+[Latin: @emph{proportio}.] Described in great detail by Gaffurius, in
+@emph{Practica musicae} (published in Milan in 1496). In mensural notation,
+proportion is:
+
+@enumerate
+
+@item A ratio that expresses the relationship between the note values that
+follow with those that precede;
+
+@item A ratio between the note values of a passage and the @q{normal}
+relationship of note values to the metrical pulse. (A special case of the
+first definition.)
+
+@end enumerate
+
+The most common proportions are:
+
+@itemize
+@item 2:1 (or simply 2), expressed by a vertical line through the
+mensuration sign (the origin of the @q{cut-time} time signature), or by
+turning the sign backwards
+@item 3:1 (or simply 3)
+@item 3:2 (@emph{sesquialtera})
+@end itemize
+
+To @q{cancel} any of these, the inverse proportion is applied. Thus:
+
+@itemize
+@item 1:2 cancels 2:1
+@item 1:3 cancels 3:1
+@item 2:3 cancels 3:2
+@item and so on.
+@end itemize
+
+Gaffurius enumerates five basic types of major:minor proportions and their
+inverses:
+
+@enumerate
+@item Multiplex, if the major number is an exact multiple of the minor (2:1,
+3:1, 4:2, 6:3); and its inverse, Submultiplex (1:2, 1:3, 2:4, 3:6)
+
+@item Epimoria or Superparticular [orig. @emph{Epimoria seu Superparticularis}],
+if the major number is one more than the minor (3:2, 4:3, 5:4); and its
+inverse, Subsuperparticular (2:3, 3:4, 4:5)
+
+@item Superpartiens, if the major number is one less than twice the minor
+(5:3, 7:4, 9:5, 11:6); and its inverse, subsuperpartiens (3:5, 4:7, 5:9, 6:11)
+
+@item Multiplexsuperparticular, if the major number is one more than twice the
+minor (5:2, 7:3, 9:4); and its inverse, Submultiplexsuperparticular (2:5, 3:7,
+4:9)
+
+@item Multiplexsuperpartiens, if the major number is one less than some other
+multiple (usually three or four) of the minor (8:3, 11:4, 14:5, 11:3); and its
+inverse, Submultiplexsuperpartiens (3:8, 4:11, 5:14, 3:11)
+
+@end enumerate
+
+He then continues to subdivide each type in various ways. For the multiplex
+proportions, for example, he indicates how many times greater the major number
+is than the minor:
+
+@itemize
+
+@item If two times greater, the proportion is @emph{dupla}. If inverted, it's
+called @emph{subdupla}. Examples: 2:1, 4:2, and 6:3.
+
+@item If three, @emph{tripla}; and its inversion, @emph{subtripla}. Example:
+3:1, 6:2, and 9:3.
+
+@item If four, @emph{quadrupla}; and its inversion, @emph{subquadrupla}.
+Example: 4:1, 8:2, and 12:3
+
+@end itemize
+
+Other proportions were possible, but whether they were frequently used is
+another question:
+
+@itemize
+
+@item 33:9, @emph{triplasuperbipartientetertias}
+@item 51:15, @emph{triplasuperbipartientequintas}
+
+@end itemize
projects / lilypond.git / blobdiff