SOLUTION: An orchestra tunes to a Frequency of 440,which sounds the A above middle c.Eech octave higher double the frequency,and each of the 12 some time in the octave increases the frequenc

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: An orchestra tunes to a Frequency of 440,which sounds the A above middle c.Eech octave higher double the frequency,and each of the 12 some time in the octave increases the frequenc      Log On


   

Question 1163598: An orchestra tunes to a Frequency of 440,which sounds the A above middle c.Eech octave higher double the frequency,and each of the 12 some time in the octave increases the frequency in the same ratio.
(a)What is this ratio?
(b)Find the Frequency of middle C.
(C)Where on the scale is a note with a Frequency of 600?
Found 2 solutions by Edwin McCravy, AnlytcPhil:
Answer by Edwin McCravy(20064) About Me  (Show Source):
Answer by AnlytcPhil(1808) About Me  (Show Source):
You can put this solution on YOUR website!
There were a few typos, so I corrected them and reposted.

Indeed, the frequencies of the musical scale are in GP, with first term 
A = a1 = 440 and the next higher A, an octave higher, is a13 = 880,
which has twice the frequency.

an = a1∙rn-1

a13 = 440∙r13-1

880 = 440∙r12

2 = r12

root%2812%2C2%29=r

That's the ratio, approximately  1.059463094

So the A above middle C vibrates 400 times a second.

A-sharp (B-flat) vibrates at

400root%2812%2C2%29=423.78522377

we multiply by 

root%2812%2C2%29=r

each time.

B vibrates at 448.9848192

C above middle C vibrates at 475.6828457

So middle C vibrates at half that

So middle C vibrates at 237.8414229  

------------------

C)Where on the scale is a note with a Frequency of 600?

matrix%282%2C1%2C%0D%0A%22%22%2C%0D%0A600=400root%2812%2C2%29%5En%29

600%2F400=%28root%2812%2C2%29%29%5En

matrix%282%2C1%2C%22%22%2C3%2F2=%282%5E%281%2F12%29%29%5En%29 

matrix%282%2C1%2C%22%22%2C3%2F2=2%5E%28n%2F12%29%29 

matrix%282%2C1%2C%22%22%2C2%2Aexpr%283%2F2%29=2%2A2%5E%28n%2F12%29%29

matrix%282%2C1%2C%22%22%2C3=2%5E%281%2Bn%2F12%29%29  

matrix%282%2C1%2C%22%22%2C3=2%5E%28%2812%2Bn%29%2F12%29%29  

matrix%282%2C1%2C%22%22%2Cln%283%29=ln%282%5E%28%2812%2Bn%29%2F12%29%29%29 

matrix%282%2C1%2C%22%22%2Cln%283%29=%28%2812%2Bn%29%2F12%29ln%282%29%29 

matrix%282%2C1%2C%22%22%2C12ln%283%29=%2812%2Bn%29ln%282%29%29

matrix%282%2C1%2C%22%22%2C12ln%283%29%2Fln%282%29=12%2Bn%29 

matrix%282%2C1%2C%22%22%2C12ln%283%29%2Fln%282%29-12=n%29

matrix%282%2C1%2C%22%22%2C7.019550009=n%29

That's a tiny bit sharper than the 7th note of the scale
starting on A-440.

A, A#, B, C, C#, D, D#, E, F, F#,  G, G#
1,  2, 3, 4,  5, 6,  7, 8, 9, 10, 11, 12

So 600 is a bit sharper than D#, the same as Eb.

[# means sharp, b means flat]

Edwin