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DANCE MUSIC 339
Its position in the manuscript leaves it uncertain whether it is to be
regarded as a coda to the whole piece. The upper part, though noted
on the stave with a Bb in the signature, has all the appearance of a
part for a percussion instrument. The notation indicates the third
rhythmic mode; but as there can be little doubt that it was meant to
be played more lascivo (in jolly style), 1 i.e. with the long equal to two
(not three) breves, a transcription in 2/4 time is preferable to 6/8.
THE RELATIONSHIP OF THE INDIVIDUAL VOICES
The intervals of the third and, to a lesser degree, the sixth were now
recognized by theorists. As early
as the latter half of the twelfth
century 2 Theinred of Dover explains why the major and minor thirds
are admitted in organa, in spite of the fact that they are not strictly
consonances:
Ditonus et semiditonus cum sibi condividentibus equisonas consonantias
propter equisonantiam cum consonantiis admittuntur in organa. 3
(The major and minor thirds, together with the intervals which combine with
them to make octaves [i.e. the minor and major sixths] are allowed in organa as
well as the consonant intervals, because of the octaves [which result from the
combination of third and sixth].)
He adds that the major third and minor sixth are more common than
the minor third and major sixth. The preference is interesting, though
it is based on faulty mathematics. It is obvious that the reluctance of
theorists to admit thirds and sixths as consonances was due to the
fact that they did not fit into the acoustic theory which they had
inherited from the Greeks. Thus, the ditonus, as its name implies, was
reckoned as two (major) tones, i.e. |x| = fj; whereas the ratio of
the major third in the harmonic series is | or fj. But as Theinred
himself says, the difference is hardly noticeable to the ear; 4 and
Odington in the late thirteenth century not only mentions that many
people regard the ditonus and the semiditonus as consonant, on
account of their close approximation to the major third and minor
third of the harmonic series, but also observes that intervals like this
which are not mathematically consonant can be made to sound so
1 See Robertus de Handlo, Regulae, in Coussemaker, Scriptores, i, pp. 388 and 402
(the words 'in maxima quinte rubrice* in the latter place are an obvious error for *in
maxima quinta quarte rubrice*), and cf. supra, p. 227, n. 2.
* For the arguments in support of this date see C. C. J. Webb in the English Hw-
torical Review, xxx (1915), pp. 658-60.
8
Oxford, Bodleian, Bodl. 842, fo. 20.
4 *
Ditonus qui sesqniquartae sonorum proportion! quae prima sequitur sesquitertiam
adeo propinquus est, ut octogesima prima tantum parte maioris termini hie
superet hanc: quod auditu percipere difficile est' (loc. cit).