Source: Nouveau Systême de Musique Theorique (Paris: 1726)
Thomas Dent’s annotated extraction of the relevant passages
Compared to the concise and straightforward descriptions of D’Alembert and Rousseau, Rameau’s description of a “tempérament ordinaire” is lengthy and convoluted. Therefore, it is open to considerable interpretation in its precise details, and every modern author may well have his or her own vision, depending on their theoretical and musical, experience with other texts, temperament theory in general, etc. I present two versions here, one of my own as well one by Mark Lindley.
Disclaimer: the attribution of comma fractions to the tempered fifths is done by an automatic table look-up function. If the exact value is not in the table, the system grabs the nearest lower value. Naturally, my table does not have all possible fractions of both commas for both narrow and wide fifths. Therefore, with all modified meantone temperaments, for those fifths which “close the circle”, i.e. fill in the gap left by the departure from pure meantone logic, the amount of tempering has been somewhat arbitrarily chosen, exactly as one does when tuning an instrument, although in this case, working with decimal values for the proportion of the fifth. Therefore, the final size may not represent any fraction of any given comma, and there is no guarantee that the amount of tempering indicated as a comma fraction in the graphic is precise. The value indicated in cents, however, is derived from the proportion, and is therefore completely trustworthy.