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musical sacle has 13 notes each hainvg a different

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musical sacle has 13 notes each hainvg a different [#permalink] New post 02 Jun 2008, 21:13
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musical sacle has 13 notes each hainvg a different frequency. In scale notes are ordered by increasing frequency and highest note is twice the lowest. For each 12 lower frequencies the ratio of freuency to next higher is fixed constant. If lower frequency = 440 then frequency of 7th note in scale is?

a) 440 Sqr(2)
b) 440 Sqr(2^7)
c) 440 Sqr(2^12)
d)440 Sqr(2^14)
e)440 Sqr(2^16)
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Re: frequencies [#permalink] New post 02 Jun 2008, 22:58
Expert's post
A

Fast solution: answer has to be between 440 and 440*2. Only A works.

Usual solution:

\(r=f_n/f_{n-1}\) - a ratio

\(f_n=r*f_{n-1}=r^2*f_{n-2}=....=r^{n-1}*f_1\)

we know that

\(f_{13}=2*f_{1}=r^{12}*f_1\) --> \(r=2^{\frac{1}{12}}\)

So, \(f_7=r^{6}*f_1=2^{\frac{6}{12}}*f_1=2^{\frac{1}{2}}*f_1=\sqrt{2}*f_1\)
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Current Student
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Joined: 28 Dec 2004
Posts: 3390
Location: New York City
Schools: Wharton'11 HBS'12
Followers: 13

Kudos [?]: 182 [0], given: 2

Re: frequencies [#permalink] New post 03 Jun 2008, 06:06
yes..i too logically knew that the ans will be A..

here is how the working is..

suppose lowest frequency is F..then the next higher frequency is r*F...the one after that is r^2*F..and so on until r^12*F

we are F=440
we are also told r^12*F=440*2

therefore r^12=2..we are asked about r^6? notice how sqrt(r^12)=r^6=sqrt(2)..

therefore the ans has to be 440sqrt(2)
Re: frequencies   [#permalink] 03 Jun 2008, 06:06
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