Chapter II: Part 2

(3) In a somewhat similar passage of the same work (c. 19) Plutarch is contending that the fewness of the notes in the scales used by the early musicians did not arise from ignorance, but was characteristic of their art, and necessary to its peculiar ethos. Among other points he notices that the tetrachord Hypatôn was not used in Dorian music ([Greek: en tois Dôriois]), and this, he says, was not because they did not know of that tetrachord--for they used it in other keys ([Greek: tonoi])--but they left it out in the Dorian key for the sake of preserving its ethos, the beauty of which they valued ([Greek: dia dê tên tou êthous phylakên aphêroun tou Dôriou tonou, timôntes to kalon autou]). Here again Westphal (_Aristoxenus_, p. 476) has to take [Greek: tonos] to mean [Greek: harmonia] or 'mode' (in his language _Tonart_, not _Transpositionsscala_). For in the view of those who distinguish [Greek: harmonia] from [Greek: tonos] it is the [Greek: harmonia] upon which the ethos of music depends. Plutarch himself had just been saying (in c. 17) that Plato preferred the Dorian [Greek: harmonia] on account of its grave and elevated character ([Greek: epei poly to semnon estin en tê Dôristi, tautên proutimêsen]). On the other hand the usual sense of [Greek: tonos] is supported by the consideration that the want of the tetrachord Hypatôn would affect the pitch of the scale rather than the succession of its intervals.

It seems to follow from a comparison of these three passages that Plutarch was not aware of any difference of meaning between the words [Greek: tonos] and [Greek: harmonia], or any distinction in the scales of Greek music such as has been supposed to be conveyed by these words. Another synonym of [Greek: tonos] which becomes very common in the later writers on music is the word [Greek: tropos][1]. In the course of the passage of Plutarch already referred to (_De Mus._ c. 17) it is applied to the Dorian mode, which Plutarch has just called [Greek: harmonia]. As [Greek: tropos] is always used in the later writers of the keys ([Greek: tonoi]) of Aristoxenus, this may be added to the places in which [Greek: harmonia] has the same meaning.

§ 13. _Modes employed on different Instruments._

In the anonymous treatise on music published by Bellermann[2] (c. 28), we find the following statement regarding the use of the modes or keys in the scales of different instruments:

'The Phrygian mode ([Greek: harmonia]) has the first place on wind-instruments: witness the first discoverers--Marsyas, Hyagnis, Olympus--who were Phrygians. Players on the water-organ ([Greek: hydraulai]) use only six modes ([Greek: 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.

[Footnote 1: Aristides Quintilianus uses [Greek: tropos] as the regular word for 'key:' e.g. in p. 136 [Greek: en tê tôn tropôn, hous kai tonous ekalesamen, ekthesei]. So Alypius (p. 2 Meib.) [Greek: dielein eis tous legomenous tropous te kai tonous, ontas pentekaideka ton arithmon]. Also Bacchius in his catechism (p. 12 Meib.) [Greek: hoi tous treis tropous adontes tinas adousi; Lydion, Phrygion, Dôrion; hoi de tous hepta tinas; Mixolydion, Lydion, Phrygion, Dôrion, Hypolydion, Hypophrygion, Hypodôrion, toutôn poios estin oxyteros? ho Mixolydios, k.t.l.] And Gaudentius (p. 21, l. 2) [Greek: kath' hekaston tropon hê tonon]. Cp. Dionys. Hal. _De Comp. Verb._ c. 19.]

[Footnote 2: _Anonymi scriptio de Musica_ (Berlin. 1841).]

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 [Greek: harmonia] is used of these keys, and with it, seemingly as an equivalent, the word [Greek: tropos].

Pollux (_Onom._ iv. 78) gives a somewhat different account of the modes used on the flute: [Greek: 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, [Greek: harmonia] in Pollux has the same meaning as [Greek: harmonia] in the _Anonymus_, and is equivalent to [Greek: tonos].

§ 14. _Recapitulation--[Greek: harmonia] and [Greek: 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 [Greek: harmonia] and [Greek: 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 [Greek: 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 [Greek: harmoniai] as gathered from the writers who treat of them, viz. Plato, Aristotle, and Heraclides Ponticus, is substantially the same as the list of [Greek: 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 [Greek: harmonia], the Hypo-phrygian; and Aristoxenus speaks of the Hypo-phrygian [Greek: tonos] as a comparatively new one. Again, the account which Aristoxenus gives of the Hypo-dorian [Greek: tonos] as a key immediately below the Dorian agrees with what Heraclides says of the Hypo-dorian [Greek: 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 [Greek: tonoi] in Aristoxenus is an exception which proves the rule: since the name of the Ionian [Greek: harmonia] is similarly absent from Aristotle.

3. The usage of the words [Greek: harmonia] and [Greek: tonos] is never such as to suggest that they refer to different things. In the earlier writers, down to and including Aristotle, [Greek: harmonia] is used, never [Greek: tonos]. In Aristoxenus and his school we find [Greek: tonos], and in later writers [Greek: tropos], but not [Greek: harmonia]. The few writers (such as Plutarch) who use both [Greek: tonos] and [Greek: harmonia] do not observe any consistent distinction between them. Those who (like Westphal) believe that there was a distinction, are obliged to admit that [Greek: harmonia] is occasionally used for [Greek: 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 [Greek: dôristi], [Greek: phrygisti], &c., or the neuter [Greek: ta dôria], [Greek: ta phrygia], &c., where there is nothing to show whether 'mode' or 'key,' [Greek: harmonia] or [Greek: tonos], is intended.

§ 15. _The Systems of Greek Music._

The arguments in favour of identifying the primitive national Modes ([Greek: harmoniai]) with the [Greek: tonoi] or keys may be reinforced by some considerations drawn from the history and use of another ancient term, namely [Greek: systêma].

A System ([Greek: systêma]) is defined by the Greek technical writers as a group or complex of intervals ([Greek: 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 ([Greek: 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_ ([Greek: 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[1].

§ 16. _The Standard Octachord System._

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 ([Greek: neatê] or [Greek: nêtê], lit. 'lowest,' our 'highest').
Paranêtê ([Greek: paranêtê], 'next to Nêtê').
Tritê ([Greek: tritê], _i.e._ 'third' string).
Paramesê ([Greek: paramesê] or [Greek: paramesos], 'next to Mesê').
Mesê ([Greek: mesê], 'middle string').
Lichanos ([Greek: lichanos], _i.e._ 'forefinger' string).
Parhypatê ([Greek: parypatê]).
Hypatê ([Greek: hypatê], lit. 'uppermost,' our 'lowest').

It will be seen that the conventional sense of high and low in the words [Greek: hypatê] and [Greek: 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_ ([Greek: tonos diazeuktikos]). Within each tetrachord the intervals depend upon the _Genus_ ([Greek: 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 ([Greek: phthongoi hestôtes, akinêtoi]), while the others vary with the genus, and are therefore 'moveable' ([Greek: pheromenoi]).

[Footnote 1: This is especially evident in the case of the Lichanos; as was observed by Aristides Quintilianus, who says (p. 10 Meib.): [Greek: hai kai tô genei lichanoi prosêgoreuthêsan, homônymôs tô plêttonti daktylô tên êchousan autas chordên onomastheisai]. But Tritê also is doubtless originally the 'third string' rather than the 'third note.']

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 ([Greek: 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 [Greek: 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:

_Diatonic._ _Enharmonic._

e =Nêtê= \ e =Nêtê= \
d Paranêtê } ( c Paranêtê }
c Tritê } +---( b* Tritê }
b =Paramesê= / | ( b =Paramesê= /
a =Mesê= \ | a =Mesê= \
g Lichanos } | ( f Lichanos }
f Parhypatê } | +-( e* Parhypatê }
e =Hypatê= / | | ( e =Hypatê= /
| |
[Greek: pyknon] [Greek: 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 [Greek: 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 ([Greek: 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 [Greek: 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.

§ 17. _Earlier Heptachord Scales._

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 [Greek: dia ti hoi archaioi heptachordous
poiountes tas harmonias tên hypatên all' ou tên nêtên
katelipon: hê ou tên] [Greek: hypatên] (leg. [Greek: nêtên]),
[Greek: 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 [Greek: pyknon]: whence the name [Greek: 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--

[Music: _e f g a a# c d_ \---- /\----- /]

In another place (_Probl._ xix. 32) the question is asked, why the interval of the octave is called [Greek: dia pasôn], not [Greek: di' oktô],--as the Fourth is [Greek: dia tessarôn], the Fifth [Greek: 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 [Greek: di oktô] was inappropriate.

Among later writers who confirm this account we may notice Nicomachus, p. 7 Meib. [Greek: mesê dia tessarôn pros amphotera en tê heptachordô kata to palaion diestôsa]: and p. 20 [Greek: 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.

§ 18. _The Perfect System._

The term 'Perfect System' ([Greek: 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 ([Greek: 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 ([Greek: 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 ([Greek: 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 ([Greek: systêma teleion ametabolon]), which may be represented in modern notation[1] as follows:

a Nêtê Hyperbolaiôn \ Tetrachord
g Paranêtê Hyperbolaiôn } Hyperbolaiôn
f Tritê Hyperbolaiôn /
e Nêtê Diezeugmenôn
d Paranêtê Diezeugmenôn \ Tetrachord
c Tritê Diezeugmenôn } Diezeugmenôn
b Paramesê /
d Nêtê Synêmmenôn \ Tetrachord
c Paranêtê Synêmmenôn } Synêmmenôn
b flat Tritê Synêmmenôn/
a Mesê \
g Lichanos Mesôn } Tetrachord
f Parhypatê Mesôn } Mesôn
e Hypatê Mesôn /
d Lichanos Hypatôn \ Tetrachord
c Parhypatê Hypatôn } Hypatôn
b Hypatê Hypatôn /
a Proslambanomenos

[Footnote 1: The correspondence between ancient and modern musical notation was first determined in a satisfactory way by Bellermann (_Die Tonleitern und Musiknoten der Griechen_), and Fortlage (_Das musicalische System der Griechen_).]

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' ([Greek: 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. [Greek: 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 [Greek: kai ha men autôn esti teleia, ha d' ou, atelê men tetrachordon, pentachordon, teleion de oktachordon.] This is a use of [Greek: 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[1]. Along with the lyre, which kept its primitive simplicity as the instrument of education and everyday use, the Greeks had the cithara ([Greek: kithara]), an enlarged and improved lyre, which, to judge from the representations on ancient monuments, was generally seen in the hands of professional players ([Greek: 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.

[Footnote 1: This observation was made by ancient writers, _e.g._ by Adrastus (Peripatetic philosopher of the second cent. A.D.): [Greek: 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).]

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[1]. 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[2]. 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).

[Footnote 1: The epigram is quoted in the pseudo-Euclidean _Introductio_, p. 19 (Meib.): [Greek: ho de] (sc. [Greek: Iôn]) [Greek: en dekachordô lyra] (_i.e._ in a poem on the subject of the ten-stringed lyre):--

[Greek: tên dekabamona taxin echousa
tas symphônousas harmonias triodous;
prin men s' heptatonon psallon dia tessara pantes
Hellênes, spanian mousan aeiramenoi.]

'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-flat c d_). This is the scale of the Lesser Perfect System before the addition of the Proslambanomenos.]

[Footnote 2: Pherecrates [Greek: 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.]

A similar or still more rapid development took place in the flute ([Greek: 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 ([Greek: 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 [Greek: 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 [Greek: 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 ([Greek: peri tês en Timaiô psychogonias]) speaks of the Proslambanomenos as having been added in comparatively recent times (p. 1029 _c_ [Greek: 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' ([Greek: 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[1]).

[Footnote 1: The term [Greek: hyperypatê] had all but disappeared from the text of Theon Smyrnaeus in the edition of Bullialdus (Paris, 1644), having been corrupted into [Greek: hypatê] or [Greek: 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 [Greek: 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 [Greek: diezeugmenê] and [Greek: hyperbolaia] in the sense of [Greek: nêtê diezeugmenôn] and [Greek: nêtê hyperbolaiôn] (Theon Smyrn. _l. c._).]

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[1]. 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.

[Footnote 1: _The Introduction to Harmonics_ ([Greek: 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 [Greek: 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_ ([Greek: kanonos katatomê]) belongs to the mathematical or Pythagorean school, dividing the tetrachord into two major tones and a [Greek: 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).]

§ 19. _Relation of System and Key._

Let us now consider the relation between this fixed or standard scale and the varieties denoted by the terms [Greek: harmonia] and [Greek: tonos].

With regard to the [Greek: 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 ([Greek: 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 [Greek: 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 [Greek: 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 ([Greek: panarmonion])[1]. '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 ([Greek: 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 [Greek: harmonia] of a lyre and the standard scale or [Greek: systêma] which (as we have seen) was based upon the lyre and its primitive gamut?

[Footnote 1: Plato, Rep. p. 399: [Greek: 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 [Greek: 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 [Greek: 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.]

If [Greek: 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[1].

[Footnote 1: 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.]

§ 20. _Tonality of the Greek musical scale._

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[1]. It is as follows:

Arist. _Probl._ xix. 20: [Greek: 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
[Greek: te] and [Greek: 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.'

[Footnote 1: _Die Lehre von den Tonempfindungen_, p. 367, ed. 1863.]

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 ([Greek: 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 ([Greek: arthentos tou aitiou tou hêrmosthai kai tou synechontos])[1].

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 ([Greek: 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):

[Greek: Dia ti euarmostoteron apo tou oxeos epi to bary ê apo
tou]

[Footnote 1: 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 ([Greek: estô gar mesê ho B k.t.l.]).]

[Greek: 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[1] 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' ([Greek: archê]), and the Hypatê an 'end'? In Aristotelian language the word [Greek: 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 ([Greek: 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 ([Greek: 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.

[Footnote 1: The term [Greek: 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 [Greek: 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 [Greek: en tois hêgoumenois topois], the semitones [Greek: 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 ([Greek: ton epitetarton hos estin hêgoumenos tou enarmoniou genous]).]

Another instance of the use of [Greek: 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:

[Greek: 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 [Greek: 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.

§ 21. _The Species of a Scale._

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_ ([Greek: schêmata, eidê]) of the interval which measured the compass of the scale in question. Thus, the interval of the Octave ([Greek: 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 ([Greek: 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 [Greek: 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.

§ 22. _The Scales as treated by Aristoxenus._

The subject of the musical scales ([Greek: 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 ([Greek: 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[1]. 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.: [Greek: ta gar
diagrammata autois tôn enarmoniôn] ([Greek: harmoniôn] MSS.)
[Greek: 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] ([Greek: harmoniôn] MSS.) [Greek:
monon elegon, peri de tôn allôn genôn te kai schêmatôn en autô
te tô genei tontô kai tois loipois oud' epecheirei oudeis
katamanthanein.]

[Footnote 1: The investigation occupies a considerable space in his _Harmonics_, viz. pp. 27-29 Meib. (from the words [Greek: peri de synecheias kai tou hexês]), and again pp. 58-72 Meib.]

'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.: [Greek: 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.
[Greek: proapod.]) [Greek: 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.: [Greek: 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 [Greek: heptachordôn] Meibomius and other editors read [Greek: hepta oktachordôn]--a correction strongly suggested by the parallel words [Greek: 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 [Greek: 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--. _Harm._ p. 28 Meib.: [Greek: 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[1].'

[Footnote 1: This point is one which Aristoxenus is fond of insisting upon: cp. p. 10, 16 [Greek: ou pros tên katapyknôsin blepontas hôsper hoi harmonikoi]: p. 38, 3 [Greek: hoti de estin hê katapyknôsis ekmelês kai panta tropon achrêstos phaneron]: p. 53, 3 [Greek: 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 ([Greek: [=kê]] for [Greek: [=kd]]). 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 [Symbols: E], _c_* was [Symbols: w], and _c_[Symbols: c] was [Symbols: 3]. Now the ancient diagrams, which divided every tone into four parts, must have had a character for _c_[Symbols: S]*, or the note three-quarters of a tone above _c_. Naturally this would be the remaining position of [Symbols: E], namely [Symbols: m]. 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: [Symbols: K] stands for _b_, and therefore [Symbols:)1] for _c_: but _c_ is a note of the original scale, and as such is written [Symbols: q]. 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 [Symbols: K 7g]) 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).]

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 [Greek: 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' ([Greek: pyknon], or a [Greek: pyknôma]), and the filling up of the scale in that way was therefore a [Greek: 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: [Greek: hautê estin hê para tois archaiois kata dieseis harmonia, heôs [=kd] dieseôn to proteron diagousa dia pasôn, to deuteron dia tôn hêmitoniôn auxêsasa]: 'this is the [Greek: 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[1].'