Monzo: Difference between revisions

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== Examples ==

For example, the interval [[15/8]] can be thought of as having ~~<math>~~5 ~~\cdot~~ 3~~</math>~~ in the numerator, and ~~<math>~~2 ~~\cdot~~ 2 ~~\cdot~~ 2~~</math>~~ in the denominator. This can be compactly represented by the expression <~~math~~>~~2^{~~-3~~} \cdot~~ 3^1 ~~\cdot~~ 5^1</~~math~~>, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the {{monzo| … }} brackets, hence yielding {{monzo| -3 1 1 }}.

To find the monzo of an interval in [[ratio]] form, factor the entire ratio as a product of primes, each raised to an exponent. For primes appearing in the denominator, these exponents will be negative. (A prime never appears in both the numerator and the denominator.) Arrange the primes in ascending order. If any primes smaller than the largest prime do not appear, include them using a zero exponent. Enter the exponents into the monzo.

For example, the interval [[15/8]] can be thought of as having 5 × 3 in the numerator, and 2 × 2 × 2 in the denominator. This can be compactly represented by the expression 2-3 × 31 × 51, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the {{monzo| … }} brackets, hence yielding {{monzo| -3 1 1 }}.

Here are some common [[5-limit]] monzos, along with their factorizations to show how to derive them:

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 == Examples ==
-For example, the interval [[15/8]] can be thought of as having <math>5 \cdot 3</math> in the numerator, and <math>2 \cdot 2 \cdot 2</math> in the denominator. This can be compactly represented by the expression <math>2^{-3} \cdot 3^1 \cdot 5^1</math>, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the {{monzo| … }} brackets, hence yielding {{monzo| -3 1 1 }}.
+To find the monzo of an interval in [[ratio]] form, factor the entire ratio as a product of primes, each raised to an exponent. For primes appearing in the denominator, these exponents will be negative. (A prime never appears in both the numerator and the denominator.) Arrange the primes in ascending order. If any primes smaller than the largest prime do not appear, include them using a zero exponent. Enter the exponents into the monzo.
+For example, the interval [[15/8]] can be thought of as having 5 × 3 in the numerator, and 2 × 2 × 2 in the denominator. This can be compactly represented by the expression 2<sup>-3</sup> × 3<sup>1</sup> × 5<sup>1</sup>, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the {{monzo| … }} brackets, hence yielding {{monzo| -3 1 1 }}.
 Here are some common [[5-limit]] monzos, along with their factorizations to show how to derive them: