Logarithmic approximants: Difference between revisions
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While bimodular approximants have historically been used as a means of estimating the sizes of very small intervals, they remain reasonably accurate as the interval size is increased to an octave or more. And being easily computable, they provide a quick means of comparing the relative sizes of intervals. For example: | While bimodular approximants have historically been used as a means of estimating the sizes of very small intervals, they remain reasonably accurate as the interval size is increased to an octave or more. And being easily computable, they provide a quick means of comparing the relative sizes of intervals. For example: | ||
Two perfect fourths (''r'' = 4/3, ''<span style="font-family: Georgia,serif; font-size: 110%;">v</span>'' = 1/7) approximate a minor seventh (''r'' = 9/5, = 2/7) | Two perfect fourths (''r'' = 4/3, ''<span style="font-family: Georgia,serif; font-size: 110%;">v</span>'' = 1/7) approximate a minor seventh (''r'' = 9/5, ''<span style="font-family: Georgia,serif; font-size: 110%;">v</span>'' = 2/7) | ||
Three major thirds (''r'' = 5/4, ''<span style="font-family: Georgia,serif; font-size: 110%;">v</span>'' = 1/9) or two <u>7/5</u>s (''<span style="font-family: Georgia,serif; font-size: 110%;">v</span>'' = 1/6) or five <u>8/7</u>s (''<span style="font-family: Georgia,serif; font-size: 110%;">v</span>'' = 1/15) approximate an octave (''r'' = 2/1,''<span style="font-family: Georgia,serif; font-size: 110%;"> v</span>'' = 1/3) | Three major thirds (''r'' = 5/4, ''<span style="font-family: Georgia,serif; font-size: 110%;">v</span>'' = 1/9) or two <u>7/5</u>s (''<span style="font-family: Georgia,serif; font-size: 110%;">v</span>'' = 1/6) or five <u>8/7</u>s (''<span style="font-family: Georgia,serif; font-size: 110%;">v</span>'' = 1/15) approximate an octave (''r'' = 2/1,''<span style="font-family: Georgia,serif; font-size: 110%;"> v</span>'' = 1/3) | ||