Mathematics of Music

My daughter Faith asked me to tune her guitar. While looking up the correct reference tone, I realized that the customary tuning of "Concert A" at 440 Hz, was different than the tuning of medical tuning forks based on octaves of C at 128 Hz, 256 Hz, 512 Hz, 1024 Hz, and 2048 Hz.  

In the West, we divide up octaves into 12 proportional semitones. The way the chromatic scale is divided is called "Temperament." There are many temperaments which have been put forth over the years, such as Pythagorean, Werckmeister, Just, Mean, Well (Bach,  "The Well-Tempered Clavier") and Even.

For the modern equal temperament scale, the frequency of each note in the chromatic scale is related to the frequency of the notes next to it by a factor of the twelfth root of 2.

F(n) = EXP ( LN (2) * n / 12 )

F(n) = e ^ ( LN  (2) * n / 12 )

F(n) = e ^ ( 0.057762265... * n )

F(n) = 2 ^ ( n / 12 ) * 256 Hz

n = steps or semitones away from the fundamental frequency beginning at n = 0.

Why do physicians use tuning forks? We use them to test hearing and vibration sense. A physician may rest the base of a ringing tuning fork on the patient's forehead, behind the ear or next to the ear to test hearing.  He may also place the base of a ringing tuning fork on a toe to test vibration sense. 

Why are medical and scientific tuning based on C4 = 256 Hz?  Scientific tuning is based on octaves of C beginning at 1 Hz and then doubling to 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048 Hz.

So, why is nearly all modern music tuned to the A4 = 440 Hz standard? An international body got together and made an arbitrary decision. Why focus on A4 and not C4?  I suppose it is because most string instruments have an A string.  Historically, early Western musicians tuned to the fundamental pitch of the organ where they were performing. Ultimately, the organ tuning determined the tuning of the orchestra. 

That said, A4 = 440 Hz is a bit sharp for my taste.  Many musicians and composers still experiment with alternative tunings and temperaments.  A4 = 432 Hz is a popular alternative tuning.  And according to Even temperament based on C4 = 256 Hz, A4 would work out to be 430.5 Hz.

Scientific Equal Temperament Tuning:

C2 = 64.0 Hz

C#2 = 67.8 Hz

D2 = 71.8 Hz

D#2 = 76.1 Hz

E2 = 80.6 Hz

F2 = 85.4 Hz

F#2 = 90.5 Hz

G2 = 95.9 Hz

G#2 = 101.6 Hz

A2 = 107.6 Hz 

A#2 = 114.0 Hz

B2 = 120.8 Hz

-------------------

C3 = 128.0 Hz

C#3 = 135.6 Hz

D3 = 143.7 Hz

D#3 = 152.2 Hz

E3 = 161.3 Hz

F3 = 170.9 Hz

F#3 = 181.0 Hz

G3 = 191.8 Hz

G#3 = 203.2 Hz

A3 = 215.3 Hz 

A#3 = 228.1 Hz

B3 = 241.6 Hz

-------------------

C4 = 256.0 Hz

C#4 = 271.2 Hz

D4 = 287.3 Hz

D#4 = 304.4 Hz

E4 = 322.5 Hz

F4 = 341.7 Hz

F#4 = 362.0 Hz

G4 = 383.6 Hz

G#4 = 406.4 Hz

A4 = 430.5 Hz  **********

A#4 = 456.1 Hz

B4 = 483.3 Hz

C5 = 512.0 Hz

Notes:

http://cnx.org/content/m11639/latest/

http://phy.mtu.edu/~suits/scales.html