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. | ||
==Contents== | ==Contents== | ||
*Book 1 discusses | *Book 1 discusses mathematical results: | ||
** | **Prime factorization | ||
**Continued fractions | **Continued fractions and mediants | ||
*Book 2 discusses basic acoustics | *Book 2 discusses basic acoustics (don't mention frequencies) | ||
**monochord; building it | |||
**Mersenne's Laws? | |||
**harmonic series | **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 | **odd- and prime-limit | ||
**chord voicings | **chord voicings | ||
* | **The tonality diamond is described as a way to "connect" overtone chords/scales over different fundamentals. | ||
==Full text (in | ==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.