Chapter III: Part 3

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.

Another consideration, which has been overlooked by Westphal and those who follow him, is the difference between the species of the Octave in the several genera, especially the difference between the Diatonic and the Enharmonic. This is not felt as a difficulty with all the species. Thus the so-called Dorian octave _e - e_ is in the Enharmonic genus _e e* f a b b* c e_, a scale which may be regarded as the Diatonic with _g_ and _d_ omitted, and the semitones divided. But the Phrygian _d-d_ cannot pass in any such way into the Enharmonic Phrygian _c e e* f a b b* c_, which answers rather to the Diatonic scale of the species _c-c_ (the Lydian). The scholars who connect the ancient Modes with the species generally confine themselves to octaves of the Diatonic genus. In this they are supported by later Greek writers--notably, as we shall see, by Ptolemy--and by the analogy of the mediaeval Modes or Tones. But on the other side we have the repeated complaints of Aristoxenus that the earlier theorists confined themselves to Enharmonic octave scales. We have also the circumstance that the writer or compiler of the pseudo-Euclidean treatise, who is our earliest authority for the names of the species, gives these names for the Enharmonic genus only. Here, once more, we feel the difference between theory and practice. To a theorist there is no great difficulty in the terms Diatonic Phrygian and Enharmonic Phrygian meaning essentially different things. But the 'Phrygian Mode' in practical music must have been a tolerably definite musical form.

§ 25. _The Ethos of Music._

From Plato and Aristotle we have learned some elements of what may be called the gamut of sensibility. Between the higher keys which in Greece, as in Oriental countries generally, were the familiar vehicle of passion, especially of the passion of grief, and the lower keys which were regarded, by Plato at least, as the natural language of ease and license, there were keys expressive of calm and balanced states of mind, free from the violent extremes of pain and pleasure. In some later writers on music we find this classification reduced to a more regular form, and clothed in technical language. We find also, what is still more to our purpose, an attempt to define more precisely the musical forms which answered to the several states of temper or emotion.

Among the writers in question the most instructive is Aristides Quintilianus. He discusses the subject of musical ethos under the first of the usual seven heads, that which deals with sounds or notes ([Greek: peri phthongôn]). Among the distinctions to be drawn in regard to notes he reckons that of ethos: the ethos of notes, he says, is different as they are higher or lower, and also as they are in the place of a Parhypatê or in the place of a Lichanos (p. 13 Meib. [Greek: hetera gar êthê tois oxyterois, hetera tois baryterois epitrechei, kai hetera men parypatoeidesin, hetera de lichanoeidesin]). Again, under the seventh head, that of [Greek: melopoiia] or composition, he treats of the 'regions of the voice' ([Greek: topoi tês phônês]). There are three kinds of composition, he tells us (p. 28), viz. that which is akin to Hypatê ([Greek: hypatoeidês]), that which is akin to Mesê ([Greek: mesoeidês]), and that which is akin to Nêtê ([Greek: nêtoeidês]). The first part of the art of composition is the choice ([Greek: lêpsis]) which the musician is able to make of the region of the voice to be employed ([Greek: lêpsis men di' hês heuriskein tô mousikô perigignetai apo poiou tês phônês to systêma topou poiêteon, poteron hypatoeidous ê tôn loipôn tinos]). He then proceeds to connect these regions, or different parts of the musical scale, with different branches of lyrical poetry. 'There are three styles of musical composition ([Greek: tropoi tês melopoiias]), viz. the Nomic, the Dithyrambic, and the Tragic; and of these the Nomic is netoid, the Dithyrambic is mesoid, and the Tragic is hypatoid.... They are called styles ([Greek: tropoi]) because according to the melody adopted they express the ethos of the mind. Thus it happens that composition ([Greek: melopoiia]) may differ in _genus_, as Enharmonic, Chromatic: in _System_, as Hypatoid, Mesoid, Netoid: in _key_, as Dorian, Phrygian: in _style_, as Nomic, Dithyrambic: in _ethos_, as we call one kind of composition "contracting" ([Greek: systaltikê]), viz. that by which we move painful feelings; another "expanding" ([Greek: diastaltikê]), that by which we arouse the spirit ([Greek: thymos]); and another "middle" ([Greek: mesê]), that by which we bring round the soul to calmness.'

This passage does not quite explicitly connect the three kinds of ethos--the diastaltic, the systaltic, the intermediate--with the three regions of the voice; but the connexion was evidently implied, and is laid down in express terms in the pseudo-Euclidean _Introductio_ (p. 21 Meib.). According to this Aristoxenean writer, 'the diastaltic ethos of musical composition is that which expresses grandeur and manly elevation of soul ([Greek: megaloprepeia kai diarma psychês andrôdes]), and heroic actions; and these are employed by tragedy and all poetry that approaches the tragic type. The systaltic ethos is that by which the soul is brought down into a humble and unmanly frame; and such a disposition will be fitting for amatory effusions and dirges and lamentations and the like. And the hesychastic or tranquilly disposed ethos ([Greek: hêsychastikon êthos]) of musical composition is that which is followed by calmness of soul and a liberal and peaceful disposition: and this temper will fit hymns, paeans, laudations, didactic poetry and the like.' It appears then that difference in the 'place' ([Greek: topos]) of the notes employed in a composition--difference, that is to say, of pitch--was the element which chiefly determined its ethos, and (by consequence) which distinguished the music appropriate to the several kinds of lyrical poetry.

A slightly different version of this piece of theory is preserved in the anonymous treatise edited by Bellermann (§§ 63, 64), where the 'regions of the voice' are said to be four in number, viz. the three already mentioned, and a fourth which takes its name from the tetrachord Hyperbolaiôn ([Greek: topos hyperboloeidês]). In the same passage the boundaries of the several regions are laid down by reference to the keys. 'The lowest or hypatoid region reaches from the Hypo-dorian Hypatê Mesôn to the Dorian Mesê; the intermediate or mesoid region from the Phrygian Hypatê Mesôn to the Lydian Mesê; the netoid region from the Lydian Mesê to the Nêtê Synemmenôn; the hyperboloid region embracing all above the last point.' The text of this passage is uncertain; but the general character of the [Greek: topoi] or regions of the voice is clearly enough indicated.

The three regions are mentioned in the catechism of Bacchius (p. 11 Meib.): [Greek: topous] (MSS. [Greek: tropous]) [Greek: de tês phônês posous legomen einai? treis. tinas? toutous; oxyn, meson, baryn.] The varieties of ethos also appear (p. 14 Meib.): [Greek: hê de metabolê kata êthos? hotan ek tapeinou eis megaloprepes; ê ex hêsychou kai synnou eis parakekinêkos.] 'What is change of ethos? when a change is made from the humble to the magnificent; or from the tranquil and sober to violent emotion.'

When we compare the doctrine of musical ethos as we find it in these later writers with the indications to be gathered from Plato and Aristotle, the chief difference appears to be that we no longer hear of the ethos of particular modes, but only of that of three or (at the most) four portions of the scale. The principle of the division, it is evident, is simply difference of pitch. But if that was the basis of the ethical effect of music in later times, the circumstance goes far to confirm us in the conclusion that it was the pitch of the music, rather than any difference in the succession of the intervals, that principally determined the ethical character of the older modes.

§ 26. _The Ethos of the Genera and Species._

Although the pitch of a musical composition--as these passages confirm us in believing--was the chief ground of its ethical character, it cannot be said that no other element entered into the case.

In the passage quoted above from Aristides Quintilianus (p. 13 Meib.) it is said that ethos depends first on pitch ([Greek: hetera êthê tois oxyterois, hetera tois baryterois]), and secondly on the moveable notes, that is to say, on the _genus_. For that is evidently involved in the words that follow: [Greek: kai hetera men parypatoeidesin, hetera de lichanoeidesin.] By [Greek: parypatoeideis] and [Greek: lichanoeideis] he means all the moveable notes ([Greek: phthongoi pheromenoi]): the first are those which hold the place of Parhypatê in their tetrachord, viz. the notes called Parhypatê or Tritê: the second are similarly the notes called Lichanos or Paranêtê. These moveable notes, then, give an ethos to the music because they determine the genus of the scale. Regarding the particular ethos belonging to the different genera, there is a statement of the same author (p. 111) to the effect that the Diatonic is masculine and austere ([Greek: arrhenôpon d' esti kai austêroteron]), the Chromatic sweet and plaintive ([Greek: hêdiston te kai goeron]), the Enharmonic stirring and pleasing ([Greek: diegertikon d' esti touto kai êpion]). The criticism doubtless came from some earlier source.

Do we ever find ethos attributed to this or that _species_ of the Octave? I can find no passage in which this source of ethos is indicated. Even Ptolemy, who is the chief authority (as we shall see) for the value of the species, and who makes least of mere difference of pitch, recognises only two forms of modulation in the course of a melody, viz. change of genus and change of pitch[1].

§ 27. _The Musical Notation._

As the preceding argument turns very much upon the practical importance of the scale which we have been discussing, first as the single octave from the original Hypatê to Nêtê, then in its enlarged form as the Perfect System, it may be worth while to show that some such scale is implied in the history of the Greek musical notation.

The use of written characters ([Greek: sêmeia]) to represent the sounds of music appears to date from a comparatively early period in Greece. In the time of Aristoxenus the art of writing down a melody ([Greek: parasêmantikê]) had come to be considered by some persons identical with the science of music ([Greek: harmonikê]),--an error which Aristoxenus is at some pains to refute. It is true that the authorities from whom we derive our knowledge of the Greek notation are post-classical. But the characters themselves, as we shall presently see, furnish sufficient evidence of their antiquity.

[Footnote 1: Ptol. _Harm._ ii. 6. After drawing a distinction between difference of key as affecting the whole of a melody or piece of music and as a means of change in the course of it--the distinction, in short, between transposition and modulation proper--he says of the latter: [Greek: hautê de hôsper ekpiptein autên] (sc. [Greek: tên aisthêsin]) [Greek: poiei tou synêthous kai prosdokômenou melous, hotan epi pleon men syneirêtai to akolouthon, metabainê de pê pros heteron eidos, êtoi kata to genos ê kata tên tasin.] That is to say, the sense of change is produced by a change of genus or of pitch. A change of _species_ is not suggested. So Dionys. Hal. _De Comp. Verb._ c. 19 [Greek: hoi de ge dithyrambopoioi kai tous tropous] (keys) [Greek: meteballon, Dôrikous te kai Phrygious kai Lydious en tô autô asmati poiountes; kai tas melôdias exêllatton, tote men enarmonious poiountes, k.t.l.]]

The Greek musical notation is curiously complicated. There is a double set of characters, one for the note assigned to the singer, the other for those of the lyre or other instrument. The notes for the voice are obviously derived from the letters of the ordinary Ionic alphabet, multiplied by the use of accents and other diacritical marks. The instrumental notes were first explained less than thirty years ago by Westphal. In his work _Harmonik und Melopöie der Griechen_ (c. viii _Die Semantik_) he showed, in a manner as conclusive as it is ingenious, that they were originally taken from the first fourteen letters of an alphabet of archaic type, akin to the alphabets found in certain parts of Peloponnesus. Among the letters which he traces, and which point to this conclusion, the most-significant are the digamma, the primitive crooked iota [Symbols: Li], and two forms of lambda, [Symbols: <] and [Symbols: F], the latter of which is peculiar to the alphabet of Argos. Of the other characters [Symbols: M], which stands for alpha, is best derived from the archaic form [Symbols: NJ]. For beta we find [Symbols: c], which may come from an archaic form of the letter[1]. The character [Symbols: El], as Westphal shows, is for [Symbols:7], or delta with part of one side left out. Similarly the ancient [Symbols: O], when the circle was incomplete, yielded the character [Symbols: C]. The crooked iota ([Symbols:'-i]) appears as [Symbols:h]. The two forms of lambda serve for different notes, thus bringing the number of symbols up to fifteen. Besides these there are two characters, [Symbols: O] and [Symbols: 6], which cannot be derived in the same way from any alphabet. As they stand for the lowest notes of the scale, they are probably an addition, later than the rest of the system. At the upper end, again, the scale is extended by the simple device of using the same characters for notes an octave higher, distinguishing them in this use by an accent. The original fifteen characters, with the letters from which they are derived, and the corresponding notes in the modern musical scale, are as follows:

[Symbols: H h E r P F C K r l < E N Z M]
[Greek: ê i e l^1 g m [digamma] th k d l^2 b n z a]
_a b c d e f g a b c d e f g a_

[Footnote 1: Since this was written I have learned from Mr. H. S. Jones that the form [Symbols:E] for beta occurs on an inscription dated about 500 B.C., viz. Count Tyszkiewicz's bronze plate, published simultaneously by Robert in the _Monumenti Antichi pubblicati per cura della reale Accademia dei Lincei_, i. pp. 593 (with plate), and Fröhner in the _Revue Archéologique_, 1891 July-August, pp. 51 ff. Pl. xix. Mr. Jones points out that this [Symbols:E] connects the crescent beta ([Symbols: C]) of Naxos, Delos, &c. with the common form, and is evidently therefore an early form of the letter.

I take this opportunity of thanking Mr. Jones for other help, especially in regard to the subject of this section.]

These notes, it will be seen, compose two octaves of the Diatonic scale, identical with the two octaves of the Greater Perfect System. They may be regarded as answering to the white notes of the modern keyboard,--those which form the complete scale in the so-called 'natural' key.

The other notes, viz. those which are required not only in different keys of the Diatonic scale, but also in all Enharmonic and Chromatic scales, are represented by the same characters modified in some simple way. Usually a character is turned half round backwards to raise it by one small interval (as from Hypatê to Parhypatê), and reversed to raise it by both (Hypatê to Lichanos). Thus the letter epsilon, [Symbols: E], stands for our _c_: and accordingly [Symbols: W] ([Symbols: E] [Greek: anestrammenon] or [Greek: hyption]) stands for _c*_, and [Symbols: 3] ([Symbols: E] [Greek: apestrammenon]) for _c[Symbols: #]_. The Enharmonic scale _c-c*-c[Symbols: #]-f_ is therefore written [Symbols: E W 3 f'], the two modifications of the letter [Symbols: E] representing the two 'moveable' notes of the tetrachord. Similarly we have the triads [Symbols: h I rl, F "q, Cup, KY>1, <V>, CUm]. As some letters do not admit of this kind of differentiation, other methods are employed. Thus [Symbols: D] is made to yield the forms [Symbols: ri] (for [Symbols: 7]) [Symbols: L A]: from [Symbols: H] (or [Symbols: B]) are obtained the forms [Symbols: Li] and [Symbols: R]: and from [Symbols: Z] (or [Symbols: i]) the forms [Symbols: A] and [Symbols: A]. The modifications of [Symbols: N] are [Symbols: /] and [Symbols: \]: those of [Symbols: 'I] are [Symbols: A] and [Symbols: N].

The method of writing a Chromatic tetrachord is the same, except that the higher of the two moveable notes is marked by a bar or accent. Thus the tetrachord _c c[Symbols: #] d f_ is written [Symbols: E W 3' /`'].

In the Diatonic genus we should have expected that the original characters would have been used for the tetrachords _b c d e_ and _e f g a_; and that in other tetrachords the second note, being a semitone above the first, would have been represented by a reversed letter ([Greek: gramma apestrammenon]). In fact, however, the Diatonic Parhypatê and Tritê are written with the same character as the Enharmonic. That is to say, the tetrachord _b c d e_ is not written [Symbols: h E H r], but [Symbols: Fix I-r]: and _d e[Symbols: b] f g_ is not [Symbols: I], but [Symbols: I-tl F].

Let us now consider how this scheme of symbols is related to the Systems already described and the Keys in which those Systems may be set ([Greek: tonoi eph' hôn tithemena ta systêmata melôdeitai]).

The fifteen characters, it has been noticed, form two diatonic octaves. It will appear on a little further examination that the scheme must have been constructed with a view to these two octaves. The successive notes are not expressed by the letters of the alphabet in their usual order (as is done in the case of the vocal notes). The highest note is represented by the first letter, [Greek: A]: and then the remaining fourteen notes are taken in pairs, each with its octave: and each of the pairs of notes is represented by two successive letters--the two forms of lambda counting as one such pair of letters. Thus:

The higher and lower _e_ are denoted by [Greek: b] and [Greek: g]
" " " _c_ " " [Greek: d] " [Greek: e]
" " " _g_ " " [Symbol: digamma] " [Greek: z]
" " " _a_ " " [Greek: ê] " [Greek: th]
" " " _b_ " " [Greek: i] " [Greek: k]
" " " _d_ " " [Greek: l^1] " [Greek: l^2]
" " " _f_ " " [Greek: m] " [Greek: n]

On this plan the alphabetical order of the letters serves as a series of links connecting the highest and lowest notes of every one of the seven octaves that can be taken on the scale. It is evident that the scheme cannot have grown up by degrees, but is the work of an inventor who contrived it for the practical requirements of the music of his time.

Two questions now arise, which it is impossible to separate. What is the scale or System for which the notation was originally devised? And how and when was the notation adapted to exhibit the several keys in which any such System might be set?

The enquiry must start from the remarkable fact that the two octaves represented by the fifteen original letters are in the _Hypo-lydian_ key--the key which down to the time of Aristoxenus was called the Hypo-dorian. Are we to suppose that the scheme was devised in the first instance for that key only? This assumption forms the basis of the ingenious and elaborate theory by which M. Gevaert explains the development of the notation (_Musique de l'Antiquité_, t. I. pp. 244 ff.). It is open to the obvious objection that the Hypo-lydian (or Hypo-dorian) cannot have been the oldest key. M. Gevaert meets this difficulty by supposing that the original scale was in the Dorian key, and that subsequently, from some cause the nature of which we cannot guess, a change of pitch took place by which the Dorian scale became a semitone higher. It is perhaps simpler to conjecture that the original Dorian became split up, so to speak, into two keys by difference of local usage, and that the lower of the two came to be called Hypo-dorian, but kept the original notation. A more serious difficulty is raised by the high antiquity which M. Gevaert assigns to the Perfect System. He supposes that the inventor of the notation made use of an instrument (the _magadis_) which 'magadised' or repeated the notes an octave higher. But this would give us a repetition of the primitive octave _e - e_, rather than an enlargement by the addition of tetrachords at both ends.

M. Gevaert regards the adaptation of the scheme to the other keys as the result of a gradual process of extension. Here we may distinguish between the recourse to the modified characters--which served essentially the same purpose as the 'sharps' and 'flats' in the signature of a modern key--and the additional notes obtained either by means of new characters ([Symbols: a] and [Symbols: e]), or by the use of accents ([Symbols:?'], &c.). The Hypo-dorian and Hypo-phrygian, which employ the new characters [Symbols: a] and [Symbols: e], are known to be comparatively recent. The Phrygian and Lydian, it is true, employ the accented notes; but they do so only in the highest tetrachord (Hyperbolaiôn), which may not have been originally used in these high keys. The modified characters doubtless belong to an earlier period. They are needed for the three oldest keys--Dorian, Phrygian, Lydian--and also for the Enharmonic and Chromatic genera. If they are not part of the original scheme, the musician who devised them may fairly be counted as the second inventor of the instrumental notation.

In setting out the scales of the several keys it will be unnecessary to give more than the standing notes ([Greek: phthongoi hestôtes]), which are nearly all represented by original or unmodified letters--the moveable notes being represented by the modified forms described above. The following list includes the standing notes, viz. Proslambanomenos, Hypatê Hypatôn, Hypatê Mesôn, Mesê, Paramesê, Nêtê Diezeugmenôn and Nêtê Hyperbolaiôn in the seven oldest keys: the two lowest are marked as doubtful:--

TABLE LEGEND:
Column A = Prosl.
Column B = Hyp. Hypatôn.
Column C = Hyp. Mesôn.
Column D = Mesê.
Column E = Par.
Column F = Nêtê Diez.
Column G = Nêtê Hyperb.

A B C D E F G

Mixo-lydian [Symbols] 4 id D > N \ = _e[Symbol: b] - e[Symbol: b]_
Lydian [Symbols] I- r c < c m = _d - d_
Phrygian [Symbols] E I- F 11 < Z = _c - c_
Dorian [Symbols] R E I' D ri N \ = _b[Symbol: b] - b[Symbol: b]_
Hypo-lydian [Symbols] H h r C I< c M = _a - a_
[Hypo-phrygian [Symbols] H I- F C < Z = _g - g_
[Hypo-dorian [Symbols] E /4 F 11 N = _f - f_

It will be evident that this scheme of notation tallies fairly well with what we know of the compass of Greek instruments about the end of the fifth century, and also with the account which Aristoxenus gives of the keys in use up to his time. We need only refer to what has been said above on p. 17 and p. 37.

It would be beyond the scope of this essay to discuss the date of the Greek musical notation. A few remarks, however, may be made, especially with reference to the high antiquity assigned to it by Westphal.

The alphabet from which it was derived was certainly an archaic one. It contained several characters, in particular [Symbols: F] for digamma, [Symbols: LI] for iota, and [Symbols: I-] for lambda, which belong to the period before the introduction of the Ionian alphabet. Indeed if we were to judge from these letters alone we should be led to assign the instrumental notation (as Westphal does) to the time of Solon. The three-stroke iota ([Symbols: I]), in particular, does not occur in any alphabet later than the sixth century B.C. On the other hand, when we find that the notation implies the use of a musical System in advance of any scale recognised in Aristotle, or even in Aristoxenus, such a date becomes incredible. We can only suppose either (1) that the use of [Symbols: Li] in the fifth century was confined to localities of which we have no complete epigraphic record, or (2) that [Symbols: i] as a form of iota was still known--as archaic forms must have been--from the older public inscriptions, and was adopted by the inventor of the notation as being better suited to his purpose than [Symbols: 1].

With regard to the place of origin of the notation the chief fact 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.