Musical Note Frequency Ratio Calculation

The frequencies of musical notes in the standard western scale follow a geometric progression. There are 12 notes in an octave, and the frequency ratio between any two adjacent notes is constant.

The problem states that the frequency multiplier between adjacent notes is $ \sqrt[12]{2} $. We need to find the ratio between the frequencies of F# and C.

Identifying Intervals Between Notes

Let's determine the number of steps (intervals) from note C to note F# in the sequence:

• C (Starting Note)
• C# (1 step)
• D (2 steps)
• D# (3 steps)
• E (4 steps)
• F (5 steps)
• F# (6 steps)

There are 6 steps from C to F#.

Calculating Frequency Ratio

Since each step corresponds to a frequency multiplication factor of $ \sqrt[12]{2} $, the ratio of the frequency of F# to the frequency of C (which is $ F_{\text{F\#}} / F_{\text{C}} $) is calculated by raising this factor to the power of the number of steps (6).

Ratio $ = (\sqrt[12]{2})^{\text{Number of steps}} $

Ratio $ = (\sqrt[12]{2})^{6} $

Using exponent rules, where $ \sqrt[n]{x} = x^{1/n} $:

Ratio $ = (2^{1/12})^{6} $

Ratio $ = 2^{(1/12) \times 6} $

Ratio $ = 2^{6/12} $

Ratio $ = 2^{1/2} $

Ratio $ = \sqrt{2} $

The frequency of C (130.8 Hz) is given but not needed for calculating the ratio.

Therefore, the ratio of frequencies of notes F# and C is $ \sqrt{2} $.