For more than 50 years, the resonant column test has been used to measure the shear modulus and damping of soils for shear strains ranging from 10−5 % to 0.5 %. For most soils, the test is non-destructive, and tests may be performed on the same specimen at multiple confining stresses simulating in situ conditions from near surface to great depths. This paper makes use of the transfer function approach for resonant column theory to obtain simple solutions for the test and applies it for two types of resonant column apparatus: the conventional fixed-base free top (including spring top) now referred to as device type 1, and for a new type of resonant column device, device type 2, where a torque transducer is mounted in the bottom platen of the device. Device type 2 uses the torque measured at the base of the specimen and the rotation measured at the top of the specimen to determine the shear modulus and damping. The advantage for taking torque measurements at the base of the specimen is because the torque that is measured is that transmitted by the specimen alone. Calibrations of top platen inertia, stiffness, damping, and torque input are not needed for device type 2. Solution of these equations with complex variables can be done with any number of programming languages. For example, simple, single page, Excel spreadsheets for each device type are provided. The paper concludes with a discussion of issues that need to be addressed before procedures involving non-resonant frequencies can be introduced into ASTM D4015 [ASTM D4015: Standard Test Method for Modulus and Damping of Soils by Fixed-Base Resonant-Column Method, Annual Book of ASTM Standards, ASTM International, West Conshohocken, PA, 2007].