Digital Sound & Music: Concepts, Applications, & Science, Chapter 4, last updated 6/25/2013

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If we multiply the voltage times 10, we get a 20 dB increase

Don’t be fooled into thinking that if we multiply the voltage by 5, we’ll get a 10 dB increase.

Instead, multiplying voltage times 5 yields about 14 dB increase in voltage.

The rest of the rows in the table related to voltage can be proven similarly.

4.3.2 Working with Critical Bands

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 that is n semitones higher than ,

√

.

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

( √ )

This equation can be simplified to find n.

Table 4.7 is included to give an idea of the twelfth root of 2 and powers of it.

√

1.0595

√

1.1225

√

1.1892

√

1.2599

√

1.3348

√

1.4142