§ 22. The Scales as treated by Aristoxenus.

The subject of the musical scales (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 (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 [20]. 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.: ta gar diagrammata autois tôn enarmoniôn (harmoniôn MSS.) 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 (harmoniôn MSS.) monon elegon, peri de tôn allôn genôn te kai schêmatôn en autô te tô genei tontô kai tois loipois oud' epecheirei oudeis katamanthanein.

'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.: 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. proapod.) 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.: 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 heptachordôn Meibomius and other editors read hepta oktachordôn—a correction strongly suggested by the parallel words 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 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—

4. Harm. p. 28 Meib.: 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 [21].'

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 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' (pyknon, or a pyknôma), and the filling up of the scale in that way was therefore a 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: hautê estin hê para tois archaiois kata dieseis harmonia, heôs κδ dieseôn to proteron diagousa dia pasôn, to deuteron dia tôn hêmitoniôn auxêsasa: 'this is the 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 [22].'

The phrase 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' (ha ekaloun harmonias). That statement has usually been taken to refer to the ancient Modes called harmoniai by Plato and Aristotle, and has been used accordingly as proof that the scales of these Modes were based upon the different species (eidê) of the Octave. But the form of the reference—'which they called 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 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 [23].

The use of harmonia as an equivalent for 'System' or 'division of the scale' appears in an important passage in Plato's Philebus (p. 17): 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 ta systêmata theôrêsai posa te esti kai poia atta, kai pôs ek te tôn diastêmatôn kai phthongôn synestêkota). Further, the use of harmonia for systêma, or rather of the plural harmoniai for the 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 (hoi prosthen paredosan), his own word being systêma. It need not be said that there is no such hesitation, either in Plato or in Aristotle, about the use of harmoniai for the modes.

The same use of harmonia is found in the Aristotelian Problems (xix. 26), where the question is asked, 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 harmoniai or Modes of early Greek music are the same as the 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 harmonia goes beyond what is probable. In itself however the word meant simply 'musical scale [24].' 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 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. tonos in one sense, systêma in another, genos enarmonion in a third. Naturally, too, the more precise terms would be first employed by technical writers.