Polishing carrier head with multiple zones

A carrier head for holding a substrate in a polishing system, comprising:

• a housing;
• a flexible membrane extending below the housing, the flexible membrane dividing a volume above the flexible membrane into a plurality of independently pressurizable chambers;
• a plurality of independently pressurizable pressure supply lines, where there is a greater number of chambers than pressure supply lines;
• a valve assembly coupled to the pressure supply lines and the multiplicity of independently pressurizable chambers, the valve assembly having a plurality of valves with each respective valve of the plurality of valves coupled to a respective pressure chamber from the plurality of independently pressurizable chambers, and wherein each respective valve is configured to selectively couple the respective pressure chamber to one of two different pressure supply lines from the plurality of pressure supply lines, wherein the multiplicity of valves comprise electromagnetic valves;
• and
• circuitry secured to the housing to receive data on a data line and configured to selectively actuate the valves of the multiplicity of valves based on the data.