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I am familiar with the origin and reasoning of the "regular" Tekiah blast that surrounds the shevarim, teru'ah and shevarim-teru'ah sounds.

What is the origin of the teki'ah gedolah in terms of its use on Rosh Hashanna? I recall that that there are various verses that allude to a "long" Shofar sound such as the one heard on Mt. Sinai, the one used for yovel (Jubilee) and the sound of the "great shofar" announcing Mashiach.

But what is the origin of using it as part of the Rosh Hashanna shofar service? I don't recall this mentioned in the Talmud anywhere.

It seems to be done as the last teki'ah blast of some type of "grouping" such as the last blast of the teki'ot d'meyushav - the group of blasts done before Musaph Amidah. Is this its purpose? To announce the ending of a grouping? Or is there some other purpose to it?

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This is mentioned in the Sefer Minhagim of Maharil (Hilchos Shofar # 13). The reason given there is that it lets people know that the set of shofar blasts has been completed:

אמר מהר"י סג"ל דלהכי מאריכין בתקיעה אחרונה ונקראת תקיעה גדולה למען ישמעו העם וידעו שסיימו הקולות ויאמרו אשרי העם יודעי תרועה

Additionally, this very question was asked to R. Chaim Elazar Shapira in Minchas Elazar 2:65. He struggled to find sources and explanations for it, eventually citing the explanation of Maharil mentioned above. He added that Maharil is the rabbi of Ashkenaz, especially when it comes to customs of prayer, and he records it as an already existing custom which indicates that it probably predates him. R. Shapira further notes that the tekiah gedolah is mentioned by R. Yaakov Emden, and he thus concludes that it is a tradition that we cannot discard:

וגם לא מצינו רמז רמיזה בשאר הפוסקים וגם בכתבי האריז"ל והמקובלים שיאמרו שום כוונה לתקיעה גדולה בסוף התר"ת אמנם מצאתי במהרי"ל בה' ראש השנה כ' וז"ל אמר במהר"י סג"ל דלהכי מאריכין בתקיעה אחרונה ונקראת תקיעה גדולה למען ישמעו העם וידעו שסיימו הקולות ויאמרו אשרי העם יודעי תרועה והראי' בפסוק במשוך היובל המה יעלו בהר נמצא שמשיכת הקול סי' סילוק שכינה ורשאין לעלות ההר ה"נ לא שנא עכ"ל ומהרי"ל שהוא רבינו רבן של בני אשכנז בפרט במנהגי התפלות ותקיעות ופיוטים בר"ה מפיו אנו חיין והוא הביא עוד זה למנהג קבוע בימיו ובודאי הי' עוד לפניו מעולם וכ' טעם לזה וכן הגאון מהר"י עמדין ז"ל בסידור שלו ע"כ בודאי איכא משום אל תטוש תורת אמך וחלילה לגרוע ולשנות

  • Interesting source. What's nice is that he alludes to a verse that suggests a supporting reason for its usage. Incidentally, what does סג"ל mean? – DanF Sep 13 '18 at 14:22
  • Briefly reading that link, he seems to have lived in the same area as Rav Amnon, Author of Unetaneh Tokef, and Mahari"l predates him, if I calculate correctly. – DanF Sep 13 '18 at 14:28
  • @DanF Maharil is later. See here for some brief history of Unetaneh Tokef and R. Amnon. – Alex Sep 13 '18 at 14:32
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    @DanF סג"ל is often an acronym that stands for "Sgan Lakohanim", which roughly translates to "assistant to the Kohanim". It was a title often given to Levi'im (who were assistants to the Kohanim), and indeed, Segal is a surname commonly found among Levi'im. – Salmononius2 Sep 13 '18 at 14:39
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The ArtScroll commentary at the end of the first set of blasts (p. 438 in my Ashkenaz volume) writes:

The purpose of the extended tekiah at the completion of the series of thirty shofar blasts is to indicate that this section of the mitzvah has been completed, and the congregation may resume its prayer service. This is analogous to the extended shofar blast (see Exodus 19:13) that marked the departure of the Shechinah (Divine Presence) from Mount Sinai after the Acceptance of the Torah (Maharil).

While this is only written in the commentary by the first set of blasts, the logic does apply to the later sets of blasts as well.

While this mostly addresses the 'why' of the question and leaves the 'when' unanswered, given that the Maharil lived at the end of the 14th/beginning of the 15th century, we can set a minimum age of this minhag to be at least the geater part of a millennium.

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