Definitions of tuning terms

© 1998 by Joseph L. Monzo

All definitions by Joe Monzo unless otherwise cited

tuning unit


    A term used by John Brombaugh to designate the tiny interval describing 1/720th part of the Pythagorean comma.

    The tuning unit is calculated as the 720th root of the ratio 531441:524288, or

    [ 2-19 * 312 ](1/720)
    thus with a ratio itself of approximately 1:1.000018821. It is an irrational number. The width of this tuning unit interval is ~0.032583348 (pretty close to 1/31) cent, and exactly 1/60 grad.

    This interval therefore divides the "octave", which is assumed to have the ratio 2:1, into ~36828.6282 equal parts. Thus the tuning unit represents one degree in ~36828.6282-EDO "non-octave" tuning, or its audibly identical "octave"-based relative 36829-EDO.

    There are just over 3069 tuning units (a more exact figure is ~3069.05235, about 30691/19) in a Semitone.

    The formula for calculating the tuning-unit-value of any ratio is:

    tuning units = log10(ratio) / log10[ 2(-19/720) * 3(1/60) ]


    A tuning-unit is ~0.817380431 (~9/11) jot, ~1.334613924 (~1 1/3) cawapus, ~5.338455696 (~5 1/3) midipus.

    Tuning-unit-sizes for some small intervals, with cents-values given for comparison: 

          interval           tuning units            cents

Pythagorean comma   720                      ~23.46001038
   syntonic comma  ~660.0392862              ~21.5062896
          kleisma  ~248.8166324 (~248 4/5)    ~8.107278862
      heptameride  ~122.3542465 (~122 1/3)    ~3.986710963
           savart  ~122.762094 (122 3/4)       4
             grad    60                       ~1.955000865
          skhisma   ~59.96071375 (~59 49/51)  ~1.953720788
         milli8ve   ~36.8286282  (~36 5/6)     1.2
        Türk cent   ~34.74398887 (~34 3/4)    ~1.132075472
             cent   ~30.6905235  (~30 2/3)     1
              jot    ~1.223420529 (~1 2/9)    ~0.039863137
           cawapu    ~0.749280359 (~3/4)      ~0.024414062
           midipu    ~0.18732009  (~1/5)      ~0.006103516

    See also Manuel Op de Coul's Logarithmic Interval Measures.

    [from Joe Monzo, JustMusic: A New Harmony]

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