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
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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