B, B flat, B natural

A little history on the B flat...

Considering the admirable variety of tonal realms afforded by the eight-mode system on a strictly diatonic basis (a variety much greater than the major-minor system was able to elicit from the much fuller material afforded by the chromatic scale), one cannot help pondering about the reasons that led to the addition of the b-flat, the single "black sheep," as it were, amond the "pure-white" flock of the Gregorian pitches... Whatever answer may be given to this question - the most obvious one being that it was added in order to avoid the tritone above f - it is interesting to notice that the b-flat is not officially recognized in the earliest treatises containing information about the tonal material of the chant. This appears most clearly from a consideration of the various sytems of letter designations advicated by the theorists of the ninth century...
The first indication of the recognition of the b-flat occurs in the Divisio monochordi ... adding the letter R for the b-flat. The tenth-century Dialogus de musica generally ascribed to Oddo of Cluny is the earliest treatise to distinguish the b-flat from the b-natural by the use of two shapes of the letter b, the b rotundum (round b) for the former, and the b durum (hard, angular b) for the latter, forms which persist in our present-day signs [flat] and [natural].

[excerpted from Gregorian Chant by Willi Apel, page 152]

The word flat (as used in music) is translated Be in German, Bémol in French, Bemolle in Italian. The sign originally came from the letter b, as its shape and its foreign names indicate. ... The flat was the first chromatic sign in music,, and was used to indicate the position of a single note - B - which was sometimes sounded as B, and often "softened" into B flat (ca. 1000 A.D.) ... The round b, called b rotundum, was b-flat, the square b, called b quadratum was b-natural. In Germany, the note b-flat is still called B, and as the square b was mistaken for an h (of the German print) it was called "H" and is called so to-day, a clerical error that has been perpetuated nearly a thousand years. After some time the two b's were used as chromatic signs (the square b became the natural sign).

[excerpted from Elson's Music Dictionary, page 112 and 58]