Return to Answer

The reason is that dividing an octave into 12 notes sounds the best for a very mathematical reason! The frequency of each semi-tone is $2^{1/12}$ away from its neighbours.

The reason is that dividing an octave into 12 notes sounds the best for a very mathematical reason! The frequency of each semi-tone is 2^1/12 away from its neighbours.

When the peaks often coincide they produce a chord, or an agreement. When the peaks rarely coincide they are discordant and the sound is disagreeable! So we can see from the table that C and G will sound the best together as C has 2 peaks for every 3 peaks that G has. The next best note for C is F, which is actually the inverse ratio of C:G. Then comes E, giving us the C-E-G chord, which we already know sounds very nice! The ratios for C-E-G are 4:5:6/4. In the minor scale we have C-E♭-G which is 6/6:5:4.

Either the numerator or the denominator must be able to be multiplied to a common, small value for the two notes to sound good together. You might think that E♭-E would sound good because they both have a 5 but it doesn't work that way. You would either get 24:25/20 or 30/25:24, neither of which would sound good because of the high numbers needed to find a common frequency.

When the peaks often coincide they produce a chord, or an agreement. When the peaks rarely coincide they are discordant and the sound is disagreeable! So we can see from the table that C and G will sound the best together as C has 2 peaks for every 3 peaks that G has. The next best note for C is F, which is actually the inverse ratio of C:G. Then comes E, giving us the C-E-G chord, which we already know sounds very nice! The ratios for C-E-G are (4:5:6)/4. In the minor scale we have C-E♭-G which is 6/(6:5:4).

Either the numerator or the denominator must be able to be multiplied to a common, small value for the two notes to sound good together. You might think that E♭-E would sound good because they both have a 5 but it doesn't work that way. You would either get (24:25)/20 or 30/(25:24), neither of which would sound good because of the high numbers needed to find a common frequency.

The reason is that dividing an octave into 12 notes sounds the best for a very mathematical reason! The frequency of each semi-tone is $2^1/12$ away from its neighbours.

Note    C × ?   Fraction    Note    C × ?   Fraction
C       1       1/1         C       2       2/1
C♯/D♭  1.059   18/17       B       1.888   17/9
D       1.122   9/8         A♯/B♭  1.782   16/9
D♯/E♭  1.189   6/5         A       1.682   5/3
E       1.260   5/4         G♯/A♭  1.587   8/5
F       1.335   4/3         G       1.498   3/2
F♯/G♭  1.414   7/5         F♯/G♭  1.414   10/7
G       1.498   3/2         F       1.335   4/3
G♯/A♭  1.587   8/5         E       1.260   5/4
A       1.682   5/3         D♯/E♭  1.189   6/5
A♯/B♭  1.782   16/9        D       1.122   9/8
B       1.888   17/9        C♯/D♭  1.059   18/17
C       2       2/1         C       1       1/1

Notice how each fraction on the right hand side (descending) is almost the inverse of the left hand side (ascending)? The difference is one of the numbers is doubled or halved each time. The smaller the two numbers are and the smaller the difference between them the better they sound to us. This is because the peaks of the waveforms they produce agree very often.

The reason is that dividing an octave into 12 notes sounds the best for a very mathematical reason! The frequency of each semi-tone is $2^{1/12}$ away from its neighbours.

Note    C × ?   Fraction    Note    C × ?   Fraction
C       1       1/1         C       2       2/1
C♯/D♭   1.059   18/17       B       1.888   17/9
D       1.122   9/8         A♯/B♭   1.782   16/9
D♯/E♭   1.189   6/5         A       1.682   5/3
E       1.260   5/4         G♯/A♭   1.587   8/5
F       1.335   4/3         G       1.498   3/2
F♯/G♭   1.414   7/5         F♯/G♭   1.414   10/7
G       1.498   3/2         F       1.335   4/3
G♯/A♭   1.587   8/5         E       1.260   5/4
A       1.682   5/3         D♯/E♭   1.189   6/5
A♯/B♭   1.782   16/9        D       1.122   9/8
B       1.888   17/9        C♯/D♭   1.059   18/17
C       2       2/1         C       1       1/1

Notice how each fraction on the right hand side (descending) is almost the inverse of the left hand side (ascending)? The difference is one of the numbers is doubled or halved each time. The smaller the two numbers are and the smaller the difference between them the better they sound to us. This is because the parts of the waveforms they produce agree very often.