Part 5

The subject of the musical scales ([Greek: systêmata]) is treated by 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.