What musical liturgical melodies could Jesus have sung in the synagogues of his day?

I posted a similar question on the Jewish Mi Yodyea site asking what religious music would have been sung in Israel after the time of David and Solomon. One commentator gave a Wiki link, part of which says this:

The Book of Psalms (/sɑːmz/ or /sɔː(l)mz/ SAW(L)MZ; Hebrew: תְּהִלִּים, Tehillim, "praises"), commonly referred to simply as Psalms , the Psalter or "the Psalms", is the first book of the Ketuvim ("Writings"), the third section of the Hebrew Bible, and thus a book of the Christian Old Testament. The title is derived from the Greek translation, ψαλμοί, psalmoi, meaning "instrumental music" and, by extension, "the words accompanying the music". The book is an anthology of individual psalms, with 150 in the Jewish and Western Christian tradition and more in the Eastern Christian churches. Psalms are used throughout traditional Jewish worship... According to the Talmud, these daily Psalms were originally recited on that day of the week by the Levites in the Temple in Jerusalem. From Rosh Chodesh Elul until Hoshanah Rabbah, Psalm 27 is recited twice daily following the morning and evening services. There is a Minhag (custom) to recite Psalm 30 each morning of Chanukkah after Shacharit: some recite this in place of the regular "Psalm for the Day", others recite this additionally. Source: The Ancient music of the Psalms

Jesus was intimately acquainted with all of the Law, the Prophets and the Wisdom Literature, and since the Psalms were set to music, it is almost certain Jewish people living 2,000 years ago would sing Psalms as part of their worship.

Many scholars believe the individual Psalms were redacted into a single collection in Second-Temple times. - Editorial agenda (Palms)

Despite the frequently heard view that their ancient music is lost, the means to reconstruct it are still extant. Fragments of temple psalmody are preserved in ancient church and synagogue chant, particularly in the tonus peregrinus melody to Psalm 114. Cantillation signs, to record the melody sung, were in use since ancient times; evidence of them can be found in the manuscripts of the oldest extant copies of Psalms in the Dead Sea Scrolls and are even more extensive in the Masoretic text, which dates to the Early Middle Ages and whose Tiberian scribes claimed to be basing their work on temple-period signs. (See Moshe ben Asher's 'Song of the Vine' colophon to the Codex Cairensis).

Several attempts have been made to decode the Masoretic cantillation, but the most successful is that of Suzanne Haïk-Vantoura (1928–2000) in the last quarter of the 20th century. Although some have dismissed Haïk-Vantoura's system, Mitchell has repeatedly defended it, showing that, when applied to the Masoretic cantillation of Psalm 114, it produces a melody recognizable as the tonus peregrinus of church and synagogue. Mitchell includes musical transcriptions of the temple psalmody of Psalms 120–134 in his commentary on the Songs of Ascents. Regardless of academic research, Sephardic Jews have retained a tradition in the Masoretic cantillation. Source: The ancient music of the Psalms

Aryeh posted an excellent answer to my question, quoting from sources which show that the Levitical priests would sing in the Temple. One quote presents a construction of a daily Temple service. The answer concludes by saying:

Scholars infer what parts of what this Levite music sounded like from the Hebrew text of the Psalms, which might show call-and-response forms or refrains, the use of an unclear instrument (such as the alamot in Psalms 46), or the name of a forgotten scale (e.g., the shminit in Psalm 6).

Here is the link to the question I asked so you can read Aryeh’s answer in full: What religious music would have been sung in Israel after the time of David and Solomon?

Ancient Echoes has done recordings with melodies done in what they believe to be sung at the time of Jesus.

On a new CD, Ancient Echoes, SAVAE has once again turned to history for inspiration, this time in the ancient music of the Middle East. The recording is a collection of tunes, songs and prayers that date from the time of Jesus and Jerusalem's Second Temple, when Jews were returning to the holy city from exile in Babylon.

Before recording Ancient Echoes, SAVAE founder Covita Moroney and her husband, artistic director Christopher Moroney, spent a year and a half learning Aramaic, a language spoken in the Middle East 2,000 years ago. They also studied the proper pronunciations of ancient dialects of Hebrew and Arabic.

The prayers and lyrics on Ancient Echoes draw on a variety of sources, including the Dead Sea Scrolls, the Torah, the New Testament and an engraving on a Greek gravestone dating from the first century. The Moroneys based most of the musical compositions on the work of Jewish musicologist Abraham Idelsohn. Living and working in Jerusalem early in the 20th century, Idelsohn studied the music and stories of the early Jews returning from exile. He then compared their songs and tales with those of Jews moving back to Palestine in the early 1900s. - Music from the Time of Jesus

One can listen to the commentary about the making of this CD here.

If you find it helpful, you might want to give Aryeh’s answer an up-vote. I don’t know anything about Gregorian or Early Byzantine Chants, but the Wiki article on the ancient music of the Psalms may be of use to you.