Definitions of tuning terms

© 1998 by Joseph L. Monzo

All definitions by Joe Monzo unless otherwise cited


trihemitone


    (Greek: "3 half-tones")

    The Pythagorean minor 3rd, composed of 3 semitones, with ratio 32/27, ~294.1349974 cents.

    The semitones are of two different sizes, because the trihemitone is composed of a tone and a limma; the tone in turn is composed of a limma and an apotome; thus the trihemitone equals 2 limmata and 1 apotome.

    The trihemitone may also be found as a perfect 4th minus a tone.


    In prime factor notation this interval is written 253-3.

    The trihemitone can be calculated thus by regular fractional math:

    
    4   9       4   8       32
    - ÷ -   =   - * -   =   --
    3   8       3   9       27
    
    

    or by vector addition:

        2   3
    
      [ 2 -1]          4/3
    - [-3  2]       ÷  9/8
    ---------   =   -------
      [ 5 -3]         32/27
    
    

    Below is a diagram illustrating these descriptions, on an approximate logarithmic scale:

    
    

    ratio vector 2 3 / A 1/1 -+- [ 0 0] / | \ 9/8 [-3 2] = tone | \ \ | 32/27 [ 5 -3] = trihemitone / G 9/8 -+- [-3 2] / 256/243 [ 8 -5] = limma | / \ F# 32/27 + [ 5 -3] \ | \ F 81/64 -+- [-6 4] 9/8 [-3 2] = tone | / E 4/3 -+- [ 2 -1] /


    [from Joe Monzo, JustMusic: A New Harmony]


    see also

    limma,
    minor 3rd,
    major 3rd,
    Tutorial on ancient Greek tetrachord-theory


updated:

2003.06.09 -- fixed error: link to "perfect 4th" was incorrectly written.
2002.10.05 -- fixed error: ratio had been given as 2187/2048 instead of 32/27.
2002.09.12 -- created


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