Pyknon

The pyknon was an important criterion in the classification of melodic genera (Greek: γένη τῶν μελῳδουμένων). The Greek word πυκνόν is an adjective meaning "close", "compact", "close-packed", or "crowded" (Liddell and Scott 1996). In Ancient Greek music theory, this term is used to describe a pair of intervals within a tetrachord, the sum of which is less than the remainder of the tetrachord (Levin 2007, 413). Although in modern usage, a tetrachord may be any four-note segment of a scale, or indeed any (unordered) collection of four pitch classes, in ancient Greek music theory a tetrachord consists of a four-note segment of the Greater and Lesser Perfect Systems bounded by the interval of a perfect fourth, the outer notes of which remain fixed in all genera and therefore are called "standing notes" (Greek: ἑστῶτες φθόγγοι). The positions of the inner notes vary from one genus to another, for which reason they are called "movable notes" (Mathiesen 1999, 301, 312, 322, 344, 350, et passim; from Greek: κινούμενοι φθόγγοι). In its basic theoretical form, the largest interval of a tetrachord is at the top, and the smallest at the bottom. The existence of a pyknon therefore depends on the uppermost interval being larger than half of a perfect fourth, which occurs only in the chromatic and enharmonic genera. Because the diatonic genus consists of two whole tones and one semitone, no single interval is larger than the other two combined, and so there is no pyknon (Barbera 1984, 229). For this reason, the enharmonic and chromatic genera are sometimes called the "pyknic genera", in order to distinguish them from the diatonic (Solomon 1984, 246).

The notes of the central tetrachord of the system in ascending order are hypate, parhypate, lichanos (or hypermese), and mese. A second tetrachord is added above, after a disjunctive tone, and the corresponding names (together with the interval ratios of the standing tones) are (Chalmers 1990, 4; Mathiesen 1999, 245):

• mese (4:3) – nete (2:1) (standing)
• lichanos – paranete (movable)
• parhypate – trite (movable)
• hypate (1:1) – paramese (3:2) (standing)

Although movable, the lichanos must remain above the parhypate, and the paranete above the trite (Mathiesen 1999, 312).

A "composite interval" is one made up of two or more smaller intervals; an "incomposite interval" has no smaller components In these terms, if the composite interval between the hypate and the lichanos (or paramese and paranete) is smaller than the incomposite interval from the lichanos to the mese (or paranete to nete), the three notes in that composite interval are together called a pyknon (Mathiesen 1999, 312). In the diatonic genus, because the composite interval from hypate to lichanos (a minor third) is larger than the remaining incomposite interval from lichanos to mese (a whole tone), the lowest three notes of the diatonic tetrachord are designated apyknon: "not close-packed" (Barbera 1977, 321n11).

Two pyknic enharmonic tetrachords, together comprising the Greek Dorian octave species in the enharmonic genus

In the enharmonic genus, the large incomposite interval was originally a ditone (the major third of Pythagorean tuning), leaving a pyknon with a total width of just a semitone. The Pythagorean ditone is equivalent to two 9:8 epogdoa, or major seconds), together an interval of 81:64, thus leaving a pyknon of 256:243—a limma (minor Pythagorean semitone), but how the pyknon was exactly (that is by exact mathematic calculation) divided into its two component intervals is not known (Chalmers 1990, 9). The tuning of Eratosthenes, as reported by Aristoxenus, uses a major third of 19:15 with the two unequal intervals of the pyknon in the ratios of 40:39 and 39:38 (Chalmers 1990, 8). Although Aristoxenus also implies that the two intervals of the pyknon in the enharmonic genus may be equal (Mathiesen 1999, 333), the anonymous author of the Euclidean Sectio Canonis (P18) is unequivocal: "The parhypatai and tritai do not divide the pyknon into equal intervals" (Barker 1981, 6).

Ptolemy reports in his Harmonics (2. 14) that two other theorists, Archytas and Didymus, replaced the ditone with the smaller, just major third with the number ratio of 5:4, making the pyknon correspondingly larger (West 1992, 170). This pyknon was divided differently by these two theorists, but in both cases the two intervals were not equal to one another. Archytas, who was the first theorist to give ratios for all of the genera, chose 28:27 and 36:35, and Didymus, some four centuries later, gave 32:31 and 31:30 (Chalmers 1990, 7–8).

Two pyknic chromatic tetrachords, together comprising the Greek Dorian octave species in the chromatic genus

In the chromatic genus, the largest interval was called a Greek: τριημιτόνιόν ἀσύνθετον, Latin: triemitonium incompositum—translated as "incomposite" (or "noncomposite") "trihemitone" (Bower 1989, 43; Hagel 2009, 105, 266, 267; Levin 1994, 125, 174; Barker 1989, 261, 267, prefers a descriptive translation, "an individed interval of three semitones"; Strunk 1998, 36, 37 uses "trisemitone"), the modern term being "minor third"—leaving a pyknon of some type of whole tone to be divided into two semitones. There is a larger number of variations in the tuning of the chromatic than in the enharmonic. Up to the beginning of the 4th century BC the chromatic pyknon spanned a major whole tone with a 9:8 ratio, and this was divided by Gaudentius into ascending semitone intervals of 256:243 and 2187:2048 (Chalmers 1990, 8). Ptolemy defined two different tunings of the chromatic genus: the "soft" chromatic with a smaller pyknon and the "intense" chromatic with a larger one. The unequal semitones dividing the pykna were in ratios of 28:27 and 15:14 for the soft chromatic and 22:21 and 12:11 for the intense. The larger remaining interval was 6:5 in the soft chromatic and 7:6 in the intense (Chalmers 1990, 9).

Greek Mixolydian octave species on E in the chromatic genus: conjunct tetrachords a and b, with note of conjunction c, and interval of disjunction d; the two pykna are separated by the larger interval between steps 3 and 4 Play 

A further refinement of tetrachordal construction, according to Aristoxenus, is that the lower interval of the pyknon must be smaller than or equal to the upper one (Barbera 1984, 229–30). Didymus in the chromatic genus and Archytas in the enharmonic broke this rule, however, and in the Harmonics (2. 13) Ptolemy criticized this feature in Didymus, holding that it is unmelodic and out of agreement with the evidence of our ears (West 1992, 170).

According to Aristoxenus (Elements of Harmony, book 2), whenever tetrachords are combined to form a scale filling an octave, "Two consecutive pycna may not occur in ascent or descent. A ditone may precede or follow [a pycnon] in ascent or descent. A tone may follow [a pycnon] only in descent" (Mathiesen 1999, 331).

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