As one feature characteristic of this time of year is the blowing of the shofar, I thought I would look at the shofar blowing this week. The month of Elul leading up to Rosh Hashana, then the first 10 days of Tishrei- the days of atonement between Rosh Hashana and Yom Kippur- we listen to the shofar. Throughout Elul, we hear the shofar blown once each day. On Rosh Hashana, we hear 100 blasts of the shofar on both days. On Yom Kippur, we hear a final shofar blast at the end of the day.
Equivalent Fractions:
Equivalent fractions are different ways of writing the same fractional piece of a whole item (or group). They are two or more fractions that represent the same amount of an item but are written with different numerators and denominators. We can find these fractions by breaking the whole item into more or fewer pieces. The important thing to look for is that when you cut it into more pieces, a certain number of those pieces has to exactly create a larger piece.
For example, [Case 1] if I have a pizza and I cut it into two pieces and eat one of the pieces, then I've eaten 1/2 of the pizza. Now, [Case 2] if I had cut the pizza into 4 pieces and eaten 2 of the pieces, then I would have eaten 2/4 pieces, which is actually the same amount of pizza as in Case 1. [Case 3] If I had cut the pizza into 6 pieces and eaten 3 of the pieces, then I would have eaten 3/6, which is, again, actually the same amount of pizza as in the first two cases. But now, what if I had cut the pizza into 3 pieces, or 5 pieces? In these two situations, we could not take a number of pieces to make them exactly 1/2 of the pizza. So, we can make equivalent fractions with 1/2 using fourths and sixths, but we can't find equivalent fractions using thirds or fifths.
Shofar Connection:
From the time that young children begin learning about the shofar blasts, they are taught that there are 3 different types of blasts: tekiah, shevarim, and teruah. They learn that tekiah is one long blast, shevarim is 3 shorter blasts, and teruah is 9 staccato blasts.
(artwork credit to my preschool daughter)
If we think about these blasts as fractions, we can consider that tekiah is one whole blast (one single unit), the 3 parts of the shevarim blast are each 1/3 of the length of the tekiah, and the 9 parts of the teruah blast are each 1/9 the length of the tekiah. If we look further, we can also compare that each group of 3 blasts of teruah is the same as 1 of the shevarim sections. In other words, we can see that 1/3 of the shevarim blast is equivalent to 3/9 of the teruah blast.
In theory, this is all nice, but how do we quantify the blasts? What measurements do we use? Are they actually equivalent fractions? I checked with my husband on this- our local shofar blower (ba'al tokeah)- to get an answer. Below is a purely theoretical example of how it should work with the blasts measured in seconds.
Note: I'm purposely using numbers here that are slightly larger than in practice to make calculations easier)
If Tekiah is set to 9 seconds, then Shevarim is also set to 9 seconds for the whole blast (so the total is equivalent to tekiah), which means that each 1/3 section must be 3 seconds to be considered kosher.
Teruah would also be set to 9 seconds for the whole blast (so the total is equivalent to tekiah and shevarim), which means that each 1/9 section must be 1 second (and 3 sections together equal 3 seconds to be equivalent to a 1/3 shevarim section) to be considered kosher.
These would work for both "tekiah-shevarim-tekiah" and "tekiah-teruah-tekiah" blasts. Since the middle section cannot be longer than the tekiah itself, we would need to adjust for the "tekiah-shevarim/teruah-tekiah". For these blasts, the tekiah would be set to 12 seconds, and the shevarim/teruah together would need to be 12 seconds, so shevarim would be 6 seconds (2 seconds for each short blast), and teruah would be 6 seconds (6/9 or 2/3 of a second for each of the staccato blasts).
Now, in practice, when my husband blows shofar, our Rabbi watches with a stop watch to make sure that the shofar blasts are, in fact, fractional pieces of each other. What does this look like in practice?
The first thing to keep in mind is that practical situations are variable, and it's better to overestimate the length of a blast in order to make sure that it's kosher. Basically, when you have a kosher blast, you want to be able to count it. In order to do this, the procedure is basically as follows:
Since we can't predict the exact time in seconds that the middle blasts will be (shevarim, teruah, and shevarim/teruah), we overestimate on the time length for the first tekiah to make sure that the total length of that middle blast is always about the same length as the tekiah, but never longer than the tekiah. Since, for the final tekiah, you know exactly how long the middle blasts took, the final tekiah is timed to be exactly as long as the middle section. Also, since holding 100 shofar blasts for 10-12 seconds is extremely wearing, the blasts are actually shorter (and do not use quick, simple numbers), which makes a lot of the calculations involve fractions of seconds.
So, in real life, the blasts look something like this:
For "tekiah-shevarim-tekiah", the first tekiah is overestimated at about 5 seconds, the goal for shevarim is to have 3 1-1/2 second sections, making the whole shevarim 4-1/2 seconds long, and the final tekiah is 4-1/2 seconds.
For "tekiah-teruah-tekiah", the first tekiah is overestimated at about 6 seconds, the goal for the teruah is to have 9 sections that are just over 1/2 second each (5/9 of a second), making the whole teruah 5 seconds long, and the final tekiah is 5 seconds.
For "tekiah-shevarim/teruah-tekiah", the first tekiah is overestimated at about 9 seconds, the goal for the shevarim/teruah is to have 3 1-1/3 second blasts for the shevarim (4 seconds total for that part) and 9 sections that are just under 1/2 second each (4/9 of a second) for the teruah (4 seconds total), making the total shevarim/teruah 8 seconds long, and the final tekiah is 8 seconds long.
What does this mean about the tekiah gedolah (the long tekiah at the end of the 100 blasts)? My husband tries to make that one 12 seconds. Why 12 seconds? Because this would be equivalent to 1 estimated tekiah length from the shevarim/teruah set, plus another 1/3 or another 3 seconds, since that additional fractional amount is enough to make it noticeably longer than the other blasts.
Best wishes for a G'mar Chatima Tova- a sealed inscription for a year filled with only good things.
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