Literature:Elements of Harmony: Difference between revisions

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'''Elements of Music''' (placeholder name) is a collection of Netagin-language textbooks covering elementary number theory.
'''Elements of Harmony''' ([[Windermere/Classical|Classical Windermere]]: ''Yămyămał clisăyfäl'') is a textbook on just intonation authored in [[Windermere/Classical|Classical Windermere]] by physicist, mathematician and composer Tsăhongtamdi covering elementary number theory, acoustics, and just intonation music theory.


==Contents==
==Contents==
*Proofs of number-theoretic results:
*Book 1 discusses mathematical results:
**Basically the ones in Euclid's Elements plus...
**Prime factorization
==Full text (in English)==
**Continued fractions and mediants
*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 by unique factorization
*Book 3 discusses just intonation scales built from notes taken from overtone and undertone series.
**odd- and prime-limit
**chord voicings
**The tonality diamond is described as a way to "connect" overtone chords/scales over different fundamentals.
 
==Full text (Classical Windermere)==
 
==Full text (English)==
 
==Sketch==
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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.
 
$\log(\frac{L_1}/{L_2})$ 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: $A(1) = 0; A(tu) > A(t) + A(u)$. (Give a geometric proof)
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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Latest revision as of 12:52, 9 December 2019

Elements of Harmony (Classical Windermere: Yămyămał clisăyfäl) is a textbook on just intonation authored in Classical Windermere by physicist, mathematician and composer Tsăhongtamdi covering elementary number theory, acoustics, and just intonation music theory.

Contents

  • Book 1 discusses mathematical results:
    • Prime factorization
    • Continued fractions and mediants
  • 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 by unique factorization
  • Book 3 discusses just intonation scales built from notes taken from overtone and undertone series.
    • odd- and prime-limit
    • chord voicings
    • The tonality diamond is described as a way to "connect" overtone chords/scales over different fundamentals.

Full text (Classical Windermere)

Full text (English)

Sketch