The mathematical sequence of powers of 3 reveals a profound connection between the 13 Attributes of Divine Compassion and the shofar blasts—the defining characteristics of the month of Elul, the month of mercy and forgiveness and of course of Rosh HaShanah. The number 3 turns out to be an important part of the laws of blowing the shofar as well.
Parts of this article appeared in the Gal Einai Hebrew Magazine from 5775 and others are taken from a class given on the 9th of Elul 5774.
The Attributes of Compassion
During the month of Elul, we frequently invoke the 13 Attributes of God’s Compassion, which we have dealt with a number of times in the passing month of Wonders.[1] Some begin rising early to recite Selichot from Rosh Chodesh Elul, while others join in during the final week of the month. The essence of Selichot is the invocation of the Attributes of Compassion—hence the name Elul, the “Month of Compassion and Forgiveness.” In the month of Elul, we also sound the shofar every day (apart from the day before Rosh HaShanah) to awaken hearts to repentance (teshuvah).
In this article, we will be looking at the mathematical relationships between the numbers 13 and 3. All shofar blasts are based on the number 3. The tekiah is 1 simple, unbroken sound. The shevarim and teruah are the result of a halachic doubt regarding the Torah’s intent with the word teruah—whether it resembles a groan (yevavah) or a sob/wail (yelalah). The difference, as our Sages explain, is that the former consists of 3 parts and the latter of 9 parts. So, all the shofar sounds are powers of 3: 1 = 30, 3 = 31, and 9 = 32.
Before moving to more complex mathematical operations, note that the names of the types of blasts are also all multiples of 3: tekiah (תְּקִיעָה) is 585, or 3 times 195; shevarim (שְׁבָרִים) equals 552, or 3 times 184; and, teruah (תְּרוּעָה) equals 681, or 3 times 227.
Now, adding the three powers of 3 from above, we find that their sum is 1 plus 3 plus 9, or 13.
This relationship is hinted at in the name of the month itself: Elul (אֱלוּל). In reduced gematria (mispar katan, which according to the Arizal signifies mochin de’katnut, and is thus suited to the cognitive capacity of young children), the values of the letters of "Elul" are: א = 1, ל = 3, ו = 6, and ל = 3, and their sum is 13 providing an allusion linking the blasts, Elul, and the 13 Attributes of Mercy.
The Mathematical Relationship Between 3 and 13
In number theory, it is known that the numbers 3 and 13 share an intrinsic connection: the sum of any 3 consecutive powers of 3 is always a multiple of 13. This is the generalization of what we just saw, the zeroth, first, and second powers of 3 total 1 times 13.
Continuing the sequence:
Let us convince ourselves of this statement’s verity by looking at the next few consecutive sets of the powers of 3:
31 ┴ 32 ┴ 33 = 39 = 3 · 13
32 ┴ 33 ┴ 34 = 117 = 9 · 13
33 ┴ 34 ┴ 35 = 351 = 27 · 13
This relationship can be readily proven mathematically (the proof is left as an exercise for the interested reader).
Sequences in Mathematics and Torah
In the writings of the Sages of Torah, Rishonim and Acharonim, up to modern times, there is almost no discussion of mathematical sequences (with the exception of a few sequences known in the classical Greek world used by Ibn Ezra and, as recently discovered, by one of the Geonim).
Today, the study of mathematical sequences is a central branch of number theory. Over the years, various reference works were compiled to catalog interesting integer sequences. One such work assigned catalog numbers to sequences for standardized reference. With the rise of the internet in the 1990s, these sequences and their catalog numbers were migrated to an online database known today as the Online Encyclopedia of Integer Sequences (OEIS) at oeis.org. The sequence of powers of 3 is officially cataloged there as A000244.
What is missing from that database is an explanation of the sequence’s meaning from a Torah perspective. From a purely mathematical angle, there is no need for additional explanation beyond investigating mathematical properties. However, everything that the Almighty created in His world, He created for His glory; every detail of creation—whether inanimate, vegetative, animal, or speaking, or from the realm of intellectual truths (mathematical, logical, or otherwise)—is intended to reflect a specific attribute of the Creator. The relationship between the powers of 3 and 13 as reflected in the themes of Elul is a Torah-based explanation for why the sum of consecutive powers of 3 is divisible by 13. When studying the properties of the sequence through a Torah lens, mathematical inquiry becomes unified with the Torah and the word of God.
The Baal Shem Tov’s Custom in the Month of Elul
The Sages of Israel, the "eyes of the congregation" who perceive from afar, may not have engaged in formal mathematics, yet their teachings rest firmly upon Torah and Divine service (avodah); their directives align naturally even within mathematical frameworks.
About two and a half centuries ago, the Ba’al Shem Tov instituted a custom received from his teacher, Achiyah HaShiloni, regarding the recitation of Psalms (Tehillim) during Elul. The Book of Psalms contains 150 chapters (a multiple of 3). The Ba’al Shem Tov’s custom is to recite 3 additional chapters sequentially on each day of Elul (over and above the standard monthly portion). On Yom Kippur, the remaining chapters are completed, 9 chapters at a time, in 4 times. The 3 chapters recited daily correspond to 31, while the 9 chapters recited on Yom Kippur correspond to 32. It is commendable to adopt this custom starting this year and complete any missed chapters.
Tekiah: The Secret of Abraham the Hebrew
There are many novel insights on the shofar found in the Shulchan Aruch[2] on the laws of blowing the shofar.
It is written that the tekiah corresponds to Abraham our father, the shevarim to Isaac, and the teruah to Jacob—this is the primary intention (kavanah) that one should have when blowing the shofar. There is a beautiful point that the Alter Rebbe makes in this regard. He writes that the tekiah is also called ha'avarah, which means passing through. This is based on the verse, “And you shall pass through [veha'avarta] a shofar of teru’ah… you shall pass through your entire land.”[3] (וְהַעֲבַרְתָּ שׁוֹפַר תְּרוּעָה… תַּעֲבִירוּ שׁוֹפָר בְּכָל אַרְצְכֶם), which implies that one must sound a simple, smooth tone described as “you shall pass through” and afterward comes the sound of the teruah followed once again with the idiom, “you shall a shofar throughout your entire land.”
In other words, in biblical language, the tekiah is called ha'avarah. This is an intrinsic connection with our patriarch, Abraham our father, because Abraham is known as “Abram the Hebrew”[4] (אַבְרָם הָעִבְרִי), where Hebrew is another form of the same word as ha’avarah. In addition, the Torah describes that, “Avram passed through the land.” He is known for passing through.
In the same vein, we find that in the Laws of the Jubilee (Hilchot Yovel), the Rambam rules that one must pass the shofar through the land on the Jubilee year and that it is not enough to just blow it. Just as Abraham our father, the Hebrew walked the land, we are required to pass the shofar through the land of Israel. He truly walks, traverses, and proclaims Godliness in the world. Among the primary intentions of the tekiah is meditating on the Patriarchs. This insight regarding the notion of traversing the land to proclaim God’s oneness, at the beginning and at the end, is an essential kavanah.
More on the Shofar and the Number Three
Earlier we saw that the names of the three sounds are multiples of 3. There are other words related to this time of year that are multiples of 3. First, we have “Rosh Hashanah” (רֹאשׁ הַשָּׁנָה) whose value is 861, or 3 times 287, where 287 is the value of “Healer” (רוֹפֵא) and then, “Yom Kippur” (יוֹם הַכִּפּוּרִים) whose value is 411, or 3 times 137, the value of “Kabbalah” (קַבָּלָה).
Where does the connection between blowing the shofar and the number 3 came from? The Alter Rebbe writes the following, “How many blasts is a person obligated to hear on Rosh HaShanah? Nine [three times three]: TaRaT, TaRaT, TaRaT.” And what is the reason for this? He writes, “Why must one hear nine blasts? Because the word "teruah" is mentioned three times in the Torah: twice regarding Rosh HaShanah, and once regarding the Yom Kippur of the Jubilee year.”
There is a very beautiful and simple allusion here: it is written, "He will revive us after two days; on the third day He will raise us up, and we shall live before Him."[5] It is taught that although biblically Rosh HaShanah is one day, even when all second days of festivals celebrated outside the Land of Israel will be cancelled, Rosh HaShanah will remain two days long; it will permanently remain "days" (plural). Thus, "He will revive us after two days" refers to the two days of Rosh HaShanah, and "on the third day" refers to Yom Kippur. This is hinted at in a straightforward way: teruah is written twice regarding Rosh Hashanah and once regarding Yom Kippur. It forms a simple structure of the two days of Rosh Hashanah and Yom Kippur—every Yom Kippur is an aspect of the Jubilee, containing the illumination of Yovel, which also means the shofar. Rosh HaShanah is also a multiple of three, as is the word Jubilee (יוֹבֵל); everything is threefold, as mentioned.
The Wailing of Sisera's Mother in the Blasts
Regarding the meaning of teruah, which are like a wail, the Alter Rebbe writes,
This teruah mentioned in the Torah is translated [in the Aramaic Targum] as yevava [meaning, a wail]. And it is written, “Through the window peered and wailed [vatayabev] the mother of Sisera,”[6] which implies that a yevava is like the sound a person makes when weeping and wailing.
The entire halachic definition of a teruah is derived from the mother of Sisera,[7] the mother of the Canaanite military commander defeated by Israel as described in the opening chapters of the Book of Judges. In the Song of Deborah, Sisera’s mother peers through her window, anxiously wailing over the delay of his chariot, unaware that he had already been killed by a woman named Yael.
What is special and almost unique about the Alter Rebbe’s version of the Shulchan Aruch is that he does not only bring the ruling, the halachah, he includes an explanation for the ruling: halachah beta’amah. Were it not for this fact, we would not be reading about Sisera's mother at all, and we wouldn’t know about her connection to blowing the shofar. Without the underlying reason, verses are not required to prove anything. The reason serves to bring proof from the verse and expound upon it—the halacha emerges from the verse. Suddenly, Sisera's mother appears here. Evidently, every time we blow the shofar, during the teruah of the Day of Remembrance (Yom Hazikaron), we must remember her.
Perhaps every time we blow the shofar, she begins to cry because through the shofar we vanquish and slay her son Sisera. Just as it was Yael, who killed him, so too we blow the shofar which is ideally a “straight ibex [Yael in Hebrew] horn,” suggesting that within us, we have a Yael.
Barak's Shofar Blast
Since the Alter Rebbe drew our attention to the mother of Sisera, we might ask, which other mothers are remembered on Rosh HaShanah? The Haftarah of the second day of Rosh HaShanah concludes with “Is Ephraim a precious son to Me…”, which is also the verse that seals the Zichronot (Remembrances) section of the special Rosh HaShanah Amidah. Rachel is the mother of Ephraim, her son Joseph’s son (for grandchildren are regarded as children). Sisera's mother is her “impure opposite.” Amazingly, the phrase, “Ephraim's mother” (אֵם אֶפְרַיִם) has the exact same numerical value as “Sisera’s mother” (אֵם סִיסְרָא). Sisera was the general of the army of the King Yavin, whose name is related to (understanding). Kabbalah identifies the shofar with the foundation of the mature sefirah of understanding, known as the Supernal Mother
When the shofar is sounded, who is primarily blowing the shofar, or in other words, who should one have in mind? The two leaders of the Israelites at the time, the ones who led the battle against Yavin and Sisera: Barak and Deborah. Can a woman blow the shofar? The Alter Rebbe rules that a woman can fulfill her own obligation. In any case, the one blowing here is Barak through the power of Deborah (moreover, Barak blows and Deborah recites/sings the shofar verses).
[1]. If you did not get these issues, you can access them either through our website, www.inner.org or through https://www.calameo.com/subscriptions/7020711.
[2]. Orach Chaim 590.
[3]. Leviticus 25:9.
[4]. Genesis 14:13.
[5]. Hosea 6:2.
[6]. Judges 5:28.
[7]. In fact, the phrase “Sisera’s mother” (אֵם סִיסְרָא) equals 372, also a multiple of 3: 3 times 127.
