Rabbi Yakov Haber
The Torah's Song; the Song of Torah
"V'ata kisvu lachem es hashira hazos, v'lam'da es B'nei Yisrael sima b'pihem, l'ma'an tihye hashira hazos l'eid b'Bnei Yisrael - And now, write for you this song, and teach it to the Children of Israel, place it in their mouth, in order that this song shall be a witness for the Children of Israel" (VaYeilech 31:19).
This verse is inherently vague. On the one hand, the context in which it is written - before Shiras Ha'azinu and immediately preceding verses predicting the Jewish people's entry into the Land of Israel, their turning to idol worship and subsequent punishment followed by: "and this song will testify as a witness" (v. 21) - indicates that the song being referenced is Shiras Ha'azinu, the song of Jewish History. On the other hand, the nearby verse, "And when Moshe finished writing down the words of this Torah in a scroll until completion" (v. 24) and the following command to the Levites, "Take this sefer Torah and place it on the side of the aron hab'ris, and it should remain there as a testimony" (v. 26) imply that the reference is to the entire Torah.
Indeed, the commentaries explain that two meanings inhere within the commandment of writing "hashira hazos": one referring to the specific portion of Shiras Ha'azinu and one to the entire Torah (see Rashi, Ralbag, Sha'arei Aharon). The Talmud (Sanhedrin 21b) teaches that even if one inherits a sefer Torah from his ancestors, he still has an obligation to write one himself and quotes the above-mentioned verse as the source for this ruling. This implies that the entire Torah is being referenced. But the Talmud (Nedarim 38a) also indicates that both Shiras Ha'azinu and the entire Torah are being referenced.
Many approaches have been suggested as to how the commandment "to write this song" can also imply the mitzvah to write the entire Torah. After all, the entire Torah is presumably not a song! Rambam (Hilchos Sefer Torah 7:1) famously explains that the commandment is primarily to write the song of Ha'azinu. But since the Torah cannot be written in separate mini scrolls (parshiyos, parshiyos), the entire Torah must be written. Rav Naftali Tzvi Yehuda Berlin zt"l (Netziv) in his Ha'ameik Davar (see also Torah Temimah) challenges this understanding since the Torah's intent might be to write Ha'azinu separately, just as there is a commandment to write the parshiyos of tefillin separately.
Netziv alternatively suggests that the phrase "this song" has a dual meaning. Firstly, it refers to the song of Ha'azinu; secondly it refers to the entire Torah. Because of this second meaning the plural "kisvu" is used since every Jew has an obligation to write a Seifer Torah. By contrast, only Moshe was commanded to write Ha'azinu. As to why the Torah is called a song, Netziv elsewhere (introduction to Ha'ameik Sh'aila, Kidmas Ha'Eimek 2:3) explains that there are two types of writing: prose and poetry, the later also being referred to as verse or song. Hashem formulated the Torah in a purposefully ambiguous manner to allow for multiple meanings all to be derived from the text. As any student of the Talmud knows, Chazal are attuned to every extra letter, every textual anomaly, and every textual similarity from which many halachos are derived. (Also see Malbim's Ayeles HaShachar.) R' Akiva is famously described as deriving "mounds and mounds of halachos" from even the crowns on top of the letters (Menachos 29b.) In the aggados, midrashim, and in sifrei chassidus verses are interpreted b'derech haremez, by way of allusion, often taking them totally out of context. All of these meanings lie within the "song" of Torah, purposefully implanted by its Author to enable those who study it to extract those multiple layers of meaning from the text. As an additional example, Malbim, in his commentary to Tanach, will often collect many different interpretations of the commentaries preceding him and show how each interpretation is alluded to in the text. The Gaon of Vilna toward the end of his life worked on demonstrating how all the halachos of the Torah are alluded to in the text of the Torah.
This important idea developed by Netziv gives us great insight as to the nature and complexity of how a finite text of the Torah can ultimately have infinite interpretations. The Torah is a manifestation of the chochmas Hashem. In the language of the Zohar, "Kudsha B'rich Hu v'Oraisa chad Hu - Hashem and the Torah are one." Learning Torah is, on a deeper plain, studying Hashem Himself. In one of the daily blessings recited over Torah study we refer to those who know Torah as "yod'ei sh'mecha - knowers of Your name." Ramban writes in his introduction to Torah that if we were to remove all the spaces between the words of the Torah and create different words by spacing them differently, we would be reading different names of Hashem. Rav Schneur Zalman Schneerson of Liadi, the first Rebbe of Lubavitch, explains, based on kabbalistic sources, that each commandment represents a different aspect of Divinity. This gives us insight as to how the Talmud could describe Hashem as "wearing tefillin" (Berachos 6a) or "wrapping Himself in a tallis" (Rosh HaShana 17b). The mitzvos which we do are an allegorical representation of Divine ideas. The real "tallis" and "tefillin" are esoteric Divine concepts; the physical tallis and tefillin allow finite human beings to connect to these infinite concepts. The "song" of Torah with its multi-dimensional and even infinite layers of meaning provides for us a window into the Eternity and All-Encompassing nature of its Author.
Yom HaKippurim was established originally as the Day of Atonement for the sins of K'lal Yisrael since it was on that day that Moshe Rabbeinu came down for the final time from Har Sinai with the second luchos. This central event was the climax of the kappara for the sin of the golden calf. In essence then, the Torah was given a second time on Yom Kippur. Since the second luchos were the ones that lasted, not the first, the Jewish people are perhaps more connected to this day of Matan Torah than even to Shavuos. The last mishna in maseches Ta'anis teaches that there were no happier holidays in Israel than Yom HaKippurim since it was the day of the giving of the Torah. Perhaps the reason that Shavuos is not listed is that since the first luchos were broken, we relate to Yom HaKippurim more intensely.
But of course, Yom Kippur is not just the day of the second Matan Torah; it also the "keitz s'licha um'chila", it is a day of national and individual repentance and return to G-d. It is a day whose very essence is infused with Divine Presence and mercy; it is a day we enter into as we would a mikveh, not just a day that we experience - "Mikveh Yisrael Hashem", just as a mikveh purifies the impure, so HaKadosh Baruch Hu purifies Israel (mishna end of Yoma). Apparently, there is a strong link between Yom Kippur's role as a day of teshuva and kappara and its role as a day of Matan Torah.
Rav Betzalel Zolti zt"l, the former chief Rabbi of Jerusalem, (Mishnas Ya'avetz 54:3-4) elaborates at length how teshuva is not just the commitment to not violate aveiros and not just the determination to perform the mitzvos properly, but to return to the totality of Torah and specifically its study - "and in it we will speak day and night". This is a manifestation of the highest form of teshuva, teshuva mei'ahava, out of love. Among his other sources for this concept, Rav Zolty quotes the haftora for Shabbos Shuva wherein Hoshei'a the prophet adjures us: "Shuva Yisrael 'ad Hashem Elokecha ki kashalta ba'avonecha; k'chu imachem d'varim v'shuvu el Hashem - Return O Israel unto G-d for you have stumbled in your sin; take words with you and return to G-d." On this, the Sifrei (Ha'azinu 306:2) comments that the phrase "take words with you" refers to words of Torah. Similarly, we implore Hashem daily in the blessing of teshuva, "Hashiveinu Avinu l'Torasecha" and only afterward do we pray, "v'hachazireinu b'teshuva sh'leima l'fanecha".
The song of Torah has to permeate our thoughts, our words as well as our actions. Perhaps demonstrating the idea of the Torah being called a song, the Torah is read with trop, musical notes, and the Jewish people have always traditionally sung the Talmud to a characteristic niggun. Just as a symphony has many different instruments each with their own purpose, all joining together to create a majestic, musical masterpiece, so too does each mitzvah elevate another limb, physical and spiritual, and each Torah idea develop another aspect of our inner personality in our quest for sh'leimus.
May Hashem grant us the assistance to realign our lives individually and collectively to play our unique role in the Divine symphony that is the Torah, the greatest gift that Hashem has bestowed upon His creation.
 In Ha'ameik Davar, Netziv references Kidmat Ha'Eimek to explain why the Torah is called a song. There, the nature of Torah being written in a manner to allow for many layers of interpretation is discussed at length. Admittedly, I did not find a direct reference there to the concept of the Torah being called a song. Above, I wrote what I thought the Netziv's intent was.
 In his introduction to Chumash Devarim, Netziv explains that the title Mishne Torah should not be translated as "Repetition of Torah" but as "Double-meaning Torah". Specifically in this Chumash, even on a p'shat level, multiple layers of meaning often apply. Ha'ameik Davar throughout Devarim, including here, applies this principle repeatedly.
 Famously, he posited that all the 613 mitzvos are alluded to in the very first word, "b'raishis".
 Refer also to the shiur referenced in footnote 4.