Pedagogy time. Vibrating bodies of any physical type will vibrate at an infinite ascending series of whole number multiples of the base frequency f (so, f, 2f, 3f, etc.) in decreasing -- but not linearly or regularly decreasing -- amplitude (the exact difference in the proportions of the various overtones, among other factors, is why different instruments sound different).
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.