In the anonymous treatise on music published by Bellermann[7] (c. 28), we find the following statement regarding the use of the modes or keys in the scales of different instruments:
'The Phrygian mode (harmonia) has the first place on wind-instruments: witness the first discoverers—Marsyas, Hyagnis, Olympus—who were Phrygians. Players on the water-organ (hydraulai) use only six modes (tropoi), viz. Hyper-lydian, Hyper-ionian, Lydian, Phrygian, Hypo-lydian, Hypo-phrygian. Players on the cithara tune their instrument to these four, viz. Hyper-ionian, Lydian, Hypo-lydian, Ionian. Flute-players employ seven, viz. Hyper-aeolian, Hyper-ionian, Hypo-lydian, Lydian, Phrygian, Ionian, Hypo-phrygian. Musicians who concern themselves with orchestic (choral music) use seven, viz. Hyper-dorian, Lydian, Phrygian, Dorian, Hypo-lydian, Hypo-phrygian, Hypo-dorian.
In this passage it is evident that we have to do with keys of the scheme attributed to Aristoxenus, including the two (Hyper-aeolian and Hyper-lydian) which were said to have been added after his time. The number of scales mentioned is sufficient to prove that the reference is not to the seven species of the octave. Yet the word harmonia is used of these keys, and with it, seemingly as an equivalent, the word tropos.
Pollux (Onom. iv. 78) gives a somewhat different account of the modes used on the flute: kai harmonia men aulêtikê Dôristi, Phrygisti, Lydios kai Iônikê, kai syntonos Lydisti hên Anthippos exeure. But this statement, as has been already pointed out (p. 22), is a piece of antiquarian learning, and therefore takes no notice of the more recent keys, as Hyper-aeolian and Hyper-ionian, or even Hypo-phrygian (unless that is the Ionian of Pollux). The absence of Dorian from the list given by the Anonymus is curious: but it seems that at that time it was equally unknown to the cithara and the water-organ. There is therefore no reason to think that the two lists are framed with reference to different things. That is to say, harmonia in Pollux has the same meaning as harmonia in the Anonymus, and is equivalent to tonos.
The inquiry has now reached a stage at which we may stop to consider what result has been reached, especially in regard to the question whether the two words harmonia and tonos denote two sets of musical forms, or are merely two different names for the same thing. The latter alternative appears to be supported by several considerations.
1. From various passages, especially in Plato and Aristotle, it has been shown that the modes anciently called harmoniai differed in pitch, and that this difference in pitch was regarded as the chief source of the peculiar ethical character of the modes.
2. The list of harmoniai as gathered from the writers who treat of them, viz. Plato, Aristotle, and Heraclides Ponticus, is substantially the same as the list of tonoi described by Aristoxenus (p. 18): and moreover, there is an agreement in detail between the two lists which cannot be purely accidental. Thus Heraclides says that certain people had found out a new harmonia, the Hypo-phrygian; and Aristoxenus speaks of the Hypo-phrygian tonos as a comparatively new one. Again, the account which Aristoxenus gives of the Hypo-dorian tonos as a key immediately below the Dorian agrees with what Heraclides says of the Hypo-dorian harmonia, and also with the mention of Hypo-dorian and Hypo-phrygian (but not Hypo-lydian) in the Aristotelian Problems. Once more, the absence of Ionian from the list of tonoi in Aristoxenus is an exception which proves the rule: since the name of the Ionian harmonia is similarly absent from Aristotle.
3. The usage of the words harmonia and tonos is never such as to suggest that they refer to different things. In the earlier writers, down to and including Aristotle, harmonia is used, never tonos. In Aristoxenus and his school we find tonos, and in later writers tropos, but not harmonia. The few writers (such as Plutarch) who use both tonos and harmonia do not observe any consistent distinction between them. Those who (like Westphal) believe that there was a distinction, are obliged to admit that harmonia is occasionally used for tonos and conversely.
4. If a series of names such as Dorian, Phrygian, Lydian and the rest were applied to two sets of things so distinct from each other, and at the same time so important in the practice of music, as what we now call modes and keys, it is incredible that there should be no trace of the double usage. Yet our authors show no sense even of possible ambiguity. Indeed, they seem to prefer, in referring to modes or keys, to use the adverbial forms dôristi, phrygisti, &c., or the neuter ta dôria, ta phrygia, &c., where there is nothing to show whether 'mode' or 'key,' harmonia or tonos, is intended.
The arguments in favour of identifying the primitive national Modes (harmoniai) with the tonoi or keys may be reinforced by some considerations drawn from the history and use of another ancient term, namely systêma.
A System (systêma) is defined by the Greek technical writers as a group or complex of intervals (to ek pleionôn ê henos diastêmatôn synkeimenon Ps. Eucl.). That is to say, any three or more notes whose relative pitch is fixed may be regarded as forming a particular System. If the notes are such as might be used in the same melody, they are said to form a musical System (systêma emmeles). As a matter of abstract theory it is evident that there are very many combinations of intervals which in this sense form a musical System. In fact, however, the variety of systems recognised in the theory of Greek music was strictly limited. The notion of a small number of scales, of a particular compass, available for the use of the musician, was naturally suggested by the ancient lyre, with its fixed and conventional number of strings. The word for string (chordê) came to be used with the general sense of a note of music; and in this way the several strings of the lyre gave their names to the notes of the Greek gamut[8].
In the age of the great melic poets the lyre had no more than seven strings: but the octave was completed in the earliest times of which we have accurate information. The scale which is assumed as matter of common knowledge in the Aristotelian Problems and the Harmonics of Aristoxenus consists of eight notes, named as follows from their place on the lyre:
Nêtê (neatê or nêtê, lit. 'lowest,' our 'highest').
Paranêtê (paranêtê, 'next to Nêtê').
Tritê (tritê, i.e. 'third' string).
Paramesê (paramesê or paramesos, 'next to Mesê').
Mesê (mesê, 'middle string').
Lichanos (lichanos, i.e. 'forefinger' string).
Parhypatê (parypatê).
Hypatê (hypatê, lit. 'uppermost,' our 'lowest').
It will be seen that the conventional sense of high and low in the words hypatê and neatê was the reverse of the modern usage.
The musical scale formed by these eight notes consists of two tetrachords or scales of four notes, and a major tone. The lower of the tetrachords consists of the notes from Hypatê to Mesê, the higher of those from Paramesê to Nêtê: the interval between Mesê and Paramesê being the so-called Disjunctive Tone (tonos diazeuktikos). Within each tetrachord the intervals depend upon the Genus (genos). Thus the four notes just mentioned—Hypatê, Mesê, Paramesê, Nêtê—are the same for every genus, and accordingly are called the 'standing' or 'immoveable' notes (phthongoi hestôtes, akinêtoi), while the others vary with the genus, and are therefore 'moveable' (pheromenoi).
In the ordinary Diatonic genus the intervals of the tetrachords are, in the ascending order, semitone + tone + tone: i.e. Parhypatê is a semitone above Hypatê, and Lichanos a tone above Parhypatê. In the Enharmonic genus the intervals are two successive quarter-tones (diesis) followed by a ditone or major Third: consequently Parhypatê is only a quarter of a tone above Hypatê, and Lichanos again a quarter of a tone above Parhypatê. The group of three notes separated in this way by small intervals (viz. two successive quarter-tones) is called a pyknon. If we use an asterisk to denote that a note is raised a quarter of a tone, these two scales may be represented in modern notation as follows:
Pynknon
πμκνον = pyknon
In the Chromatic genus and its varieties the division is of an intermediate kind. The interval between Lichanos and Mesê is more than one tone, but less than two: and the two other intervals, as in the enharmonic, are equal.
The most characteristic feature of this scale, in contrast to those of the modern Major and Minor, is the place of the small intervals (semitone or pyknon), which are always the lowest intervals of a tetrachord. It is hardly necessary to quote passages from Aristotle and Aristoxenus to show that this is the succession of intervals assumed by them. The question is asked in the Aristotelian Problems (xix. 4), why Parhypatê is difficult to sing, while Hypatê is easy, although there is only a diesis between them (kaitoi diesis hekateras). Again (Probl. xix. 47), speaking of the old heptachord scale, the writer says that the Paramesê was left out, and consequently the Mesê became the lowest note of the upper pyknon, i.e. the group of 'close' notes consisting of Mesê, Tritê, and Paranêtê. Similarly Aristoxenus (Harm. p. 23) observes that the 'space' of the Lichanos, i.e. the limit within which it varies in the different genera, is a tone while the space of the Parhypatê is only a diesis, for it is never nearer Hypatê than a diesis or further off than a semitone.
Regarding the earlier seven-stringed scales which preceded this octave our information is scanty and somewhat obscure. The chief notice on the subject is the following passage of the Aristotelian Problems:
Probl. xix. 47 dia ti hoi archaioi heptachordous poiountes tas harmonias tên hypatên all' ou tên nêtên katelipon: hê ou tên hypatên (leg. nêtên), alla tên nyn paramesên kaloumenên aphêroun kai to toniaion diastêma; echrônto de tê eschatê mesê tou epi to oxy pyknou; did kai mesên autên prosêloreusan [hê] oti ên tou men anô tetrachordon teleutê, tou de katô archê, kai meson eiche logon tonô tôn akrôn?
'Why did the ancient seven-stringed scales include Hypatê but not Nêtê? Or should we say that the note omitted was not Nêtê, but the present Paramesê and the interval of a tone (i.e. the disjunctive tone)? The Mesê, then, was the lowest note of the upper pyknon: whence the name mesê, because it was the end of the upper tetrachord and beginning of the lower one, and was in pitch the middle between the extremes.'
This clearly implies two conjunct tetrachords—
tetrachord
In another place (Probl. xix. 32) the question is asked, why the interval of the octave is called dia pasôn, not di' oktô,—as the Fourth is dia tessarôn, the Fifth dia pente. The answer suggested is that there were anciently seven strings, and that Terpander left out the Tritê and added the Nêtê. That is to say, Terpander increased the compass of the scale from the ancient two tetrachords to a full Octave; but he did not increase the number of strings to eight. Thus he produced a scale like the standard octave, but with one note wanting; so that the term di oktô was inappropriate.
Among later writers who confirm this account we may notice Nicomachus, p. 7 Meib. mesê dia tessarôn pros amphotera en tê heptachordô kata to palaion diestôsa: and p. 20 tê toinyn archaiotropô lyra toutesti tê heptachordô, kata synaphên ek duo tetrachordôn synestôsê k.t.l.
It appears then that two kinds of seven-stringed scales were known, at least by tradition: viz. (1) a scale composed of two conjunct tetrachords, and therefore of a compass less than an octave by one tone; and (2) a scale of the compass of an octave, but wanting a note, viz. the note above Mesê. The existence of this incomplete scale is interesting as a testimony to the force of the tradition which limited the number of strings to seven.
The term 'Perfect System' (systêma teleion) is applied by the technical writers to a scale which is evidently formed by successive additions to the heptachord and octachord scales explained in the preceding chapter. It may be described as a combination of two scales, called the Greater and Lesser Perfect System.
The Greater Perfect System (systêma teleion meizon) consists of two octaves formed from the primitive octachord System by adding a tetrachord at each end of the scale. The new notes are named like those of the adjoining tetrachord of the original octave, but with the name of the tetrachord added by way of distinction. Thus below the original Hypatê we have a new tetrachord Hypatôn (tetrachordon hypatôn), the notes of which are accordingly called Hypatê Hypatôn, Parhypatê Hypatôn, and Lichanos Hypatôn: and similarly above Nêtê we have a tetrachord Hyperbolaiôn. Finally the octave downwards from Mesê is completed by the addition of a note appropriately called Proslambanomenos.
The Lesser Perfect System (systêma teleion elasson) is apparently based upon the ancient heptachord which consisted of two 'conjunct' tetrachords meeting in the Mesê. This scale was extended downwards in the same way as the Greater System, and thus became a scale of three tetrachords and a tone.
These two Systems together constitute the Perfect and 'unmodulating' System (systêma teleion ametabolon), which may be represented in modern notation [9] as follows:
| a | Nêtê Hyperbolaiôn | } | Tetrachord |
| g | Paranêtê Hyperbolaiôn | } | Hyperbolaiôn |
| f | Tritê Hyperbolaiôn | } | |
| e | Nêtê Diezeugmenôn | ||
| d | Paranêtê Diezeugmenôn | } | |
| c | Tritê Diezeugmenôn | } | Tetrachord |
| b | Paramesê | } | Diezeugmenôn |
| d | Nêtê Synêmmenôn | } | |
| c | Paranêtê Synêmmenôn | } | Tetrachord |
| b♭ | Tritê Synêmmenôn | } | Synêmmenôn |
| a | Mesê | } | |
| g | Lichanos Mesôn | } | Tetrachord |
| f | Parhypatê Mesôn | } | Mesôn |
| e | Hypatê Mesôn | ||
| d | Lichanos Hypatôn | } | |
| c | Parhypatê Hypatôn | } | Tetrachord |
| b | Hypatê Hypatôn | } | Hypatôn |
| a | Proslambanomenos |
No account of the Perfect System is given by Aristoxenus, and there is no trace in his writings of an extension of the standard scale beyond the limits of the original octave. In one place indeed (Harm. p. 8, 12 Meib.) Aristoxenus promises to treat of Systems, 'and among them of the perfect System' (peri te tôn allôn kai tou teleiou). But we cannot assume that the phrase here had the technical sense which it bore in later writers. More probably it meant simply the octave scale, in contrast to the tetrachord and pentachord—a sense in which it is used by Aristides Quintilianus, p. 11 Meib. synêmmenôn de eklêthê to holon systêma hoti tô prokeimenô teleiô tô mechri mesês synêptai, 'the whole scale was called conjunct because it is conjoined to the complete scale that reaches up to Mesê' (i.e. the octave extending from Proslambanomenos to Mesê). So p. 16 kai ha men autôn esti teleia, ha d' ou, atelê men tetrachordon, pentachordon, teleion de oktachordon. This is a use of teleios which is likely enough to have come from Aristoxenus. The word was doubtless applied in each period to the most complete scale which musical theory had then recognised.
Little is known of the steps by which this enlargement of the Greek scale was brought about. We shall not be wrong in conjecturing that it was connected with the advance made from time to time in the form and compass of musical instruments [10]. Along with the lyre, which kept its primitive simplicity as the instrument of education and everyday use, the Greeks had the cithara (kithara), an enlarged and improved lyre, which, to judge from the representations on ancient monuments, was generally seen in the hands of professional players (kitharôdoi). The development of the cithara showed itself in the increase, of which we have good evidence even before the time of Plato, in the number of the strings. The poet Ion, the contemporary of Sophocles, was the author of an epigram on a certain ten-stringed lyre, which seems to have had a scale closely approaching that of the Lesser Perfect System [11]. A little later we hear of the comic poet Pherecrates attacking the musician Timotheus for various innovations tending to the loss of primitive simplicity, in particular the use of twelve strings [12]. According to a tradition mentioned by Pausanias, the Spartans condemned Timotheus because in his cithara he had added four strings to the ancient seven. The offending instrument was hung up in the Scias (the place of meeting of the Spartan assembly), and apparently was seen there by Pausanias himself (Paus. iii. 12, 8).
A similar or still more rapid development took place in the flute (aulos). The flute-player Pronomus of Thebes, who was said to have been one of the instructors of Alcibiades, invented a flute on which it was possible to play in all the modes. 'Up to his time,' says Pausanias (ix. 12, 5), 'flute-players had three forms of flute: with one they played Dorian music; a different set of flutes served for the Phrygian mode (harmonia); and the so-called Lydian was played on another kind again. Pronomus was the first who devised flutes fitted for every sort of mode, and played melodies different in mode on the same flute.' The use of the new invention soon became general, since in Plato's time the flute was the instrument most distinguished by the multiplicity of its notes: cp. Rep. p. 399 ti de? aulopoious ê aulêtas paradexei eis tên polin? ê ou touto polychordotaton? Plato may have had the invention of Pronomus in mind when he wrote these words.
With regard to the order in which the new notes obtained a place in the schemes of theoretical musicians we have no trustworthy information. The name proslambanomenos, applied to the lowest note of the Perfect System, points to a time when it was the last new addition to the scale. Plutarch in his work on the Timaeus of Plato (peri tês en Timaiô psychogonias) speaks of the Proslambanomenos as having been added in comparatively recent times (p. 1029 c hoi de neôteroi ton proslambanomenon tonô diapheronta tês hypatês epi to bary taxantes to men holon diastêma dis dia pasôn epoiêsan). The rest of the Perfect System he ascribes to 'the ancients' (tous palaious ismen hypatas men dyo, treis de nêtas, mian de mesên kai mian paramesên tithemenous). An earlier addition—perhaps the first made to the primitive octave—was a note called Hyperhypatê, which was a tone below the old Hypatê, in the place afterwards occupied on the Diatonic scale by Lichanos Hypatôn. It naturally disappeared when the tetrachord Hypatôn came into use. It is only mentioned by one author, Thrasyllus (quoted by Theon Smyrnaeus, cc. 35-36 [13]).
The notes of the Perfect System, with the intervals of the scale which they formed, are fully set out in the two treatises that pass under the name of the geometer Euclid, viz. the Introductio Harmonica and the Sectio Canonis. Unfortunately the authorship of both these works is doubtful [14]. All that we can say is that if the Perfect System was elaborated in the brief interval between the time of Aristotle and that of Euclid, the materials for it must have already existed in musical practice.
Let us now consider the relation between this fixed or standard scale and the varieties denoted by the terms harmonia and tonos.
With regard to the tonoi or Keys of Aristoxenus we are not left in doubt. A system, as we have seen, is a series of notes whose relative pitch is fixed. The key in which the System is taken fixes the absolute pitch of the series. As Aristoxenus expresses it, the Systems are melodies set at the pitch of the different keys (tous tonous, eph' hôn tithemena ta systêmata melôdeitai). If then we speak of Hypatê or Mesê (just as when we speak of a moveable Do), we mean as many different notes as there are keys: but the Dorian Hypatê or the Lydian Mesê has an ascertained pitch. The Keys of Aristoxenus, in short, are so many transpositions of the scale called the Perfect System.
Such being the relation of the standard System to the key, can we suppose any different relation to have subsisted between the standard System and the ancient 'modes' known to Plato and Aristotle under the name of harmoniai?
It appears from the language used by Plato in the Republic that Greek musical instruments differed very much in the variety of modes or harmoniai of which they were susceptible. After Socrates has determined, in the passage quoted above (p. 7), that he will admit only two modes, the Dorian and Phrygian, he goes on to observe that the music of his state will not need a multitude of strings, or an instrument of all the modes (panarmonion) [15]. 'There will be no custom therefore for craftsmen who make triangles and harps and other instruments of many notes and many modes. How then about makers of the flute (aulos) and players on the flute? Has not the flute the greatest number of notes, and are not the scales which admit all the modes simply imitations of the flute? There remain then the lyre and the cithara for use in our city; and for shepherds in the country a syrinx (pan's pipes).' The lyre, it is plain, did not admit of changes of mode. The seven or eight strings were tuned to furnish the scale of one mode, not of more. What then is the relation between the mode or harmonia of a lyre and the standard scale or systêma which (as we have seen) was based upon the lyre and its primitive gamut?
If harmonia means 'key,' there is no difficulty. The scale of a lyre was usually the standard octave from Hypatê to Nêtê: and that octave might be in any one key. But if a mode is somehow characterised by a particular succession of intervals, what becomes of the standard octave? No one succession of intervals can then be singled out. It may be said that the standard octave is in fact the scale of a particular mode, which had come to be regarded as the type, viz. the Dorian. But there is no trace of any such prominence of the Dorian mode as this would necessitate. The philosophers who recognise its elevation and Hellenic purity are very far from implying that it had the chief place in popular regard. Indeed the contrary was evidently the case [16].
It may be said here that the value of a series of notes as the basis of a distinct mode—in the modern sense of the word—depends essentially upon the tonality. A single scale might yield music of different modes if the key-note were different. It is necessary therefore to collect the scanty notices which we possess bearing upon the tonality of Greek music. The chief evidence on the subject is a passage of the Problems, the importance of which was first pointed out by Helmholtz [17].
It is as follows:
Arist. Probl. xix. 20: Dia ti ean men tis tên mesên kinêsê hêmôn, harmosas tas allas chordas, kai chrêtai tô organô, ou monon hotan kata ton tês mesês genêtai phthongon lypei kai phainetai anarmoston, alla kai kata tên allên melôdian, ean de tên lichanon ê tina allon phthongon, tote phainetai diapherein monon hotan kakeinê tis chrêtai? ê eulogôs touto symbainei? panta gar ta chrêsta melê pollakis tê mesê chrêtai, kai pantes hoi agathoi poiêtai pykna pros tên mesên apantôsi, kan apelthôsi tachy epanerchontai, pros de allên houtôs oudemian. kathaper ek tôn logôn eniôn exairethentôn syndesmôn ouk estin ho logos Hellênikos, hoion to te kai to kai, enioi de outhen lypousi, dia to tois men anankaion einai chrêsthai pollakis, ei estai logos, tois de mê, houtô kai tôn phthongôn hê mesê hôsper syndesmos esti, kai malista tôn kalôn, dia to pleistakis enyparchein ton phthongon autês.
'Why is it that if the Mesê is altered, after the other strings have been tuned, the instrument is felt to be out of tune, not only when the Mesê is sounded, but through the whole of the music,—whereas if the Lichanos or any other note is out of tune, it seems to be perceived only when that note is struck? Is it to be explained on the ground that all good melodies often use the Mesê, and all good composers resort to it frequently, and if they leave it soon return again, but do not make the same use of any other note? just as language cannot be Greek if certain conjunctions are omitted, such as te and kai, while others may be dispensed with, because the one class is necessary for language, but not the other: so with musical sounds the Mesê is a kind of 'conjunction,' especially of beautiful sounds, since it is most often heard among these.'
In another place (xix. 36) the question is answered by saying that the notes of a scale stand in a certain relation to the Mesê, which determines them with reference to it (hê taxis hê hekastês êdê di' ekeinên): so that the loss of the Mesê means the loss of the ground and unifying element of the scale (arthentos tou aitiou tou hêrmosthai kai tou synechontos) [18].
These passages imply that in the scale known to Aristotle, viz. the octave e-e, the Mesê a had the character of a Tonic or key-note. This must have been true a fortiori of the older seven-stringed scale, in which the Mesê united the two conjunct tetrachords. It was quite in accordance with this state of things that the later enlargement completed the octaves from Mesê downwards and upwards, so that the scale consisted of two octaves of the form a-a. As to the question how the Tonic character of the Mesê was shown, in what parts of the melody it was necessarily heard, and the like, we can but guess. The statement of the Problems is not repeated by any technical writer, and accordingly it does not appear that any rules on the subject had been arrived at. It is significant, perhaps, that the frequent use of the Mesê is spoken of as characteristic of good melody (panta ta chrêsta melê pollakis tê mesê chrêtai), as though tonality were a merit rather than a necessity.
Another passage of the Problems has been thought to show that in Greek music the melody ended on the Hypatê. The words are these (Probl. xix. 33):
Dia ti euarmostoteron apo tou oxeos epi to bary ê apo tou
bareos epi to oxy; poteron hoti to apo tês archês ginetai archesthai? hê gar mesê kai hêgemôn oxytatê tou tetrachordou; to de ouk ap' archês all' apo teleutês.
'Why is a descending scale more musical than an ascending one? Is it that in this order we begin with the beginning,—since the Mesê or leading note [19] is the highest of the tetrachord,—but with the reverse order we begin with the end?'
There is here no explicit statement that the melody ended on the Hypatê, or even that it began with the Mesê. In what sense, then, was the Mesê a 'beginning' (archê), and the Hypatê an 'end'? In Aristotelian language the word archê has various senses. It might be used to express the relation of the Mesê to the other notes as the basis or ground-work of the scale. Other passages, however, point to a simpler explanation, viz. that the order in question was merely conventional. In Probl. xix. 44 it is said that the Mesê is the beginning (archê) of one of the two tetrachords which form the ordinary octave scale (viz. the tetrachord Mesôn); and again in Probl. xix. 47 that in the old heptachord which consisted of two conjunct tetrachords (e-a-d) the Mesê (a) was the end of the upper tetrachord and the beginning of the lower one (hoti ên tou men anô tetrachordou teleutê, tou de katô archê). In this last passage it is evident that there is no reference to the beginning or end of the melody.
Another instance of the use of archê in connexion with the musical scale is to be found in the Metaphysics (iv. 11, p. 1018 b 26), where Aristotle is speaking of the different senses in which things may be prior and posterior:
Ta de kata taxin; tauta d' estin hosa pros ti hen hôrismenon diestêke kata ton logon, hoion parastatês tritostatou proteron, kai paranêtê nêtês; entha men gar ho koryphaios, entha de hê mesê archê.
'Other things [are prior and posterior] in order: viz. those which are at a varying interval from some one definite thing; as the second man in the rank is prior to the third man, and the Paranêtê to the Nêtê: for in the one case the coryphaeus is the starting-point, in the other the Mesê.'
Here the Mesê is again the archê or beginning, but the order is the ascending one, and consequently the Nêtê is the end. The passage confirms what we have learned of the relative importance of the Mesê: but it certainly negatives any inference regarding the note on which the melody ended.
It appears, then, that the Mesê of the Greek standard System had the functions of a key-note in that System. In other words, the music was in the mode (using that term in the modern sense) represented by the octave a-a of the natural key—the Hypo-dorian or Common Species. We do not indeed know how the predominant character of the Mesê was shown—whether, for example, the melody ended on the Mesê. The supposed evidence for an ending on the Hypatê has been shown to be insufficient. But we may at least hold that as far as the Mesê was a key-note, so far the Greek scale was that of the modern Minor mode (descending). The only way of escape from this conclusion is to deny that the Mesê of Probl. xix. 20 was the note which we have understood by the term—the Mesê of the standard System. This, as we shall presently see, is the plea to which Westphal has recourse.
The object of the preceding discussion has been to make it clear that the theory of a system of modes—in the modern sense of the word—finds no support from the earlier authorities on Greek music. There is, however, evidence to show that Aristoxenus, and perhaps other writers of the time, gave much thought to the varieties to be obtained by taking the intervals of a scale in different order. These varieties they spoke of as the forms or species (schêmata, eidê) of the interval which measured the compass of the scale in question. Thus, the interval of the Octave (dia pasôn) is divided into seven intervals, and these are, in the Diatonic genus, five tones and two semitones, in the Enharmonic two ditones, four quarter-tones, and a tone. As we shall presently see in detail, there are seven species of the Octave in each genus. That is to say, there are seven admissible octachord scales (systêmata emmelê), differing only in the succession of the intervals which compose them.
Further, there is evidence which goes to connect the seven species of the Octave with the Modes or harmoniai. In some writers these species are described under names which are familiar to us in their application to the modes. A certain succession of intervals is called the Dorian species of the Octave, another succession is called the Phrygian species, and so on for the Lydian, Mixo-lydian, Hypo-dorian, Hypo-phrygian, and Hypo-lydian. It seems natural to conclude that the species or successions of intervals so named were characteristic in some way of the modes which bore the same names, consequently that the modes were not keys, but modes in the modern sense of the term.
In order to estimate the value of this argument, it is necessary to ask, (1) how far back we can date the use of these names for the species of the Octave, and (2) in what degree the species of the Octave can be shown to have entered into the practice of music at any period. The answer to these questions must be gathered from a careful examination of all that Aristoxenus and other early writers say of the different musical scales in reference to the order of their intervals.
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.
In the Harmonics of Aristoxenus an account of the seven species of the Octave followed the elaborate theory of Systems already referred to (p. 48), and doubtless exhibited the application of that general theory to the particular cases of the Fourth, Fifth, and Octave. Unfortunately the existing manuscripts have only preserved the first few lines of this chapter of the Aristoxenean work (p. 74, ll. 10-24 Meib.).
The next source from which we learn anything of this part of the subject is the pseudo-Euclidean Introductio Harmonica. The writer enumerates the species of the Fourth, the Fifth, and the Octave, first in the Enharmonic and then in the Diatonic genus. He shows that if we take Fourths on a Diatonic scale, beginning with Hypatê Hypatôn (our b), we get successively b c d e (a scale with the intervals ½ 1 1), c d e f (1 1 ½) and d e f g (1 ½ 1). Similarly on the Enharmonic scale we get—
| Hypatê Hypatôn to Hypatê Mesôn | b b* c e | (¼ ¼ 2 ) | ||
| Parhypatê " " Parhypatê " | b* c e e* | (¼ 2 ¼) | ||
| Lichanos " " Lichanos " | c e e* f | (2 ¼ ¼) |
In the case of the Octave the species is distinguished on the Enharmonic scale by the place of the tone which separates the tetrachords, the so-called Disjunctive Tone (tonos diazeuktikos). Thus in the octave from Hypatê Hypatôn to Paramesê (b-b) this tone (a-b) is the highest interval; in the next octave, from Parhypatê Hypatôn to Tritê Diezeugmenôn (c-c), it is the second highest; and so on. These octaves, or species of the Octave, the writer goes on to tell us, were anciently called by the same names as the seven oldest Keys, as follows:
| Mixo-lydian | b - b | ¼ ¼ 2 ¼ ¼ 2 1 | ||
| Lydian | b*- b* | ¼ 2 ¼ ¼ 2 1 ¼ | ||
| Phrygian | c - c | 2 ¼ ¼ 2 1 ¼ ¼ | ||
| Dorian | e - e | ¼ ¼ 2 1 ¼ ¼ 2 | ||
| Hypo-lydian | e*- e* | ¼ 2 1 ¼ ¼ 2 ¼ | ||
| Hypo-phrygian | f - f | 2 1 ¼ ¼ 2 ¼ ¼ | ||
| Hypo-dorian | a - a | 1 ¼ ¼ 2 ¼ ¼ 2 |
On the Diatonic scale, according to the same writer, the species of an Octave is distinguished by the places of the two semitones. Thus in the first species, b-b, the semitones are the first and fourth intervals (b-c and e-f): in the second, c-c, they are the third and the seventh, and so on. He does not however say, as he does in the case of the Enharmonic scale, that these species were known by the names of the Keys. This statement is first made by Gaudentius (p. 20 Meib.), a writer of unknown date. If we adopt it provisionally, the species of the Diatonic octave will be as follows:
| [Mixo-lydian] | b - b | ½ 1 1 ½ 1 1 1 | ||
| [Lydian] | c - c | 1 1 ½ 1 1 1 ½ | ||
| [Phrygian] | d - d | 1 ½ 1 1 1 ½ 1 | ||
| [Dorian] | e - e | ½ 1 1 1 ½ 1 1 | ||
| [Hypo-lydian] | f - f | 1 1 1 ½ 1 1 ½ | ||
| [Hypo-phrygian] | g - g | 1 1 ½ 1 1 ½ 1 | ||
| [Hypo-dorian] | a - a | 1 ½ 1 1 ½ 1 1 |
Looking at the octaves which on our key-board, as on the Greek scale, exhibit the several species, we cannot but be struck with the peculiar relation in which they stand to the Keys. In the tables given above the keys stand in the order of their pitch, from the Mixo-lydian down to the Hypo-dorian: the species of the same names follow the reverse order, from b-b upwards to a-a. This, it is obvious, cannot be an accidental coincidence. The two uses of this famous series of names cannot have originated independently. Either the naming of the species was founded on that of the keys, or the converse relation obtained between them. Which of these two uses, then, was the original and which the derived one? Those who hold that the species were the basis of the ancient Modes or harmoniai must regard the keys as derivative. Now Aristoxenus tells us, in one of the passages just quoted, that the seven species had long been recognised by theorists. If the scheme of keys was founded upon the seven species, it would at once have been complete, both in the number of the keys and in the determination of the intervals between them. But Aristoxenus also tells us that down to his time there were only six keys,—one of them not yet generally recognised,—and that their relative pitch was not settled. Evidently then the keys, which were scales in practical use, were still incomplete when the species of the Octave had been worked out in the theory of music.
If on the other hand we regard the names Dorian, &c. as originally applied to keys, we have only to suppose that these names were extended to the species after the number of seven keys had been completed. This supposition is borne out by the fact that Aristoxenus, who mentions the seven species as well known, does not give them names, or connect them with the keys. This step was apparently taken by some follower of Aristoxenus, who wished to connect the species of the older theorists with the system of keys which Aristoxenus had perfected.
The view now taken of the seven species is supported by the whole treatment of musical scales (systêmata) as we find it in Aristoxenus. That treatment from first to last is purely abstract and theoretical. The rules which Aristoxenus lays down serve to determine the sequence of intervals, but are not confined to scales of any particular compass. His Systems, accordingly, are not scales in practical use: they are parts taken anywhere on an ideal unlimited scale. And the seven species of the Octave are regarded by Aristoxenus as a scheme of the same abstract order. They represent the earlier teaching on which he had improved. He condemned that teaching for its want of generality, because it was confined to the compass of the Octave and to the Enharmonic genus, and also because it rested on no principles that would necessarily limit the species of the Octave to seven. On the other hand the diagrams of the earlier musicians were unscientific, in the opinion of Aristoxenus, on the ground that they divided the scale into a succession of quarter-tones. Such a division, he urged, is impossible in practice and musically wrong (ekmeles). All this goes to show that the earlier treatment of Systems, including the seven Species, had the same theoretical character as his own exposition. The only System which he recognises for practical purposes is the old standard octave, from Hypatê to Nêtê: and that System, with the enlargements which turned it into the Perfect System, kept its ground with all writers of the Aristoxenean school.
Even in the accounts of the pseudo-Euclid and the later writers, who treat of the Species of the Octave under the names of the Keys, there is much to show that the species existed chiefly or wholly in musical theory. The seven species of the Octave are given along with the three species of the Fourth and the four species of the Fifth, neither of which appear to have had any practical application. Another indication of this may be seen in the seventh or Hypo-dorian species, which was also called Locrian and Common (ps. Eucl. p. 16 Meib.). Why should this species have more than one name? In the Perfect System it is singular in being exemplified by two different octaves, viz. that from Proslambanomenos to Mesê, and that from Mesê to Nêtê Hyperbolaiôn. Now we have seen that the higher the octave which represents a species, the lower the key of the same name. In this case, then, the upper of the two octaves answers to the Hypo-dorian key, and the lower to the Locrian. But if the species has its two names from these two keys, it follows that the names of the species are derived from the keys. The fact that the Hypo-dorian or Locrian species was also called Common is a further argument to the same purpose. It was doubtless 'common' in the sense that it characterised the two octaves which made up the Perfect System. Thus the Perfect System was recognised as the really important scale.
Another consideration, which has been overlooked by Westphal and those who follow him, is the difference between the species of the Octave in the several genera, especially the difference between the Diatonic and the Enharmonic. This is not felt as a difficulty with all the species. Thus the so-called Dorian octave e-e is in the Enharmonic genus e e* f a b b* c e, a scale which may be regarded as the Diatonic with g and d omitted, and the semitones divided. But the Phrygian d-d cannot pass in any such way into the Enharmonic Phrygian c e e* f a b b* c, which answers rather to the Diatonic scale of the species c-c (the Lydian). The scholars who connect the ancient Modes with the species generally confine themselves to octaves of the Diatonic genus. In this they are supported by later Greek writers—notably, as we shall see, by Ptolemy—and by the analogy of the mediaeval Modes or Tones. But on the other side we have the repeated complaints of Aristoxenus that the earlier theorists confined themselves to Enharmonic octave scales. We have also the circumstance that the writer or compiler of the pseudo-Euclidean treatise, who is our earliest authority for the names of the species, gives these names for the Enharmonic genus only. Here, once more, we feel the difference between theory and practice. To a theorist there is no great difficulty in the terms Diatonic Phrygian and Enharmonic Phrygian meaning essentially different things. But the 'Phrygian Mode' in practical music must have been a tolerably definite musical form.