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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Revision as of 12:57, 9 June 2017

Elements of Harmony (Netagin: letter/[TRANSFIX]-PL.CONST harmony) is a collection of Netagin-language textbooks by physicist, mathematician and composer Tsâhoŋ-Tamdi covering elementary number theory, acoustics, and just intonation music theory.

Contents

  • Book 1 discusses mathematical results:
    • Basically the number theory results in Euclid's Elements plus...
    • Continued fractions
  • Book 2 discusses basic acoustics (don't mention frequencies)
    • monochord; building it
    • Mersenne's Laws?
    • harmonic series
    • intervals as rational string length ratios (given equal thickness and tension); these can be written as tuples/monzos by unique factorization
  • Book 3 discusses harmonic properties of various scales.
    • odd- and prime-limit
    • chord voicings
    • otonal and utonal chords

Full text (Netagin)

Full text (English)

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