﻿406 
  

  

  Mr. 
  R. 
  H. 
  M. 
  Bosanquet 
  on 
  the 
  

  

  If 
  c=l, 
  c# 
  = 
  10, 
  /c#=12, 
  d=21, 
  

  

  It 
  would 
  be 
  possible 
  to 
  construct 
  a 
  key-board 
  on 
  the 
  principles 
  already 
  

   explained, 
  which 
  would 
  give 
  complete 
  control 
  over 
  the 
  notes 
  of 
  the 
  system 
  

   of 
  118. 
  A 
  portion 
  of 
  such 
  a 
  key-board 
  would 
  be 
  practically 
  indistin- 
  

   guishable 
  from 
  one 
  tuned 
  to 
  the 
  positive 
  system 
  of 
  perfect 
  thirds, 
  as 
  

   the 
  error 
  of 
  the 
  thirds 
  of 
  the 
  system 
  of 
  118 
  is 
  too 
  small 
  to 
  be 
  perceived 
  

   by 
  the 
  ear. 
  

  

  Application 
  of 
  the 
  Negative 
  System 
  of 
  Perfect 
  Thirds 
  {Mean-Tone 
  System) 
  

   to 
  the 
  "Generalized 
  Key-hoard? 
  

  

  If 
  the 
  thirds, 
  such 
  as 
  c-e, 
  are 
  made 
  perfect, 
  and 
  the 
  fifths 
  -05376 
  flat, 
  

   we 
  have 
  the 
  mean-tone 
  system. 
  The 
  forms 
  of 
  scales 
  and 
  chords 
  in 
  nega- 
  

   tive 
  systems 
  are 
  different 
  from 
  those 
  in 
  positive 
  systems. 
  The 
  scales 
  are 
  

   very 
  easy 
  to 
  play, 
  and 
  the 
  chords 
  also. 
  It 
  is 
  expected 
  that 
  this 
  applica- 
  

   tion 
  may 
  prove 
  of 
  practical 
  importance. 
  

  

  Following 
  the 
  scale 
  of 
  unmarked 
  naturals 
  on 
  the 
  plan, 
  we 
  can 
  realize 
  

   the 
  nature 
  of 
  the 
  fingering. 
  It 
  is 
  the 
  same 
  as 
  that 
  of 
  the 
  Pytha- 
  

   gorean 
  scale 
  with 
  the 
  system 
  of 
  perfect 
  fifths. 
  The 
  tones 
  are 
  all 
  2-fifths 
  

   tones, 
  and 
  the 
  semitones 
  both 
  5-fifths 
  semitones. 
  

  

  Application 
  of 
  the 
  Negative 
  System 
  of 
  31 
  to 
  the 
  " 
  Generalized 
  

   Key-board." 
  

  

  The 
  fifths 
  are 
  a 
  little 
  better 
  than 
  in 
  the 
  last 
  case, 
  viz. 
  -05181 
  flat 
  ; 
  the 
  

   thirds 
  -00783 
  sharp. 
  The 
  only 
  difference 
  in 
  the 
  employment 
  of 
  the 
  

   system 
  is 
  that 
  the 
  arrangement 
  is 
  cyclical. 
  The 
  tones 
  all 
  consist 
  of 
  

   five 
  units, 
  semitones 
  of 
  three. 
  

  

  The 
  Investigation 
  of 
  Cycles 
  of 
  the 
  Higher 
  Orders 
  — 
  the 
  new 
  Cycle 
  of 
  643 
  

  

  and 
  others. 
  

  

  The 
  system 
  of 
  301 
  is 
  of 
  interest, 
  as 
  combining 
  the 
  properties 
  of 
  a 
  

   tolerably 
  good 
  positive 
  cyclical 
  system 
  with 
  the 
  representation 
  of 
  intervals 
  

   accurately 
  to 
  three 
  places 
  by 
  means 
  of 
  logarithms. 
  This 
  system 
  has 
  been 
  

   lately 
  used, 
  in 
  particular 
  by 
  Mr. 
  Ellis, 
  for 
  approximate 
  calculations. 
  It 
  

   appears 
  to 
  be 
  of 
  some 
  interest 
  to 
  investigate 
  generally 
  what 
  systems 
  of 
  

   higher 
  orders 
  do 
  represent 
  either 
  of 
  the 
  systems 
  with 
  perfect 
  thirds, 
  and 
  

   with 
  what 
  degree 
  of 
  accuracy 
  they 
  do 
  so. 
  

  

  First, 
  with 
  respect 
  to 
  positive 
  systems. 
  If 
  a 
  system 
  n 
  of 
  the 
  Hh 
  order 
  

  

  be 
  a 
  close 
  approximation 
  to 
  the 
  system 
  of 
  perfect 
  thirds, 
  then 
  will 
  — 
  8- 
  

  

  n 
  

  

  (the 
  departure 
  of 
  its 
  third) 
  approximate 
  in 
  value 
  to 
  — 
  -13686 
  ; 
  or 
  

  

  r 
  -13686 
  1 
  

   n=-S- 
  = 
  58-4526 
  near1 
  ^' 
  

  

  or 
  

  

  n 
  = 
  r 
  58*4526 
  nearly. 
  

  

  