The four alternately conjunct and disjunct tetrachords Hypaton, Meson, Diezeugmenon, and Hyperbolaion together with the tone Proslambanomenos comprise the Disjunct or Greater Perfect (or Complete) System. It spans two octaves and a tone and runs from A to a' in modern notation. see "systema".
The GPS, together with the Lesser Perfect System (LPS), made up the "complete" Perfect Immutable System (PIS).
Shown below is my very general schematic illustrating the Greek names for the GPS in the diatonic genus, in descending order, along with approximate letter-names to help the Western reader to comprehend.
- A nete hyperbolaion | tetrachord | G paranete hyperbolaion hyperbolaion | | F trite hyperbolaion - E nete diezeugmenon | tetrachord | D paranete diezeugmenon diezeugmenon | | C trite diezeugmenon - B paramese tone of disjunction - A mese | tetrachord | G lichanos meson meson | | F parhypate meson - E hypate meson | tetrachord | D lichanos hypaton hypaton | | C parhypate hypaton - B hypate hypaton tone of disjunction A Proslambanomenos
Note that the intervallic structure of all tetrachords is identical.
The tuning of the diatonic genus was almost always given by theorists as Pythagorean, i.e., 3-limit, as shown in the following lattice-diagram:
F --- C --- G --- D --- A --- E --- B -4 -3 -2 -1 0 1 2 exponent of 3
Note that the two inner notes in each tetrachord could be retuned to other pitches in the chromatic and enharmonic genera, and even in different shades of the diatonic genus.
After the use of the other genera ceased, sometime during the "dark ages" -- i.e., between the German invasions of the Roman Empire (400s AD) and the emergence of the oldest surviving Frankish music-theory treatises (c. 750 AD) -- this conception of the Pythagorean diatonic scale as the only "proper" tuning lasted at least until the writings of Marchetto of Padua (1318).
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