Literature:Elements of Harmony: Difference between revisions

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 ==Contents==
-*Book 1 discusses and proves number-theoretic results:
+*Book 1 discusses mathematical results:
 **Basically the ones in Euclid's Elements plus...
 **Continued fractions
-*Book 2 discusses basic acoustics and chords using the number theory proved in Book 1.
+**Calculus?
+*Book 2 discusses basic acoustics using the number theory proved in Book 1.
 **harmonic series
+**intervals as string length ratios (given equal thickness and tension)
+**motivates logs; gives computation of logs
+*Book 3 discusses harmonic properties of various scales.
 **odd- and prime-limit
+**
+**chord voicings
 ***writes the intervals in a given prime-limit as tuples/monzos
 **otonal and utonal chords
-**chord voicings
-*Book 3 discusses harmonic properties of various scales.
+'''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, i.e. is an isomorphism from the multiplicative group of positive real numbers to the additive group of real numbers.
+$\log(\frac{L_1}/{L_2})$ gives the subjective intervallic change when moving from string 1 to string 2.
+Thus: if $r_1$ and $r_2$ are string length ratios, then we want
+$\log(1) = 0;$ $\log(a) > 0$ when $a > 1;$ $\log(r_1 r_2) = \log(r_1) + \log(r_2).$
+The area $A(t)$ under the hyperbola $y = \frac{1}/{x}$ from $x = 1$ to $x = t$ satisfies the desired properties: $A(1) = 0; A(tu) = A(t) + A(u)$.
+Ts-T derives 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 $|x| < 1$. (Need calculus?)
 ==Full text (in English)==