Which church mode is it now?

In order to continue my invistigations to continue the reconstruction of the missing melodic lines, today I have digged myself in the system of Chruch Modes. My companion was (this time) Richard Hoppin's Medieval Music from which I would like to quote a few fragments as well, using them as footsteps towards my opinion on the "tonality" of Syon trure (note: however using the term "tonality" is definately not the best one to indicate the character of the range in the case of Medieval compositions, but I would stick to modern theoretical terms, so even non-early musicians will be able to understand more or less what I am talking about).

I have been thinking about this recently: which Church mode could the remaining melodic lines belong to? After all it would be essential to know this if I want to make a more or less valid reconstruction of the missing melodic lines. People were thinking about Church modes in the Middle Ages the same way as we think now our system of tonality, therefore Medieval composers were influenced by these modes. The Modality and the solmization system is definately much younger than the whole chant repertory, but at the time of Mesiter Alexander the system was already in use and his chant composer contemporaries used it: it was the music theory of the time. Therefore it is difficult to believe that vernacular composers could make themselves independent from the achievements of the earlier times. People listened to the music based on this system for two centuries in the church from before they would even be born until the day they died. This had undoubtedly an influence on their way of thinking and feeling about music. Just think about how much we are influenced by the music interpretations of the nineteenth-century in the beginning of the twentyfirst-century: we listened to it in our mothers womb and we (as well-educated musicians at the Conservatory) are listening to these interpretations, our teachers are showing us them as the best possible examples to follow.

I do not really wish to give here an extensive explanation of the Church Modes, but shortly: the system of Church modes consist of four pair of modes, these groups are identified by the Greek ordinal numbers: Protus, Deuterus, Tritus  and Tetrardus (see illustration below). Each pair has their own finalis (for example the finalis mode 1 and mode 2 is the "d"), and they consist of an authentic and a plagal scale with a range of an octave. These terms (plagal and authentic) are indicating the position of the finalis in the range of the particular modus (or tonus).

As a matter of range, according to Hoppin: "Not all melodies, of course, have a range of exactly one octave. Many of the chants, especially the older ones, move within the limits of a fifth or a sixth. Later melodies frequently have a range of an octave plus a third or a fourth. Even in melodies of restricted range, the note below the final is commonly used in all authentic modes except Mode 5. For listeneres accustomed to the leading tone in melodies associated with tonal harmony, it is curious to discover that plainchant composers evidently disliked the semitone below the final. In contrast to the usage in authentic modes, plagal melodies frequently extend one note above their basic octave. [...] Further extensions of the range can usually be explained as combinations of authentic and plagal forms, known to medieval theorists as 'mixed' modes."

The last possibility actually caught my attention. I was analysing the music material of Syon trure and I couldn't figure out which mode could this melody fit in? First I was suspecting that it might not fit in the modal system at all, but after reading Hoppin's chapter about the Church modes, but if I think about the last sentence of the fragment above, I have the feeling that in the case of Syon trure this might be the case: 'mixed mode'.
The range of "A" is f-C', and then "B" goes even higher, from C' to B' flat, which makes the whole range from f up to B' flat, and this is broader than an octave of a range, actually an octave plus a fourth. Although the use of B flats are not that often in Mode 5, if we assume, that Syon trure is in a "Tritus tonality", and we know that Meister Alexander knew the rules of composing very well, but he definately broke these rules, or rather to say he bends them... Then why not use a B flat in the melody he composed for Syon trure?

Well, this theory of mine definately needs further investigation, but at least I have a point that I might be able to use when I will finally recontruct the melody. I've taken a breve look at some other melodies by him and he does bend the rules so the result might suit his taste.
And one should not forget that composers (especially the ones with strong character, tending to free themselves from cliches) like to bend the rules of the certain compositional technique (see for example at the later composers Bach or Mozart).