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The Mathematics of Various Tuning Systems

  • 1.  The Mathematics of Various Tuning Systems

    Posted 07-12-2018 12:44
    Edited by Roshan Kakiya 07-12-2018 12:53

    Just Intonation

    Formula: 1200 × log₂(Frequency ratio).

    Example: Pure fifth = 1200 × log₂(3/2) = 701.96 cents.

    Alternative formula for calculators: 1200 × log₁₀(Frequency ratio)/log₁₀(2).

    Example: Pure fifth = 1200 × log₁₀(3/2)/log₁₀(2) = 701.96 cents.



    Equal Temperament


    Formula: 1200 × (Number of semitones within a specific interval/Number of semitones within an octave).

    Example 1: 12-TET fifth = 1200 × (7/12) = 700 cents.

    Example 2: 53-TET fifth = 1200 × (31*/53) = 701.89 cents.

    * 53 × 7/12 = 31.



    Well Temperament


    [Linked Image]



     Flattened Fifths


    Formula: Flattened fifth = 701.96 cents − (the amount by which the Pythagorean comma is being divided × the value of the Pythagorean comma in cents).

    Example: Pure fifth flattened by 1/6 of the Pythagorean comma = 701.96 cents − (1/6 × 23.46 cents) = 701.96 cents − 3.91 cents = 698.05 cents.


    Sharpened Fifths


    Formula: Sharpened fifth = 701.96 cents + (the amount by which the Pythagorean comma is being divided × the value of the Pythagorean comma in cents).

    Example: Pure fifth sharpened by 1/12 of the Pythagorean comma = 701.96 cents + (1/12 × 23.46 cents) = 701.96 cents + 1.96 cents = 703.92 cents.



    Pythagorean Comma


    12 Pure fifths = 701.955 cents × 12 = 8423.46 cents.

    7 Octaves = 1200 cents × 7 = 8400 cents.

    Pythagorean comma = 12 pure fifths − 7 octaves = 8423.46 cents − 8400 cents = 23.46 cents.

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    Roshan Kakiya
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