Music and Math
- ref:
- interactive circle: https://pianolit.com/tools/circle-of-fifths
- cn topic (音乐、音律与数学): https://www.youtube.com/watch?v=wdHyPtchVPQ
- cn topic (science in music): https://www.youtube.com/watch?v=v5QlocAclXY
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=1/2); 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 |
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 (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%
Temperament Approach (no ratio)
- 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
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): https://www.youtube.com/watch?v=qkt5X_4FJBY&t=1s
Music Theory
- ref:
- cn topic (basics): https://www.youtube.com/watch?v=bzpEUJXtvfI
- cn topic (Circle of Fifths): https://www.youtube.com/watch?v=EyF7EBnmdEs
- cn topic (major scale, minor scale): https://www.youtube.com/watch?v=T70L-t60j5c