139,285
edits
Literature:Elements of Harmony: Difference between revisions
Literature:Elements of Harmony (view source)
Revision as of 12:52, 9 December 2019
, 9 December 2019no edit summary
mNo edit summary |
mNo edit summary Tags: Mobile edit Mobile web edit |
||
| (21 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
'''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 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 (don't mention frequencies) | ||
**monochord; building it | **monochord; building it | ||
**Mersenne's Laws? | **Mersenne's Laws? | ||
**harmonic series | **harmonic series | ||
**intervals as rational 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 | *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 ( | ==Full text (Classical Windermere)== | ||
==Full text (English)== | ==Full text (English)== | ||
==Sketch== | ==Sketch== | ||
<!-- | |||
'''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. | ||
| Line 33: | Line 32: | ||
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 $|x| < 1$. | ||
--> | |||