Türk cent

Turkish classical music uses Pythagorean tuning. A very good approximation to this is provided by 53edo, one degree of which is nearly the same as both the Pythagorean comma and the syntonic comma. Turkish theorists have further subdivided the 53edo degree into 20 parts, making 1060 to the "8ve".

The Türk cent is thus calculated as the 1060th root of 2, or 2^(1/1060), with a ratio of approximately 1:1.000654126. It is an irrational number. The width of the Türk cent is ~1.132075472 (~1 ¹/₈) cents, ~28.3990566 (~28 ²/₅) jots, ~34.74398887 (~34 ³/₄) tuning-units, ~46.36981132 (~46 ¹/₃) cawapus, and ~185.4792453 (~185 ¹/₂) midipus.

There are exactly 88 ¹/₃ Türk cents in a Semitone.

The formula for calculating the Türk-cents-value of any ratio is:

Türk-cents = log₁₀(ratio) * [ 1060 / log₁₀(2) ]


Türk-cent-sizes for some small intervals, with regular (Ellis) cents-values given for comparison:

      interval           Türk-cents            (Ellis) cents

Pythagorean comma   ~20.72300917 (~20 5/7)      ~23.46001038
     53edo degree    20                         ~22.64150943
   syntonic comma   ~18.99722248 (~18 359/360)   ~21.5062896
          kleisma    ~7.161429661 (~7 1/6)       ~8.107278862
      heptameride    ~3.521594684 (~3 1/2)       ~3.986710963
           savart    ~3.533333333 (3 8/15)        4
             grad    ~1.726917431 (~1 5/7)       ~1.955000865
          skhisma    ~1.725786696 (~1 5/7)       ~1.953720788
         milli8ve     1.06 (1 3/50)               1.2
             cent    ~0.883333333 (53/60 = ~8/9)  1
         monzisma    ~0.258282877 (~1/4)         ~0.29239571
              jot    ~0.035212437 (~1/28)        ~0.039863137
      tuning-unit    ~0.028781957 (~1/35)        ~0.032583348
           cawapu    ~0.021565755 (~1/46)        ~0.024414062
           midipu    ~0.005391439 (~1/185)       ~0.006103516

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

[from Joe Monzo, JustMusic: A New Harmony]

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