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Thread: Musical notes question - Pythagorean Tuning

  1. #1
    May 2012

    Musical notes question - Pythagorean Tuning

    You may use the following formulae that a0 = 1 and a-n= 1 / an for any a ℚ \ {0} and n .

    We also need to define the "closeness to 1" of a positive rational number, as follows. For any q Q+, let q' = q if q 1, but let q' = 1/q if q < 1 (so that q' 1 in all cases). Then, given q, r Q+ with 1 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 P0, then intorducing a note with the pitch P1 = 3/2(P0), and further notes with PK+1 = 3/2(Pk) where k=1, 2, ... , N-1) for some N . The objective is to choose N such that PN P0 (under the equivalence relation that is x y if x = 2m*y for some m ).

    Using the fundamental Theorem of Arithmetic, prove that it is impossible for PN P0for any N with this system of tuning.
    Last edited by pete078421; May 15th 2012 at 03:19 AM.
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  2. #2
    MHF Contributor

    Apr 2005

    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.)
    Last edited by HallsofIvy; May 15th 2012 at 11:18 AM.
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