Binary Linescore

I have been wondering about the highest score of a game in which each half-inning had either 0 runs or 1 run. A binary linescore, in other words.

In looking at linescores from the 1920s, I found this American League game from September 6, 1927:

St. Louis - 011 111 011 - 7 15  1
Cleveland - 100 010 101 - 4 12  1

11 runs! I imagine that would be hard to top. (Has there been a game with eight one-run innings?)

Also, I like this one from September 16, 1923:

Giants - 111 111 400 - 10 12  1
Cubs   - 002 103 000 -  6 14  3

Has a team ever scored one run in each of the first seven innings?