Details about Malki Tzedek, Heiman, Yeduthun and Assaf. When did they live? Who were they? etc
Malki-tzedek isn't explicitly mentioned in Psalms as authoring one of the psalms, but he is mentioned in psalm 110. There are different views on who Malki-tzedek of tehillim was. According to the Ibn Ezra, this is a title King David gives Hashem. According to the Malbim and Radak, this is a title Hashem gives to King David. According to Prof. Yehudah Elitzur, this was a title given by the author of the psalm to King Yehoshafat.
Asaf was a prophet (see also here) and the head of the meshorerim from the Levite branch of Gershon during the time of King David. When comparing the number of generations of ancestors of the three heads of the meshorerim - based on the assumption that each name represents one generation - it seems that Asaf was the oldest of the three. Asaf's descendants continued serving both in the First Temple and in the Second.
Heiman was a prophet (see also here) and the head of the meshorerim from the Levite branch of Kehat during the time of King David. He was the grandson of Shmuel the prophet. When comparing the number of generations of ancestors of the three heads of the meshorerim - based on the assumption that each name represents one generation - it seems that Heiman was the youngest of the three. Heiman had fourteen sons and three daughters. His descendants continued to serve at least in the First Temple.
Yedutun's original name was Eitan (this is based on the fact that whenever Asaf and Heiman appear together, the third is either Yedutun or Eitan). As such, it's possible that he's "Eitan Ha'Ezrachi" mentioned in Kings and the author of psalm 89 (compare psalm 88, one psalm prior, where Heiman is call Ha'Ezrachi)1. He was a prophet (see also here) and the head of the meshorerim of the Levite branch of Merari during the time of King David. When comparing the number of generations of ancestors of the three heads of the meshorerim - based on the assumption that each name represents one generation - it seems that Yedutun was somewhere in-between Asaf and Heiman. Yedutun had eight sons, one of whom was Oved-Edom, by whom the Holy Ark was left by for safe-keeping for a certain period. His descendants continued serving both in the First Temple and in the Second.
1 Other views are that Eitan Ha'Ezrachi was either Avraham Avinu or Eitan, son of Zerach, son of Yehudah.
So @Harel13 has provided a truly excellent answer! Whereas he has focused on the pesukim in Nach I will try and provide some of the commentaries to add to the scriptural background he has already provided:
By way of introduction, it is worth noting the Gemara in Bava Basra 14b where the people you seek to learn more about are mentioned by name:
David wrote the book of Psalms by means of ten elders of previous generations, assembling a collection that included compositions of others along with his own. He included psalms authored by Adam the first man, by Melchizedek king of Salem, and by Abraham, and by Moses, and by Heman, and by Jeduthun, and by Asaph...and three sons by Korach. (Sefaria translation)
Asaph - [as mentioned for example in Tehillim 50:1]
- The Midrash in Vayikra Rabbah 17:1 questions his genealogy and asks whether he is the son of Korach or it is in fact another Asaph.
- However the view that he is the son of Korach is supported in the Midrash Shir HaShirim Rabbah 4:4 which counts him as one of the sons, and due to the fact that he was a Talmud Chacham he was zoche (merited) to both sing with his brothers and in solitude. Although, Rav counters that he couldn't have been one of Korach's sons.
- Either way, in Tanna D'Vei Eliyahu 30 it notes:
ת"ל מזמור לאסף אלא אסף היה נביא ושמח בדבר
The Psalm of Asaph comes to teach you that Asaph was a prophet and happy about the matter.
The text goes on to qualify that when he prophesises about the destruction of the Temple Asaph notes that whoever is going to build Jerusalem and ascend its gates will raise Korach, "my father's father" from the ground which would potentially point to him being the grandson of Korach and not son.
The Midrash Eicha Rabbasi 4:14 (top right paragraph) notes how the people asked Asaph why he continues to sing when the Beis Hamikdash was destroyed and he responded that Hashem invoked His anger only against the trees and stone and not on His people.
Aside from Tehillim 50, he also authored Tellim 78-83, which according to the Artscroll commentary makes him, "the most prolific psalmist after Dovid himself".
Yedusun - (e.g. Tehillim 62:1)
Returning back to the above cited Shir HaShirim Rabbah 4:4, and to make things a bit more confusing, Rabbi Yochanan suggests that Asaph, Heiman and Yedusun are all the same people.
In another instance when he is mentioned (Tehillim 39:1), it is worth noting the Radak who says that Dovid Hamelech composed the psalm and gave it to Yedusun who was one of the singers in the Beis Hamikdash.
Heiman - (e.g. Tehillim 88:1)
- As far as working out his genealogy the Gemara in Bava Basra 15a tries to qualify who he is and concludes that there were two Heimans - one who was Moshe (refer to the Ben Yehoyada who points out how it is hinted in the name) and one who was the grandson of Shmuel (as referenced by Rabbeinu Gershom)
- Based on the Targum Yonasan on Shmuel 1 2:1 we can posit that the Heiman of Tehillim fame is indeed the grandson of Shmuel, the son of Yoel. It notes there that when Chanah prayed, she did so with a 'ruach nevuah' - a spirit of prophecy. And it writes there:
הֵימָן בַּר יוֹאֵל בַּר בְּרִי שְׁמוּאֵל עָתִיד דִיקוּם הוּא וְאַרְבַּעַת עֲשַׂר בְּנוֹהִי לְמֶהֱוֵי אָמְרִין בְּשִׁירָה עַל יְדֵי נִבְלִין וְכִנוֹרִין עִם אֲחֵיהוֹן לֵיוָאֵי לְשַׁבָּחָא בְּבֵית מַקְדְשָׁא
Heiman the son of Yoel, son of my son Shmuel, together with his fourteen sons, are destined to sing songs with lyres and harps with their Levite brethren in the Beis Hamikdash.
- Alternatively, the Heiman of Tehillim could possibly be Heiman ben Zerach. Rashi on the above cited Tehillim writes:
of Heman the Ezrahite - He was one of the sons of Zerah the son of Judah, for his lineage is delineated in (I Chron. 2:6): “And the sons of Zerah were Zimri, Ethan, Heman, Chalcol, and Darda [sic].” All five were great sages, as is said in reference to Solomon (I Kings 5: 11): “And he was wiser than all men, than Ethan the Ezraite, and Heman, and Chalcol, and Darda, the sons of Mahol,” and they established the Psalms, which were set down in the Book of Psalms. Therefore, they are called the sons of Mahol. So have I found in exact commentaries.
Malki-Tzedek - [as mentioned in Tehillim 110:4]
As has already been mentioned, the commentaries point to the fact that this is a reference to David HaMelech. However, it is known that Malki-Tzedek was Shem (refer to Bereishis Rabbah 56:10 and Midrash Shocher Tov 37:1) and the Gemara in Nedarim 32b writes that Hashem intended to bring the priesthood out of Shem as the verse says in Bereishis 14:18:
וְה֥וּא כֹהֵ֖ן לְאֵ֥-ל עֶלְיֽוֹן
"He was a priest of G-d Most High"
But because Shem placed precedence on Avraham over Hashem, the priesthood was transferred to Avraham. The Gemara brings the line from Tehillim where Malki-Tzedek is mentioned, as a reference to this exchange:
אמר לו אברהם וכי מקדימין ברכת עבד לברכת קונו מיד נתנה לאברהם שנאמר נאם ה' לאדני שב לימיני עד אשית אויביך הדום לרגליך (תהלים קי, א) ובתריה כתיב נשבע ה' ולא ינחם אתה כהן לעולם על דברתי מלכי צדק (תהלים קי, ד) על דיבורו של מלכי צדק
Abraham said to him: And does one place the blessing of the servant before the blessing of his master? You should have blessed God first. Immediately the Holy One, Blessed be He, gave the priesthood to Abraham, as it is stated: “The Lord says to my lord: Sit at My right hand, until I make your enemies your footstool” (Psalms 110:1), and afterward it is written: “The Lord has sworn, and will not repent: you shall be a priest forever, because you are a king of righteousness [al divrati malki tzedek]” (Psalms 110:4), which is explained homiletically to mean: Due to the improper words [divrati] of Melchizedek, the offspring of Abraham shall be priests of G-d forever. (Sefaria translation & notation)