|F.J Budden in An Introduction to Number Scales and Computers, Longmans, 1965, page 65, says he thinks the record number of dots is 4 in Verdi's Requiem in the Rex Tremenda. It is useful when a long note is followed by a quick note and the next note is "on the beat".|
Binary fractions are written using column headings as follows:
... 8 4 2 1 · 1/2 1/4 1/8 ...So 1/4 = 0·012 and 3/8 = 0·0112 since it is 1/4+1/8.
In binary, a dot after a crotchet adds a one in a fractional column:
crotchet = 1and so on.
crotchet dot = 1·12
crotchet dot dot = 1·112
crotchet dot dot dot = 1·1112