The Modes of Ancient Greek Music Part 11

The considerations set forth in the last chapter seem to show that if difference of mode or species cannot be entirely denied of the cla.s.sical period of Greek music, it occupied a subordinate and almost unrecognised place.

The main elements of the art were, (1) difference of _genus_,--the sub-divisions of the tetrachord which Aristoxenus and Ptolemy alike recognise, though with important discrepancies in detail; (2) difference of pitch or _key_; and (3) _rhythm_. Pa.s.sing over the last, as not belonging to the subject of _Harmonics_, we may now say that genus and key are the only grounds of distinction which are evidently of practical importance. No others were a.s.sociated with the early history of the art, with particular composers or periods, with particular instruments, or with the ethos of music. This, however, is only true in the fullest sense of Greek music before the time of Ptolemy. The main object of Ptolemy's reform of the keys was to provide a new set of scales, each characterised by a particular succession of intervals, while the pitch was left to take care of itself. And it is clear, especially from the specimens which Ptolemy gives of the scales in use in his time, that he was only endeavouring to systematise what already existed, and bring theory into harmony with the developments of practice. We must suppose, therefore, that the musical feeling which sought variety in differences of key came to have less influence on the practical art, and that musicians began to discover, or to appreciate more than they had done, the use of different 'modes' or forms of the octave scale. Along with this change we have to note the comparative disuse of the Enharmonic and Chromatic divisions of the tetrachord. The Enharmonic, according to Ptolemy, had ceased to be employed. Of the three varieties of Chromatic given by Aristoxenus only one remains on Ptolemy's list, and that the one which in the scheme of Aristoxenus involved no interval less than a semitone. And although Ptolemy distinguished at least three varieties of Diatonic, it is worth notice that only one of these was admitted in the tuning of the lyre,--the others being confined to the more elaborate cithara. In Ptolemy's time, therefore, music was rapidly approaching the stage in which all its forms are based upon a single scale--the natural diatonic scale of modern Europe.

In the light of these facts it must occur to us that Westphal's theory of seven modes or species of the Octave is really open to an _a priori_ objection as decisive in its nature as any of the testimony which has been brought against it. Is it possible, we may ask, that a system of modes a.n.a.logous to the ecclesiastical Tones can have subsisted along with a system of scales such as the genera and 'colours' of early Greek music? The reply may be that Ptolemy himself combines the two systems. He supposes five divisions of the tetrachord, and seven modes based upon so many species of the Octave--in all thirty-five different scales (or seventy, if we bring in the distinction of octaves [Greek: apo netes] and [Greek: apo meses]). But when we come to the scales actually used on the chief Greek instrument, the cithara, the number falls at once to six.

Evidently the others, or most of them, only existed on paper, as the mathematical results of certain a.s.sumptions which Ptolemy had made.

And if this can be said of Ptolemy's theory, what would be the value of a similar scheme combining the modes with the Enharmonic and the different varieties of the Chromatic genus? The truth is, surely, that such a scheme tries to unite elements which belong to different times, which in fact are the fundamental ideas of different stages of art.

The most striking characteristic of Greek music, especially in its earlier periods, is the multiplicity and delicacy of the intervals into which the scale was divided. A sort of frame-work was formed by the division of the octave into tetrachords, completed by the so-called disjunctive tone; and so far all Greek music was alike. But within the tetrachord the reign of diversity was unchecked. Not only were there recognised divisions containing intervals of a fourth, a third, and even three-eighths of a tone, but we gather from several things said by Aristoxenus that the number of possible divisions was regarded as theoretically unlimited. Thus he tells us that there was a constant tendency to flatten the 'moveable' notes of the Chromatic genus, and thus diminish the small intervals, for the sake of 'sweetness' or in order to obtain a plaintive tone[1];--that the Lichanos of a tetrachord may in theory be any note between the Enharmonic Lichanos (_f_ in the scale _e-e*-f-a_) and the Diatonic (_g_ in the scale _e-f-g-a_)[2];--and that the magnitude of the smaller intervals and division of the tetrachord generally belongs to the indefinite or indeterminate element in music[3]. Moreover, in spite of the disuse of several of the older scales, much of this holds good for the time of Ptolemy. The modern diatonic scale is fully recognised by him, but only as one of several different divisions. And the division which he treats as the ordinary or standard form of the octave is not the modern diatonic scale, but one of the so-called 'soft' or flattened varieties. It is clear that in the best periods of Greek music these refinements of melody, which modern musicians find scarcely conceivable, were far from being accidental or subordinate features. Rather, they were as much bound up with the fundamental nature of that music as complex harmony is with the music of modern Europe.

[Footnote 1: Aristox. _Harm._ p. 23 Meib. [Greek: hoi men gar te nun katechouse melopoiia ounetheis monon ontes eiktos ten ditonon lichanon] (_f_ in the scale _e-a_) [Greek: exorizousi; suntonoterais gar chrontai schedon hoi pleistoi ton nun. toutou d' aition to boulesthai glukainein aei. semeion de hoti toutou stochazontai, malista men gar kai pleiston chronon en to chromati diatribousin.

hotan d' aphikontai pote eis ten harmonian engus tou chromatos prosagousi, sunepismomenou tou ethous.]]

[Footnote 2: Ibid. p. 26 [Greek: noeteon gar apeirous ton arithmon tas lichanous. hou gar an steses ten phonen apodedeigmenon lichano topou lichanos estai; diakenon de ouden esti tou lichanoeidous topou, oude toiouton hoste me dechisthai lichanon]. And p. 48 [Greek: epeide per ho tes lichanou topos eis apeirous temnetai tomas].]

[Footnote 3: Aristox. _Harm._ p. 69 Meib. [Greek: kata men oun ta megethe ton diastematon kai tas ton phthongon taseis apeira pos phainetai einai ta peri to melos, kata de tas dynameis kai kata ta eide kai kata tas theseis peperasmena te kai tetagmena.]]

The mediaeval modes or Tones, on the other hand, are essentially based on the diatonic scale,--the scale that knows only of tones and semitones. To suppose that they held in the earliest Greek music the prominent place which we find a.s.signed to the ancient Modes or [Greek: harmoniai] is to suppose that the art of music was developed in Greece in two different directions, under the influence of different and almost opposite ideas. Yet nothing is more remarkable in all departments of Greek art than the strictness with which it confines itself within the limits given once for all in the leading types, and the consequent harmony and consistency of all the forms which it takes in the course of its growth.

The dependence of artistic forms in their manifold developments upon a central governing idea or principle has never been more luminously stated than by the ill.u.s.trious physicist Helmholtz, in the thirteenth chapter of his _Tonempfindungen_. I venture to think that in applying that truth to the facts of Greek music he was materially hindered by the accepted theory of the Greek modes. The scales which he a.n.a.lyses under that name were certainly the basis of all music in the Middle Ages, and are much more intelligible as such than in relation to the primitive Greek forms of the art[1].

[Footnote 1: The ecclesiastical Modes received their final shape in the _Dodecachordon_ of Glarea.n.u.s (Bale, 1547). They are substantially the Greek modes of Westphal's theory, although the Greek names which Glarea.n.u.s adopted seem to have been chosen at haphazard. But the ecclesiastical Modes, as Helmholtz points out, were developed under the influence of polyphonic music from the earlier stages represented by the Ambrosian and Gregorian scales. It would be a singular chance if they were also, as Greek modes, the source from which the Ambrosian and Gregorian scales were themselves derived.

Some further hints on this part of the subject may possibly be derived from the musical scales in use among nations that have not attained to any form of harmony, such as the Arabians, the Indians, or the Chinese. A valuable collection of these scales is given by Mr.

A. J. Ellis at the end of his translation of Helmholtz (Appendix XX.

Sect. K, _Non-harmonic Scales_). Among the most interesting for our purpose are the eight mediaeval Arabian scales given on the authority of Professor Land (nos. 54-61). The first three of these--called 'Ochaq, Nawa and Boas[=i]li--follow the Pythagorean intonation, and answer respectively to the Hypo-phrygian, Phrygian, and Mixo-lydian species of the octave. The next two--Rast and Zenkouleh--are also Hypo-phrygian in species, but the Third and Sixth are flatter by about an eighth of a tone (the Pythagorean comma). In Zenkouleh the Fifth also is similarly flattened. The last two scales--Hhosa[=i]ni and Hhidjazi--are Phrygian: but the Second and Fifth, and in the case of Hhidjazi also the Sixth, are flatter by the interval of a comma.

The remaining scale, called Rahawi, does not fall under any species, since the semitones are between the Third and Fourth, and again between the Fifth and Sixth. It will be seen that in general character--though by no means in details--this series of scales bears a considerable resemblance to the 'scales of the cithara' as given by Ptolemy (_supra_, p. 85). In both cases the several scales are distinguished from each other partly by the order of the intervals (_species_), partly by the intonation, or magnitude of the intervals employed (_genus_). This latter element is conspicuously absent from the ecclesiastical Modes.]

-- 37. _Epilogue--Speech and Song._

Several indications combine to make it probable that singing and speaking were not so widely separated from each other in Greek as in the modern languages with which we are most familiar.

(1) The teaching of the grammarians on the subject of accent points to this conclusion. Our habit of using Latin translations of the terms of Greek grammar has tended to obscure the fact that they belong in almost every case to the ordinary vocabulary of music. The word for 'accent' ([Greek: tonos]) is simply the musical term for 'pitch' or 'key.' The words 'acute' ([Greek: oxys]) and 'grave'

([Greek: barys]) mean nothing more than 'high' and 'low' in pitch. A syllable may have two accents, just as in music a syllable may be sung with more than one note. Similarly the 'quant.i.ty' of each syllable answers to the time of a musical note, and the rule that a long syllable is equal to two short ones is no doubt approximately correct. Consequently every Greek word (enc.l.i.tics being reckoned as parts of a word) is a sort of musical phrase, and every sentence is a more or less definite melody--[Greek: loG.o.des ti melos], as it is called by Aristoxenus (p. 18 Meib.). Moreover the accent in the modern sense, the _ictus_ or stress of the voice, appears to be quite independent of the pitch or 'tonic' accent: for in Greek poetry the _ictus_ ([Greek: arsis]) is determined by the metre, with which the tonic accent evidently has nothing to do. In singing, accordingly, the tonic accents disappear; for the melody takes their place, and gives each syllable a new pitch, on which (as we shall presently see) the spoken pitch has no influence. The rise and fall of the voice in ordinary speaking is perceptible enough in English, though it is more marked in other European languages. Helmholtz tells us--with tacit reference to the speech of North Germany--that an affirmative sentence generally ends with a drop in the tone of about a Fourth, while an interrogative is marked by a rise which is often as much as a Fifth[1]. In Italian the interrogative form is regularly given, not by a particle or a change in the order of the words, but by a rise of pitch. The Gregorian church music, according to a series of rules quoted by Helmholtz (_l. c._), marked a comma by a rise of a Tone, a colon by a fall of a Semitone; a full stop by a Tone above, followed by a Fourth below, the 'reciting note'; and an interrogation by a phrase of the form _d b c d_ (_c_ being the reciting note).

These examples, however, do little towards enabling modern scholars to form a notion of the Greek system of accentuation. In these and similar cases it is the _sentence as a whole_ which is modified by the tonic accent, whereas in Greek it is the individual _word_. It is true that the accent of a word may be affected by its place in the sentence: as is seen in the loss of the accent of oxytone words when not followed by a pause, in the anastrophe of prepositions, and in the treatment of the different cla.s.ses of enc.l.i.tics. But in all these instances it is the intonation of the word as such, not of the sentence, which is primarily concerned. What they really prove is that the musical accent is not so invariable as the stress accent in English or German, but may depend upon the collocation of the word, or upon the degree of emphasis which it has in a particular use.

[Footnote 1: _Tonempfindungen_, p. 364 (ed. 1863).]

(2) The same conclusion may be drawn from the terms in which the ancient writers on music endeavour to distinguish musical and ordinary utterance.

Aristoxenus begins his _Harmonics_ by observing that there are two movements of the voice, not properly discriminated by any previous writer; namely, the _continuous_, which is the movement characteristic of speaking, and the _discrete_ or that which proceeds by _intervals_, the movement of singing. In the latter the voice remains for a certain time on one note, and then pa.s.ses by a definite interval to another. In the former it is continually gliding by imperceptible degrees from higher to lower or the reverse[1]. In this kind of movement the rise and fall of the voice is marked by the _accents_ ([Greek: prosodiai]), which accordingly form the melody, as it may be called, of spoken utterance[2]. Later writers state the distinction in much the same language. Nicomachus tells us that the two movements were first discriminated by the Pythagoreans. He dwells especially on the ease with which we pa.s.s from one to the other. If the notes and intervals of the speaking voice are allowed to be separate and distinct, the form of utterance becomes singing[3].

Similarly Aristoxenus says that we do not rest upon a note, unless we are led to do so by the influence of feeling ([Greek: an me dia pathos pote eis toiauten kinesin anankasthomen elthein]).

[Footnote 1: Aristox., _Harm._ p. 3 Meib. [Greek: kineitai men gar kai dialegomenon hemon kai melodounton ten eiremenen kinesin; oxy gar kai bary delon hos en amphoterois toutois enestin.] Also p. 8 [Greek: dyo tines eisin ideai kineseos, he te syneches kai he diastematike; kata men oun ten syneche topon tina diexienai phainetai he phone te aisthesei houtos hos an medamou histamene, k.t.l.] And p. 9 [Greek: ten oun syneche logiken einai phanen, k.t.l.]]

[Footnote 2: Ibid. p. 18 Meib. [Greek: tou ge loG.o.dous kechoristai taute to mousikon melos; legetai gar de kai loG.o.des ti melos, to synkeimenon ek ton prosodion ton en tois onomasin; physikon gar to epiteinein kai anienai en to dialegesthai.]]

[Footnote 3: Nicomachus, _Enchiridion_, p. 4 [Greek: ei gar tis e dialegomenos e apologoumenos tini e anaginoskon ge ekdela metaxy kath' hekaston phthongon poiei ta megethe, diistanon kai metaballon ten phonen ap' allou eis allon, ouketi legein ho toioutos oude anaginoskein alla meleazein legetai.]]

According to the rhetorician Dionysius of Halicarna.s.sus the interval used in the melody of spoken utterance is approximately a Fifth, or three tones and a half ([Greek: dialektou men oun melos heni metreitai diastemati to legomeno dia pente, hos engista; kai oute epiteinetai pera ton trion tonon kai hemitoniou epi to oxy oute anietai tou choriou toutou pleion epi to bary][1]). He gives an interesting example (quoted above on p. 91) from the _Orestes_ of Euripides, to show that when words are set to music no account is taken of the accents, or spoken melody. Not merely are the intervals varied (instead of being nearly uniform), but the rise and fall of the notes does not answer to the rise and fall of the syllables in ordinary speech. This statement is rendered the more interesting from the circ.u.mstance that the inscription discovered by Mr. Ramsay (_supra_, p. 89), which is about a century later, does exhibit precisely this correspondence. Apparently, then, the melody of the inscription represents a new idea in music,--an attempt to bring it into a more direct connexion with the tones of the speaking voice.

The fact of such an attempt being made seems to indicate that the divergence between the two kinds of utterance was becoming more marked than had formerly been the case. It may be compared with the invention of recitative in the beginning of the seventeenth century.

Aristides Quintilia.n.u.s (p. 7 Meib.) recognises a third or intermediate movement of the voice, viz. that which is employed in the recitation of poetry. It is probable that Aristides is one of the latest writers on the subject, and we may conjecture that in his time the Greek

[Footnote 1: _De Compositione Verborum_, c. 11, p. 58 Reisk.]

language had in great measure lost the original tonic accents, and with them the quasi-melodious character which they gave to prose utterance.

In the view which these notices suggest the difference between speaking and singing is reduced to one of degree. It is a.n.a.lysed in language such as we might use to express the difference between a monotonous and a varied manner of speaking, or between the sounds of an Aeolian harp and those of a musical instrument.

(3) What has been said of melody in the two spheres of speech and song applies also _mutatis mutandis_ to rhythm. In English the time or quant.i.ty of syllables is as little attended to as the pitch. But in Greek the distinction of long and short furnished a prose rhythm which was a serious element in their rhetoric. In the rhythm of music, according to Dionysius, the quant.i.ty of syllables could be neglected, just as the accent was neglected in the melody[1]. This, however, does not mean that the natural time of the syllables could be treated with the freedom which we see in a modern composition. The regularity of lyric metres is sufficient to prove that the increase or diminution of natural quant.i.ty referred to by Dionysius was kept within narrow limits, the nature of which is to be gathered from the remains of the ancient system of Rhythmic. From these sources we learn with something like certainty that the rhythm of ordinary speech, as determined by the succession of long or short syllables, was the basis not only of metres intended for recitation, such as the hexameter and the iambic trimeter, but also of lyrical rhythm of every kind.

[Footnote 1: _De Comp._ c. 11, p. 64 [Greek: to de auto ginetai kai peri tous rhythmous; he men gar peze lexis oudenos oute onomatos oute rhematos biazetai tous chronous oude metat.i.thesin, all' oias pareilephe te physei tas syllabas, tas te makras kai tas bracheias, toiautas phylattei; he de mousike te kai rhythmike metaballousin autas meiousai kai parauxousai, oite pollakis eis tanantia metach.o.r.ein; ou gar tais syllabais apeuthynousi tous chronous, alla tois chronois tas syllabas.]]

(4) As to the use of the stress accent in Greek prose we are without direct information. In verse it appears as the metrical _ictus_ or _arsis_ of each foot, which answers to what English musicians call the 'strong beat' or accented part of the bar[1]. In the Homeric hexameter the ictus is confined to long syllables, and appears to have some power of lengthening a short or doubtful syllable. In the Attic poetry which was written in direct imitation of colloquial speech, viz. the tragic and comic trimeter, there is no necessary connexion between the ictus and syllabic length: but on the other hand a naturally long syllable which is without the ictus may be rhythmically short. In lyrical versification the ictus does not seem to have any connexion with quant.i.ty: and on the whole we may gather that it was not until the Byzantine period of Greek that it came to be recognised as a distinct factor in p.r.o.nunciation. The chief elements of utterance--pitch, time and stress--were independent in ancient Greek speech, just as they are in music. And the fact that they were independent goes a long way to prove our main contention, viz. that ancient Greek speech had a peculiar quasi-musical character, consequently that the difficulty which modern scholars feel in understanding the ancient statements on such matters as accent and quant.i.ty is simply the difficulty of conceiving a form of utterance of which no examples can now be observed.

[Footnote 1: The metrical accent or ictus was marked in ancient notation by points placed over the accented syllable. These points have been preserved in Mr. Ramsay's musical inscription (see the Appendix, p. 133) and in one or two places of the fragment of the _Orestes_ (p. 130). Hence Dr. Crusius has been able to restore the rhythm with tolerable certainty, and has made the interesting discovery that in both pieces the ictus falls as a rule on a short syllable. The only exceptions in the inscription are circ.u.mflexed syllables, where the long vowel or diphthong is set to two notes, the first of which is short and accented. The accents on the short first syllables of the dochmiacs of Euripides are a still more unexpected evidence of the same rhythmical tendency.]

The conception which we have thus been led to form of ancient Greek as it was spoken is not without bearing on the main subject of these pages. For if the language even in its colloquial form had qualities of rhythm and intonation which gave it this peculiar half musical character, so that singing and speaking were more closely akin than they ever are in our experience, we may expect to find that music was influenced in some measure by this state of things. What is there, then, in the special characteristics of Greek music which can be connected with the exceptional relation in which it stood to language?

Greek music was primarily and chiefly vocal. Instrumental music was looked upon as essentially subordinate,--an accompaniment or at best an imitation of singing. For in the view of the Greeks the words ([Greek: lexis]) were an integral part of the whole composition. They contained the ideas, while the music with its variations of time ([Greek: rhythmos]) and pitch ([Greek: harmonia]) furnished a natural vehicle for the appropriate feelings. Purely instrumental music could not do this, because it could not convey the ideas or impressions fitted to be the object of feeling. Hence we find Plato complaining on this ground of the separation of poetry and music which was beginning to be allowed in his time. The poets, he says, rend asunder the elements of music; they separate rhythm and dance movements from melody, putting unmusical language into metre, and again make melody and rhythm without words, employing the lyre and the flute without the voice: so that it is most difficult, when rhythm and melody is produced without language, to know what it means, or what subject worthy of the name it represents ([Greek: kai hoto eoike ton axiologon mimematon]). It is utterly false taste, in Plato's opinion, to use the flute or the lyre otherwise than as an accompaniment to dance and song[1]. Similarly in the Aristotelian _Problems_ (xix. 10) it is asked why, although the human voice is the most pleasing, singing without words, as in humming or whistling, is not more agreeable than the flute or the lyre. Shall we say, the writer answers, 'that the human voice too is comparatively without charm if it does not _represent_ something? ([Greek: e oud' ekei, ean me mimetai, h.o.m.oios hedy?]) That is to say, music is expressive of _feeling_, which may range from acute pa.s.sion to calm and lofty sentiment, but feeling must have an object, and this can only be adequately given by language. Thus language is, in the first instance at least, the matter to which musical treatment gives artistic form.

In modern times the tendency is to regard instrumental music as the highest form of the art, because in instrumental music the artist creates his work, not by taking ideas and feelings as he finds them already expressed in language, but directly, by forming an independent vehicle of feeling,--a new language, as it were, of pa.s.sion and sentiment,--out of the absolute relations of movement and sound.

The intimate connexion in Greek music between words and melody may be shown in various particulars. The modern practice of basing a musical composition--a long and elaborate chorus, for example--upon a few words, which are repeated again and again as the music is developed, would have been impossible in Greece.

[Footnote 1: Plato, _Legg._ p. 669.]

It becomes natural when the words are not an integral part of the work, but only serve to announce the idea on which it is based, and which the music brings out under successive aspects. The same may be said of the use of a melody with many different sets of words. Greek writers regard even the repet.i.tion of the melody in a strophe and antistrophe as a concession to the comparative weakness of a chorus.

With the Greeks, moreover, the union in one artist of the functions of poet and musician must have tended to a more exquisite adaptation of language and music than can be expected when the work of art is the product of divided labour. In Greece the principle of the interdependence of language, metre, and musical sound was carried very far. The different recognised styles had each certain metrical forms and certain musical scales or keys appropriated to them, in some cases also a certain dialect and vocabulary. These various elements were usually summed up in an ethnical type, one of those which played so large a part in their political history. Such a term as Dorian was not applied to a particular scale at random, but because that scale was distinctive of Dorian music: and Dorian music, again, was one aspect of Dorian temper and inst.i.tutions, Dorian literature and thought.

Whether the Greeks were acquainted with harmony--in the modern sense of the word--is a question that has been much discussed, and may now be regarded as settled[1]. It is clear that the Greeks were acquainted with the phenomena on which harmony depends, viz. the effect produced by sounding certain notes together. It appears also that they made some use of harmony,--and of dissonant as well as consonant intervals,--in instrumental accompaniment ([Greek: krousis]). On the other hand it was unknown in their vocal music, except in the form of ba.s.s and treble voices singing the same melody.

In the instrumental accompaniment it was only an occasional ornament, not a necessary or regular part of the music. Plato speaks of it in the _Laws_ as something which those who learn music as a branch of liberal education should not attempt[1]. The silence of the technical writers, both as to the use of harmony and as to the tonality of the Greek scale, points in the same direction. Evidently there was no _system_ of harmony,--no notion of the effect of _successive_ harmonies, or of two distinct _parts_ or progressions of notes harmonising with each other.

[Footnote 1: On this point I may refer to the somewhat fuller treatment in Smith's _Dictionary of Antiquities_, art. MUSICA (Vol.

II, p. 199, ed. 1890-91).]

The want of harmony is to be connected not only with the defective tonality which was probably characteristic of Greek music,--we have seen (p. 42) that there is some evidence of tonality,--but still more with the non-harmonic quality of many of the intervals of which their scales were composed. We have repeatedly dwelt upon the variety and strangeness (to our apprehension) of these intervals. Modern writers are usually disposed to underrate their importance, or even to explain them away. The Enharmonic, they point out, was produced by the interpolation of a note which may have been only a pa.s.sing note or _appoggiatura_. The Chromatic also, it is said, was regarded as too difficult for ordinary performers, and most of its varieties went out of use at a comparatively early period. Yet the accounts which we find in writers so remote in time and so opposed in their theoretical views as Aristoxenus and Ptolemy, bear the strongest testimony to the reality and persistence of

[Footnote 1: Plato, _Legg_. p. 812 d [Greek: panta oun ta toiauta me prospherein tois mellousin en trisin etesi to tes mousikes chresimon eklepsesthai dia tachous.]]

these non-diatonic scales. And we have the decisive fact that of the six scales of the cithara given by Ptolemy (see p. 85) not one is diatonic in the modern sense of the word. It may be alleged on the other side that the ideal scale in the _Timaeus_ of Plato is purely diatonic, and exhibits the strictest Pythagorean division. But that scale is primarily a framework of mathematical ratios, and could not take notice of intervals which had not yet been identified with ratios. It is not certain when the discovery of Pythagoras was extended to the non-diatonic scales. Even in the _Sectio Canonis_ of Euclid there is no trace of knowledge that any intervals except those of the Pythagorean diatonic scale had a numerical or (as we should say) physical basis[1].

[Footnote 1: In Euclid's _Sectio Canonis_ the Pythagorean division is a.s.sumed, and there is no hint of any other ratio than those which Pythagoras discovered. Prop. xvii shows how to find the Enharmonic Lichanos and Paranete by means of the Fourth and Fifth. Prop. xviii proves against Aristoxenus (of course without naming him), that a [Greek: pyknon] cannot be divided into two equal intervals; but there is no attempt to explain the nature of the Enharmonic diesis. It is worth notice that in these propositions the Lichanos and Paranete of the Enharmonic scale are called [Greek: lichanos] and [Greek: paranete] simply, as though the Enharmonic were the only genus--a usage which agrees with that of the Aristotelian _Problems_ (supra, p. 33).