You can determine the frequency of the notes between C1 and C2 (to be exact, of those corresponding to the white keys of the piano) by using only the intervals of octave and fifth. Denote the frequencies of C1, D1, E1... by ν(C1), ν(D1), ν(E1)...
- C1 has frequency 1 (it is our unit of measure).
- C2 has frequency 2 (second harmonic).
- G1 has frequency 3/2 (fifth C1-G1): ν(G1)= ν(C1)·(3/2)=1·(3/2)=3/2.
- D1 has frequency 9/8. In fact G1-D2 is a fifth, then ν(D2)=ν(G1)·(3/2)=(3/2)·(3/2)=9/4. Since 9/4=2,25>2, D2 is outside the octave C1-C2. But you can descend by an octave dividing by two the frequency, obtaing ν(D1)=ν(D2):2=(9/4):2=(9/4)·(1/2)=9/8.
- A1 has frequency 27/16: D1-A1 is a fifth, and then ν(A1)=ν(D1)·(3/2)=(9/8)·(3/2)=27/16.
- E1 has frequency 81/64: A1-E2 is a fifth, and then ν(E2)=ν(A1)·(3/2)=(27/16)·(3/2)=81/32. Since 81/32=2,53125>2, you must descend by an octave: ν(E1)=ν(E2):2=(81/32):2=(81/32)·(1/2)=81/64.
- B1 has frequency 243/128: E1-B1 is a fifth, and then ν(B1)=ν(E1)·(3/2)=(81/64)·(3/2)=243/128.
- F1 has frequency 4/3, since F1-C2 is a fifth (descending if you start from C2), and then ν(F1)=ν(C2):(3/2)=2:(3/2)=2·(2/3)=4/3.
Below you can see the notes of
C major 
scale with their respective frequencies:
| NOTE |
C1 |
D1 |
E1 |
F1 |
G1 |
A1 |
B1 |
C2 |
| FREQUENCY |
1 |
9/8 |
81/64 |
4/3 |
3/2 |
27/16 |
243/128 |
2 |
Observe that the characteristic values of the intervals that you used (2 for the octave and 3/2 for the fifth), have been obtained from the physical phenomenon of harmonic sounds. However, while for some notes, such as G1 or F1, the frequency is a simple fraction (in the sense that the numerator and denominator are small numbers), and then their sound waves are very similar to those of the fundamental note C1, not just can be said for others, such as E1 and, above all, B1. In addition, C1, E1 and G1 form the
chord of C major; but just E1 has frequency 81/64, and 81 and 64 are too big to harmonize well this note with the other two.
| This table summarizes the steps of the calculation of the frequencies of the notes of Pythagorean scale. |
