1. You search into the numbers of geometry as follows: the extremes
being multiplied, amount to as much as the means multiplied; as for
example, VI and XII being multiplied, make LXXII; the means VIII and
IX being multiplied, amount to the same.
ON MUSIC
INTRODUCTION
As an educational subject music is the oldest of those grouped under
the heading of the seven liberal arts. In Plato’s time music and
gymnastic were the staples of education, and the former term meant
chiefly the study of poetry, with music in the proper sense of the
word as a mere adjunct. As the different subjects, such as grammar,
rhetoric, geometry, arithmetic, appeared in the curriculum, the field
of music narrowed and it held a less commanding place. Conflicting
points of view in regard to it appear to have arisen. The older
educational tradition connected music with grammar and the other
literary studies. On the other hand, the influence of the Pythagorean
theory of number and of its application to music tended to dissociate
grammar and music, and to place the latter in relation to the
mathematical sciences. It has been noticed that among the older
Roman writers from whom evidence on this matter can be drawn—Cicero,
Varro, Seneca, Quintilian, and others—the association of music and
grammar appears the natural one, while in the Roman writers of the
second, third, and fourth centuries both traditions prevail, with
an increasing preference for placing music among the mathematical
sciences, where it finally found itself when the canon of the seven
liberal arts was formed, and where it remained to the end of the
middle ages.[254]
[254] Schmidt, _Questiones de musicis scriptoribus Romanis,
imprimis de Cassiodoro et Isidoro_ (Darmstadt, 1899). This
dissertation is in part an examination of the question whether
the Roman writers associated music with grammar or the
mathematical sciences in their enumerations of educational
subjects. It contains a useful list of passages bearing on the
seven liberal arts.
In Isidore little is to be found to justify the mathematical
environment of music. It is true that at times he defines it as a
mathematical science[255] and he insists on the musical view of the
universe as a necessary complement to other views. “Without music,”
he says, “there can be no perfect knowledge, for there is nothing
without it. For even the universe itself is said to have been formed
under the guidance of harmony.”[256] But, with the exception of a
paragraph on the musical mean, his treatment is entirely taken up
with the non-mathematical aspect of the subject, and the definition
“music is the practical knowledge of melody”[257] is the one that
more closely fits the occasion.
[255] Five definitions of music are given by Isidore, two making
no allusion to its mathematical character. They are as follows:
“Musica est peritia modulationis sono cantuque consistens.”
_Etym._, 3, 15, 1.
“Musica est disciplina quae de numeris loquitur qui inveniuntur
in sonis.” _Etym._, 3, Preface.
“Musica est disciplina quae de numeris loquitur qui ad aliquid
sunt his qui inveniuntur in sonis.” _Etym._, 2, 24, 15.
“Musica quae in carminibus cantibusque consistit.” _Etym._, 1, 2,
2.
“Musica est ars spectabilis voce vel gestu, habens in se
numerorum ac soni certam dimensionem cum scientia perfectae
modulationis. Haec constat ex tribus modis, id est, sono, verbis,
numeris.” _Diff._, ii, cap. 39.
[256] _Etym._, 3, 17, 1.
[257] _Etym._, 3, 15, 1.
The treatment[258] of music is of about the same length as that of
arithmetic, and is devoted mainly to definitions of musical terms and
brief descriptions of wind and stringed instruments. It appears that
Isidore knew nothing of music in a technical sense.[259]
[258] C. Schmidt, _op. cit._, after a detailed comparison of
passages, concludes that Isidore did not obtain his material for
_De Musica_ from Cassiodorus or Augustine, but that all three go
back independently to an original work produced by an unknown
Christian writer. However, the numerous identical passages in
Cassiodorus and Isidore would indicate that the latter had used
the former at least as a guide in plagiarism. See Schmidt, pp.
26–52, and compare Dressel, _De Isidori Originum Fontibus_
(Turin, 1874), pp. 5 and 6.
[259] Woodridge in the _Oxford History of Music_ (Oxford,
1901), vol. i, p. 33, note, says of Isidore’s _De Musica_, that
it “clearly reveals the complete ignorance of his time. His
dicta upon music are chiefly crude and misleading paraphrases
from Cassiodorus and others, from which it is evident that the
signification of the terms employed had completely escaped him.
Modes are not mentioned by him [but _cf._ 3, 20, 7] and keys and
genera are confounded together.”
EXTRACTS
Book III, Chapter 15. On music and its name.
1. Music is the practical knowledge of melody, consisting of sound
and song; and it is called music by derivation from the Muses. And
the Muses were so-called ἀπὸ τοῦ μῶσθαι, that is, from inquiring,
because it was by them, as the ancients had it, that the potency of
songs and the melody of the voice were inquired into.
2. Since sound is a thing of sense it passes along into past time,
and it is impressed on the memory. From this it was pretended by the
poets that the Muses were the daughters of Jupiter and Memory. For
unless sounds are held in the memory by man they perish, because they
cannot be written.