Critical and Historical Essays
On the other hand, the monochord, with which students in physics
are familiar, was his invention; and the first mathematical
demonstrations of the effect on musical pitch of length of
cord and tension, as well as the length of pipes and force of
breath, were his.
These mathematical divisions of the monochord, however,
eventually did more to stifle music for a full thousand years
than can easily be imagined. This division of the string
made what we call harmony impossible; for by it the major
third became a larger interval than our modern one, and the
minor third smaller. Thus thirds did not sound well together,
in fact were dissonances, the only intervals which _did_
harmonize being the fourth, fifth, and octave. This system
of mathematically dividing tones into equal parts held good
up to the middle of the sixteenth century, when Zarlino, who
died in 1590, invented the system in use at the present time,
called the _tempered scale_, which, however, did not come into
general use until one hundred years later.
Aristoxenus, a pupil of Aristotle, who lived more than a
century after Pythagoras, rejected the monochord as a means for
gauging musical sounds, believing that the ear, not mathematical
calculation, should be the judge as to which interval sounds
"perfect." But he was unable to formulate a system that
would bring the third (and naturally its inversion the sixth)
among the harmonizing intervals or consonants. Didymus (about
30 B.C.) first discovered that two different-sized whole
tones were necessary in order to make the third consonant;
and Ptolemy (120 A.D.) improved on this system somewhat. But
the new theory remained without any practical effect until
nearly the seventeenth century, when the long respected theory
of the perfection of mathematical calculation on the basis of
natural phenomena was overthrown in favour of actual effect. If
Aristoxenus had had followers able to combat the crushing
influence of Euclid and his school, music might have grown up
with the other arts. As it is, music is still in its infancy,
and has hardly left its experimental stage.
Thus Pythagoras brought order into the music as well as
into the lives of people. But whereas it ennobled the
people, it killed the music, the one vent in life through
which unbounded utterance is possible; its essence is so
interwoven with spirituality that to tear it away and fetter
it with human mathematics is to lower it to the level of mere
utilitarianism. And so it was with Greek music, which was held
subordinate to metre, to poetry, to acting, and finally became
a term of contempt. Pythagoras wished to banish the flute,
as Plato also did later, and the name of flute player was used
as a reproach. I fancy this was because the flute, on account
of its construction, could ignore the mathematical divisions
prescribed for the stringed instruments, and therefore could
indulge in purely emotional music. Besides, the flute was
the chosen instrument of the orgiastic Bacchic cult, and its
associations were those of unbridled license. To be sure, the
voice was held by no mathematical restrictions as to pitch;
but its music was held in check by the words, and its metre
by dancing feet.
Having measured the musical intervals, there still remained
the task of classifying the different manners of singing which
existed in Greece, and using all their different notes to form
a general system. For just as in different parts of Greece
there existed different dances, the steps of which were known
as Lydian, Ionian, Locrian, and Dorian feet, and so on, so the
melodies to which they were danced were known as being in the
Lydian, Ionian, Locrian, or Dorian scale or mode. In speaking
of Hindu music, I explained that what we call a mode consists
of a scale, and that one mode differs from another _only_ in
the position of the semitones in this scale. Now in ancient
Greece there were in use over fifteen different modes, each one
common to the part of the country in which it originated. At the
time of Pythagoras there were seven in general use: the Dorian,
Lydian, Aeolian or Locrian, Hypo- (or low) Lydian, Phrygian,
Hypo- (or low) Phrygian, and Mixolydian or mixed Lydian. The
invention of the latter is attributed to Sappho by Plutarch,
quoting Aristoxenus.
These modes were all invested with individual characters
by the Greeks, just as in the present day we say our major
mode is happy, the minor sad. The Dorian mode was considered
the greatest, and, according to Plato, the only one worthy of
men. It was supposed to have a dignified, martial character. The
Lydian, on the other hand, was all softness, and love songs
were written in it. The Phrygian was of a violent, ecstatic
nature, and was considered as being especially appropriate for
dithyrambs, the metre for the wild bacchanalian dances. For
instance, Aristotle tells how Philoxenus attempted to set
dithyrambic verse to the Dorian mode, and, failing, had to
return to the Phrygian. The Mixolydian, which was Sappho's mode,
was the mode for sentiment and passion. The Dorian, Phrygian,
and Lydian were the oldest modes.
Each mode or scale was composed of two sets of four notes,
called tetrachords, probably derived from the ancient form
of the lyre, which in Homer's time is known to have had four
strings.
Leaving the matter of actual pitch out of the question (for
these modes might be pitched high or low, just as our major
or minor scale may be pitched in different keys), these three
modes were constructed as follows:
Greek Dorian (E F) G A (B C) D E,
that is, semitone, tone, tone.
/
| Phrygian D (E F) G A (B C) D,
| or F[#] (G[#] A) B C[#] (D[#] E) F[#],
Asiatic | that is, tone, semitone, tone.
|
| Lydian C D (E F) G A (B C),
\ that is, tone, tone, semitone.
Thus we see that a tetrachord commencing with a half-tone and
followed by two whole tones was called a Dorian tetrachord;
one commencing with a tone, followed by a half-tone, and again
a tone, constituted a Phrygian tetrachord. The other modes
were as follows: In the Aeolian or Locrian the semitones occur
between the second and third notes, and the fifth and sixth:
[F: b, (c+ d) e (f+ g) a b]
Theraclides Ponticus identifies the Hypodorian with the Aeolian,
but says that the name "hypo-" merely denoted a likeness to
Doric, not to pitch. Aristoxenus denies the identity, and
says that the Hypodorian was a semitone below the Dorian or
Hypolydian. In the Hypophrygian, the semitones occur between
the third and fourth, and sixth and seventh degrees:
[F: c+ d+ (e+ f+) g+ (a+ b) c+']
In the Hypolydian, the semitones occur between the fourth and
fifth, and seventh and eighth: [F: e- f g (a b-) c' (d' e-')]
The Dorian (E), Phrygian (commencing on F[sharp] with the fourth
sharped), and the Lydian (A[flat] major scale) modes we have
already explained. In the Mixolydian, the semitones occur
between the first and second, and fourth and fifth degrees:
[G: (a b-) c' (d' e-') f' g' a']
According to the best evidence (in the works of Ptolemy,
"Harmonics," second book, and Aristides), these were
approximately the actual pitch of the modes as compared one
to another.
And now the difficulty was to weld all these modes together
into one scale, so that all should be represented and 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 down almost unchanged to our times; for we have but
two modes, major and minor, which are used on every pitch,
constituting various keys. These Greek modes are the basis
on which all our modern ideas of tonality rest; for our major
mode is simply the Greek Lydian, and our minor mode the Aeolian.
LIST OF NOTES IN THE GREEK SCALE
disjunct
Aeolian. [G: a'] +- A. Nete, or highest. ---+
Hypophrygian. +-| G. Paranete, next highest. |
Hypolydian. +-| | F. Trite, third. |
Dorian. +-| | | E. Nete, highest. ---+ conjunct
Phrygian. +-| | | | D. Paranete, next highest. ---+ ---+
Lydian. +-| | | | | C. Trite, third. | |
Mixolydian. +-| | | | | | B. Paramese, next to central tone | |
| | | | | | +- A. Mese, central tone. ---+ ===+
| | | | | +--- G. Lichanos, index finger. |
| | | | +----- F. Parhypate, next to lowest. |
| | | +------- E. Hypate, lowest. ===+
| | +--------- D. Lichanos, index. |
| +----------- C. Parhypate, next to lowest. |
+------------- B. Hypate, lowest. ---+
[F: a,] A. Proslambanomenos, added tone.
To go into detailed explanation of the Greek enharmonic
and chromatic pitch will scarcely be worth while, and I will
therefore merely add that the instruments were sometimes tuned
differently, either to relieve the inevitable monotony of this
purely diatonic scale or for purposes of modulation. A Dorian
tetrachord is composed of semitone, tone, tone; to make it
chromatic, it was changed as follows: [G: e' f' g-' a'] the
_lichanos_, or index finger string, being lowered a semitone.