Sound shift, although an important factor, is not the primary reason that Fanqie doesn't fully work for determining the sound of characters in modern varieties of Chinese. In actuality, Fanqie never fully worked for any variety of Chinese, modern or ancient. This is because the phonology recorded in rime dictionaries wasn't for any particular language, but at the very outset was a compromise between Northern and Southern varieties - that is, Middle Chinese was itself artificial, never being a natural language.
Some time ago, during the first year of the 開皇 era, I was host to 劉臻 and seven other ministers (蕭該, 顏之推, 盧思道, 李若, 辛德源, 薛道衡, and 魏彥淵). After an evening of joyous drinking and feasting, the discussion turned to the subject of phonology.
The sounds of previous eras and the other geographical regions are different, and the subject had different schools of discipline: the 吳楚 (southern) topolects were light and gentle; the 燕趙 (northern) topolects were heavy and thick; the 秦隴 topolects (now around Shaanxi and Gansu) read the departing tone as the checked tone; the 梁益 topolects (now around Sichuan) read the level tone as the departing tone. Some schools treat「支」and「脂」,「魚」and「虞」with the same rhyme; some treat「先」and「仙」,「尤」and「侯」with the same initial. If one wished to indulge in a wide variety of literature, then one can personally interchange light and heavy sounds; however, if one wished to understand phonology, then one must clearly differentiate them.
The subject matter given in 呂靜《韻集》, 夏侯該《韻略》, 陽休之《韻略》, 周思言《音韻》, 李季節《音譜》, 杜臺卿《韻略》 and others were all different to the point of being contradictory. The phonology of 江東 (Jiangnan) is completely different to that of 河北 (northern regions). Thus, we discussed differences between the northern and southern topolects and between past and present, leading on to the choice of a precise and refined phonology standard, freed from clumsy and inelegant representations of sound, with many of the choices made by 蕭該 and 顏之推. 魏彥淵 later wrote to me: "That was a challenging discussion, but the most difficult of problems have been solved. Why don't you record the results of our conference? We'll treat the points that we have agreed upon as the established rules [of phonology]." Holding a brush under the candlelight, I proceeded to summarise the phonological rules decided, the studious questions asked, and their incisive responses of that evening, attempting to capture their quintessence.
Over the next ten to twenty years, the knowledge established that evening underwent ceaseless additions from knowledgeable scholars and officials. Today, I've regressed to my roots, and have taken on students; as long as they compose works, they must understand the rules of phonology. I'm living among the mountains; ceasing to travel on far journeys for academic pursuits, the questions that I harbour have no place to be discussed. As for the old friends and colleagues of mine: those who have long since passed away, I impossibly wish for their resurrection; and those who are still alive have gone down different paths in their lives, and I wish to sever ties with them. From the phonological studies taken from the different schools and literature on the subject matter from the past to present, the body of knowledge amassed is composed into the five volumes of Qieyun. Minute differences are stressed and fine details are emphasised, whilst not being obstinate and conceited [in its opinions]. It is truly an illustrious composition, worthy as a treasured legacy for many generations. I previously criticised Sima Qian for the veneration of his own work, but now I'm fond of teasing those who gaze on, open-mouthed in awe [at such grand creations]. However, I don't dare to presume the work's propensity towards widespread popularity, and only wish for it to be used within the households.
Preface to Qieyun, by 陸法言
辛酉 year, 仁壽 era of the Sui Dynasty