Literature:Elements of Harmony: Difference between revisions

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'''Elements of Harmony''' (placeholder name) is a collection of [[Netagin]]-language textbooks covering elementary number theory, acoustics, and just intonation music 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.
 
Supporters of [[Verse:Tricin/Plud Schrog-Hahn|Soha Plu]] believe that the text was written in [[Schlaub]] before it was translated into Classical Windermere, and that Tsăhongtamdi is simply a Windermere pseudonym of a (probably) Hlou composer.


==Contents==
==Contents==
*Book 1 discusses mathematical results:
*Book 1 discusses mathematical results:
**Basically the ones in Euclid's Elements plus...
**Prime factorization
**Continued fractions
**Continued fractions and mediants
**Calculus?
*Book 2 discusses basic acoustics (don't mention frequencies)
*Book 2 discusses basic acoustics using the number theory proved in Book 1.
**monochord; building it
**Mersenne's Laws?
**harmonic series
**harmonic series
**intervals as string length ratios (given equal thickness and tension)
**intervals as rational string length ratios (given equal thickness and tension); these can be written as tuples by unique factorization
**motivates logs; gives computation of logs
*Book 3 discusses just intonation scales built from notes taken from overtone and undertone series.
*Book 3 discusses harmonic properties of various scales.
**odd- and prime-limit
**odd- and prime-limit
**
**chord voicings
**chord voicings
***writes the intervals in a given prime-limit as tuples/monzos
**The tonality diamond is described as a way to "connect" overtone chords/scales over different fundamentals.
**otonal and utonal chords
 
Tsăhongtamdi's most influential recommendation was against using fixed-pitch instruments; he argued that they were expressively limited. This recommendation was lasting in influence - most instruments used in traditional Hlou-Shum music are flexible-pitch instruments.
 
==Full text (Schlaub)==
 
==Full text (Classical Windermere)==


'''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.  
==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.
$\log(\frac{L_1}/{L_2})$ gives the subjective intervallic change when moving from string 1 to string 2.
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$\log(1) = 0;$ $\log(a) > 0$ when $a > 1;$ $\log(r_1 r_2) = \log(r_1) + \log(r_2).$
$\log(1) = 0;$ $\log(a) > 0$ 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: $A(1) = 0; A(tu) = A(t) + A(u)$.
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 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?)
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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==Full text (in English)==
[[Category:Tricin]]

Latest revision as of 11:57, 27 January 2025

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.

Supporters of Soha Plu believe that the text was written in Schlaub before it was translated into Classical Windermere, and that Tsăhongtamdi is simply a Windermere pseudonym of a (probably) Hlou composer.

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.

Tsăhongtamdi's most influential recommendation was against using fixed-pitch instruments; he argued that they were expressively limited. This recommendation was lasting in influence - most instruments used in traditional Hlou-Shum music are flexible-pitch instruments.

Full text (Schlaub)

Full text (Classical Windermere)

Full text (English)

Sketch