Automatic music transcription, a central topic in music signal analysis, is typically limited to equal-tempered music and evaluated on a quartertone tolerance level. A system is proposed to automatically transcribe microtonal and heterophonic music as applied to the makam music of Turkey. Specific traits of this music that deviate from properties targeted by current transcription tools are discussed, and a collection of instrumental and vocal recordings is compiled, along with aligned microtonal reference pitch annotations. An existing multi-pitch detection algorithm is adapted for transcribing music with 20 cent resolution, and a method for converting a multi-pitch heterophonic output into a single melodic line is proposed. Evaluation metrics for transcribing microtonal music are applied, which use various levels of tolerance for inaccuracies with respect to frequency and time. Results show that the system is able to transcribe microtonal instrumental music at 20 cent resolution with an F-measure of 56.7%, outperforming state-of-the-art methods for the same task. Case studies on transcribed recordings are provided, to demonstrate the shortcomings and the strengths of the proposed method.