﻿1874.] 
  Mr. 
  A. 
  J. 
  Ellis 
  on 
  Musical 
  Duodenes. 
  7 
  

  

  my 
  name 
  for 
  Since 
  log 
  f 
  =88 
  log 
  a- 
  very 
  nearly, 
  one 
  comma 
  may 
  

  

  be 
  said 
  to 
  contain 
  88 
  skhists. 
  Skhistic 
  temperament 
  is 
  indistinguishable 
  

  

  from 
  just 
  intonation, 
  and 
  I 
  shall 
  use 
  it 
  in 
  the 
  theory 
  of 
  constructing 
  

  

  instruments. 
  If 
  in 
  Table 
  I. 
  we 
  suppose 
  the 
  horizontal 
  intervals 
  to 
  be 
  still 
  

  

  5 
  3 
  81 
  

  

  ^, 
  but 
  the 
  vertical 
  intervals 
  to 
  be 
  and 
  the 
  sign 
  f 
  to 
  stand 
  for 
  ^ 
  V 
  Jk» 
  

  

  while 
  tj=l 
  as 
  before, 
  then 
  this 
  Table 
  represents 
  skhistic 
  relations 
  ; 
  so 
  that 
  

   if 
  from 
  any 
  note, 
  as 
  EJ? 
  (col-. 
  5, 
  line 
  cc), 
  we 
  proceed 
  by 
  8 
  skhistic 
  Fifths 
  

   up 
  to 
  fB 
  and 
  then 
  one 
  major 
  Third 
  to 
  the 
  right, 
  we 
  find 
  a 
  tone 
  D#, 
  

   which, 
  when 
  reduced 
  to 
  the 
  same 
  Octave, 
  is 
  identical 
  with 
  E£>. 
  We 
  thus 
  

   find 
  a 
  number 
  of 
  schistic 
  synonyms 
  shown 
  in 
  Tables 
  II. 
  & 
  III. 
  

  

  We 
  may 
  express 
  the 
  errors 
  in 
  the 
  temperaments 
  just 
  discussed 
  in 
  terms 
  

   of 
  skhists 
  thus, 
  using 
  #56o- 
  to 
  mean 
  " 
  too 
  sharp 
  by 
  56 
  skhists," 
  and 
  so 
  

   on, 
  and 
  to 
  mean 
  " 
  no 
  error 
  ": 
  — 
  

  

  Just. 
  

  

  1. 
  Quintal. 
  

  

  2. 
  Tertian. 
  

  

  3. 
  

  

  Equal. 
  

  

  4. 
  Skhismic. 
  

  

  5. 
  Skhistic. 
  

  

  Minor 
  Third 
  

  

  t>88<r 
  

  

  t>22(7 
  

  

  

  ?64(7 
  

  

  #8(7 
  

  

  

  Major 
  Third 
  

  

  #88(7 
  

  

  

  

  

  £56(7 
  

  

  \>8<r 
  

  

  

  

  Fourth 
  

  

  

  

  #22 
  ff 
  

  

  

  t 
  8(7 
  

  

  

  

  #1(7 
  

  

  Fifth 
  

  

  

  

  \>22<r 
  

  

  

  ? 
  8(7 
  

  

  

  

  

  Minor 
  Sixth 
  

  

  t>88(7 
  

  

  

  

  

  ?56(7 
  

  

  #8(7 
  

  

  

  

  Major 
  Sixth 
  

  

  #88(7 
  

  

  #22(7 
  

  

  #64(7 
  

  

  \>$<r 
  

  

  #1(7 
  

  

  The 
  error 
  of 
  one 
  skhist 
  is 
  quite 
  inappreciable 
  by 
  the 
  most 
  practised 
  

   ears 
  in 
  melody, 
  and 
  can 
  be 
  detected 
  harmonically 
  only 
  by 
  very 
  slow 
  beats 
  

   for 
  tones 
  in 
  the 
  highest 
  Octaves 
  used 
  in 
  music. 
  

  

  6. 
  Cyclic 
  Temperaments. 
  The 
  following 
  method 
  is 
  far 
  more 
  general 
  

   than 
  that 
  given 
  in 
  my 
  previous 
  paper 
  (Proc. 
  vol. 
  xiii. 
  p. 
  412). 
  Put 
  

  

  m 
  . 
  log 
  Y=v 
  . 
  log 
  2, 
  m 
  . 
  log 
  Jc=q 
  . 
  log 
  2, 
  

  

  m 
  . 
  log 
  T 
  = 
  t 
  . 
  log 
  2, 
  m 
  . 
  log 
  s 
  = 
  z 
  . 
  log 
  2, 
  

  

  and, 
  after 
  substituting 
  these 
  values 
  for 
  log 
  Y, 
  log 
  T, 
  log 
  Jc, 
  log 
  s 
  in 
  the 
  

   logarithms 
  of 
  equations 
  (5) 
  and 
  (6), 
  divide 
  out 
  by 
  log 
  2, 
  and 
  multiply 
  up 
  

   by 
  m. 
  Then 
  

  

  12v 
  — 
  7m 
  = 
  q 
  + 
  z, 
  m—3t 
  = 
  2q—z 
  (9,10) 
  

  

  Take 
  any 
  integral 
  values 
  for 
  q 
  and 
  z, 
  and 
  find 
  the 
  integral 
  values 
  which 
  

   satisfy 
  one 
  of 
  these 
  indeterminate 
  equations 
  for 
  v, 
  m, 
  or 
  t, 
  m, 
  and 
  substi- 
  

   tute 
  in 
  the 
  other, 
  taking 
  the 
  resulting 
  integral 
  values 
  of 
  t 
  or 
  v 
  respec- 
  

   tively. 
  The 
  five 
  integral 
  values 
  determine 
  a 
  cycle 
  in 
  which 
  the 
  Octave 
  is 
  

   divided 
  into 
  m 
  aliquot 
  parts, 
  which 
  may 
  be 
  termed 
  octs, 
  v 
  of 
  which 
  make 
  

   a 
  Fifth, 
  t 
  a 
  major 
  Third, 
  q 
  a 
  comma, 
  and 
  z 
  a 
  skhisma 
  of 
  " 
  the 
  cycle 
  of 
  m." 
  

   Most 
  of 
  the 
  results 
  are 
  valueless, 
  but 
  the 
  following 
  present 
  either 
  theo- 
  

   retical 
  convenience 
  or 
  historical 
  interest 
  : 
  — 
  

  

  