Aristoxenus as a general problem, without reference to the scales in
actual use. He complains that his predecessors dealt only with the
octave scale, and only with the Enharmonic genus, and did not address
themselves to the real question of the melodious sequence of
intervals. Accordingly, instead of beginning with a particular scale,
such as the octave, he supposes a scale of indefinite compass,--just
as a mathematician postulates lines and surfaces of unlimited
magnitude. His problem virtually is, given any interval known to the
particular genus supposed, to determine what intervals can follow it
on a musical scale, either ascending or descending. In the Diatonic
genus, for example, a semitone must be followed by two tones, so as
to make up the interval of a Fourth. In the Enharmonic genus the
dieses or quarter-tones can only occur two together, and every such
pair of dieses ([Greek: pyknon]) must be followed in the ascending
order by a ditone, in the descending order by a ditone or a tone. By
these and similar rules, which he deduces mathematically from one or
two general principles of melody, Aristoxenus in effect determines
all the possible scales of each genus, without restriction of compass
or pitch[1]. But whenever he refers for the purpose of illustration
to a scale in actual use, it is always the standard octave already
described (from Hypatê to Nêtê), or a part of it. Thus nothing can be
clearer than the distinction which he makes between the theoretically
infinite scale, subject only to certain principles or laws
determining the succession of intervals, and the eight notes, of
fixed relative pitch, which constituted the gamut of practical music.
The passages in which Aristoxenus dwells upon the advance which he
has made upon the methods of his predecessors are of considerable
importance for the whole question of the species of the Octave. There
are three or four places which it will be worth while to quote.
1. Aristoxenus, _Harm._ p. 2, 15 Meib.: [Greek: ta gar
diagrammata autois tôn enarmoniôn] ([Greek: harmoniôn] MSS.)
[Greek: ekkeitai monon systêmatôn, diatonôn d' ê chrômatikôn
oudeis pôpoth' heôraken; kaitoi ta diagrammata g' autôn edêlou
tên pasan tês melôdias taxin, en hois peri systêmatôn
oktachordôn enarmoniôn] ([Greek: harmoniôn] MSS.) [Greek:
monon elegon, peri de tôn allôn genôn te kai schêmatôn en autô
te tô genei tontô kai tois loipois oud' epecheirei oudeis
katamanthanein.]
[Footnote 1: The investigation occupies a considerable space in his
_Harmonics_, viz. pp. 27-29 Meib. (from the words [Greek: peri de
synecheias kai tou hexês]), and again pp. 58-72 Meib.]
'The diagrams of the earlier writers set forth Systems in the
Enharmonic genus only, never in the Diatonic or Chromatic: and
yet these diagrams professed to give the whole scheme of their
music, and in them they treated of Enharmonic octave Systems
only; of other genera and other forms of this or any genus no
one attempted to discover anything.'
2. Ibid. p. 6, 20 Meib.: [Greek: tôn d' allôn katholou men
kathaper emprosthen eipomen oudeis hêptai, henos de systêmatos
Eratoklês epecheirêse kath' hen genos exarithmêsai ta schêmata
tou dia pasôn apodeiktikôs tê periphora tôn diastêmatôn
deiknys; ou katamathôn hoti, mê prosapodeichthentôn] (qu.
[Greek: proapod.]) [Greek: tôn de tou dia pente schêmatôn kai
tôn tou dia tessarôn pros de toutois kai tês syntheseôs autôn
tis pot' esti kath' hên emmelôs syntithentai, pollaplasia tôn
hepta symbainein gignesthai deiknytai.]
'The other Systems no one has dealt with by a general method:
but Eratocles has attempted in the case of one System, in one
genus, to enumerate the forms or _species_ of the Octave, and
to determine them mathematically by the periodic recurrence of
the intervals: not perceiving that unless we have first
demonstrated the forms of the Fifth and the Fourth, and the
manner of their melodious combination, the forms of the Octave
will come to be many more than seven.'
The 'periodic recurrence of intervals' here spoken of may be
illustrated on the key-board of a piano. If we take successive
octaves of white notes, _a-a_, _b-b_, and so on, we obtain each time
a different order of intervals (_i.e._ the semitones occur in
different places), until we reach _a-a_ again, when the series begins
afresh. In this way it is shown that only seven species of the Octave
can be found on any particular scale. Aristoxenus shows how to prove
this from first principles, viz. by analysing the Octave as the
combination of a Fifth with a Fourth.
3. Ibid. p. 36, 29 Meib.: [Greek: tôn de systêmatôn tas diaphoras hoi
men holôs ouk epecheiroun exarithmein, alla peri autôn monon tôn
heptachordôn ha ekaloun harmonias tên episkepsin epoiounto, hoi de
epicheirêsantes oudena tropon exêrithmounto.]
For [Greek: heptachordôn] Meibomius and other editors read [Greek:
hepta oktachordôn]--a correction strongly suggested by the parallel
words [Greek: systêmatôn oktachordôn] in the first passage quoted.
'Some did not attempt to enumerate the differences of the Systems,
but confined their view to the seven octachord Systems which they
called [Greek: harmoniai]; others who did make the attempt did not
succeed.'
It appears from these passages that before the time of Aristoxenus
musicians had framed diagrams or tables showing the division of the
octave scale according to the Enharmonic genus: and that a certain
Eratocles--of whom nothing else is known--had recognised seven forms
or species of the octachord scale, and had shown how the order of the
intervals in the several species passes through a sort of cycle.
Finally, if the correction proposed in the third passage is right,
the seven species of the Octave were somehow shown in the diagrams of
which the first passage speaks. In what respect Eratocles failed in
his treatment of the seven species can hardly be conjectured.
Elsewhere the diagrams are described by Aristoxenus somewhat
differently, as though they exhibited a division into Enharmonic
dieses or quarter-tones, without reference to the melodious character
of the scale. Thus we find him saying--. _Harm._ p. 28 Meib.: [Greek:
zêtêteon de to syneches ouch hôs hoi harmonikoi en tais tôn
diagrammatôn katapyknôsesin apodidonai peirôntai, toutous
apophainontes tôn phthongôn hexês allêlôn keisthai hois symbebêke to
elachiston diastêma diechein aph' hautôn. ou gar to mê dynasthai
dieseis oktô kai eikosin hexês melôdeisthai tês phônês estin, alla
tên tritên diesin panta poiousa ouch hoia t' esti prostithenai.]
'We must seek continuity of succession, not as theoretical musicians
do in filling up their diagrams with small intervals, making those
notes successive which are separated from each other by the least
interval. For it is not merely that the voice cannot sing
twenty-eight successive dieses: with all its efforts it cannot sing a
third diesis[1].'
[Footnote 1: This point is one which Aristoxenus is fond of insisting
upon: cp. p. 10, 16 [Greek: ou pros tên katapyknôsin blepontas hôsper
hoi harmonikoi]: p. 38, 3 [Greek: hoti de estin hê katapyknôsis
ekmelês kai panta tropon achrêstos phaneron]: p. 53, 3 [Greek: kata
tên tou melous physin zêtêteon to hexês kai ouch hôs hoi eis tên
katapyknôsin blepontes eiôthasin apodidonai to hexês].
The statement that the ancient diagrams gave a series of twenty-eight
successive dieses or quarter-tones has not been explained. The number
of quarter-tones in an octave is only twenty-four. Possibly it is a
mere error of transcription ([Greek: [=kê]] for [Greek: [=kd]]). If
not, we may perhaps connect it with the seven intervals of the
ordinary octave scale, and the simple method by which the enharmonic
intervals were expressed in the instrumental notation. It has been
explained that raising a note a quarter of a tone was shown by
turning it through a quarter of a circle. Thus, our _c_ being denoted
by [Symbols: E], _c_* was [Symbols: w], and _c_[Symbols: c] was
[Symbols: 3]. Now the ancient diagrams, which divided every tone into
four parts, must have had a character for _c_[Symbols: S]*, or the
note three-quarters of a tone above _c_. Naturally this would be the
remaining position of [Symbols: E], namely [Symbols: m]. Again, we
have seen that when the interval between two notes on the diatonic
scale is only a semitone, the result of the notation is to produce a
certain number of duplicates, so to speak. Thus: [Symbols: K] stands
for _b_, and therefore [Symbols:)1] for _c_: but _c_ is a note of the
original scale, and as such is written [Symbols: q]. It may be that
the diagrams to which Aristoxenus refers made use of these
duplicates: that is to say, they may have made use of all four
positions of a character (such as [Symbols: K 7g]) whether the
interval to be filled was a tone or a semitone. If so, the seven
intervals would give twenty-eight characters (besides the upper
octave-note), and apparently therefore twenty-eight dieses. Some
traces of this use of characters in four positions have been noticed
by Bellermann (_Tonleitern_, p. 65).]
This representation of the musical diagrams is borne out by the
passage in the _Republic_ in which Plato derides the experimental
study of music:
_Rep._ p. 531 a [Greek: tas gar akouomenas au symphônias kai
phthongous allêlois anametrountes anênyta, hôsper hoi astronomoi,
ponousin. Nê tous theous, ephê, kai geloiôs ge, pyknômat' atta
onomazontes kai paraballontes ta ôta, hoion ek geitonôn phônên
thêreuomenoi, hoi men phasin eti katakouein en mesô tina êchên kai
smikrotaton einai touto diastêma, hô metrêteon, hoi de k.t.l.]
Here Socrates is insisting that the theory of music should be studied
as a branch of mathematics, not by observation of the sounds and
concords actually heard, about which musicians spend toil in vain.
'Yes,' says Glaucon, 'they talk of the close-fitting of intervals,
and put their ears down to listen for the smallest possible interval,
which is then to be the measure.' The smallest interval was of course
the Enharmonic diesis or quarter of a tone, and this accordingly was
the measure or unit into which the scale was divided. A group of
notes separated by a diesis was called 'close' ([Greek: pyknon], or a
[Greek: pyknôma]), and the filling up of the scale in that way was
therefore a [Greek: katapyknôsis tou diagrammatos]--a filling up with
'close-set' notes, by the division of every tone into four equal
parts.
An example of a diagram of this kind has perhaps survived in a
comparatively late writer, viz. Aristides Quintilianus, who gives a
scale of two octaves, one divided into twenty-four dieses, the next
into twelve semitones (_De Mus._ p. 15 Meib.). The characters used
are not otherwise known, being quite different from the ordinary
notation: but the nature of the diagram is plain from the
accompanying words: [Greek: hautê estin hê para tois archaiois kata
dieseis harmonia, heôs [=kd] dieseôn to proteron diagousa dia pasôn,
to deuteron dia tôn hêmitoniôn auxêsasa]: 'this is the [Greek:
harmonia] (division of the scale) according to dieses in use among
the ancients, carried in the case of the first octave as far as
twenty-four dieses, and dividing the second into semitones[1].'
The phrase [Greek: hê kata dieseis harmonia], used for the division
of an octave scale into quarter-tones, serves to explain the
statement of Aristoxenus (in the third of the passages above quoted)
that the writers who treated of octave Systems called them
'harmonies' ([Greek: ha ekaloun harmonias]). That statement has
usually been taken to refer to the ancient Modes called [Greek:
harmoniai] by Plato and Aristotle, and has been used accordingly as
proof that the scales of these Modes were based upon the different
species ([Greek: eidê]) of the Octave. But the form of the
reference--'which _they called_ [Greek: harmoniai]'--implies some
forgotten or at least unfamiliar use of the word by the older
technical writers. It is very much more probable that the [Greek:
harmoniai] in question are divisions of the octave scale, as shown in
theoretical diagrams, and had no necessary connexion with the Modes.
Apparently some at least of these diagrams were not musical scales,
but tables of all the notes in the compass of an octave; and the
Enharmonic diesis was used, not merely on account of the importance
of that genus, but because it was the smallest interval, and
therefore the natural unit of measurement[2].
[Footnote 1: The fullest account of this curious fragment of notation
is that given by Bellermann in his admirable book, _Die Tonleitern
und Musiknoten der Griechen_, pp. 61-65. His conjectures as to its
origin do not claim a high degree of probability. See the remarks on
pp. 97-99.]
[Footnote 2: Cp. Plato, _Rep._ p. 531: [Greek: kai smikrotaton einai
touto diastêma, hô metrêteon.] It may even be that this sense of
[Greek: harmonia] was connected with the use for the Enharmonic
genus. It is at least worth notice that the phrase [Greek: ha ekaloun
harmonias] in this passage answers to the adjective [Greek:
enarmoniôn] in the passage first quoted (compare the words [Greek:
peri autôn monon tôn hepta oktachordôn ha ekaloun harmonias] with
[Greek: peri systêmatôn oktachordôn enarmoniôn monon]).]
The use of [Greek: harmonia] as an equivalent for 'System' or
'division of the scale' appears in an important passage in Plato's
_Philebus_ (p. 17): [Greek: all', ô phile, epeidan labês ta
diastêmata hoposa esti ton arithmon tês phônês oxytêtos te peri kai
barytêtos, kai hopoia, kai tous horous tôn diastêmatôn, kai ta ek
toutôn hosa systêmata gegonen, ha katidontes hoi prosthen paredosan
hêmin tois hepomenois ekeinois kalein auta harmonias, k.t.l.] In this
passage,--which has an air of technical accuracy not usual in Plato's
references to music (though perhaps characteristic of the
_Philebus_),--there is a close agreement with the technical writers,
especially Aristoxenus. The main thought is the application of limit
or measure to matter which is given as unlimited or indefinite--the
distinction drawn out by Aristoxenus in a passage quoted below (p.
81). The treatment of the term 'System' is notably Aristoxenean (cp.
_Harm._ p. 36 [Greek: ta systêmata theôrêsai posa te esti kai poia
atta, kai pôs ek te tôn diastêmatôn kai phthongôn synestêkota]).
Further, the use of [Greek: harmonia] for [Greek: systêma], or rather
of the plural [Greek: harmoniai] for the [Greek: systêmata] observed
by the older musical theorists, is exactly what is noticed by
Aristoxenus as if it were more or less antiquated. Even in the time
of Plato it appears as a word of traditional character ([Greek: hoi
prosthen paredosan]), his own word being [Greek: systêma]. It need
not be said that there is no such hesitation, either in Plato or in
Aristotle, about the use of [Greek: harmoniai] for the modes.
The same use of [Greek: harmonia] is found in the Aristotelian
_Problems_ (xix. 26), where the question is asked, [Greek: dia ti
mesê kaleitai en tais harmoniais, tôn de oktô ouk esti meson], _i.e._
how can we speak of the Mesê or 'middle note' of a scale of eight
notes?
We have now reviewed all the passages in Aristoxenus which can be
thought to bear upon the question whether the [Greek: harmoniai] or
Modes of early Greek music are the same as the [Greek: tonoi] or Keys
discussed by Aristoxenus himself. The result seems to be that we have
found nothing to set against the positive arguments for the
identification already urged. It may be thought, perhaps, that the
variety of senses ascribed to the word [Greek: harmonia] goes beyond
what is probable. In itself however the word meant simply 'musical
scale[1].' The Pythagorean use of it in the sense of 'octave scale,'
and the very similar use in reference to diagrams which represented
the division of that scale, were antiquated in the time of
Aristoxenus. The sense of 'key' was doubtless limited in the first
instance to the use in conjunction with the names Dorian, &c., which
suggested a distinction of pitch. From the meaning 'Dorian scale' to
'Dorian key' is an easy step. Finally, in reference to genus [Greek:
harmonia] meant the Enharmonic scale. It is not surprising that a
word with so many meanings did not keep its place in technical
language, but was replaced by unambiguous words, viz. [Greek: tonos]
in one sense, [Greek: systêma] in another, [Greek: genos enarmonion]
in a third. Naturally, too, the more precise terms would be first
employed by technical writers.
[Footnote 1: So in Plato, _Leg._ p. 665 a: [Greek: tê dê tês kinêseôs
taxei rhythmos onoma eiê, tê d' au tês phônês, tou te oxeos hama kai
bareos synkerannymenôn, harmonia onoma prosagoreuoito.]]
§ 23. _The Seven Species._
(See the Appendix, Table I.)
In the _Harmonics_ of Aristoxenus an account of the seven species of
the Octave followed the elaborate theory of Systems already referred
to (p. 48), and doubtless exhibited the application of that general
theory to the particular cases of the Fourth, Fifth, and Octave.
Unfortunately the existing manuscripts have only preserved the first
few lines of this chapter of the Aristoxenean work (p. 74, ll. 10-24
Meib.).
The next source from which we learn anything of this part of the
subject is the pseudo-Euclidean _Introductio Harmonica_. The writer
enumerates the species of the Fourth, the Fifth, and the Octave,
first in the Enharmonic and then in the Diatonic genus. He shows that
if we take Fourths on a Diatonic scale, beginning with Hypatê Hypatôn
(our _b_), we get successively _b c d e_ (a scale with the intervals
1/2 1 1), _c d e f_ (1 1 1/2) and _d e f g_ (1 1/2 1). Similarly on
the Enharmonic scale we get--
Hypatê Hypatôn to Hypatê Mesôn _b b* c e_ (1/4 1/4 2 )
Parhypatê " " Parhypatê " _b* c e e*_ (1/4 2 1/4)
Lichanos " " Lichanos " _c e e* f_ (2 1/4 1/4)
In the case of the Octave the species is distinguished on the
Enharmonic scale by the place of the tone which separates the
tetrachords, the so-called Disjunctive Tone ([Greek: tonos
diazeuktikos]). Thus in the octave from Hypatê Hypatôn to Paramesê
(_b-b_) this tone (_a-b_) is the highest interval; in the next
octave, from Parhypatê Hypatôn to Tritê Diezeugmenôn (_c-c_), it is
the second highest; and so on. These octaves, or species of the
Octave, the writer goes on to tell us, were anciently called by the
same names as the seven oldest Keys, as follows:
Mixo-lydian _b - b_ 1/4 1/4 2 1/4 1/4 2 1
Lydian _b*- b*_ 1/4 2 1/4 1/4 2 1 1/4
Phrygian _c - c_ 2 1/4 1/4 2 1 1/4 1/4
Dorian _e - e_ 1/4 1/4 2 1 1/4 1/4 2
Hypo-lydian _e*- e*_ 1/4 2 1 1/4 1/4 2 1/4
Hypo-phrygian _f - f_ 2 1 1/4 1/4 2 1/4 1/4
Hypo-dorian _a - a_ 1 1/4 1/4 2 1/4 1/4 2
On the Diatonic scale, according to the same writer, the species of
an Octave is distinguished by the places of the two semitones. Thus
in the first species, _b-b_, the semitones are the first and fourth
intervals (_b-c_ and _e-f_): in the second, _c-c_, they are the third
and the seventh, and so on. He does not however say, as he does in
the case of the Enharmonic scale, that these species were known by
the names of the Keys. This statement is first made by Gaudentius (p.
20 Meib.), a writer of unknown date. If we adopt it provisionally,
the species of the Diatonic octave will be as follows:
[Mixo-lydian] _b - b_ 1/2 1 1 1/2 1 1 1
[Lydian] _c - c_ 1 1 1/2 1 1 1 1/2
[Phrygian] _d - d_ 1 1/2 1 1 1 1/2 1
[Dorian] _e - e_ 1/2 1 1 1 1/2 1 1
[Hypo-lydian] _f - f_ 1 1 1 1/2 1 1 1/2
[Hypo-phrygian] _g - g_ 1 1 1/2 1 1 1/2 1
[Hypo-dorian] _a - a_ 1 1/2 1 1 1/2 1 1
§ 24. _Relation of the Species to the Keys._
Looking at the octaves which on our key-board, as on the Greek scale,
exhibit the several species, we cannot but be struck with the
peculiar relation in which they stand to the Keys. In the tables
given above the keys stand in the order of their pitch, from the
Mixo-lydian down to the Hypo-dorian: the species of the same names
follow the reverse order, from _b-b_ upwards to _a-a_. This, it is
obvious, cannot be an accidental coincidence. The two uses of this
famous series of names cannot have originated independently. Either
the naming of the species was founded on that of the keys, or the
converse relation obtained between them. Which of these two uses,
then, was the original and which the derived one? Those who hold that
the species were the basis of the ancient Modes or [Greek: harmoniai]
must regard the keys as derivative. Now Aristoxenus tells us, in one
of the passages just quoted, that the seven species had long been
recognised by theorists. If the scheme of keys was founded upon the
seven species, it would at once have been complete, both in the
number of the keys and in the determination of the intervals between
them. But Aristoxenus also tells us that down to his time there were
only six keys,--one of them not yet generally recognised,--and that
their relative pitch was not settled. Evidently then the keys, which
were scales in practical use, were still incomplete when the species
of the Octave had been worked out in the theory of music.
If on the other hand we regard the names Dorian, &c. as originally
applied to keys, we have only to suppose that these names were
extended to the species after the number of seven keys had been
completed. This supposition is borne out by the fact that
Aristoxenus, who mentions the seven species as well known, does not
give them names, or connect them with the keys. This step was
apparently taken by some follower of Aristoxenus, who wished to
connect the species of the older theorists with the system of keys
which Aristoxenus had perfected.
The view now taken of the seven species is supported by the whole
treatment of musical scales ([Greek: systêmata]) as we find it in
Aristoxenus. That treatment from first to last is purely abstract and
theoretical. The rules which Aristoxenus lays down serve to determine
the sequence of intervals, but are not confined to scales of any
particular compass. His Systems, accordingly, are not scales in
practical use: they are parts taken anywhere on an ideal unlimited
scale. And the seven species of the Octave are regarded by
Aristoxenus as a scheme of the same abstract order. They represent
the earlier teaching on which he had improved. He condemned that
teaching for its want of generality, because it was confined to the
compass of the Octave and to the Enharmonic genus, and also because
it rested on no principles that would necessarily limit the species
of the Octave to seven. On the other hand the diagrams of the earlier
musicians were unscientific, in the opinion of Aristoxenus, on the
ground that they divided the scale into a succession of
quarter-tones. Such a division, he urged, is impossible in practice
and musically wrong ([Greek: ekmeles]). All this goes to show that
the earlier treatment of Systems, including the seven Species, had
the same theoretical character as his own exposition. The only System
which he recognises for practical purposes is the old standard
octave, from Hypatê to Nêtê: and that System, with the enlargements
which turned it into the Perfect System, kept its ground with all
writers of the Aristoxenean school.
Even in the accounts of the pseudo-Euclid and the later writers, who
treat of the Species of the Octave under the names of the Keys, there
is much to show that the species existed chiefly or wholly in musical
theory. The seven species of the Octave are given along with the
three species of the Fourth and the four species of the Fifth,
neither of which appear to have had any practical application.
Another indication of this may be seen in the seventh or Hypo-dorian
species, which was also called Locrian and Common (ps. Eucl. p. 16
Meib.). Why should this species have more than one name? In the
Perfect System it is singular in being exemplified by two different
octaves, viz. that from Proslambanomenos to Mesê, and that from Mesê
to Nêtê Hyperbolaiôn. Now we have seen that the higher the octave
which represents a species, the lower the key of the same name. In
this case, then, the upper of the two octaves answers to the
Hypo-dorian key, and the lower to the Locrian. But if the species has
its two names from these two keys, it follows that the names of the
species are derived from the keys. The fact that the Hypo-dorian or
Locrian species was also called Common is a further argument to the
same purpose. It was doubtless 'common' in the sense that it
characterised the two octaves which made up the Perfect System. Thus
the Perfect System was recognised as the really important scale.