It has been suggested that the inscriptions in the Psalms may bear the names of tunes or modes. As a test, I wondered how many there are and if they would map to the 7 modes inherent in the theory of Suzanne Haik-Vantoura. Maybe each named psalm provides a type to be chosen for the psalms that have no such inscription? Possibly there are only 7 'tunes' in the list below. Thoughts?
Psalms 4, 6, 54, 55, 61, 67, 76, בנגינות on strings
Psalm 5, אֶל הַנְּחִילוֹת, on the flutes
the above may be related to real instruments - i.e. not the name of a tune
Psalms 6, 12, עַל הַשְּׁמִינִית over octaves
possibly referring to a drone
1 Psalms 8, 81, 84, עַל הַגִּתִּית on musing
2 Psalm 9, עַל מוּת לַבֵּן on death of the son
3 Psalm 22, עַל אַיֶּלֶת הַשַּׁחַר to the hart of the dawn
Psalm (39, 62,) 77, עַל יְדוּתוּן (for) on Jeduthun
maybe always a person
4 Psalms 45, 60 (singular), 69, 80, עַל שֹׁשַׁנִּים on lilies
5 Psalms 53, 88, עַל מָחֲלַת in illness
6 Psalm 56, עַל יוֹנַת אֵלֶם רְחֹקִים on the dove dumbed by distance
7 Psalms 57, 58, 59, 75, אַל תַּשְׁחֵת do not destroy
After completion of a detailed study of the reconstructed music of all the psalms, perhaps this will be an answerable question.