Mensuration was not well understood. As a consequence,
mensuration had different interpretations over time,
country and music theorist. The result is that with
multiple interpretations, there is no "one", special
Correct interpretation. Thus what is presented above
would agree with some interpretations, but disagree
with others. The state of "mathematics" during the
Renaissance: Euclid's Geometry was new. Music
accompanied dance, thus mensuration did double-duty:
it acted as a basis for musical theory, but to some
degree had also to be useful for dance. Mensuration
wasn't quite as useful to dance as desired, thus
"misura" worked along with mensuration. Dance required
different "steps" that could easily and quickly be
changed, but also had to be easy to learn, as well
as provide a theory based upon harmonic ratios.
References below should be consulted for more detailed
information about mensuration.
Changing the mensuration from imperfect to perfect
meant that the longa and breve would be divided by
three instead of two, thus the music might speed up
by one-third. "Medieval and Renaissance Music: A
Performer's Guide", McGee, Timothy J., Univ. of
Toronto Press, 1985, p. 41, footnote 3.
References
Aldrich, Putnam; "Rhythm in Seventeenth-Century
Italian Monody with an Anthology of Songs and Dances",
W. W. Norton & Co., 1966, Chapters 3 and four.
.
Apel, Willi; "Harvard Dictionary of Music", Harvard
Univ. Press, 1962, pp. 439-441
.
Berger, Anna Maria Busse; "Mensuration and Proportion
Signs: Origins and Evolution", Oxford Univ. Press, 2000
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