With the increasing importance of smart gadgets in our daily lives, there is a need for an automatic piano transcription system in various multimedia services. For automatic piano transcription, the string inharmonicity coefficient ( ) and fundamental frequency () should be detected robustly and accurately. The proposed triplet-sequentially additive partial (SAP) algorithm improves the current estimation algorithm in terms of both performance and speed with less prior knowledge. Additionally, this joint ( , ) estimation algorithm is applied directly to the transcription of real piano recordings, and the 4.41% improvement of accuracy was achieved over another transcription system that had both similar processing steps and feature extraction method.