﻿8 
  

  

  Mr. 
  A. 
  J. 
  Ellis 
  on 
  Musical 
  Duodenes. 
  [Nov. 
  19, 
  

  

  No. 
  

  

  C^vpl 
  P 
  CiT 
  

  

  on. 
  

  

  Fifth, 
  

   v. 
  

  

  ilXtlJUI 
  -LJLi.il 
  U.; 
  

  

  t. 
  

  

  KyKJlxXLLLCXj 
  

  

  q. 
  

  

  z. 
  

  

  1 
  

  

  30103 
  

  

  17609 
  

  

  9691 
  

  

  539 
  

  

  48 
  

  

  2 
  

  

  3010 
  

  

  1761 
  

  

  969 
  

  

  55 
  

  

  7 
  

  

  3 
  

  

  301 
  

  

  176 
  

  

  97 
  

  

  5 
  

  

  

  

  4 
  

  

  53 
  

  

  31 
  

  

  17 
  

  

  1 
  

  

  

  

  5 
  

  

  53 
  

  

  31 
  

  

  18 
  

  

  

  

  1 
  

  

  6 
  

  

  31 
  

  

  18 
  

  

  10 
  

  

  

  

  -1 
  

  

  7 
  

  

  12 
  

  

  7 
  

  

  4 
  

  

  

  

  

  

  Of 
  these, 
  the 
  first 
  three 
  are 
  here, 
  I 
  believe, 
  for 
  the 
  first 
  time 
  shown 
  to 
  

   be 
  true 
  cyclic 
  temperaments. 
  

  

  1. 
  The 
  cycle 
  of 
  30103 
  is 
  such 
  an 
  excellent 
  representative 
  of 
  just 
  

   intonation 
  (giving 
  even 
  6 
  octs 
  for 
  the 
  skhist), 
  that 
  it 
  can 
  be 
  used 
  without 
  

   sensible 
  error, 
  in 
  place 
  of 
  ordinary 
  logarithms, 
  to 
  reduce 
  the 
  relations 
  of 
  

   intervals 
  to 
  addition 
  and 
  subtraction, 
  for 
  general 
  use 
  among 
  musicians 
  

   or 
  learners 
  unacquainted 
  with 
  higher 
  arithmetic. 
  By 
  dividing 
  out 
  by 
  

   100,000 
  we 
  obtain 
  almost 
  precisely 
  the 
  five-figure 
  logarithms 
  of 
  the 
  

   intervals. 
  

  

  2. 
  The 
  cycle 
  of 
  3010 
  is 
  almost 
  as 
  Correct, 
  with 
  smaller 
  numbers. 
  

  

  3. 
  The 
  cycle 
  of 
  301 
  is 
  almost 
  a 
  perfect 
  representation 
  of 
  skhistic 
  

   temperament, 
  in 
  which 
  the 
  skhisma 
  is 
  eliminated, 
  and 
  for 
  that 
  reason 
  

   becomes 
  perhaps 
  the 
  most 
  practical 
  representation 
  of 
  general 
  musical 
  

   intonation. 
  

  

  4. 
  The 
  first 
  cycle 
  of 
  53 
  is 
  Nicholas 
  Mercator's 
  representation 
  of 
  just 
  

   intonation, 
  but 
  it 
  is 
  more 
  correctly 
  a 
  representation 
  of 
  skhistic 
  tempera- 
  

   ment, 
  and 
  not 
  so 
  good 
  as 
  No. 
  3 
  1 
  . 
  

  

  5. 
  The 
  second 
  cycle 
  of 
  53 
  is 
  a 
  very 
  accurate 
  representation 
  of 
  Py- 
  

   thagorean 
  intonation, 
  and 
  has 
  actually 
  been 
  proposed 
  for 
  the 
  violin 
  by 
  

   Drobisch. 
  

  

  6. 
  The 
  cycle 
  of 
  31 
  is 
  Huyghens's 
  Cyclus 
  Harmonicus, 
  and 
  closely 
  repre- 
  

   sents 
  the 
  tertian 
  or 
  mean 
  temperament. 
  

  

  7. 
  The 
  cycle 
  of 
  12 
  is 
  the 
  ordinary 
  equal 
  temperament, 
  and 
  its 
  principal 
  

   convenience 
  consists 
  in 
  the 
  very 
  small 
  number 
  of 
  its 
  octs, 
  here 
  called 
  

   Semitones. 
  

  

  Unequal 
  Temperaments, 
  whether 
  they 
  consist 
  of 
  12 
  selected 
  tones 
  from 
  

   uniform 
  temperaments, 
  or 
  of 
  12 
  tones 
  turned 
  intentionally 
  false 
  (see 
  my 
  

   former 
  paper, 
  Proc. 
  vol. 
  xiii. 
  pp. 
  414-417, 
  for 
  their 
  theory), 
  are 
  now 
  aban- 
  

   doned. 
  But 
  the 
  difficulty 
  of 
  tuning 
  equal 
  temperament 
  by 
  estimation 
  of 
  

   ear, 
  or 
  even 
  by 
  the 
  monochord, 
  and 
  of 
  retaining 
  the 
  intonation 
  of 
  the 
  

   piano 
  or 
  organ 
  unchanged 
  for 
  even 
  an 
  hour, 
  makes 
  all 
  temperaments 
  in 
  

   actual 
  use 
  really 
  unequal. 
  The 
  difficulty 
  of 
  original 
  tuning 
  by 
  estimation 
  

  

  1 
  When 
  this 
  paper 
  was 
  read, 
  I 
  mentioned 
  that 
  this 
  was 
  the 
  cycle 
  used 
  by 
  Mr. 
  

   jBosanquet 
  in 
  his 
  paper 
  read 
  before 
  the 
  Eoyal 
  Society 
  on 
  30th 
  January, 
  1873. 
  

  

  