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Fig. 1.2
Arithmetic diagram from Boëthius's
De Arithmetica
(MS Cod. Sang. 248,
fol. 10a, St. Gallen, Stiftsbibliothek).
Arcs are an important part of Boëthius's graphical schema. He uses
them in his discussions of arithmetic and geometric ratios, and they are
integral to his treatise on music, which has its foundation in ancient Greek
musical theory. Fig. 1.3 displays an arc diagram taken from a twelfth
century version of Boëthius's
De institutione musica
(MS VadSlg. 296, fol.
53r, St. Gallen, Kantonsbibliothek, Vadianische Sammlung) [24]. It shows
the ratios: 4:2 (
dupla
) that yields the octave (
diapason
); 6:2 (
tripla
), that
represents the twelfth (
diapason ac diapente
); and 6:4, a ratio that governs
the fifth (
diapente
).
In Fig. 1.4 [24] Boëthius goes beyond this simple arc diagram by
charting three tetrachord genera for the monochord. The monochord is a
single stringed musical instrument invented by Pythagoras to study ratios
that eventually became an integral part of medieval music education. A
tetrachord is a four note series having a specific pattern of whole and half
steps that served as a basis for Greek and Medieval melodic construction.
Here, the three monochord genera are plotted from top to bottom: the
diatonic genus (e.g. whole, whole, half steps), chromatic genus, and enharmonic
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