which we have to deal with is the use of the character [Symbols: I-]
for lambda, which is distinctive of the alphabet of Argos, along with
the commoner form [Symbols: <]. Westphal indeed asserts that both
these forms are found in the Argive alphabet. But the inscription (C.
I. 1) which he quotes[1] for [Symbols: <] really contains only
[Symbols: t-] in a slightly different form. We cannot therefore say
that the inventor of the notation derived it entirely from the
alphabet of Argos, but only that he shows an acquaintance with that
alphabet. This is confirmed by the fact that the form [Symbols: Li]
for iota is not found at Argos. Probably therefore the inventor drew
upon more than one alphabet for his purpose, the Argive alphabet
being one.
[Footnote 1: _Harmonik und Melopöie_, p. 286 (ed. 1863). The true
form of the letter is given by Mr. Roberts, _Greek Epigraphy_, p.
109.]
The special fitness of the notation for the scales of the Enharmonic
genus may be regarded as a further indication of its date. We shall
see presently that that genus held a peculiar predominance in the
earliest period of musical theory--that, namely, which was brought to
an end by Aristoxenus.
If the author of the notation--or the second author, inventor of the
modified characters--was one of the musicians whose names have come
down to us, it would be difficult to find a more probable one than
that of Pronomus of Thebes. One of the most striking features of the
notation, at the time when it was framed, must have been the
adjustment of the keys. Even in the time of Aristoxenus, as we know
from the passage so often quoted, that adjustment was not universal.
But it is precisely what Pronomus of Thebes is said to have done for
the music of the flute (_supra_, p. 38). The circumstance that the
system was only used for instrumental music is at least in harmony
with this conjecture. If it is thought that Thebes is too far from
Argos, we may fall back upon the notice that Sacadas of Argos was the
chief composer for the flute before the time of Pronomus[1], and
doubtless Argos was one of the first cities to share in the advance
which Pronomus made in the technique of his art.
[Footnote 1: Pausanias (iv. 27, 4) says of the founding of Messene:
[Greek: eirgazonto de kai hypo mousuiês allês men oudemias, aulôn de
Boiôtiôn kai Argeiôn; ta te Sakada kai Pronomou melê tote dê
proêchthê malista eis hamillan.]]
§ 28. _Traces of the Species in the Notation._
Before leaving this part of the subject it will be well to notice the
attempt which Westphal makes to connect the species of the Octave
with the form of the musical notation.
The basis of the notation, as has been explained (p. 69), is formed
by two Diatonic octaves, denoted by the letters of the alphabet from
[Greek: a] to [Greek: n], as follows:
[Greek: ê i e l g m [digamma] th k d l b n z a]
_ a b c d e f g a b c d e f g a_
In this scale, as has been pointed out (p. 71), the notes which are
at the distance of an octave from each other are always expressed by
two _successive_ letters of the alphabet. Thus we find--
[Greek: b - g] is the octave _e - e_, the Dorian species.
[Greek: d - e] " " _c - c_, the Lydian species.
[Greek: [digamma] - z]" " _g - g_, the Hypo-phrygian species.
[Greek: ê - th] " " _a - a_, the Hypo-dorian species.
Westphal adopts the theory of Boeckh (as to which see p. 11) that the
Hypo-phrygian and Hypo-dorian species answered to the ancient Ionian
and Aeolian modes. On this assumption he argues that the order of the
pairs of letters representing the species agrees with the order of
the Modes in the historical development of Greek music. For the
priority of Dorian, Ionian, and Aeolian he appeals to the authority
of Heraclides Ponticus, quoted above (p. 9). The Lydian, he supposes,
was interposed in the second place on account of its importance in
education,--recognised, as we have seen, by Aristotle in the
_Politics_ (viii. 7 _ad fin._). Hence he regards the notation as
confirming his theory of the nature and history of the Modes.
The weakness of this reasoning is manifold. Granting that the
Hypo-dorian and Hypo-phrygian answer to the old Aeolian and Ionian
respectively, we have to ask what is the nature of the priority which
Heraclides Ponticus claims for his three modes, and what is the value
of his testimony. What he says is, in substance, that these are the
only kinds of music that are truly Hellenic, and worthy of the name
of modes ([Greek: harmoniai]). It can hardly be thought that this is
a criticism likely to have weighed with the inventor of the notation.
But if it did, why did he give an equally prominent place to Lydian,
one of the modes which Heraclides condemned? In fact, the
introduction of Lydian goes far to show that the coincidence--such as
it is--with the views of Heraclides is mere accident. Apart, however,
from these difficulties, there are at least two considerations which
seem fatal to Westphal's theory:
1. The notation, so far as the original two octaves are concerned,
must have been devised and worked out at some one time. No part of
these two octaves can have been completed before the rest. Hence the
order in which the letters are taken for the several notes has no
historical importance.
2. The notation does not represent only the _species_ of a scale,
that is to say, the relative pitch of the notes which compose it, but
it represents also the absolute pitch of each note. Thus the octaves
which are defined by the successive pairs of letters, [Greek:b-g,
d-e], and the rest, are octaves of definite notes. If they were
framed with a view to the ancient modes, as Westphal thinks, they
must be the actual scales employed in these modes. If so, the modes
followed each other, in respect of pitch, in an order exactly the
reverse of the order observed in the keys. It need hardly be said
that this is quite impossible. § 29. _Ptolemy's Scheme of Modes._
The first writer who takes the Species of the Octave as the basis of
the musical scales is the mathematician Claudius Ptolemaeus (fl.
140-160 A.D.). In his _Harmonics_ he virtually sets aside the scheme
of keys elaborated by Aristoxenus and his school, and adopts in their
place a system of scales answering in their main features to the
mediaeval Tones or Modes. The object of difference of key, he says,
is not that the music as a whole may be of a higher or lower pitch,
but that a melody may be brought within a certain compass. For this
purpose it is necessary to vary the succession of intervals (as a
modern musician does by changing the signature of the clef). If, for
example, we take the Perfect System ([Greek: systêma ametabolon]) in
the key of _a_ minor--which is its natural key,--and transpose it to
the key of _d_ minor, we do so, according to Ptolemy, not in order to
raise the general pitch of our music by a Fourth, but because we wish
to have a scale with _b_ flat instead of _b_ natural. The flattening
of this note, however, means that the two octaves change their
species. They are now of the species _e - e_. Thus, instead of
transposing the Perfect System into different keys, we arrive more
directly at the desired result by changing the species of its
octaves. And as there are seven possible species of the Octave, we
obtain seven different Systems or scales. From these assumptions it
follows, as Ptolemy shows in some detail, that any greater number of
keys is useless. If a key is an octave higher than another, it is
superfluous because it gives us a mere repetition of the same
intervals[1].
[Footnote 1: _Harm._ ii. 8 [Greek: hoi de hyperekpiptontes tou dia
pasôn tous ap' autou tou dia pasôn apôterô parelkontôs hypotithentai,
tous autous aei ginomenous tois proeilêmmenois.]]
If we interpose a key between (_e.g._) the Hypo-dorian and the
Hypo-phrygian, it must give us over again either the Hypo-dorian or
the Hypo-phrygian scale[1]. Thus the fifteen keys of the
Aristoxeneans are reduced to seven, and these seven are not
transpositions of a single scale, but are all of the same pitch. See
the table at the end of the book.
With this scheme of Keys Ptolemy combined a new method of naming the
individual notes. The old method, by which a note was named from its
relative place in the Perfect System, must evidently have become
inconvenient. The Lydian Mesê, for example, was two tones higher than
the Dorian Mesê, because the Lydian scale as a whole was two tones
higher than the Dorian. But when the two scales were reduced to the
same compass, the old Lydian Mesê was no longer in the middle of the
scale, and the name ceased to have a meaning. It is as though the
term 'dominant' when applied to a Minor key were made to mean the
dominant of the relative Major key. On Ptolemy's method the notes of
each scale were named from their places in it. The old names were
used, Proslambanomenos for the lowest, Hypatê Hypatôn for the next,
and so on, but without regard to the intervals between the notes.
Thus there were two methods of naming, that which had been in use
hitherto, termed 'nomenclature according to _value_' ([Greek:
onomasia kata dynamin]), and the new method of naming from the
various scales, termed 'nomenclature according to _position_'
([Greek: onomasia kata thesin]). The former was in effect a retention
of the Perfect System and the Keys: the latter put in their place a
scheme of seven different standard Systems.
[Footnote 1: _Harm._ ii. 11 [Greek: hôste mêd' an heteron eti doxai
tô eidei ton tonon para ton proteron, all' hypodôrion palin, ê ton
auton hypophrygion, oxyphônoteron tinos ê baryphônoteron monon.]]
In illustration of his theory Ptolemy gives tables showing in numbers
the intervals of the octaves used in the different keys and genera.
He shows two octaves in each key, viz. that from Hypatê Mesôn
([Greek: kata thesin]) to Nêtê Diezeugmenôn (called the octave
[Greek: apo nêtês]), and that from Proslambanomenos to Mesê (the
octave [Greek: apo mesês]). As he also gives the divisions of five
different 'colours' or varieties of genus, the whole number of
octaves is no less than seventy.
Ptolemy does not exclude difference of pitch altogether. The whole
instrument, he says, may be tuned higher or lower at pleasure[1].
Thus the pitch is treated by him as modern notation treats the
_tempo_, viz. as something which is not absolutely given, but has to
be supplied by the individual performer.
Although the language of Ptolemy's exposition is studiously
impersonal, it may be gathered that his reduction of the number of
keys from fifteen to seven was an innovation proposed by himself[2].
If this is so, the rest of the scheme,--the elimination of the
element of pitch, and the 'nomenclature by position,'--must also be
due to him. Here, however, we find ourselves at issue with Westphal
and those who agree with him on the main question of the Modes.
According to Westphal the nomenclature by position is mentioned by
Aristoxenus, and is implied in at least one important passage of the
Aristotelian _Problems_. We have now to examine the evidence which he
adduces to support his contention.
[Footnote 1: _Harm._ ii. 7 [Greek: pros tên toiautên diaphoran hê tôn
organôn holôn epitasis ê palin anesis aparkei.]]
[Footnote 2: This may be traced in the occasionally controversial
tone; as _Harm._ ii. 7 [Greek: hoi men ep' elatton tou dia pasôn
phthasantes, hoi d' ep' auto monon, hoi de epi to meizon toutou,
prokopên tina schedon toiautên aei tôn neôterôn para tous
palaioterous thêrômenôn, anoikeion tês peri to hêrmosmenon physeôs te
kai apokatastaseôs; hê monê perainein anankaion esti tên tôn esomenôn
akrôn tonôn diastasin]. We may compare c. 11.]
§ 30. _Nomenclature by Position._
Two passages of Aristoxenus are quoted by Westphal in support of his
contention. The first (p. 6 Meib.) is one in which Aristoxenus
announces his intention to treat of Systems, their number and nature:
'setting out their differences in respect of compass ([Greek:
megethos]), and for each compass the differences in form and
composition and position ([Greek: tas te kata schêma kai kata
synthesin kai kata thesin]), so that no element of melody,--either
compass or form or composition or position,--may be unexplained.' But
the word [Greek: thesis], when applied to Systems, does not mean the
'position' of single notes, but of groups of notes. Elsewhere (p. 54
Meib.) he speaks of the position of tetrachords towards each other
([Greek: tas tôn tetrachordôn pros allêla theseis]), laying it down
that any two tetrachords in the same System must be consonant either
with each other or with some third tetrachord. The other passage
quoted by Westphal (p. 69 Meib.) is also in the discussion of
Systems. Aristoxenus is pointing out the necessity of recognising
that some elements of melodious succession are fixed and limited,
others are unlimited:
[Greek: kata men oun ta megethê tôn diastêmatôn kai tas tôn
phthongôn taseis apeira pôs phainetai einai ta peri melos,
kata de tas dynameis kai kata ta eidê kai kata tas theseis
peperasmena te kai tetagmena.]
'In the size of the intervals and the pitch of the notes the
elements of melody seem to be infinite; but in respect of the
values (_i.e._ the relative places of the notes) and in
respect of the forms (_i.e._ the succession of the intervals)
and in respect of the positions they are limited and settled.'
Aristoxenus goes on to illustrate this by supposing that we wish to
continue a scale downwards from a [Greek: pyknon] or pair of small
intervals (Chromatic or Enharmonic). In this case, as the [Greek:
pyknon] forms the lower part of a tetrachord, there are two
possibilities. If the next lower tetrachord is disjunct, the next
interval is a tone; if it is conjunct, the next interval is the large
interval of the genus ([Greek: hê men gar kata tonon eis diazeuxin
agei to tou systêmatos eidos, hê de kata thateron diastêma ho ti
dêpot' echei megethos eis synaphên]). Thus the succession of
intervals is determined by the relative position of the two
tetrachords, as to which there is a choice between two definite
alternatives. This then is evidently what is meant by the words
[Greek: kata tas theseis][1]. On the other hand the [Greek: thesis]
of Ptolemy's nomenclature is the absolute pitch (_Harm._ ii. 5
[Greek: pote men par' autên tên thesin, to oxyteron haplôs ê
baryteron, onomazomen]), and this is one of the elements which
according to Aristoxenus are indefinite.
[Footnote 1: So Bacch. p. 19 Meib. [Greek: theseis de tetrachordôn
hois to melos horizetai eisin hepta? synaphê, diazeuxis,
hypodiazeuxis, k.t.l.] (see the whole passage).]
Westphal also finds the nomenclature by position implied in the
passage of the Aristotelian _Problems_ (xix. 20) which deals with the
peculiar relation of the Mesê to the rest of the musical scale. The
passage has already been quoted and discussed (_supra_, p. 43), and
it has been pointed out that if the Mesê of the Perfect System
([Greek: mesê kata dynamin]) is the key-note, the scale must have
been an octave of the _a_-species. If octaves of other species were
used, as Westphal maintains, it becomes necessary to take the Mesê of
this passage to be the [Greek: mesê kata thesin], or Mesê by
position. That is, Westphal is obliged by his theory of the Modes to
take the term Mesê in a sense of which there is no other trace before
the time of Ptolemy. But--
(1) It is highly improbable that the names of the notes--Mesê,
Hypatê, Nêtê and the rest--should have had two distinct meanings.
Such an ambiguity would have been intolerable, and only to be
compared with the similar ambiguity which Westphal's theory implies
in the use of the terms Dorian, &c.
(2) If the different species of the octave were the practically
important scales, as Westphal maintains, the position of the notes in
these scales must have been correspondingly important. Hence the
nomenclature by position must have been the more usual and familiar
one. Yet, as we have shown, it is not found in Aristotle, Aristoxenus
or Euclid--to say nothing of later writers.
(3) The nomenclature by position is an essential part of the scheme
of Keys proposed by Ptolemy. It bears the same relation to Ptolemy's
octaves as the nomenclature by 'value' bears to the old standard
octave and the Perfect System. It was probably therefore devised
about the time of Ptolemy, if not actually by him.
§ 31. _Scales of the Lyre and Cithara._
The earliest evidence in practical music of any octaves other than
those of the standard System is to be found in the account given by
Ptolemy of certain scales employed on the lyre and cithara. According
to this account the scales of the lyre (the simpler and commoner
instrument) were of two kinds. One was Diatonic, of the 'colour' or
variety which Ptolemy recognises as the prevailing one, viz. the
'Middle Soft' or 'Tonic' ([Greek: diatonon toniaion])[1].
[Footnote 1: We may think of this as a scale in which the semitones
are considerably smaller, _i.e._ in which _c_ and _f_ are nearly a
quarter of a tone flat.]
The other was a 'mixture' of this Diatonic with the standard
Chromatic ([Greek: chrôma suntonon]): that is to say, the octave
consisted of a tetrachord of each genus. These octaves apparently
might be of any _species_, according to the key chosen[1]. On the
cithara,--which was a more elaborate form of lyre, confined in
practice to professional musicians,--six different octave scales were
employed, each of a particular species and key. They are enumerated
and described by Ptolemy in two passages (_Harm._ i. 16 and ii. 16),
which in some points serve to correct each other.[2]
[Footnote 1: Ptol. _Harm._ ii. 16 [Greek: periechetai de ta men en tê
lyra kaloumena sterea tonou tinos hypo tôn tou toniaiou diatonou
arithmôn tou autou tonou, ta de malaka hypo tôn en tô migmati tou
malakou chrômatos apithmôn tou autou tonou]. Here [Greek: tonou
tinos] evidently means 'of any given key,' and [Greek: tou autou
tonou] 'of that key.' There is either no restriction, or none that
Ptolemy thought worth mentioning, in the choice of the key and
species.]
[Footnote 2: The two passages enumerate the scales in a slightly
different manner. In i. 16 they are arranged in view of the genus or
colour into--
Pure Middle Soft Diatonic, viz.--
[Greek: sterea], of the lyre.
[Greek: tritai] } of the cithara.
[Greek: hypertropa] }
Mixture of Chromatic, viz.--
[Greek: malaka], of the lyre.
[Greek: tropika], of the cithara.
Mixture of Soft Diatonic, viz.--
[Greek: parypatai], of the cithara.
Mixture of [Greek: diatonon syntonon], viz.--
[Greek: lydia] } of the cithara.
[Greek: iastia] }
It is added, however, that in their use of this last 'mixture'
musicians are in the habit of tuning the cithara in the Pythagorean
manner, with two Major tones and a [Greek: leimma] (called [Greek:
diatonon ditoniaion]).
In the second passage (ii. 16) the scales of the lyre are given
first, then those of the cithara with the key of each. The order is
the same, except that [Greek: parypatai] comes before [Greek:
tropika] (now called [Greek: tropoi]), and [Greek: lydia] is placed
last. The words [Greek: ta de lydia hoi tou toniaiou diatonou] [sc.
[Greek: arithmoi periechousi]] [Greek: tou dôriou] cannot be correct,
not merely because they contradict the statement of the earlier
passage that [Greek: lydia] denoted a mixture with [Greek: diatonon
syntonon] (or in practice [Greek: diatonon ditoniaion]), but also
because the scales that do not admit mixture are placed first in the
list in both passages. Hence we should doubtless read [Greek: ta de
lydia hoi <tou migmatos> tou <di>toniaiou diatonou tou Dôriou].]
Of the six scales two are of the Hypo-dorian or Common species
(_a-a_). One of these, called [Greek: tritai], is purely Diatonic of
the Middle Soft variety; the intervals expressed by fractions are as
follows:
_a_ 9/8 _b_ 28/27 _c_ 8/7 _d_ 9/8 _e_ 28/27 _f_ 8/7 _g_ 9/8 _a_
The other, called [Greek: tropoi] or [Greek: tropika], is a mixture,
Middle Soft Diatonic in the upper tetrachord, and Chromatic in the
lower:
_a_ 9/8 _b_ 22/21 _c_ 12/11 _c_[Symbols: sharp] 7/6 _e_ 28/27 _f_ 8/7
_g_ 9/8 _a_
Two scales are of the Dorian or _e_-species, viz. [Greek: parypatai],
a combination of Soft and Middle Soft Diatonic:
_e_ 21/20 _f_ 10/9 _g_ 8/7 _a_ 9/8 _b_ 28/27 _c_ 8/7 d 9/8 _e_
and [Greek: lydia], in which the upper tetrachord is of the strict or
'highly strung' Diatonic ([Greek: diatonon syntonon]--our 'natural'
temperament):
_e_ 28/27 _f_ 8/7 _g_ 9/8 _a_ 9/8 _b_ 16/15 _c_ 9/8 _d_ 10/9 _e_
Westphal (_Harmonik und Melopöie_, 1863, p. 255) supposes a
much deeper corruption. He would restore [Greek: ta de lydia
[kai iastia hoi tou migmatos tou syntonou diatonou tou ... ta
de ...] hoi tou toniaiou diatonou tou Dôriou]. This introduces
a serious discrepancy between the two passages, as the number
of scales in the second list is raised to eight (Westphal
making [Greek: iastia] and [Greek: iastiaioliaia] distinct
scales, and furthermore inserting a new scale, of unknown
name). Moreover the (unknown) scale of unmixed [Greek:
diatonon toniaion] is out of its place at the end of the list.
Westphal's objection to [Greek: lydia] as the name of a scale
of the _Dorian_ species of course only holds good on his
theory of the Modes.
The only other differences between the two passages are:
(1) In the scales of the lyre called [Greek: malaka] the
admixture, according to i. 16, is one of [Greek: chrômatikon
syntonon], according to ii. 16 of [Greek: chr. malakon]. But,
as Westphal shows, Soft Chromatic is not admitted by Ptolemy
as in practical use. It would seem that in the second passage
the copyist was led astray by the word [Greek: malaka] just
before.
(2) The [Greek: iastia] of i. 16 is called [Greek:
iastiaioliaia] in ii. 16. We need not suppose the text to be
faulty, since the two forms may have been both in use.
Another point overlooked in Westphal's treatment is that
[Greek: diatonon syntonon] and [Greek: d. ditoniaion] are not
really distinguished by Ptolemy. In one passage (i. 16) he
gives his [Greek: lydia] and [Greek: iastia] as a mixture with
[Greek: d. syntonon], adding that in practice it was [Greek:
d. ditoniaion]. In the other (ii. 16) he speaks at once of
[Greek: d. ditoniaion]. This consideration brings the two
places into such close agreement that any hypothesis involving
discrepancy is most improbable.
In practice it appears that musicians tuned the tetrachord _b-e_ of
this scale with the Pythagorean two Major tones and [Greek: leimma].
Of the remaining scales one, called [Greek: hypertropa], is Phrygian
in species (_d-d_), and of the standard genus:
_d_ 9/8 _e_ 28/27 _f_ 8/7 _g_ 9/8 _a_ 9/8 _b_ 28/27 _c_ 8/7 _d_
One, called [Greek: iastia], or [Greek: iastiaioliaia], is of the
Hypo-phrygian or _g_-species, the tetrachord _b-e_ being 'highly
strung' Diatonic or (in practice) Pythagorean, viz.:
_g_ 9/8 _a_ 9/8 _b_ 256/243 _c_ 9/8 _d_ 9/8 _e_ 28/27 _f_ 8/7 _g_
Regarding the tonality of these scales there is not very much to be
said. In the case of the Hypo-dorian and Dorian octaves it will be
generally thought probable that the key-note is _a_ (the [Greek: mesê
kata dynamin]). If so, the difference between the two species is not
one of 'mode,'--in the modern sense,--but consists in the fact that
in the Hypo-dorian the compass of the melody is from the key-note
upwards, while in the Dorian it extends a Fourth below the key-note.
It is possible, however, that the lowest note (_e_) of the Dorian
octave was sometimes the key-note: in which case the _mode_ was
properly Dorian. In the Phrygian octave of Ptolemy's description the
key-note cannot be the Fourth or Mesê [Greek: kata thesin] (_g_),
since the interval _g-c_ is not consonant (9/8 × 9/8 × 28/27 being
less than 4/3). Possibly the lowest note (_d_) is the key-note; if so
the scale is of the Phrygian mode (in the modern sense). In the
Hypo-phrygian octave there is a similar objection to regarding the
Mesê [Greek: kata thesin] (_c_) as the key-note, and some probability
in favour of the lowest note (_g_). If the Pythagorean division of
the tetrachord _g-c_ were replaced by the natural temperament, which
the language used by Ptolemy[1] leads us to regard as the true
division, the scale would exhibit the intervals--
_g_ 5/4 _b_ 6/5 _d_ 7/6 _f_ 8/7 _g_
which give the natural chord of the Seventh. This however is no more
than a hypothesis.
It evidently follows from all this that Ptolemy's octaves do not
constitute a system of _modes_. They are merely the groups of notes,
of the compass of an octave, which are most likely to be used in the
several keys, and which Ptolemy or some earlier theorist chose to
call by the names of those keys.