The "dominant" is a secondary pitch that usually serves as a reciting tone in the melody. "Ambitus" refers to the range of pitches used in the melody. Melodies whose final is in the middle of the ambitus, or which have only a limited ambitus, are categorized as "plagal", while melodies whose final is in the lower end of the ambitus and have a range of over five or six notes are categorized as "authentic". Although corresponding plagal and authentic modes have the same final, they have different dominants. The existent pseudo-Greek names of the modes, rarely used in medieval times, derive from a misunderstanding of the Ancient Greek modes; the prefix "hypo-" (under, Gr.) indicates a plagal mode, where the melody moves below the final. In contemporary Latin manuscripts the modes are simply called Protus authentus /plagalis, Deuterus, Tritus and Tetrardus: the 1st mode, authentic or plagal, the 2nd mode etc. In the Roman Chantbooks the modes are indicated by Roman numerals. Although the modes with melodies ending on A, B, and C are sometimes referred to as Aeolian, Locrian, and Ionian, these are not considered distinct modes and are treated as transpositions of whichever mode uses the same set of hexachords. The actual pitch of the Gregorian chant is not fixed, so the piece can be sung in whichever range is most comfortable. Certain classes of Gregorian chant have a separate musical formula for each mode, allowing one section of the chant to transition smoothly into the next section, such as the psalm verses that are sung between the repetition of antiphons, or the Gloria Patri. Thus we find models for the recitation of psalmverses, Alleluia and Gloria Patri for all eight modes. Not every Gregorian chant fits neatly into Guido's hexachords or into the system of eight modes. For example, there are chants – especially from German sources – whose neumes suggest a warbling of pitches between the notes E and F, outside the hexachord system, or in other words, employing a form of chromatism. Early Gregorian chant, like Ambrosian and Old Roman chant, whose melodies are most closely related to Gregorian, did not use the modal system. The great need for a system of organizing chants lies in the need to link antiphons with standard tones, as in for example, the psalmody at the Office.