limma, leimma

(Greek: "remnant", plural: leimmata, limmata)

• 1. The Pythagorean diatonic semitone, the interval left after subtracting two Whole Tones from the Perfect Fourth. It has a ratio of 256/243 and 90 cents.
  
  

  [from John Chalmers, Divisions of the Tetrachord]

  . . . . . . . . . . . . . . . . . . . .

  
  In prime factor notation this interval is written 2⁸3^-5.

  The limma can be calculated thus by regular fractional math:

  
  4   81       4   64       256
  - ÷ --   =   - * --   =   ---
  3   64       3   81       243
  
  

  or by vector addition:

      2  3
  
    [ 2 -1]           4/3
  - [-6  4]       ÷  81/64
  ---------   =   ---------
    [ 8 -5]         256/243
  
  

  Below is a diagram illustrating this description, on an approximate logarithmic scale:

  
  
  ratio      vector
              2  3
  
  A  1/1  -+- [ 0  0]   
           |            \
           |             \
           |              \
  G  9/8  -+- [-3  2]      81/64   [-6  4]  =  ditone
           |              /
           |             /
           |            /
  F 81/64 -+- [-6  4]   \
           |              256/243  [ 8 -5]  =  limma
  E  4/3  -+- [ 2 -1]   /
  
  

  A more accurate logarithmic value for it is ~90.22499567 cents.

  
  

• 2. The term used by W. S. B. Woolhouse (in his Essay on Musical Intervals, Harmonics, and Temperament) to refer 16/15 [= ~ 111.731 cents], the ratio of the 5-limit diatonic semitone.

  As the size of this 16/15 interval resembles another Pythagorean semitone -- the apotome -- much more closely, Woolhouse perhaps should have used that name instead. Apparently he based his terminology on the function of this semitone, for the apotome is the Pythagorean chromatic semitone while the limma is the Pythagorean diatonic semitone.

  
  [from Joe Monzo, JustMusic: A New Harmony]

  
  see also

  apotome,
  anomaly,
  diesis,
  comma,
  kleisma,
  skhisma,
  5-limit intervals, 100 cents and under
  Tutorial on ancient Greek tetrachord-theory

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updated:

2002.09.12
2002.01.05

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