Dear Basil
It's simply the fourth soft chromatic mode (using the scale of the 2nd mode) used for Apolytikia, Kathismata, Kontakia (and Hypakoai, if any) for both syllabic and short melismatic chants. The syllabic chants in this mode are more known and familiar, but the short melismatic (2 beats/syllable on average) are not unknown. Examples: the slow versions of "Tachy prokatalave" and "O ypsotheis" in Petros's or Ioannis's Eirmologia, and their prosomoia in the first days of Holy Week (to be found in these Eirmologia or other books). See also the short melismatic "Theos Kyrios" in this mode in Petros or Ioannis.
The old "Phos ilaron" is also ascribed to this mode (but also to 2nd mode, because of the similarity of the scale), which is long melismatic according to the exegesis of Petros, transcribed in the New method by the Three Teachers, or could also be short melismatic, as Karas has transcribed it.
There are also the "Allelouia" of the Great Lent Orthros (long melismatic, and in fact coinciding to the corresponding Alleiouia of the 2nd mode) and the Triadikon, found in the transcriptions of Chourmouzios.
Various Apolytikia of great feasts in short melismatic style (like the one you uploaded) can possibly be found in various books.
Happy New Year!