WASHINGTON (AP) — Home prices rose in May from April in every city tracked by a leading index, a sign that increasing sales and tight inventories are supporting a modest housing recovery.

The Standard & Poor’s/Case-Shiller home price index released Tuesday showed increases in all of the 20 cities tracked. And a measure of national prices rose 2.2 percent from April to May, the second increase after seven months of flat or declining readings.

Chicago, Atlanta and San Francisco posted the biggest monthly increases. Detroit, San Diego and Charlotte posted the smallest gains.

Phoenix, one of the hardest-hit cities in the housing slump, posted the strongest year-over-year gain in home prices. Still, prices there remain more than 50 percent below their peak, reached in summer 2006.

The increases partly reflect the impact of seasonal buying. The month-to-month prices aren’t adjusted for seasonal factors.

In the past year, the 20-city price index has dropped 0.7 percent, the smallest decline since September 2010. That’s much lower than the 1.8 percent year-over-year decline in April.

Many economists were encouraged by the widespread nature of the increases.

“The fact that most regions … have seen gains in recent months is a positive sign that the gradual improvement in housing conditions is becoming broader based,” Peter Newland, an economist at Barclays Capital, said in a note to clients.

David Blitzer, chairman of the S&P’s index committee, cautioned that the trend would need to continue into the summer and fall to ensure that it isn’t just a reflection of strong springtime and early summer sales.

“The housing market seems to be stabilizing, but we are definitely in wait and see mode for the next few months,” he said.