musicwiki:concepts

# Music and Math

• 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=1/2); its half frequency C3 (1 octave lower note, length=1*2 =2);
• 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^?)/(3^12), a factorial never perfectly matching 1/2, however, its pattern maintaining 2/3 ratio
• 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 combine 1 2 3 4 5 6 7 8 9 10 11 12 C G D A E B F# Gb C# Db C# Db G# Ab D# Eb D# Eb A# Bb F F C G C C C# Db D D# Eb E F F# Gb G G# Ab A A# Bb B
• 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 (major 2nd), so are some other notes
• 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%
• well temperament: 1/2 maintain ratio, a set of defined frequency of each note
• equal temperament: 1/2 maintain ratio, half step is x, solving x^12 = 1/2
• 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 Theory

• musicwiki/concepts.txt