If you are already familiar with the piano keyboard, you'll find the following explanation not necessary. I include it just in case.
It is a little difficult to visualize without a keyboard in front of you. Here's a link to an image to which you can refer.
http://www.mouthmusic.com/pianokey.htm
The standard piano keyboard begins with an A key (A0, 27.5 Hz) and finishes with a C-key. There are 12 keys to an octave, named C,D,E,F,G,A,B (white keys) and C#,D#,F#,G#,A# (black keys to the right of the lettered key, pronounced C-sharp, D-sharp, etc.).
As your teacher said, the frequency is doubled as we progress through octaves to the right. For example, the leftmost A-key has a frequency of 27.5, and the next A-key to its right has a frequency of 55, and so on.
What you teacher stated is that the frequencies of all the (11) keys between the two A's are scaled geometrically.
If we denote f(A0) the frequency of the first A-key, and f(A1) the frequency of the next A-key, then f(A1)/f(A0)=2.
Since they are in geometric progression, then
f(A1)/f(G#) = f(G$)/f(G) = f(G)/f(F#) = f(F#)/f(F) .... = f(C)/f(B) = f(B)/f(A#) = f(A#)/f(A) = Ratio, R
He would like to see a calculation of all the intermediate frequencies between keys A1 (55 Hz) and A2 (110 Hz).
There are 12 spaces between the two keys. So f(A1)*R^12=f(A2), or
R^12=f(A2)/f(A1)=2
R=2^(1/12)=1.0595 approximately.
That is to say,
f(A#)=f(A1)*1.0595=58.27 Hz
f(B)=f(A#)*1.0595=61.74 Hz
f(C)=f(B)*1.0595 = 65.41 Hz
... and so on.
Post your calculations for a check if you wish.
There are 7 notes of "A" on a standard piano. When the first A is struck, the string associated with the key vibrates 55 times per second. The next A above the given A vibrates twice as fast. An exponential function with a base of two is used to determine the frequency of the 11 notes between the two A's. Find this function.
I have no clue what they mean or how to determine this, it is a question the teacher gave us and not in the book.
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