[10] This observation was made by ancient writers, e.g. by Adrastus (Peripatetic philosopher of the second cent. A.D.): epÊuxÊmenÊs de tÊs mousikÊs kai polychordÔn kai polyphthongÔn gegonotÔn organÔn tÔ proslÊphthÊnai kai epi to bary kai epi to oxy tois pro[:y]parchousin oktÔ phthongois allous pleionas, homÔs k.t.l. (Theon Smyrn. c. 6).

[11] The epigram is quoted in the pseudo-Euclidean Introductio, p. 19 (Meib.): ho de (sc. IÔn) en dekachordÔ lyra (i.e. in a poem on the subject of the ten-stringed lyre):—

'The triple ways of music that are in concord' must be the three conjunct tetrachords that can be formed with ten notes (b c d e f g a b? c d). This is the scale of the Lesser Perfect System before the addition of the Proslambanomenos.

[12] Pherecrates cheirÔn fr. 1 (quoted by Plut. de Mus. c. 30). It is needless to refer to the other traditions on the subject, such as we find in Nicomachus (Harm. p. 35) and Boethius.

[13] The term hyperypatÊ had all but disappeared from the text of Theon Smyrnaeus in the edition of Bullialdus (Paris, 1644), having been corrupted into hypatÊ or parypatÊ in every place except one (p. 141, 3). It has been restored from MSS. in the edition of Hiller (Teubner, Leipzig, 1878). The word occurs also in Aristides Quintilianus (p. 10 Meib.), where the plural hyperypatai is used for the notes below HypatÊ, and in Boethius (Mus. i. 20).

It may be worth noticing also that Thrasyllus uses the words diezeugmenÊ and hyperbolaia in the sense of nÊtÊ diezeugmenÔn and nÊtÊ hyperbolaiÔn (Theon Smyrn. l. c.).

[14] The Introduction to Harmonics (eisagÔgÊ harmonikÊ) which bears the name of Euclid in modern editions (beginning with J. Pena, Paris, 1557) cannot be his work. In some MSS. it is ascribed to Cleonides, in others to Pappus, who was probably of the fourth century A. D. The author is one of the harmonikoi or Aristoxeneans, who adopt the method of equal temperament. He may perhaps be assigned to a comparatively early period on the ground that he recognises only the thirteen keys ascribed to Aristoxenus—not the fifteen keys given by most later writers (Aristides Quint., p. 22 Meib.). For some curious evidence connecting it with the name of the otherwise unknown writer Cleonides, see K. von Jan, Die Harmonik des Aristoxenianers Kleonides (Landsberg, 1870). The Section of the Canon (kanonos katatomÊ) belongs to the mathematical or Pythagorean school, dividing the tetrachord into two major tones and a leimma which is somewhat less than a semitone. In point of form it is decidedly Euclidean: but we do not find it referred to by any writer before the third century A. D.—the earliest testimony being that of Porphyry (pp. 272-276 in Wallis' edition).

[15] Plato, Rep. p. 399: ouk ara, Ên d' egÔ, polychordias ge oude panarmoniou hÊmin deÊsei en tais Ôdais te kai melesin. Ou moi, ephÊ, phainetai. TrigÔnÔn ara kai pÊktidÔn kai pantÔn organÔn hosa polychorda kai polyarmonia dÊmiourgous ou threpsomen. Ou phainometha. Ti de? aulopoious Ê aulÊtas paradexei eis tÊn polin? Ê ou touto polychordotaton, kai auta ta panarmonia aulou tynchanei onta mimÊma? DÊla dÊ, Ê d' hos. Lyra dÊ soi, Ên d' egÔ, kai kithara leipetai, kai kata polin chrÊsima; kai au kat' agrous tois nomeusi syrinx an tis eiÊ.

The aulos was not exactly a flute. It had a mouthpiece which gave it the character rather of the modern oboe or clarinet: see the Dictionary of Antiquities, s. v. tibia. The panarmonion is not otherwise known, and the passage in Plato does not enable us to decide whether it was a real instrument or only a scale or arrangement of notes.

[16] The passage quoted above from the Knights of Aristophanes (p. 7) is sufficient to show that a marked preference for the Dorian mode would be a matter for jest.

[17] Die Lehre von den Tonempfindungen, p. 367, ed. 1863.

[18] So in the Euclidean Sectio Canonis the propositions which deal with the 'movable' notes, viz. ParanÊtÊ and Lichanos (Theor. xvii) and ParhypatÊ and TritÊ (Theor. xviii), begin by postulating the MesÊ (estÔ gar mesÊ ho B k.t.l.).

[19] The term hÊgemÔn or 'leading note' of the tetrachord MesÔn, here applied to the MesÊ, is found in the same sense in Plutarch, De Mus. c. 11, where ho peri ton hÊgemona keimenos tonos] means the disjunctive tone. Similarly Ptolemy (Harm. i. 16) speaks of the tones in a diatonic scale as being en tois hÊgoumenois topois, the semitones en tois hepomenois (sc. of the tetrachord): and again of the ratio 5:4 (the major Third) as the 'leading' one of an Enharmonic tetrachord (ton epitetarton hos estin hÊgoumenos tou enarmoniou genous).

[20] The investigation occupies a considerable space in his Harmonics, viz. pp. 27-29 Meib. (from the words peri de synecheias kai tou hexÊs), and again pp. 58-72 Meib.

[21] This point is one which Aristoxenus is fond of insisting upon: cp. p. 10, 16 ou pros tÊn katapyknÔsin blepontas hÔsper hoi harmonikoi: p. 38, 3 hoti de estin hÊ katapyknÔsis ekmelÊs kai panta tropon achrÊstos phaneron: p. 53, 3 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 (?? for ?d). 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 epsilon02, c* was epsilon04, and c? was epsilon03. Now the ancient diagrams, which divided every tone into four parts, must have had a character for c?*, or the note three-quarters of a tone above c. Naturally this would be the remaining position of epsilon02, namely epsilon05. 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: kappa01 stands for b, and therefore kappa02 for c: but c is a note of the original scale, and as such is written gamma01. 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 4kappas) 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).]

[22] 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.

[23] Cp. Plato, Rep. p. 531: kai smikrotaton einai touto diastÊma, hÔ metrÊteon. It may even be that this sense of harmonia was connected with the use for the Enharmonic genus. It is at least worth notice that the phrase ha ekaloun harmonias in this passage answers to the adjective enarmoniÔn in the passage first quoted (compare the words peri autÔn monon tÔn hepta oktachordÔn ha ekaloun harmonias with peri systÊmatÔn oktachordÔn enarmoniÔn monon).

[24] So in Plato, Leg. p. 665 a: tÊ dÊ tÊs kinÊseÔs taxei rhythmos onoma eiÊ, tÊ d' au tÊs phÔnÊs, tou te oxeos hama kai bareos synkerannymenÔn, harmonia onoma prosagoreuoito.

[25] 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: hautÊ de hÔsper ekpiptein autÊn (sc. tÊn aisthÊsin) 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 hoi de ge dithyrambopoioi kai tous tropous (keys) meteballon, DÔrikous te kai Phrygious kai Lydious en tÔ autÔ asmati poiountes; kai tas melÔdias exÊllatton, tote men enarmonious poiountes, k.t.l.

[26] Since this was written I have learned from Mr. H. S. Jones that the form archaic_beta 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 archaic_beta connects the crescent beta ( 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.

[27] Harmonik und MelopÖie, p. 286 (ed. 1863). The true form of the letter is given by Mr. Roberts, Greek Epigraphy, p. 109.

[28] Pausanias (iv. 27, 4) says of the founding of Messene: 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.

[29] Harm. ii. 8 hoi de hyperekpiptontes tou dia pasÔn tous ap' autou tou dia pasÔn apÔterÔ parelkontÔs hypotithentai, tous autous aei ginomenous tois proeilÊmmenois.

[30] Harm. ii. 11 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.

[31] Harm. ii. 7 pros tÊn toiautÊn diaphoran hÊ tÔn organÔn holÔn epitasis Ê palin anesis aparkei.

[32] This may be traced in the occasionally controversial tone; as Harm. ii. 7 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.

[33] So Bacch. p. 19 Meib. theseis de tetrachordÔn hois to melos horizetai eisin hepta? synaphÊ, diazeuxis, hypodiazeuxis, k.t.l. (see the whole passage).

[34] We may think of this as a scale in which the semitones are considerably smaller, i.e. in which c and f are nearly a quarter of a tone flat.

[35] Ptol. Harm. ii. 16 periechetai de ta men en tÊ lyra kaloumena sterea tonou tinos hypo tÔn tou toniaiou diatonou arithmÔn tou autou tonou, ta de malaka hypo tÔn en tÔ migmati tou malakou chrÔmatos apithmÔn tou autou tonou. Here tonou tinos evidently means 'of any given key,' and tou autou tonou 'of that key.' There is either no restriction, or none that Ptolemy thought worth mentioning, in the choice of the key and species.

[36] The two passages enumerate the scales in a slightly different manner. In i. 16 they are arranged in view of the genus or colour into—

Pure Middle Soft Diatonic, viz.—
sterea, of the lyre.
tritai } of the cithara.
hypertropa }

Mixture of Chromatic, viz.—
malaka, of the lyre.
tropika, of the cithara.

Mixture of Soft Diatonic, viz.—
parypatai, of the cithara.

Mixture of diatonon syntonon, viz.—
lydia } of the cithara.
iastia }

It is added, however, that in their use of this last 'mixture' musicians are in the habit of tuning the cithara in the Pythagorean manner, with two Major tones and a leimma (called diatonon ditoniaion).

In the second passage (ii. 16) the scales of the lyre are given first, then those of the cithara with the key of each. The order is the same, except that parypatai comes before tropika (now called tropoi), and lydia is placed last. The words ta de lydia hoi tou toniaiou diatonou [sc. arithmoi periechousi tou dÔriou cannot be correct, not merely because they contradict the statement of the earlier passage that lydia denoted a mixture with diatonon syntonon (or in practice diatonon ditoniaion), but also because the scales that do not admit mixture are placed first in the list in both passages. Hence we should doubtless read ta de lydia hoi [tou migmatos] tou [di]toniaiou diatonou tou DÔriou.]

[37] Harm. i. 16 plÊn kathoson adousi men akolouthÔs tÔ dedeigmenÔ syntonÔ diatonikÔ, kathaper exestai skopein apo tÊs tÔn oikeiÔn autou logÔn parabolÊs, harmozontai de heteron ti genos (sc. the Pythagorean), xynengizon men ekeinÔ, k.t.l.

[38] It seems needless to set out these melodies here. The first satisfactory edition of them is that of Bellermann, Die Hymnen des Dionysius und Mesomedes (Berlin, 1840). They are given by Westphal in his Musik des griechischen Alterthumes (1883), and by Gevaert, Musique de l'AntiquitÉ, vol. i. pp. 445 ff.; also in Mr. W. Chappell's History of Music (London, 1874), where the melodies of the first and third hymns will be found harmonised by the late Sir George Macfarren.

The melody published by Kircher (Musurgia, i. p. 541) as a fragment of the first Pythian ode of Pindar has no attestation, and is generally regarded as a forgery.

[39] Of the discovery made at Delphi, after most of this book was in type, I hope to say something in the Appendix.

[40] Harm. p. 18 Meib. legetai gar dÊ kai logÔdes ti melos, to synkeimenon ek tÔn prosÔdiÔn, to en tois onomasi; physikon gar to epiteinein kai anienai en tÔ dialegesthai.

[41] I need not repeat what is said by Dr. Wessely and M. Ruelle in defence of the genuineness of our fragment. They justly point to the remarkable coincidence that the music of this very play is quoted by Dionysius of Halicarnassus (l. c.). It would almost seem as if it was the only well-known specimen of music of the classical period of tragedy.

The transcription of Dr. Crusius, with his conjectural restorations, will be found in the Appendix. I have only introduced one of his corrections here, viz. the note on the second syllable of kateklysen.

[42] Dr. Crusius, however, detects a F; (the sign for g) over the first syllable of kateklusen and the second syllable of pontou. There is little trace of them in his facsimile.

[43] This argument is used, along with some others not so cogent, in Mr. W. Chappell's History of Music (p. 130).

[44] Ps. Eucl. Introd. p. 20 Meib. kata systÊma de hotan ek synaphÊs eis diazeuxin Ê anapalin metabolÊ ginÊtai. Anonym. § 65 systÊmatikai de (sc. metabolai) hopotan ek diazeuxeÔs eis synaphÊn Ê empalin metelthÊ to melos.

[45] As represented primarily by the analysis of Helmholtz, Die Tonempfindungen, p. 467, ed. 1863.

[46] Harmonik und MelopÖie, p. 356 (ed. 1863): 'Die Älteste griechische Tonart ist demnach eine Molltonart.... Aus Kleinasien wurden zunÄchst zwei Durtonarten nach Griechenland eingefÜhrt, die lydische und phrygische.' In the 1886 edition of the same book (p. 189) Westphal discovers a similar classification of modes implied in the words of Plato, Rep. p. 400 a tri' atta estin eidÊ ex hÔn hai baseis plekontai, hÔsper en tois phthongois tettara hothen hai pasai harmoniai. But Plato is evidently referring to some matter of common knowledge. The three forms or elements of which all rhythms are made up are of course the ratios 1: 1, 2: 1 and 3: 2, which yield the three kinds of rhythm, dactylic, iambic and cretic (answering to common, triple, and quintuple time). Surely the four elements of all musical scales of which Plato speaks are not four kinds of scale (Harmonien-Klassen), but the four ratios which give the primary musical intervals—viz. the ratios 2: 1, 3: 2, 4: 3 and 9: 8, which give the Octave, Fifth, Fourth and Tone.

[47] If Hypo-phrygian is the same as the older Ionian (p. 11), the coincidence is complete for the time of Aristotle. Plato treats the claim of Ionian to rank among the Hellenic modes as somewhat doubtful (Laches, p. 188).

[48] Aristox. Harm. p. 23 Meib. hoi men gar tÊ nun katechousÊ melopoiia ounÊtheis monon ontes eiktÔs tÊn ditonon lichanon (f in the scale e-a) exorizousi; suntonÔterais gar chrÔntai schedon hoi pleistoi tÔn nun. toutou d' aition to boulesthai glukainein aei. sÊmeion de hoti toutou stochazontai, malista men gar kai pleiston chronon en tÔ chrÔmati diatribousin. hotan d' aphikÔntai pote eis tÊn harmonian engus tou chromatos prosagousi, sunepismÔmenou tou Êthous.

[49] Ibid. p. 26 noÊteon gar apeirous ton arithmon tas lichanous. hou gar an stÊsÊs tÊn phÔnÊn apodedeigmenon lichanÔ topou lichanos estai; diakenon de ouden esti tou lichanoeidous topou, oude toiouton hÔste mÊ dechisthai lichanon. And p. 48 epeidÊ per ho tÊs lichanou topos eis apeirous temnetai tomas.

[50] Aristox. Harm. p. 69 Meib. kata men oun ta megethÊ tÔn diastÊmatÔn kai tas tÔn phthongÔn taseis apeira pÔs phainetai einai ta peri to melos, kata de tas dynameis kai kata ta eidÊ kai kata tas theseis peperasmena te kai tetagmena.

[51] The ecclesiastical Modes received their final shape in the Dodecachordon of Glareanus (BÂle, 1547). They are substantially the Greek modes of Westphal's theory, although the Greek names which Glareanus 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 Boasili—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—Hhosaini 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.

[52] Tonempfindungen, p. 364 (ed. 1863).

[53] Aristox., Harm. p. 3 Meib. kineitai men gar kai dialegomenÔn hÊmÔn kai melÔdountÔn tÊn eirÊmenÊn kinÊsin; oxy gar kai bary dÊlon hÔs en amphoterois toutois enestin. Also p. 8 dyo tines eisin ideai kinÊseÔs, hÊ te synechÊs kai hÊ diastÊmatikÊ; kata men oun tÊn synechÊ topon tina diexienai phainetai hÊ phÔnÊ tÊ aisthÊsei houtÔs hÔs an mÊdamou histamenÊ, k.t.l. And p. 9 tÊn oun synechÊ logikÊn einai phanen, k.t.l.

[54] Ibid. p. 18 Meib. tou ge logÔdous kechÔristai tautÊ to mousikon melos; legetai gar dÊ kai logÔdes ti melos, to synkeimenon ek tÔn prosÔdiÔn tÔn en tois onomasin; physikon gar to epiteinein kai anienai en tÔ dialegesthai.

[55] Nicomachus, Enchiridion, p. 4 ei gar tis Ê dialegomenos Ê apologoumenos tini Ê anaginÔskÔn ge ekdÊla metaxy kath' hekaston phthongon poiei ta megethÊ, diistanÔn kai metaballÔn tÊn phÔnÊn ap' allou eis allon, ouketi legein ho toioutos oude anaginÔskein alla meleazein legetai.

[56] De Compositione Verborum, c. 11, p. 58 Reisk.

[57] De Comp. c. 11, p. 64 to de auto ginetai kai peri tous rhythmous; hÊ men gar pezÊ lexis oudenos oute onomatos oute rhÊmatos biazetai tous chronous oude metatithÊsin, all' oias pareilÊphe tÊ physei tas syllabas, tas te makras kai tas bracheias, toiautas phylattei; hÊ de mousikÊ te kai rhythmikÊ metaballousin autas meiousai kai parauxousai, Ôite pollakis eis tanantia metachÔrein; ou gar tais syllabais apeuthynousi tous chronous, alla tois chronois tas syllabas.

[58] 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 circumflexed 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.

[59] Plato, Legg. p. 669.

[60] 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).

[61] Plato, Legg. p. 812 d panta oun ta toiauta mÊ prospherein tois mellousin en trisin etesi to tÊs mousikÊs chrÊsimon eklÊpsesthai dia tachous.

[62] In Euclid's Sectio Canonis the Pythagorean division is assumed, and there is no hint of any other ratio than those which Pythagoras discovered. Prop. xvii shows how to find the Enharmonic Lichanos and ParanÊtÊ by means of the Fourth and Fifth. Prop. xviii proves against Aristoxenus (of course without naming him), that a 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 ParanÊtÊ of the Enharmonic scale are called lichanos and paranÊtÊ simply, as though the Enharmonic were the only genus—a usage which agrees with that of the Aristotelian Problems (supra, p. 33).

According to Ptolemy (i. 13) the Pythagorean philosopher Archytas was the author of a new division of the tetrachord for each of the three genera. In it the natural Major Third (5: 4) was given for the large interval of the Enharmonic, in place of the Pythagorean ditone (81: 64); and the Diatonic was the same as the Middle Soft Diatonic of Ptolemy. But, as Westphal long ago pointed out (Harmonik und MelopÖie, p. 230, ed. 1863), this scheme is probably the work of the later Pythagorean school. It seems to be unknown to Plato and Aristoxenus,—the latter wrote a life of Archytas—and also to Euclid, as we have seen. The next scheme of musical ratios is that of Eratosthenes, who makes no use of the natural Major Third.

[63] The two schools distinguished by Plato seem to be those which were afterwards known as the harmonikoi or Aristoxeneans, and the mathÊmatikoi, who carried on the tradition of Pythagoras. The harmonikoi regarded a musical interval as a quantity which could be measured directly by the ear, without reference to the numerical ratio upon which it might be based. They practically adopted the system of equal temperament. The mathÊmatikoi sought for ratios, but by experiment 'among the consonances which are heard,' as Plato says. Hence they failed equally with those whose method never rose above the facts of sense.