The Mishnah (6:7) teaches that thirteen sifters were used to when sift the flour for the omer offering. For the flour use for the shtei ha'lechem, twelve sifters were used, whereas for the lechem ha'panim it was eleven. R' Shimon however argued that there was no fixed number of sifters; they simply continued until the flour was sifted sufficiently.
The Gemara (66b) cites a Beraita, where the first opinion is that they would use a fine sieve, then a course sieve and keep alternating between the two different grains. According to Rashi there were only two utensils that were needed to achieve this. R' Shimon ben Elazar however argues that there were thirteen with each one finer than the next.
In the Rashi ktav Yad however, Rashi explains that R' Shimon ben Elazar was simply explaining the first opinion -- there is no argument. In other words, there were thirteen different sifter each with different grades, but it would alternate between the finest and the coarsest ones remaining. The intention of the fine sieves was to sift out the dust-like elements, while the coarse sieves were intended to let through only the fine flour and remove the coarse components.
The Rambam in his commentary on the Mishnah explains that the number of sifters was simply for a mitzvah, and even without that exact number of utensils, the korbanot would still be valid. The Bartenura adds that the Chachamim required this number of utensils lechatchila. He explains that this is just like the amount of barley and wheat listed in the previous Mishnah that is initially required, from which the required quantities of flour is derived. Those initial quantities are also lechatchila and if the required amount is derived from less, it would be valid.
The Minchat Chinnuch (307:6) explains that the Rambam understands that the two Mishnayot are connected. The difference in the starting amounts of raw grain and the number of sifters is driven by the difference in the grain from which the korbanot are brought. The omer comes from barley and the shtei ha'lechem comes from new produce, while old wheat is used for the lechem ha'panim. He adds that, if it was necessary to bring the shtei ha'lechem from old produce (yashan) then the process would be much like the lechem ha'panim and (ideally) required only eleven sifters.
The Tosfot Yom Tov however notes that Rashi explains that the number of sieves is halacha le'moshe mi'sinai. The Tosfot Yom Tov notes that there is only one sifter difference between the omer, shtei ha'lechem and lechem ha'panim. He explains that Rashi must maintain that the reasons provided above for the difference between them, is not enough to explain that minor difference in the number of keilim. He explains that this is perhaps what lead Rashi to the understanding that the number of keilim is halacha le'moshe mi'sinai. That being that case, he explains that it is possible that if less than the number utensils listed in the Mishnah were used, the korban would be invalid. The Minchat Chinnuch (307:6) adds that even if the shtei ha'lechem came from yashan, twelve sifters must nonetheless be used.
In closing, as we are in the asseret yemei teshuva, perhaps we can glean some inspiration from the omer -- the korban that was intended to permit new produce for the coming year. It is for this korban that thirteen sifters were required -- equal in number to the attributes of mercy. We have seen above the dispute regarding the technique used to sift (according to Rashi). According to R' Shimon, each sifter was progressively finer than the next. According to this understanding, refinement results from waves of finer discernment and analysis -- picking out all the waste. According to the first opinion however, the sifters had a different pattern. A coarse sifter achieved a similar purpose of letting through the finest flour and retaining the waste to be discard. The alternate fine sifter however let through the fine dust that was not of interest and retained that which was valuable. The extracting in this phase was to pull out that which was desirable. It is this dual focus that is necessary for true refinement. Not just critically picking away at the bad but also recognizing and raising up the right and good.
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