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M'nachos 76 amud 2, describes how flour was sifted to yield the good-quality soles flour for the omer offering that was brought on the second day of Pesach. It was sifted thirteen times, the mishna says, and the Bavli clarifies:

In a fine [sifter], in a coarse one, in a fine one, in a coarse one. Rabbi Shim'on (ben El'azar) says: There were thirteen sifters in the Temple, one above another, one above another. The top one collects bran; the bottom one collects soles.

There are several different explanations of this on the page of Vilna shas itself (besides whatever others may have done with it), and I'll focus on three:

  • The commentary attributed to Rashi (printed atop the page): R. Shim'on disagrees with the first statement (and I'll omit the explanation of the first statement). According to R. Shim'on, there are thirteen sifters. The coarsest one catches bran, and the second one sifts out too-fine flour. The resultant coarse flour is then sifted in eleven sifters that are even finer (progressively so) to make sure all the too-fine flour is gone.
  • The manuscript-based Rashi (printed at the bottom of the page): R. Shim'on agrees with and clarifies the first statement. They hold that there are thirteen sifters. The finest one sifts out too-fine flour; then the coarsest one catches bran; then the second-finest sifts out too-fine flour; then the second-coarsest catches bran; then the third-finest… you get the picture. The middlemost sifter yields the good stuff.
  • R. Peretz's version of Rashi, quoted within the manuscript version of Rashi in square brackets: R. Shim'on agrees with and clarifies the first statement. They hold that there are thirteen sifters, and the sifting is done as above (the finest, then the coarsest, then the second-finest, etc.). What the coarse ones catch is discarded as bran; but what the fine sifters emit is collected for use in the mincha.

I have three questions on the explanations of R. Shim'on:

  1. According to the commentary attributed to Rashi, progressively finer sifters sift out more and more too-fine flour. That doesn't seem like it would work: surely if something didn't pass through a sifter, you'd use a coarser one to sift it out, not a finer one.
  2. According to the manuscript-based Rashi (not R. Peretz's version), the middlemost sifter yields the correct flour. Is that what's caught by the middlemost sifter, or what it lets out?
  3. According to R. Peretz's version of Rashi, several sifters (besides the last-used sifter) emit flour that was then used. Was it used for the omer or for other offerings?
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If I find the source for this I'll turn it into an answer: I've read that there was an initial coarse grind. Some of the wheat would crumble into fine dust; the bran would be coarse; and what was left could be ground again into more refined flour. –  J. C. Salomon Jul 15 at 4:30

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