﻿1874.] 
  

  

  Mr. 
  A. 
  J. 
  Ellis 
  on 
  Musical 
  Duodenes. 
  

  

  9 
  

  

  of 
  ear 
  in 
  the 
  case 
  of 
  skhistic 
  temperament, 
  where 
  the 
  Fifths 
  have 
  to 
  be 
  

   flattened 
  by 
  an 
  almost 
  inaudible 
  skhist, 
  is 
  so 
  much 
  enhanced 
  as 
  to 
  be 
  

   insuperable 
  except 
  by 
  Scheibler's 
  method 
  l 
  . 
  Hence 
  it 
  is 
  necessary 
  to 
  find 
  

   a 
  practical 
  substitute. 
  This 
  I 
  term 
  

  

  Unequally 
  Just 
  Intonation. 
  — 
  Suppose 
  that 
  the 
  48 
  tones 
  marked 
  on 
  2 
  by 
  

   a 
  dotted 
  line 
  in 
  Table 
  I. 
  have 
  to 
  be 
  tuned 
  in 
  this 
  substitute 
  for 
  skhistic 
  

   intonation. 
  Tune 
  C 
  to 
  the 
  fork. 
  Take 
  4 
  just 
  Fifths 
  up 
  (or 
  Fourths 
  

   down), 
  C 
  to 
  Gr, 
  Gr 
  to 
  D, 
  D 
  to 
  f 
  A, 
  fA 
  to 
  f 
  E, 
  without 
  beats 
  ; 
  and 
  three 
  just 
  

   Fifths 
  down 
  (or 
  Fourths 
  up), 
  C 
  to 
  F, 
  F 
  to 
  B[>, 
  B\> 
  to 
  E[>. 
  Then 
  tune 
  C 
  

   to 
  E 
  as 
  a 
  just 
  major 
  Third 
  up, 
  without 
  beats, 
  and 
  from 
  E 
  proceed 
  to 
  its 
  

   just 
  Fifth 
  B, 
  verifying 
  the 
  result 
  by 
  determining 
  that 
  it 
  is 
  a 
  just 
  major 
  

   Third 
  above 
  Gr, 
  and 
  so 
  on 
  to 
  Gr# 
  up 
  and 
  $Gr 
  down. 
  Then 
  if 
  from 
  G# 
  we 
  

   proceeded 
  to 
  the 
  just 
  Fifth, 
  D#, 
  the 
  resulting 
  tone 
  would 
  be 
  exactly 
  one 
  

   skhisma 
  sharper 
  than 
  E|?, 
  whereas 
  in 
  skhistic 
  intonation 
  it 
  would 
  be 
  

   identical 
  with 
  E|?, 
  as 
  already 
  shown. 
  It 
  is 
  needless 
  to 
  say 
  that 
  no 
  tuner 
  

   could 
  effect 
  this 
  exact 
  difference 
  of 
  a 
  skhisma, 
  but 
  he 
  will 
  come 
  practically 
  

   near 
  it, 
  and 
  the 
  error 
  is 
  that 
  of 
  the 
  Fifth 
  (the 
  least 
  of 
  the 
  errors) 
  in 
  equal 
  

   temperament. 
  If 
  we 
  were 
  to 
  proceed 
  in 
  this 
  way 
  for 
  all 
  the 
  six 
  columns 
  

   of 
  8 
  tones 
  marked 
  off 
  in 
  Table 
  I., 
  we 
  should 
  have 
  just 
  major 
  Thirds 
  

   throughout, 
  and 
  just 
  Fifths 
  also 
  in 
  all 
  but 
  5 
  cases 
  — 
  namely, 
  £D## 
  to 
  JB, 
  

   B# 
  to 
  i&, 
  Gr# 
  to 
  Efc>, 
  fE 
  to 
  C\), 
  and 
  fC 
  to 
  fA\>\>, 
  each 
  of 
  which 
  would 
  be 
  

   too 
  flat 
  by 
  one 
  skhisma. 
  Since 
  iGt 
  (Table 
  I. 
  col. 
  6, 
  line 
  as) 
  is 
  a 
  just 
  major 
  

   Third 
  below 
  +B, 
  and 
  a 
  just 
  minor 
  Third 
  above 
  +E, 
  but 
  JE 
  is 
  a 
  whole 
  

   skhisma 
  flatter 
  than 
  +D## 
  (col. 
  8, 
  line 
  p), 
  which 
  would 
  be 
  played 
  for 
  JE, 
  

   it 
  follows 
  that 
  the 
  minor 
  Third, 
  +D#lf 
  to 
  iGt, 
  would 
  be 
  a 
  skhisma 
  too 
  flat 
  

   or 
  close 
  ; 
  and 
  similarly 
  that 
  the 
  minor 
  Thirds, 
  B# 
  to 
  E[>, 
  Gr# 
  to 
  Op, 
  and 
  

   f 
  E 
  to 
  tA^, 
  or 
  4 
  minor 
  Thirds 
  on 
  the 
  whole, 
  wOuld 
  be 
  a 
  skhisma 
  too 
  

   flat 
  2 
  . 
  Hence 
  this 
  style 
  of 
  tuning 
  gives 
  5 
  Fifths 
  and 
  4 
  minor 
  Thirds, 
  as 
  

  

  1 
  Calculate 
  the 
  logarithms 
  of 
  the 
  ratios 
  of 
  all 
  the 
  skhistic 
  tones 
  by 
  perpetual 
  addition 
  

   or 
  subtraction 
  of 
  04 
  760300 
  (=log 
  § 
  — 
  log 
  <r) 
  to 
  or 
  from 
  0, 
  continually 
  adding 
  or 
  sub- 
  

   tracting 
  0*3010300 
  (=log 
  2) 
  to 
  make 
  the 
  results 
  positive 
  and 
  lie 
  between 
  this 
  and 
  0. 
  

   Add 
  the 
  logarithm 
  of 
  the 
  vibrational 
  number 
  of 
  C, 
  and 
  then 
  find 
  the 
  numbers 
  (to 
  three 
  

   places 
  of 
  decimals) 
  corresponding 
  to 
  these 
  logarithms. 
  This 
  gives 
  the 
  vibrational 
  

   numbers 
  of 
  all 
  the 
  skhistic 
  tones 
  in 
  the 
  Octave, 
  of 
  which 
  48 
  will 
  be 
  required. 
  Subtract 
  

   4 
  from 
  each 
  of 
  these 
  values, 
  and 
  procure 
  tuning-forks 
  giving 
  exactly 
  the 
  tones 
  thus 
  

   determined 
  to 
  at 
  least 
  the 
  hundredth 
  of 
  a 
  vibration 
  in 
  a 
  second. 
  These 
  maybe 
  obtained 
  

   of 
  the 
  great 
  manufacturer 
  of 
  acoustic 
  apparatus, 
  Mons. 
  R. 
  Koenig, 
  of 
  Paris 
  ; 
  but 
  it 
  is 
  

   necessary 
  to 
  state 
  that 
  the 
  English 
  and 
  German 
  (not 
  the 
  French) 
  system 
  of 
  counting 
  

   vibrations 
  is 
  to 
  be 
  used. 
  Then 
  tune 
  each 
  tone 
  roughly 
  to 
  the 
  corresponding 
  fork, 
  and 
  

   afterwards 
  sharpen 
  it 
  until 
  it 
  beats 
  4 
  times 
  in 
  a 
  second 
  with 
  the 
  fork. 
  By 
  this 
  means, 
  

   and 
  by 
  this 
  means 
  only, 
  with 
  great 
  care 
  and 
  attention, 
  the 
  pitch 
  may 
  probably 
  be 
  ob- 
  

   tained 
  with 
  sufficient 
  accuracy 
  to 
  distinguish 
  skhistic 
  from 
  just 
  intonation. 
  And 
  simi- 
  

   larly 
  for 
  equal 
  and 
  tertian 
  intonation. 
  

  

  2 
  Taking 
  the 
  cycle 
  of 
  30103, 
  +G 
  contains 
  17070, 
  +B 
  26761, 
  and 
  +D#tf 
  9200 
  octs. 
  

   Hence 
  the 
  Fifth, 
  JD#Jf 
  to 
  JB, 
  has 
  only 
  17561 
  in 
  place 
  of 
  17609 
  octs, 
  and 
  the 
  minor 
  

   Third, 
  |D#$ 
  to 
  jG, 
  only 
  7870, 
  in 
  place 
  of 
  7918 
  octs— 
  that 
  is, 
  in 
  each 
  case 
  48 
  octs 
  too 
  

   little 
  ; 
  that 
  is, 
  these 
  intervals 
  are 
  one 
  skhisma 
  too 
  flat. 
  

  

  