between 
  notes 
  3 
  

   and 
  *i, 
  which 
  are 
  seen 
  to 
  be 
  a 
  musical 
  fifth 
  apart. 
  This 
  

   mechanism 
  accordingly 
  more 
  than 
  bridges 
  all 
  the 
  smaller 
  

   gaps, 
  occurring 
  among 
  the 
  higher 
  open 
  notes, 
  but 
  fails 
  to 
  

   bridge 
  the 
  larger 
  gap 
  of 
  an 
  octave 
  occurring 
  between 
  notes 
  

   2 
  and 
  1. 
  But, 
  if 
  the 
  note 
  of 
  frequency 
  4/3 
  cjin 
  be 
  obtained 
  

   without 
  the 
  valves, 
  then 
  the 
  valves 
  can 
  give 
  the 
  chromatic 
  

   intervals 
  in 
  the 
  gap 
  between 
  notes 
  2 
  and 
  4/3, 
  and 
  in 
  that 
  

   between 
  the 
  notes 
  4/3 
  and 
  1. 
  Thus 
  the 
  chromatic 
  scale 
  is 
  

   obtainable 
  throughout 
  what 
  is 
  called 
  the 
  pedal 
  octave. 
  

  

  Xow, 
  though 
  it 
  might 
  be 
  very 
  difficult 
  to 
  account 
  for 
  a 
  

   full 
  but 
  isolated 
  resonance 
  of 
  the 
  pitch 
  referred 
  to 
  (4/3 
  or 
  

   low 
  F), 
  it 
  is 
  vet 
  easy 
  to 
  see 
  that 
  the 
  resonance 
  at 
  this 
  pitch 
  

   may 
  not 
  be 
  isolated 
  at 
  all, 
  but 
  may 
  actually 
  spread 
  continuously 
  

   down 
  from 
  the 
  pitch 
  in 
  question 
  to 
  the 
  pedal 
  of 
  true 
  pitch; 
  

   and 
  yet 
  this 
  fact 
  escape 
  notice 
  among 
  musicians. 
  For 
  they, 
  

   having 
  got 
  this 
  one 
  note 
  without 
  valves, 
  might 
  then 
  use 
  them 
  

   to 
  extend 
  the 
  scale. 
  

  

  The 
  view 
  here 
  put 
  forward 
  is 
  that, 
  on 
  the 
  instruments 
  

   which 
  show 
  best 
  this 
  low 
  F, 
  the 
  true 
  pitch 
  of 
  best 
  resonance 
  

   of 
  the 
  pedal 
  usually 
  lies 
  between 
  this 
  F 
  and 
  the 
  C 
  below, 
  

   and 
  that, 
  by 
  appropriate 
  forcing, 
  the 
  notes 
  F 
  and 
  C 
  can 
  

   each 
  be 
  sounded, 
  but 
  also 
  every 
  other 
  note 
  between, 
  and 
  pro- 
  

   bably 
  some 
  of 
  these 
  easier 
  than 
  the 
  extremes 
  of 
  pitch 
  to 
  

   which 
  the 
  resonance 
  can 
  be 
  pushed. 
  And 
  this 
  is 
  indicated 
  

   in 
  the 
  above 
  schemes 
  by 
  the 
  dots 
  linking 
  up 
  the 
  nominal 
  

   pitch 
  of 
  the 
  pedal 
  with 
  the 
  anomalous 
  note. 
  The 
  low 
  F 
  may 
  

   occur 
  on 
  some 
  instruments 
  of 
  a 
  different 
  form 
  of 
  tubing 
  

   by 
  a 
  downward 
  spread 
  of 
  continuous 
  resonance 
  from 
  note 
  

   No. 
  2. 
  

  

  The 
  convictions 
  just 
  stated 
  were 
  arrived 
  at 
  by 
  an 
  alter- 
  

   nation 
  of 
  theory 
  and 
  experiment 
  which 
  need 
  not 
  be 
  detailed 
  

   here. 
  

  

  