====== 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 ([[https://en.wikipedia.org/wiki/Major_second|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