Well, it all depends on how one wants to sing Byzantine Music (B.M.):
The simplest way to sing it, is like straight staff music as performed by a piano etc with absolutely "flat" notes, e.g. the natural Ke (La) would be a flat sound fixed on 440Hz. Period. In that case, staff music (in the West) is the natural notation to use. We forget the exact intervals for the moment.
On the other hand, one might want to sing it like the widely accepted beautiful traditional chanters (psaltai), some of them from recordings of the previous century, or other chanters of our time. The accepted beautiful traditional B.M. interpretation uses practically
no flat notes. Each note has a non linear pitch depending on the orthography of the note e.g. petasti, antikenoma, psifiston, vareia, klasma, kentimata. The exact interpretation of these neumes depends on the length of the note and on the chanter's personal style within some limits. It would be a very tedious (if not practically impossible) task to properly write and use these detailed interpretations in staff music, and what about the exact intervals? In this case, then, the natural thing to do seems to be the use of the Byzantine notation.
By way of illustrating (in 2017) what we are talking about, I examined the beautiful interpretations of a specific hymn by five well known and accepted chanters (Stanitsas,Emmanouilidis,Fortomas,Karampasis,Vasilikos, plus BZQ parallagi and flat) and I produced the spectral views:
https://bzquality.wordpress.com/ὓφος/
You will see that most notes have an elaborated analysis, including a kind of varying vibrato.
Each of the five chanters has taken the freedom of his own qualitative analysis as well as tempo. To quote Britannica (1951):
"...the total effect is a rather lively and melodious recitative rather than a tune in the modern sense".
The last part of the first colour image is a flat interpretation by the BZQ software, using a midi violin sound. The case before that, is also generated by BZQ and includes a "typical" qualitative analysis (style Y4), and using a built-in human (pre-recorded) voice of chanter "FCHAR", plus portamento.
By the way, BZQ examines the score and calculates/advises "where to pitch your first note".
It also calculates and lists an approximate score in Western notes.