Gaifman's normal form theorem showed that every ÿrst-order sentence of quantiÿer rank n is equivalent to a Boolean combination of "scattered local sentences", where the local neighborhoods have radius at most 7 n−1 . This bound was improved by Lifsches and Shelah to 3 × 4 n−1 . We use Ehrenfeucht-Fra ssà e type games with a "shrinking horizon" to get a spectrum of normal form theorems of the Gaifman type, depending on the rate of shrinking. This spectrum includes the result of Lifsches and

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... h, with a more easily understood proof and with the bound on the radius improved to 4 n−1 . We also obtain bounds for a normal form theorem of Schwentick and Barthelmann.