Tetrachord
A tetrachord is some division of the perfect fourth into three parts, creating four distinct notes (hence tetra-). A tetrachordal scale is a scale constructed by joining two tetrachords with an interval of 9/8 (because 4/3 * 9/8 * 4/3 is an octave), and is resultantly a heptatonic scale (because the unison and octave are equivalent). Often, the additional assumption is made that the two tetrachords are identical. The diatonic scale is a tetrachordal scale, because its step pattern LLsLLLs can be broken into two tetrachords of LLs separated by a large step representing 9/8. The simplest otonal tetrachord is 9:10:11:12.
Ancient Greek theory
In Ancient Greek, the notes we consider "lower" were considered "higher"; for the sake of consistency, the English-language meanings of "low" and "high" will be used throughout this section.
The notes of a tetrachord are named in Ancient Greek music theory. They are, from lowest to highest in pitch, hypate, parhypate, lichanos, and mese. The hypate and mese are always a perfect fourth apart, and the parhypate and lichanos are considered "movable", and vary between tunings.
In Ancient Greek theory, the largest interval of a tetrachord is never between the hypate and parhypate, and is in fact usually between the lichanos and mese (that is, the version of the diatonic scale constructed from tetrachords would be in what we today call the Phrygian mode, sLLLsLL); in fact, the interval between the hypate and parhypate is usually the smallest of the three intervals of the tetrachord. Ideally, both of these always hold true, but in practice it was variable especially in certain types of scales.
Genera
The Ancient Greek tetrachords are divided into three genera, based on the size of the largest interval (the "characteristic interval").
- In the diatonic genus, the characteristic interval is less than half of the span of the tetrachord (less than about 249 cents). The characteristic interval may be the middle interval of the tetrachord rather than the highest interval. The MOS diatonic scale and the utonal counterpart of 9:10:11:12 fall into this category, and so does the Zarlino diatonic if its largest interval is interpreted as 9/8. A diatonic tetrachord may be seen as Semitone - Whole Tone - Whole Tone.
- In the chromatic genus, the characteristic interval is larger than half of the span of the tetrachord. It is also always the step between the lichanos and mese. The upper bound of the chromatic genus is vague, but chromatic tetrachords generally have a characteristic interval of a minor third. In this case, the middle interval may be the smaller of the two remaining intervals, but is usually still the larger of the two. A chromatic tetrachord may be seen as Semitone - Semitone - Minor 3rd.
- In the enharmonic genus, the characteristic interval is larger than that of the chromatic genus, generally a major third. Similarly to the chromatic genus, the middle interval (between the parhypate and lichanos) may be the smaller of the two remaining intervals. An enharmonic tetrachord may be seen as Quartertone - Quartertone - Major 3rd.
Modes
A tetrachordal scale has seven modes like any other heptatonic scale. The systems described by different Ancient Greek theorists varied, but in the original system, three of these modes were used; if the notes of a tetrachordal scale in its basic arrangement described in the page's header are labeled EFGABCDE (like the modern Phrygian mode) such that A-B is the 9/8 interval separating the two tetrachords, then that arrangement is called Dorian in ancient Greek theory; the rotation DEFGABCD is Phrygian, and CDEFGABC is Lydian.
By extension (and by Aristoxenos' theory), BCDEFGAB may be called Mixolydian, ABCDEFGA may be called Hypodorian or Locrian, GABCDEFG may be called Hypophrygian, and FGABCDEF may be called Hypolydian. Note that the order of these mode names is loosely opposite their order in modern theory.
Modern xenharmonic interpretation
Complete section
