Literature:Elements of Harmony

Elements of Harmony (placeholder name) is a collection of Netagin-language textbooks covering elementary number theory, acoustics, and just intonation music theory.

Contents

• Book 1 discusses mathematical results:
  • Basically the ones in Euclid's Elements plus...
  • Continued fractions
  • 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); these can be written as tuples/monzos
  • motivates logs; gives computation of logs
• Book 3 discusses harmonic properties of various scales.
  • odd- and prime-limit
  • chord voicings
  • otonal and utonal chords

Sketch

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?)