The expression of a relation between two numerical quantities. The two parts of a fraction are called the numerator and the denominator.
Fractions are important in tuning theory because a ratio may be expressed as a fraction, and the mathematical procedures used in manipulating fractions may thus be used to calculate musical ratios.
Fractions are also another way to express the operation of division: the numerator is the quantity being divided, and the denominator is the divisor.
As long as both the numerator and denominator are multiplied or divided by the same number, the relationship does not change. The resulting fractions thus express the same numerical quantity as the original fraction.
This kind of manipulation is important in the addition and subtraction of fractions, because to add or subtract fractions, the denominators must be the same. Thus, if the two fractions to be added have different denominators, both the numerator and denominator of one or both of the fractions must be multiplied by some number until both denominators are the same. The numerators may then be added, and the resulting fraction is the quantity resulting from the addition. Subtraction works in exactly the same way.
Multiplication of fractions is more straightforward: the numerators are multiplied together, and the denominators are multiplied together, and the resulting fraction is the quantity resulting from the multiplication. Division of fractions is performed by inverting the terms of the second fraction and then multiplying the numerators and denominators.
(An alternative and much simpler method of calculating intervals is to use vector addition.)
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