Musical notes question - Pythagorean Tuning

__Notes__

You may use the following formulae that a^{0 }= 1 and a^{-n}= 1 / a^{n} for any a http://upload.wikimedia.org/wikipedi...e1a4c7cff0.png ℚ \ {0} and n http://upload.wikimedia.org/wikipedi...e1a4c7cff0.png http://upload.wikimedia.org/wikipedi...07f00175bc.png.

We also need to define the "closeness to 1" of a positive rational number, as follows. For any q http://upload.wikimedia.org/wikipedi...e1a4c7cff0.png Q+, let q' = q if q http://upload.wikimedia.org/wikipedi...6b8f7959dd.png1, but let q' = 1/q if q < 1 (so that q' http://upload.wikimedia.org/wikipedi...6b8f7959dd.png 1 in all cases). Then, given q, r http://upload.wikimedia.org/wikipedi...e1a4c7cff0.png Q+ with 1http://upload.wikimedia.org/wikipedi...5c0f8773f9.png q' < r', we say that q is closer to 1 than r is.

**The Question**

The pitch P of a musical note is a number (actually the frequency of a sound wave, in some given units). Pythagoras is said to have discovered that two notes played together sound harmonious if their pitches are related by the ratio 3/2. Pythagorean tuning aims to generate a musical scale (a nite set of musical notes) by starting with a note with some arbitrary pitch P_{0}, then intorducing a note with the pitch P_{1 = }3/2(P_{0}), and further notes with P_{K+1 = }3/2(P_{k}) where k=1, 2, ... , N-1) for some N http://upload.wikimedia.org/wikipedi...e1a4c7cff0.png http://upload.wikimedia.org/wikipedi...07f00175bc.png. The objective is to choose N such that P_{N }http://upload.wikimedia.org/wikipedi...e4889baf49.png P_{0} (under the equivalence relation that is x http://upload.wikimedia.org/wikipedi...e4889baf49.png y if x = 2^{m}*y for some m http://upload.wikimedia.org/wikipedi...e1a4c7cff0.png http://upload.wikimedia.org/wikipedi...0628d7836a.png).

Using the fundamental Theorem of Arithmetic, prove that it is impossible for P_{N }http://upload.wikimedia.org/wikipedi...e4889baf49.png P_{0}for any N http://upload.wikimedia.org/wikipedi...e1a4c7cff0.png http://upload.wikimedia.org/wikipedi...07f00175bc.png with this system of tuning.

Re: Musical notes question - Pythagorean Tuning

So $\displaystyle P_1= (3/2)P_0$, $\displaystyle P_2= (3/2)P_1= (3/2)^2P_0$, $\displaystyle P_3= (3/2)P_2= (3/2)^3P_0$, etc. It is easy to see that

$\displaystyle P_n= (3/2)^nP_0$. It should be just as easy to see that $\displaystyle (3/2)^nP_0\ne 2^mP_0$ for any m.

(Technical point- you must have $\displaystyle P_0\ne 0$ in order that this be true.)