﻿392 
  On 
  Air-Pressures 
  used 
  in 
  playing 
  Brass 
  Instruments. 
  

  

  Conclusion. 
  — 
  In 
  reviewing 
  the 
  results 
  of 
  the 
  experiments 
  

   it 
  is 
  clear 
  that 
  the 
  following 
  general 
  inferences 
  may 
  be 
  

   drawn 
  : 
  — 
  

  

  1. 
  Other 
  things 
  being 
  equal, 
  the 
  louder 
  the 
  note 
  the 
  greater 
  

   the 
  pressure, 
  as 
  was 
  to 
  be 
  expected. 
  See 
  Tables 
  III., 
  IV., 
  

   V., 
  and 
  VI. 
  

  

  2. 
  Again, 
  all 
  else 
  being 
  retained 
  the 
  same, 
  the 
  higher 
  the 
  

   pitch 
  of 
  the 
  note 
  played 
  on 
  a 
  given 
  instrument 
  the 
  greater 
  

   the 
  air-pressure 
  used. 
  See 
  Tables 
  III., 
  V., 
  and 
  VI. 
  

  

  In 
  the 
  scales 
  (Table 
  II. 
  and 
  Pis. 
  VII. 
  & 
  VIII.) 
  this 
  is 
  some- 
  

   times 
  apparently 
  violated, 
  not 
  only 
  on 
  the 
  trombone 
  played 
  by 
  

   the 
  writer 
  but 
  also 
  in 
  the 
  professional 
  playing 
  of 
  the 
  trumpet 
  

   and 
  cornet. 
  This 
  seems 
  to 
  be 
  due 
  to 
  a 
  slight 
  unintentional 
  

   difference 
  in 
  intensity 
  or 
  in 
  the 
  manner 
  of 
  producing 
  the 
  

   notes. 
  (See 
  first 
  footnote 
  to 
  Table 
  III.) 
  In 
  the 
  higher 
  

   parts 
  of 
  the 
  scale 
  it 
  is 
  hard 
  to 
  avoid 
  increasing 
  the 
  loudness. 
  

   This 
  may 
  account 
  for 
  the 
  upward 
  turn 
  of 
  the 
  curves 
  at 
  this 
  

   part. 
  

  

  3. 
  In 
  the 
  preliminary 
  experiments, 
  on 
  plotting 
  the 
  frequency 
  

   of 
  the 
  notes 
  of 
  the 
  scale 
  as 
  abscissae 
  and 
  the 
  pressures 
  as 
  

   ordinates 
  fairly 
  smooth 
  curves 
  were 
  obtained 
  convex 
  upwards. 
  

   When, 
  however, 
  quantities 
  proportional 
  to 
  the 
  logarithms 
  of 
  

   the 
  frequencies 
  are 
  taken 
  for 
  abscissae, 
  the 
  curves 
  become 
  

   almost 
  straight 
  lines. 
  See 
  Pis. 
  VII. 
  & 
  VIII., 
  in 
  which 
  the 
  

   abscissae 
  are 
  Mr. 
  Ellis's 
  logarithmic 
  cents, 
  100 
  to 
  the 
  equally- 
  

   tempered 
  semitone, 
  represented 
  by 
  the 
  large 
  squares, 
  and 
  the 
  

   intonation 
  is 
  assumed 
  to 
  be 
  in 
  equal 
  temperament. 
  

  

  4. 
  Now 
  if 
  the 
  logarithms 
  of 
  the 
  frequencies 
  be 
  taken 
  to 
  

   measure 
  pitches 
  of 
  notes, 
  then 
  the 
  difference 
  of 
  logarithms 
  

   measures 
  the 
  ratios 
  of 
  frequencies, 
  i. 
  e. 
  measures 
  the 
  intervals 
  

   between 
  them. 
  And, 
  as 
  is 
  well 
  known, 
  this 
  is 
  the 
  only 
  way 
  

   which 
  admits 
  of 
  simple 
  addition 
  of 
  numbers 
  representing 
  the 
  

   component 
  intervals 
  giving 
  the 
  number 
  representing 
  the 
  re- 
  

   sultant 
  interval. 
  Thus 
  the 
  logarithm 
  of 
  the 
  frequency 
  is 
  in 
  

   a 
  certain 
  very 
  real 
  and 
  important 
  sense 
  the 
  best 
  physical 
  

   measure 
  of 
  the 
  pitch 
  of 
  a 
  note. 
  Hence, 
  adopting 
  this 
  method, 
  

   and 
  noting 
  that 
  the 
  curves 
  in 
  Pis. 
  VII. 
  & 
  VIII. 
  are 
  nearly 
  

   straight 
  lines, 
  we 
  may 
  say 
  that 
  the 
  air-pressure 
  required 
  to 
  

   sound 
  any 
  note 
  with 
  given 
  intensity 
  is 
  approximately 
  pro- 
  

   portional 
  to 
  its 
  pitch 
  defined 
  logarithmically. 
  

  

  5. 
  Where 
  alternative 
  positions 
  or 
  fingerings 
  are 
  used 
  for 
  

   the 
  same 
  note 
  the 
  pressures 
  are 
  practically 
  the 
  same, 
  see 
  

   Table 
  IV., 
  the 
  results 
  in 
  which 
  were 
  confirmed 
  also 
  by 
  

   Mr. 
  Pickerill 
  with 
  his 
  instruments. 
  In 
  other 
  words, 
  it 
  is 
  

   almost 
  indifferent 
  as 
  regards 
  pressure 
  required, 
  whether 
  a 
  

   note 
  is 
  played 
  on 
  a 
  given 
  instrument 
  as 
  the 
  third 
  harmonic 
  

  

  