Ratio math: Difference between revisions
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This page describes the way we manipulate [[ratios]] arithmetically. Some of it is just math, and some of it is specific to music. | This page describes the way we manipulate [[ratios]] arithmetically. Some of it is just math, and some of it is specific to music. | ||
[[Jake Freivald|I]] am not trying to format this particularly nicely, since a lot of these kinds of discussions come up on Facebook and similar places, and you may as well see it ugly here so you understand it when it's ugly there. | [[Jake Freivald|I]] am not trying to format this particularly nicely, since a lot of these kinds of discussions come up on Facebook and similar places, and you may as well see it ugly here so you understand it when it's ugly there. | ||
== About | == About ratios == | ||
Ratios are just one way of writing various kinds of numbers. | Ratios are just one way of writing various kinds of numbers. | ||
1/1, 2/1, 3/1, 4/1, 5/1, and so on are the same as 1, 2, 3, 4, 5, and so on. Put differently, when there is no bottom number (denominator), it | 1/1, 2/1, 3/1, 4/1, 5/1, and so on are the same as 1, 2, 3, 4, 5, and so on. Put differently, when there is no bottom number (denominator), it is as if there was a denominator equal to 1. | ||
Ratios can be reduced by dividing both top and bottom by the same numbers until you | Ratios can be reduced by dividing both top and bottom by the same numbers until you cannot reduce them anymore. So: | ||
54/24 | 54/24 | ||
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= 9 / 4 [dividing top and bottom by 3] | = 9 / 4 [dividing top and bottom by 3] | ||
… so 54/24 can be reduced to 9/4. | |||
(You can do this faster if you get used to prime factorization: | (You can do this faster if you get used to prime factorization: | ||
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All of these numbers mean the same thing: 2, 2/1, 4/2, 8/4, 16/8. The top (numerator) is double the bottom (denominator). | All of these numbers mean the same thing: 2, 2/1, 4/2, 8/4, 16/8. The top (numerator) is double the bottom (denominator). | ||
It | It is legitimate to write things like | ||
2/1 = 2 | 2/1 = 2 | ||
or | or | ||
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---- | ---- | ||
When we | When we are finding the difference between intervals, we're actually dividing their ratios. | ||
Remember that dividing by a ratio is the same as multiplying by its reciprocal (flip, then multiply). | Remember that dividing by a ratio is the same as multiplying by its reciprocal (flip, then multiply). | ||
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= 6/5. | = 6/5. | ||
== Octave | == Octave reduction == | ||
{{Main| Octave reduction }} | |||
When a number is larger than the octave, which is 2/1 (or equivalently just "2"), we octave-reduce it by dividing by 2/1. | When a number is larger than the octave, which is 2/1 (or equivalently just "2"), we octave-reduce it by dividing by 2/1. | ||
9/4 is greater than 8/4, so it | 9/4 is greater than 8/4, so it is greater than 2. | ||
To octave-reduce, divide by 2. | To octave-reduce, divide by 2. | ||
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---- | ---- | ||
Let | Let us do it again, using 8/3. | ||
(8/3) / (2/1) | (8/3) / (2/1) | ||
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---- | ---- | ||
Let | Let us do it again, using 5/12. | ||
5/12 * 2 = 10/12, which is still less than 1. | 5/12 * 2 = 10/12, which is still less than 1. | ||
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---- | ---- | ||
Whichever way you | Whichever way you are going, keep going until the ratio is between 1 and 2. | ||
[[Category: | [[Category:Ratio]] | ||
[[Category:Elementary math]] | [[Category:Elementary math]] | ||