Falling Bass

An element present in some classical and popular music pieces is wherein the bass line (the lowest line of notes, usually played by the bass guitar, the string basses, or the left hand of a keyboardist) slowly descends.

Often used as a type of Ostinato—i.e., as a repeated figure against which other music is set, such as a varying Melody line.

There are two versions:
 * Diatonic: along the scale in the key of the music. Usually, this starts on "do", so we're talking "do, ti, la, so, etc." in the bass line.
 * Chromatic: along the Chromatic Scale (i.e., including the notes between ti and la, la and so, etc.) from the home note of the key to some other note. Usually, it goes down to the fifth scale degree (i.e. "so").

In both cases, there are usually a standard Chord Progressions that go along with such sequences.
 * For diatonic sequences: I, V6, vi, V or iii6, IV, I6, ii, V in a major key, and i, VII or v6, VI, v or III6, iv, I6, ii° or ii or II, V, possibly with sevenths on the chords. Note that this is actually very close to the Pachelbel's Canon Progression, which may have evolved from it. The bass line goes "do, ti, la, so, fa, mi, re, so"; yes, the last note isn't quite "falling", but that's how it usually works.
 * For chromatic sequences: I, V6, Ib7 or bVII, IV6, iv6, and I64 or V in major keys, substitute i for I and the last chord could also be III6 in minor keys. In pop chord notation, in the key of C major, that would be C, G/B, C7/Bb; or Bb, F/A, Fm/Ab, C/G or G . You can see the bottom note going C-B-Bb-A-Ab-G; that's the essence of this trope. After hitting the fifth scale degree ("so"), other bass motion usually happens, but it's usually chromatic down to the "so". Back in the Baroque era, the chromatic Falling Bass was often used to depict (and interpreted as depicting) suffering. But back then, and during the Classical Era and Romantic Era, another progression for this same bass line was popular, which involved a lot of diminished Seventh chords.

No, not an instrument dropped out a 10-story window. That would just be "crashing bass".

It is requested that you add examples in chronological order.

Diatonic examples

 * Procol Harum's "A Whiter Shade of Pale"
 * John Lennon, "Mind Games"
 * Queen: a lot of their songs feature bass going progressively downward from root to seventh to sixth in an I-V-vi progression. See Bohemian Rhapsody, the ballad section, for an example of this.
 * Billy Joel, "Piano Man". I, V6, IV6, I64, IV, I6, II7, V in C major.
 * The intro to Mega Man 4 has basically this sequence, except that the sixth note is not scale degree 3 but scale degree 1.
 * In Mega Man ZX, "En-trance Code", the music in trans server rooms, starts with this sequence, in C minor.

Chromatic examples

 * The example in every music theory student's textbook: the aria "Dido's Lament" from Henry Purcell's opera Dido And Aeneas. In G minor.
 * The "Crucifixus" from the Mass in B minor by Johann Sebastian Bach uses this as a four-bar ostinato bass-line pattern, in E minor.
 * Ludwig Van Beethoven's "Waldstein Sonata", at the very beginning of the first movement. In C major.
 * Fryderyk Chopin's Prelude in E minor from his set of 24 preludes in each key uses this, though it starts on the third scale degree and has a bunch of unusual chords.
 * "These Boots Are Made for Walkin'", written by Lee Hazlewood and recorded by Nancy Sinatra.
 * The verse of The Beatles' "Lucy In The Sky With Diamonds" has a version that skips from the first to the third note in the usual sequence.
 * The Eagles' famous song "Hotel California" has an interesting subversion of this: The verse's chord progression is typical of the Falling Bass, but it keeps all the chords in root position. The result is this: Bm, F#, A, E7, G, D, Em, F#.
 * Same with Wings's "Mamunia."
 * Schoolhouse Rock: "Figure Eight", the middle major-key section. (Also an example of an A-B-A form song.) This goes down to scale degree two ("re"), giving us C, G/B, C7/Bb, F/A, Fm/Ab, C/G, F#hdim7 (F#m7b5), Bdim7/F, C/E, D#dim7 Dmin7 G7 C.
 * The opening theme of MOTHER, "Mother Earth", only uses four steps of the chromatic sequence faithfully, but it's still using the technique for its poignance.
 * The third opening theme of Magic Knight Rayearth is similar to Hotel California in that it features a chord progression typical of the Falling Bass but has the chords in root position.
 * Ritsuko Okazaki's song "fay", in the verse. This song is in G major, and the progression is G, D/F#, G7/F, C/E, Cm/Eb, G/D, D.
 * The bassline of the "Alternia" theme (see Homestuck) takes the five incremental half-steps down to so before taking a half-step up and a whole step up. The composer fully admits to the influence of MOTHER and EarthBound, and this song is likely inspired by the "Mother Earth" theme (search above).