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— | musicwiki:concepts [2021/05/27 18:01] – [Music and Physics] ying | ||
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+ | ====== Music and Math ====== | ||
+ | |||
+ | * ref: | ||
+ | * interactive circle: https:// | ||
+ | * cn topic (音乐、音律与数学): | ||
+ | * cn topic (science in music): https:// | ||
+ | |||
+ | |||
+ | ===== A single Ratio Approach ===== | ||
+ | |||
+ | * a single Note, such as C4, as frequency of a string (length unit as L=1), its 2x frequency C5 (1 octave higher note, length=1/ | ||
+ | * since short length string making higher frequency sound, long string making lower frequency sound | ||
+ | * **we can define**: 2/3 length of that C4 string is G4, (which is a note in-between C4 and C5, L=2/3) | ||
+ | * then, start from G4, as frequency of a new string, 2/3 of that length, we get D5 (L=2/3 * 2/3 = 4/9), so if we start with C3 as 2, then D4 = 2*4/9 = 8/9 | ||
+ | * then, same circle of C4 (1), G4 (2/3), D5 (4/9) -> D4 (8/9), A5 (8/27) -> A4 (16/27), E6 (16/81) -> E4 (64/81) continues | ||
+ | * that way of push making C5 (2^? | ||
+ | * however: the frequency difference between each step, like C to C#/Db, is around 105.3% to 106.8% | ||
+ | |||
+ | ^ cycle | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | | ||
+ | ^ 1 | C | | | | | | | G | | | | | | | | | | | | | | ||
+ | ^ 2 | | | D | | | | | | | A | | | | | | | | | | | | ||
+ | ^ 3 | | | | | E | | | | | | | B | | | | | | | | | | ||
+ | ^ 4 | | | | | | | F# \\ Gb | | | | | | | C# \\ Db | | | | | | | | ||
+ | ^ 5 | | C# \\ Db | | | | | | | G# \\ Ab | | | | | | | D# \\ Eb | | | | | | ||
+ | ^ 6 | | | | D# \\ Eb | | | | | | | A# \\ Bb | | | | | | | F | | | | ||
+ | ^ 7 | | | | | | F | | | | | | | C | | | | | | | G | | ||
+ | ^ 8 | C | | | | | | | | | | | | | | | | | | | | | ||
+ | ^^ | ||
+ | ^ combine | C | C# \\ Db | D | D# \\ Eb | E | F | F# \\ Gb | G | G# \\ Ab | A | A# \\ Bb | B | | ||
+ | |||
+ | ===== A multi Ratio Approach ===== | ||
+ | |||
+ | * string length ratio can determine the frequency ratio, so simple ratio makes string simple to make. | ||
+ | * a perfect ratio like 1x,2x, 1/2 is simplest. | ||
+ | * other simple ratio ranking as: 1/2 > 2/3 > 3/4 > 4/5 | ||
+ | * compared to above definition, G4 is 2/3 from C4 | ||
+ | * compared to above definition, E4 (64/81) is mostly near 4/5 (x16 up/down = 64/80) | ||
+ | |||
+ | * **using 2/3 and 4/5 way of pushing** (Pure intonation) | ||
+ | * C4 is 1 | ||
+ | * E4 is 4/5 (48/60) | ||
+ | * G4 is 2/3 => B5 = 4/5 x 2/3 = 8/15 | ||
+ | * G4 is 2/3 => Eb4 = (2/3) / (4/5) = 10/12 (50/60) | ||
+ | * C5 is 1/2 => F4 = (1/2) / (2/3) = 3/4 | ||
+ | * F4 is 3/4 => A4 = 3/4 x 4/5 = 3/5 | ||
+ | * A4 is 3/5 => D4 = (3/5) / (2/3) = 9/10 (use 2/3 ratio from A4) | ||
+ | * G5 is 2/3 => D4 = (2/3) / (3/4) = 8/9 (use 3/4 ratio from G4) | ||
+ | * as you see, there is 2 way to get D4 ([[https:// | ||
+ | * the half step of each next note, like from E to Eb, are not the same, half step difference range from 104.1% to 108% | ||
+ | |||
+ | ===== Temperament Approach (no ratio) ===== | ||
+ | |||
+ | * **well temperament**: | ||
+ | * **equal temperament**: | ||
+ | |||
+ | ===== Conclusion on Music Note Tuning ===== | ||
+ | |||
+ | * different music to math definition means different music tuning system | ||
+ | * different time, different culture, different history, different instrument means their default sounding may be slightly different, based on their choice of taste of sound. | ||
+ | * modern equal temperament tuning is a balance of all for flexibility of sound adapting | ||
+ | |||
+ | ====== Music and Physics ====== | ||
+ | |||
+ | * string frequency: | ||
+ | * longer string, lower frequency | ||
+ | * shorter string, higher frequency | ||
+ | * more tense string tightens, higher frequency | ||
+ | * more loose string tightens, lower frequency | ||
+ | * heavier string, lower frequency | ||
+ | * lighter string, higher frequency | ||
+ | |||
+ | ====== Music Notation ====== | ||
+ | |||
+ | * ref: | ||
+ | * cn topic (五線譜): | ||