to transcribe the melodic endings or terminations of psalmody. 11th-century theorists like Guido of Arezzo (Regulae rhythmicae) or Hermann of Reichenau (Musica) refused the Dasia tone system, because it displayed tetraphonic tone system and not the systema teleion (corresponding to the white keys of the keyboard) which had all the pitches needed for the "melos of the echoi" (ex sonorum copulatione in "Musica enchiriadis", emmelis sonorum in the compilation "alia musica"). Nevertheless, the first example of the eighth chapter in Musica enchiriadis, called "Quomodo ex quatuor Sonorum vi omnes toni producantur", already used the fifth of the Protus (D—a) for an illustration, how alleluia melodies are developed by the use of the intonation formula for the "Autentus protus". The different forms of a tonary Tonaries can differ substantially in length and shape: As a treatise they usually describe the octave, fifth and fourth species of each tone, but also their modal characteristics like microtonal shifts or the change to another melodic frame. It can also be an abridged form or breviary, which just show the sacramentary (for mass chants) or antiphonary (for the office chant of the Vigils and the Hours) according to the liturgical year. The tonus of the antiphonal chant genres is indicated by later added rubrics as "ATe" for "Autentus Tetrardus" (see the Gradual-Sacramentaries of Corbie and Saint-Denis) or the Roman Ordinals I-VIII according to Hucbald's system, as we can find it in the early Troper-Sequentiary of St. Géraud in Aurillac (F-Pn lat. 1084) and the abridged Antiphonary of St. Martial (F-Pn lat. 1085). The most common form was the shortest one which had no theoretical explanation. Since the late 9th century each section started with an intonation formula and the psalmody of the mode, its pitches represented by letters or later by diastematic neume notation. Subsections followed the different chant genres quoted as examples for the represented tone. Antiphonal refrains in psalm recitation (antiphons like introits and communiones), usually represented by its text incipit, were sorted according to various terminations used in psalmody, the so-called "differentiae". A very rare form of tonary is a fully notated one, which shows every chant genre (not only the antiphonal ones with psalmody as introit and communio of the proper mass) ordered according to its tonus. A very famous example is the full tonary for mass chant by Abbot William of Volpiano, written for his Abbey St. Benignus of Dijon