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Math Help - Representing Ratio's in Algebraic Formulae

  1. #1
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    Representing Ratio's in Algebraic Formulae

    Okay so i'm looking to create algebraic formula's specific ratio's. This relates to interval relationships in a musical scale for example:

    An octave has ratio of 2:1 which is represented by the following formula:
    <br />
\displaystyle \frac {f2}{f1}=2^n

    where n = the number of octaves.

    So if i wanted to find the value of the frequency 10 octaves above 20 hertz i would say:

    \displaystyle \frac {f2}{20}=2^{10}

    \displaystyle f2=(20)(1,024)

    \displaystyle f2=20,480

    So i'm wondering how you would formulate this for other frequency ratios. i.e:

    3:2, 5:4 etc.

    Any help would be great.

    Cheers

    BIOS
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  2. #2
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    any fraction can be related to a ratio as follows:

     \frac{a}{b} = a:b-a
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  3. #3
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    What does a:b represent mathematically there in terms of an operation?

    Say i wanted to find the frequency a fifth above 20 hertz. And a fifth has a relationship represented by the ration 3:2. How would i formulate that as i have done above with the octave?

    For a fifth:

    \displaystyle \frac {f2}{20}=?
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  4. #4
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    Please be careful using terms like 'fifth' on this forum as you are likely to be misinterpreted. try 'musical fifth'.

    You want the following

    f2 : 20 = 3:2

    if you have had practice on ratios, you can rearrange as follows
    0.5 * f2:10 = 1.5:1
    0.5 * f2 = 1.5*10
    f2 = 1.5 * 20


    Otherwise, you can convert to fractions and solve like any other equation:
    Step1:
    convert them both to fractions using the relationship
    a:b = \frac{a}{a+b}

    Step 2
    \frac{f2}{f2+20} = \frac{3}{5}

    \frac{f2 + 20}{f2} = \frac{5}{3}

    1 + \frac{20}{f2} = \frac{5}{3}

    \frac{20}{f2} = \frac{2}{3}

    \frac{f2}{20} = \frac{3}{2}

    f2 = 20 * \frac{3}{2}


    rearrange and solve for f2.
    Last edited by SpringFan25; October 1st 2010 at 10:33 AM.
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  5. #5
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    Sorry mate. Should have explained. A fifth is the term for an interval above a particular frequency that is a ratio of 3:2 with the original frequency.

    Interval ratio - Wikipedia, the free encyclopedia

    So just looking to create formula's for each interval ratio.
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  6. #6
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    Is there a way to do it logarithmically as with the original formula? So i find two fifths above, three fifths above etc...
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  7. #7
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    if each step increases the frequency by a factor of 1.5, then

    \frac{f_s}{f_t} = f_t * 1.5^{s-t}

    Or, starting from a frequency of 20, and going up "t" fifths

    f_{new} = 20 * 1.5^{t}
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  8. #8
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    Sweet man. I understand where to go from there. Thanks for all the help!
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