Monzo: Difference between revisions
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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 }}. | 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 }}. | ||
Here are some common [[5-limit]] monzos, along with their factorizations to show how to derive them: | |||
Here are some common [[5-limit]] monzos, | |||
{| class="wikitable center-1" | {| class="wikitable center-1" | ||
|- | |- | ||
! Ratio | ! Ratio | ||
! Factors | |||
! Monzo | ! Monzo | ||
|- | |- | ||
| [[3/2]] | | [[3/2]] | ||
| <math>2^{-1} \cdot 3</math> | |||
| {{monzo| -1 1 0 }} | | {{monzo| -1 1 0 }} | ||
|- | |- | ||
| [[5/4]] | | [[5/4]] | ||
| <math>2^{-2} \cdot 5</math> | |||
| {{monzo| -2 0 1 }} | | {{monzo| -2 0 1 }} | ||
|- | |- | ||
| [[9/8]] | | [[9/8]] | ||
| <math>2^{-3} \cdot 3^2</math> | |||
| {{monzo| -3 2 0 }} | | {{monzo| -3 2 0 }} | ||
|- | |- | ||
| [[81/80]] | | [[81/80]] | ||
| <math>2^{-4} \cdot 3^4 \cdot 5^{-1}</math> | |||
| {{monzo| -4 4 -1 }} | | {{monzo| -4 4 -1 }} | ||
|} | |} | ||
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|- | |- | ||
! Ratio | ! Ratio | ||
! Factors | |||
! Monzo | ! Monzo | ||
|- | |- | ||
| [[7/4]] | | [[7/4]] | ||
| <math>2^{-2} \cdot 7</math> | |||
| {{monzo| -2 0 0 1 }} | | {{monzo| -2 0 0 1 }} | ||
|- | |- | ||
| [[7/6]] | | [[7/6]] | ||
| <math>2^{-1} \cdot 3^{-1} \cdot 7</math> | |||
| {{monzo| -1 -1 0 1 }} | | {{monzo| -1 -1 0 1 }} | ||
|- | |- | ||
| [[7/5]] | | [[7/5]] | ||
| <math>5^{-1} \cdot 7</math> | |||
| {{monzo| 0 0 -1 1 }} | | {{monzo| 0 0 -1 1 }} | ||
|} | |} | ||
:'''Practical hint:''' On the wiki, the monzo template helps you getting correct brackets ([[Template:Monzo|read more…]]). | |||
== Relationship with vals == | == Relationship with vals == | ||