[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.