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Literature:Elements of Harmony: Difference between revisions
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'''Motivation for log''': Assume strings 1 and 2 have equal tension and thickness, and have lengths $L_1$ and $L_2$ respectively. Find a function $\log$ that, given the ratio between $L_1$ and $L_2$, measures the corresponding subjective difference in pitch. | '''Motivation for log''': Assume strings 1 and 2 have equal tension and thickness, and have lengths $L_1$ and $L_2$ respectively. Find a function $\log$ that, given the ratio between $L_1$ and $L_2$, measures the corresponding subjective difference in pitch. | ||
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Ts-T derives using calculus that the desired function is well-approximated by Taylor polynomials: namely, $log(x+1) = A(x+1) ≈ \sum_{n=1}^{N} (-1)^n \frac{x^n}/{n}$, when $|x| < 1$. | Ts-T derives using calculus that the desired function is well-approximated by Taylor polynomials: namely, $log(x+1) = A(x+1) ≈ \sum_{n=1}^{N} (-1)^n \frac{x^n}/{n}$, when $|x| < 1$. | ||
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