Musical system of ancient Greece This article concerns itself with the music theory and musical intervals used since Ancient Greece (see also musical tuning). For a discussion of the cultural aspects and history of ancient Greek music, see Music of ancient Greece. 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, to The Per- fect Immutable System, encompassing a span of fifteen pitch keys (see tonoi below) (Chalmers 1993, chapt. 6, p. 99) Any discussion of ancient Greek music, theoretical, philosophical or aesthetic, is fraught with two problems: there are few examples of written music, and there are many, sometimes fragmentary theoretical and philosoph- Depiction of the ancient Greek Tone system ical accounts. This article provides an overview that in- cludes examples of different kinds of classification while also trying to show the broader form evolving from the simple tetrachord to system as a whole. The central octave of the ancient Greek system 1 Systêma ametabolon, an overview of the tone system The Greek note symbols originate from the work of Egert Pöhlmann (1970). At about the turn of the 5th to 4th century BCE the tonal The Greater Perfect System (systêma teleion meizon) was composed of four stacked tetrachords called the system, systema teleion, had been elaborated in its en- tirety. As an initial introduction to the principal names of (from bottom to top) Hypatôn, Mesôn, Diezeugmenôn and Hyperbolaiôn tetrachords (see the right hand side of the divisions of the system and the framing tetrachords, a depiction of notes and positional terms follows. Please the diagram). Each of these tetrachords contains the two fixed notes that bound it. note, this is an as yet not completely translated version of a German illustration, hence, b in the illustration is B♭ The octaves are each composed of two like tetrachords and h, B♮. (1–1–½) connected by one common tone, the ''Synaphé. Greek theorists conceived of scales from higher pitch to At the position of the Paramése, which should be the con- lower (the opposite of modern practice), and the largest necting (Synaphé) tone, the continuation of the system intervals were always at the top of the tetrachord, with the encounters a boundary (at b-flat, b). To retain the logic smallest at the bottom. The 'characteristic interval' of a of the internal divisions of the tetrachords (see below for tetrachord is the largest one (or the 'tone' in the case of more detail) such that méson not consist of three whole the 'tense/hard diatonic' genus). The image shows the de- tone steps (b-a-g-f), an interstitial note, the diázeuxis scending two octaves of tones with corresponding modern ('dividing') was introduced between Paramése and Mése. note symbols and ancient Greek vocal, and instrumental, The tetrachord diezeugménon is the 'divided'. To bridge note symbols in the central columns. The modern note this inconsistency, the system allowed moving the Néte names are merely there for an orientation as to the inter- one step up permitting the construction of the synem- vals. They do not correspond to ancient Greek pitches or menón ('connecting') tetrachord (see the far left of the note names. The section delimited by a blue brace is the diagram). range of the central octave. The range is approximately The use of the synemmenón tetrachord effected a modu- what we today depict as follows: lation of the system, hence the name systema metabolon, 1 2 2 THE PYTHAGOREANS the modulating system also the Lesser Perfect Sys- rigorous proof that the basic musical intervals cannot be tem. It was considered apart, built of three stacked divided in half, or in other words, that there is no mean tetrachords—the Hypatôn, Mesôn and Synêmmenôn. proportional between numbers in super-particular ratio The first two of these are the same as the first two tetra- (octave 2:1, fourth 4:3, fifth 3:2, 9:8) (Huffman 2011; chords of the Greater Perfect (right side diagram), with Barker 1984–89, 2:46–52). Euclid in his The Division a third tetrachord placed above the Mesôn (left side di- of the Canon (Katatomē kanonos, the Latin Sectio Cano- agram). When viewed together, with the Synêmmenôn nis) further developed Archytas’s theory, elaborating the tetrachord placed between the Mesôn and Diezeugmenôn acoustics with reference to the frequency of vibrations (or tetrachords, they make up the Immutable (or Unmodu- movements) (Levin 1990, ). lating) System (systêma ametabolon). The three divisions of the tetrachords of Archytas were: In sum, it is clear that the ancient Greeks conceived of the enharmonic 5:4, 36:35, and 28:27; the chromatic a unified system with the octave as the unifying struc- 32:27, 243:224, and 28:27; and the diatonic 9:8, 8:7, and ture (interval). The very last (deepest) tone no longer be- 28:27 (Huffman 2011). The three tunings of Archytas longs to the system of tetrachords reflected in its name, appear to have corresponded to the actual musical prac- the Proslambanomenós, the adjoined. tice of his day (Barker 1984–89, 2:46–52). Below elaborates the mathematics that led to the logic of Tetrachords were classified in ancient Greek theory into the system of tetrachords just described. genera depending on the position of the third note lichanos (the indicator) from the bottom of the lower tetrachord (in the upper tetrachord, referred to as the 2 The Pythagoreans paranete). The interval between this note and the upper- most define the genus. A lichanos a minor third from the bottom and one whole (major second) from the top, Main article: Pythagorean interval genus diatonic. If the interval was a minor third, about one whole tone from the bottom, genus chromatic. If the After the discovery of the fundamental intervals (oc- interval was a major third with the 4/3 (or a semitone tave, fourth and fifth), the first systematic divisions of the from the bottom), genus enharmonic (Chalmers 1993, octave we know of were those of Pythagoras to whom chapt. 5, p. 47). In Archytas’s case, only the lichanos was often attributed the discovery that the frequency of varies. a vibrating string is inversely proportional to its length. More generally, depending on the positioning of the in- Pythagoras construed the intervals arithmetically, allow- terposed tones in the tetrachords, three genera of all seven ing for 1:1 = Unison, 2:1 = Octave, 3:2 = Fifth, 4:3 = octave species can be recognized. The diatonic genus is Fourth within the octave. Pythagoras’s scale consists of a composed of tones and semitones. The chromatic genus stack of perfect fifths, the ratio 3:2 (see also Pythagorean is composed of semitones and a minor third. The enhar- Interval and Pythagorean Tuning). monic genus consists of a major third and two quarter- tones or diesis (Cleonides 1965, 35–36). After the in- The earliest such description of a scale is troduction of the Aristoxenos system (see below), the found in Philolaus fr. B6. Philolaus recog- framing interval of the fourth is fixed, while the two nizes that, if we go up the interval of a fourth internal (lichanoi and parhypate) pitches are movable. from any given note, and then up the inter- Within the basic forms the intervals of the chromatic val of a fifth, the final note is an octave above and diatonic genera were varied further by three and two the first note. Thus, the octave is made up “shades” (chroai), respectively (Cleonides 1965, 39–40; of a fourth and a fifth. ... Philolaus’s scale Mathiesen 2001a, 6(iii)(e)). thus consisted of the following intervals: 9:8, The elaboration of the tetrachords was also accompanied 9:8, 256:243 [these three intervals take us up by penta- and hexachords. As stated above, the union of a fourth], 9:8, 9:8, 9:8, 256:243 [these four in- tetra- and pentachords yields the octachord, or the com- tervals make up a fifth and complete the octave plete heptatonic scale. However, there is sufficient evi- from our starting note]. This scale is known as dence that two tetrachords where initially conjoined with the Pythagorean diatonic and is the scale that an intermediary or shared note. The final evolution of Plato adopted in the construction of the world the system did not end with the octave as such but with soul in the Timaeus (36a-b). (Huffman 2011) Systema teleion (above), a set of five tetrachords linked by conjunction and disjunction into arrays of tones spanning The next notable Pythagorean theorist we know of is two octaves (Chalmers 1993, chapt. 6, p. 99). Archytas, contemporary and friend of Plato, who ex- After elaborating the Systema teleion in light of empir- plained the use of arithmetic, geometric and harmonic ical studies of the division of the tetrachord (arithmeti- means in tuning musical instruments. Archytas is the first cal, geometrical and harmonious means) and composi- ancient Greek theorist to provide ratios for all 3 genera tion of tonoi/harmoniai, we examine the most significant (Chalmers 1993, chapt. 6, p. 99). Archytas provided a 4.1 Dorian 3 individual system, that of Aristoxenos, which influenced to use both a d and either a d♭ or d♭♭ because much classification well into the Middle Ages. of the non-heptatonic nature of these scales. The empirical research of scholars like Richard Crocker C and F are synonyms for d♭♭ a g♭♭. The ap- (1963) (also Crocker 1964 Crocker 1966), C. André propriate tunings for these scales are those of Barbera (1977) and Barbera (1984), and John Chalmers Archytas (Mountford 1923) and Pythagoras.