The musical scale used in most music in the Western tradition, however, does not use anything like a harmonic series. Rather, it (presently) uses an equal-tempered scale, such that each note is the same distance from the next. This is a convention adopted to make keyboard music in many different styles and keys more practical to play, but has almost no musical basis per se. To a sensitive ear, a lot of the intervals in an equal-tempered system (most notably the major third) are starkly out of tune from their harmonic manifestations.
Bach did not use, nor attempt to use, equal-tempered scales. This is an error of historical writing that was introduced by a poorly-informed musicologist into the 1890 edition of the Grove Dictionary of Music, and has persisted ever since. Bach not only could not have tuned his instruments to a truly equal temperament (the technology to do so was not available until the 1820s), almost everybody of his time agreed that more-equal temperaments sounded generally awful and unmusical. Bach used "Well Temperament," which is a distinct system of temperament (of which there are many variants; just which one he used is subject to debate), that kept most intervals in most keys acceptably approximate, while allowing each key to have a slightly different flavor/color.
I imagine the birds sing notes out of a harmonic series because the intervals are much easier to hear.