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 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
  • 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 Notation

Music Theory

  • musicwiki/concepts.txt
  • Last modified: 2021/05/27 18:06
  • by ying