Literature:Elements of Harmony: Difference between revisions
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'''Elements of Harmony''' ( | '''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: | ||
** | **Prime factorization | ||
**Continued fractions | **Continued fractions and mediants | ||
*Book 2 discusses basic acoustics (don't mention frequencies) | |||
*Book 2 discusses basic acoustics | **monochord; building it | ||
**Mersenne's Laws? | |||
**harmonic series | **harmonic series | ||
**intervals as string length ratios (given equal thickness and tension); these can be written as tuples | **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. | |||
*Book 3 discusses | |||
**odd- and prime-limit | **odd- and prime-limit | ||
**chord voicings | **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 ( | ==Full text (Classical Windermere)== | ||
==Full text (English)== | ==Full text (English)== | ||
==Sketch== | ==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. | '''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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$\log(1) = 0;$ $\log(a) > 0$ 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) | 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 $|x| < 1$. | 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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[[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.