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stroy the consonance of the interval. In fact, when a keyed instrument is tuned according to the equal temperament, the major thirds are too great by an interval little more than half of this (^ff- nearly), and yet are only just tolerable. This subject is important, because it bears immediately upon the question whether harmony was used in the Greek music.

x An aggregate of two or more intervals, or rather a series of sounds separated from one another by intervals, constituted a system. Systems were named from the number of sounds which they com­prehended. Thus an octachord was a system of eight sounds, a pentachord of five, and so on : and usually, though not necessarily, the number of sounds corresponded to the interval between the two extreme sounds.

The fundamental system in ancient music was the tetrachord, or system of four sounds, of which the extremes were at an interval of a fourth. In modern music it is the octachord, and comprehends an octave between the extremes. The important and peculiar property of the latter system, namely, the completeness of its scale, was fully understood, as the name of the interval Sia Trav&v sufficiently indicates (see also^Aristides, pp. 16, 17), but it was not taken in theory for the foundation of the scale; or at any rate was considered as made up of two tetrachords.

The Genus of a system depended upon the dis­tribution of the two intermediate sounds of the tetrachord. The Greek musicians used three Ge­nera : —

I. The Diatonic, in which the intervals between the four sounds were (ascending), semitone, tone9 tone: —

II. The Chromatic ; semitone, semitone, tone, and half: —

III. The Enharmonic ; diesis, diesis, double tone: —



(The second note is meant to represent a sound half way between E and F, for which the modern system supplies no notation.)

Of these genera the Diatonic was allowed to be the most ancient and natural, and the Enharmonic the most modern and difficult; the latter however seems soon to have become the favourite with theorists at least, for Aristoxenus complains that all writers before his time had devoted their treatises almost entirely to it, to the neglect of the two others. (Aristox. pp.2 and 19.)

The only difference between the ancient and modern Diatonic is, that in the former all the tones are major tones, whereas in the latter, according to the theory generally admitted, major and minor


tones occur alternately. (See Crotch's Elements of Musical Composition, chap, ix.) The interval called a semitone in the above descriptions is therefore strictly neither equal to the modern major semitone, nor to half a major tone, but the ear would hardly appreciate the difference in melody.

Besides these genera, certain Colours (xp6ai) or specific modifications of them are enumerated. (Eucl. p. 10.)

The Enharmonic had only one XPoai namely, the genus itself as described above : it is commonly called simply ap/j.oviaf

The Chromatic had three : 1st. XP^A101 rovicuov, or simply xpco/xa, the same as the genus ; 2nd. Xpw^ua yfJLtohiov., in which intervals of three-eighths of a tone were substituted for the two semitones ; 3rd. XP^A"* jUaAa/cd*/, in which intervals of one third of a tone were similarly employed.

The Diatonic had two XP°a/ • 1st. Staroz/ozJ ffvvrovov, or simply si&tovov, the same as the genus*; 2nd. Si&rovov juaAaicoV, in which an in­terval of three-fourths of a tone was substituted for the second semitone (ascending).

The following table will exhibit at one view the intervals between the sounds of the tetrachord, taken in the ascending order, according to each of these XPo«{, the tone being represented by unity, and two tones and a half being supposed to make up a fourth, a supposition which is not exactly true, but is commonly adopted by the ancient writers as sufficiently accurate for their purpose. (See Eucl. Sectio Canonis Theor. xv.)

I. Diatonic ... 1. Siarovov (gvvtqvqv) -i, 1, 1.

2. SidTovov {JLaXaKov . i, -^, -^.

II. Chromatic . . 1. xP&/JLa (tovicuov) . i. i, ~.

2. XP^A"

3. XP^A"

III. Enharmonic . ap/j.ovia ...... jl, ^, 2.

There seems to be little evidence that any of the XP^al were practically used, except the three principal ones, didrovov, xp&V^a, apfJ-ovia. But it would be wrong to conclude hastily that the others would be impossible in practice, or necessarily un-pleasing. In the soft Diatonic for instance, the interval which is roughly described as five-fourths of a tone would be greater than a major tone, but less than a minor third ; now there are two in­tervals of this kind corresponding to the superpar-ticular ratios f and f, which ought therefore by analogy to be consonant, or at any rate capable of being employed as well as the tone and semitone ; and although they are not used in modern mnsic, or at least not admitted in theory *, nothing but experiment can determine how far the ear might become accustomed to them. Ifj. this view be cor­rect, the intervals of the tetrachord in the $io.tovov would probably correspond to the ratios

. f, f,

v> similar considerations might be ap­plied to the other XP°ai-

The four sounds of the tetrachord were distin­guished by the following names : virdrif] (sc. X°P^) was the lowest ; vi\TT} or vedrr) the highest ; irapv-

* See Smith's Harmonics, sect. iv. art. 1 0. These intervals exist in the natural scales of the horn, trumpet, &c., and are in fact used, instead of the minor third and tone, in the harmony of the domi­nant seventh, both by stringed instruments and voices when unaccompanied by tempered instru­ments, .

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