Recall from Section 1 that critical bands are areas in the human ear that are sensitive to certain bandwidths of frequencies. The presence of critical bands in our ears is responsible for the masking of frequencies that are close to other louder ones that are received by the same critical band.
In most sources, tables that estimate the widths of critical bands in human hearing give the bandwidths only in Hertz. In Table 4.4, we added two additional columns. Column 5 of Table 4.4 derives the number of semitones n in a critical band based on the beginning and ending frequencies in the band. Column 6 is the approximate size of the critical band in octaves. Let’s look at how we derived these two columns.
First, consider column 5, which gives the critical bandwidth in semitones. Chapter 3 explains that there are 12 semitones in an octave. The note at the high end of an octave has twice the frequency of a note at the low end. Thus, for frequency $$f_{2}$$ that is n semitones higher than $$f_{1}$$,
$$!f_{2}=\sqrt[12]{2}^{n}\ast f_{1}$$
To derive column 5 for each row, let b be the beginning frequency of the band, and let e be the end frequency of the band in that row. We want to find n such that
$$!e=b\ast\left ( \sqrt[12]{2} \right )^{n}$$
This equation can be simplified to find n.
$$!e=b\ast 2^{\frac{n}{12}}$$
$$!\frac{e}{b}=2^{\frac{n}{12}}$$
Table 4.7 is included to give an idea of the twelfth root of 2 and powers of it.
[table th=”0″ width=”40%”]
$$\sqrt[12]{2}^{1}=2^{\frac{1}{12}}$$,1.0595
$$\sqrt[12]{2}^{2}=2^{\frac{2}{12}}$$,1.1225
$$\sqrt[12]{3}^{3}=2^{\frac{3}{12}}$$,1.1892
$$\sqrt[12]{4}^{4}=2^{\frac{4}{12}}$$,1.2599
$$\sqrt[12]{5}^{5}=2^{\frac{5}{12}}$$,1.3348
$$\sqrt[12]{6}^{6}=2^{\frac{6}{12}}$$,1.4142
$$\sqrt[12]{7}^{7}=2^{\frac{7}{12}}$$,1.4983
$$\sqrt[12]{8}^{8}=2^{\frac{8}{12}}$$,1.5874
$$\sqrt[12]{9}^{9}=2^{\frac{9}{12}}$$,1.6818
$$\sqrt[12]{10}^{10}=2^{\frac{10}{12}}$$,1.7818
$$\sqrt[12]{11}^{11}=2^{\frac{11}{12}}$$,1.8877
$$\sqrt[12]{12}^{12}=2^{\frac{12}{12}}$$,2
[/table]
Table 4.7 Powers of $$\sqrt[12]{2}$$
Column 5 is an estimate for n rounded to the nearest integer, which is the approximate number of semitone steps from the beginning to the end of the band.
Column 6 is derived based on the n computed for column 5. If n is the number of semitones in a critical band and there are 12 semitones in an octave, then $$\frac{n}{12}$$ is the size of the critical band in octaves. Column 6 is $$\frac{n}{12}$$.