The musical system of ancient Greece evolved over a period of more than 500 years from simple scales of tetrachords, or divisions of the perfect fourth, into several complex systems encompassing tetrachords and octaves, as well as octave scales divided into seven to thirteen intervals.
Any discussion of the music of ancient Greece, theoretical, philosophical or aesthetic, is fraught with two problems: there are few examples of written music, and there are many, sometimes fragmentary, theoretical and philosophical accounts. The empirical research of scholars like Richard Crocker, C. André Barbera, and John Chalmers has made it possible to look at the ancient Greek systems as a whole without regard to the tastes of any one ancient theorist. The primary genera they examine are those of Pythagoras and the Pythagorean school, Archytas, Aristoxenos, and Ptolemy (including his versions of the genera of Didymos and Eratosthenes).
Overview of the first complete tone system As an initial introduction to the principal names and divisions of the Ancient Greek tone system this article will give a depiction of the "perfect system" or systema teleion, which was elaborated in its entirety by about the turn of the 5th to 4th century BCE.
The diagram at the right reproduces information from . It shows the common ancient harmoniai, the tonoi in all genera, and the system as a whole in one complete map. (Half-sharp and double-sharp notes not used with the depicted notes are omitted.)
The central three columns of the diagram show, first the modern note-names, then the two systems of symbols used in ancient Greece: the vocal symbols (favoured by singers) and instrumental symbols (favoured by instrument players). The modern note-names are given in the Helmholtz pitch notation, and the Greek note symbols are as given in the work of . The pitches of the notes in modern notation are conventional, going back to the time of a publication by in 1840; in practice the pitches would have been somewhat lower.
The section spanned by a blue brace is the range of the central octave. The range is approximately what is now depicted on a modern music staff and is given in the graphic below, left. Note that Greek theorists described scales as descending from higher pitch to lower, which is the opposite of modern practice and caused considerable confusion among Renaissance interpreters of ancient musicological texts.
The earliest Greek scales were organized in tetrachords, which were series of four descending tones, with the top and bottom tones being separated by an interval of a fourth, in modern terms. The sub-intervals of the tetrachord were unequal, with the largest intervals always at the top, and the smallest at the bottom. The 'characteristic interval' of a tetrachord is its largest one.
The Greater Perfect System (systema teleion meizon) was composed of four stacked tetrachords called (from lowest to highest) the Hypaton, Meson, Diezeugmenon and Hyperbolaion tetrachords. These are shown on the right hand side of the diagram. Octaves were composed from two stacked tetrachords connected by one common tone, the synaphe.
At the position of the paramese, the continuity of the system encounters a boundary (at b-flat, b). To retain the logic of the internal divisions of the tetrachords and avoid the Meson being forced into three whole tone steps (b–a–g–f), an interstitial note, the diazeuxis ('dividing'), was introduced between the paramese and mese. This procedure gives its name to the tetrachord diezeugmenon, which means the 'divided'.
To bridge the inconsistency of the diazeuxis, the system allowed moving the nete one step up, permitting the construction of the Synemmenon ('conjunct') tetrachord, shown at the far left of the diagram.
The use of the Synemmenon tetrachord effected a modulation of the system, hence the name systema metabolon, the modulating system, also called the Lesser Perfect System. This was considered apart, built of three stacked tetrachords—the Hypaton, Meson and Synemmenon. The first two of these are the same as the first two tetrachords of the Greater Perfect System, with a third tetrachord placed above the Meson. When all these are considered together, with the Synemmenon tetrachord placed between the Meson and Diezeugmenon tetrachords, they make up the Immutable (or Unmodulating) System (systema ametabolon). The lowest tone does not belong to the system of tetrachords, as is reflected in its name, the Proslambanomenos, the adjoined.
In sum, it is clear that the Ancient Greeks conceived of a unified system with the tetrachord as the basic structure, but the octave as the principle of unification.
Below is an elaboration of the mathematics that led to the logic of the system of tetrachords just described.
The Pythagoreans
After the discovery of the fundamental intervals (octave, fourth and fifth), the first known systematic divisions of the octave were those of Pythagoras to whom was often attributed the discovery that the frequency of a vibrating string is inversely proportional to its length. Pythagoras construed the intervals arithmetically, allowing for 1:1 = Unison, 2:1 = Octave, 3:2 = Fifth, 4:3 = Fourth. Pythagoras's scale consists of a stack of perfect fifths, the ratio 3:2 (see also Pythagorean Interval and Pythagorean Tuning).
The earliest such description of a scale is found in Philolaus fr. B6. Philolaus recognizes that, if the interval of a fourth goes up from any given note, and then up the interval of a fifth, the final note is an octave above the first note. Thus, the octave is made up of a fourth and a fifth. ... Philolaus's scale thus consisted of the following intervals: 9:8, 9:8, 256:243 [these three intervals take us up a fourth], 9:8, 9:8, 9:8, 256:243 [these four intervals make up a fifth and complete the octave from our starting note]. This scale is known as the Pythagorean diatonic and is the scale that Plato adopted in the construction of the world soul in the Timaeus (36a-b).
The next notable Pythagorean theorist known today is Archytas, contemporary and friend of Plato, who explained the use of arithmetic, geometric and harmonic means in tuning musical instruments. Euclid further developed Archytas's theory in his The Division of the Canon (Katatomē kanonos, the Latin Sectio Canonis). He elaborated the acoustics with reference to the frequency of vibrations (or movements).
Archytas provided a rigorous proof that the basic musical intervals cannot be divided in half, or in other words, that there is no mean proportional between numbers in super-particular ratio (octave 2:1, fourth 4:3, fifth 3:2, 9:8).
Archytas was also the first ancient Greek theorist to provide ratios for all 3 genera. The three genera of tetrachords recognized by Archytas have the following ratios: The enharmonic – 5:4, 36:35, and 28:27 The chromatic – 32:27, 243:224, and 28:27; The diatonic – 9:8, 8:7, and 28:27. These three tunings appear to have corresponded to the actual musical practice of his day.
The genera arose after the framing interval of the tetrachord was fixed, because the two internal notes (called lichanoi and parhypate) still had variable tunings. Tetrachords were classified into genera depending on the position of the lichanos (thus the name lichanos, which means "the indicator"). For instance a lichanos that is a minor third from the bottom and a major second from the top, defines the genus diatonic. The other two genera, chromatic and enharmonic, were defined in similar fashion.
More generally, three genera of seven octave species can be recognized, depending on the positioning of the interposed tones in the component tetrachords:
The diatonic genus is composed of tones and semitones, the chromatic genus is composed of semitones and a minor third, the enharmonic genus consists of a major third and two quarter-tones or diesis.
Within these basic forms, the intervals of the chromatic and diatonic genera were varied further by three and two "shades" (chroai), respectively .
The elaboration of tetrachords was also accompanied by penta- and hexachords. The joining of a tetrachord and a pentachord yields an octachord, i.e. the complete seven-tone scale plus a higher octave of the base note. However, this was also produced by joining two tetrachords, which were linked by means of an intermediary or shared note. The final evolution of the system did not end with the octave as such but with the Systema teleion, a set of five tetrachords linked by conjunction and disjunction into arrays of tones spanning two octaves, as explained above.
The system of Aristoxenus Having elaborated the Systema teleion, the most significant individual system, that of Aristoxenos, which influenced much classification well into the Middle Ages, will be examined.
Aristoxenus was a disciple of Aristotle who flourished in the 4th century BC. He introduced a radically different model for creating scales, and the nature of his scales deviated sharply from his predecessors. His system was based on seven "octave species" named after Greek regions and ethnicities – Dorian, Lydian, etc. This association of the ethnic names with the octave species appears to have preceded Aristoxenus, and the same system of names was revived in the Renaissance as names of musical modes according to the harmonic theory of that time, which was however quite different from that of the ancient Greeks. Thus the names Dorian, Lydian etc. should not be taken to imply a historical continuity between the systems.
In contrast to Archytas who distinguished his "genera" only by moving the lichanoi, Aristoxenus varied both lichanoi and parhypate in considerable ranges. Instead of using discrete ratios to place the intervals in his scales, Aristoxenus used continuously variable quantities: as a result he obtained scales of thirteen notes to an octave, and considerably different qualities of consonance.
The octave species in the Aristoxenian tradition were:
Mixolydian: hypate hypaton–paramese (b–b′) Lydian: parhypate hypaton–trite diezeugmenon (c′–c″) Phrygian: lichanos hypaton–paranete diezeugmenon (d′–d″) Dorian: hypate meson–nete diezeugmenon (e′–e″) Hypolydian: parhypate meson–trite hyperbolaion (f′–f″) Hypophrygian: lichanos meson–paranete hyperbolaion (g′–g″) Common, Locrian, or Hypodorian: mese–nete hyperbolaion or proslambanomenos–mese (a′–a″ or a–a′)
These names are derived from: an Ancient Greek subgroup (Dorians), a small region in central Greece (Locris), certain neighboring (non-Greek) peoples from Asia Minor (Lydia, Phrygia). The prefixes myxo and hypo were added to these to form associated scales above and below.
Aristoxenian tonoi The term tonos (pl. tonoi) was used in four senses, for it could designate a note, an interval, a region of the voice, and a pitch.
The ancient writer Cleonides attributes thirteen tonoi to Aristoxenus, which represent a transposition of the tones of the Pythagorean system into a more uniform progressive scale over the range of an octave.
According to Cleonides, these transpositional tonoi were named analogously to the octave species, supplemented with new terms to raise the number of degrees from seven to thirteen. In fact, Aristoxenus criticized the application of these names by the earlier theorists, whom he called the "Harmonicists".
According to the interpretation of at least two modern authorities, in the Aristoxenian tonoi the Hypodorian is the lowest, and the Mixolydian is next-to-highest: the reverse of the case of the octave species. The nominal base pitches are as follows (in descending order, after Mathiesen; Solomon uses the octave between A and a instead):