d yet not
be complicated by what we should call accidentals. This was
accomplished in the following manner, by simple mathematical
means:

We remember that the Dorian, which was the most greatly
favoured mode in Greece, was divided into two tetrachords of
exactly the same proportions, namely, semitone, tone, tone. By
taking the lowest note of the Mixolydian, B, and forming a
Dorian tetrachord on it, B C D E were acquired. Adding to this
another Dorian tetrachord, E F G A (commencing on the last note
of the first), and repeating the same series of tetrachords
an octave higher, we have in all four Dorian tetrachords,
two of which overlap the others. The two middle tetrachords,
constituting the original Dorian mode, were called _disjunct_,
the two outer ones which overlap the middle ones were called
_conjunct_ or _synemmenon_ tetrachords.

If we consider this new scale from octave to octave, commencing
with the lowest note, that is to say from B to B, we find that
it coincides exactly with the Mixolydian mode; therefore this
was called the Mixolydian octave. The octave in this scale
from the second note, C to C, coincides exactly with the Lydian
mode, and was called the Lydian octave; from the third note, D,
up to its octave gives the Phrygian; from the fourth note, E,
the Dorian; from the fifth, F, the Hypolydian; from the sixth,
G, the Hypophrygian; and from the seventh, A, the Aeolian
or Hypodorian octave. Add one note to the lower end of this
universal Greek scale, as it was called, and we see that the
whole tonal system was included within two octaves. To each of
the notes comprising it was given a name partly derived from
its position in the tetrachords, and partly from the fingering
employed in lyre playing, as shown in the diagram on page 87.

The fifteen strings of the _kithara_ were tuned according to
this scale, and the A, recurring three times in it, acquired
something of the importance of a tonic or key note. As yet,
however, this scale allowed of no transposition of a mode to
another pitch; in order to accomplish this the second tetrachord
was used as the first of another similar system. Thus,
considering the second tetrachord, E F G A, as first of the
new scale, it would be followed by A B[flat] C D, and the
two disjunct tetrachords would be formed. Followed by the two
upper conjunct tetrachords, and the _proslambanomenos_ added,
our system on a new pitch would be complete. This procedure
has come do