Before applying this innovative fitting method to ATP, we tested the method on our room temperature ACP sample from Figure to demonstrate that our new method works consistently and at the minimum reproduces the results obtained by the Fourier Transform technique. In Figure , MQCOs of 7 and 8 can be seen above the error bars using the multi-cosine fitting method for odd and even spin counting profiles respectively. In addition, we applied the same fitting method by considering only even or odd frequencies with respect to the even or odd spin counting profile, with a maximum integer multiples of 8 or 7 respectively. Remarkably, the fitting convergence was identical when both even and odd frequencies were considered. These extracted MQCOs are similar to the conventional Fourier transform technique which extracted 9 and 10 respectively. We conclude that the fitting method with integer multiple of cosine waves provides a more dependable lower bound of MQCOs above the error value. We adopted the same multi-cosine waveform fitting method to extract the MQCOs 13 present in ATP spin counting profiles. High CSA in ATP made extracting the MQCOs more challenging using Equation 5 in Supporting Information Section VI. To account for imperfect time reversal and high CSA values, we introduced the phase shift in Equation in the limit of cosine approximation value close to zero. Odd spin counting experiments on ATP yielded the expected coherence orders of 3 for the alpha, beta, and gamma phosphorous chemical shift environments at room temperature (Figure , top half) above the error value by considering all frequencies 0 through 3 in the fitting procedure based on Equation in Supporting Information Section VI. When cosine frequencies 4 and 5 were considered in Figure , they did not exceed the error. This observation is consistent with the selective buildup of only intramolecular coherence orders in ATP and was validated by a sodium phosphate control sample (Figure ). Due to a combination of factors, including high CSA (Figure ) and imperfection in the glassing matrix, the MQCO extraction from ATP under DNP conditions was more challenging than that from HAp and ACP, as will be discussed in the following paragraph. We can currently conclude that as the number of intramolecular phosphate bonds and long range order increases, the extracted MQCOs increase, as seen with increasing MQCOs in the order of ATP, ACP, and HAp. We find that the process of vitrification does not drastically affect the extracted coherence orders and any loss of MQCO can be attributed to a decreased signal to noise ratio due to shortened T 2 relaxation at low temperature. We also find that by using multi-cosine curve fitting method instead of the Fourier transform, we can more confidently determine the lower bound of extracted MQCOs.