Esther M. Zimmer Lederberg
Mensuration (in Detail)

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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.
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Apel, Willi; "Harvard Dictionary of Music", Harvard Univ. Press, 1962, pp. 439-441
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Berger, Anna Maria Busse; "Mensuration and Proportion Signs: Origins and Evolution", Oxford Univ. Press, 2000

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