Historical temperaments analysis system

This paper describes in a succinct form an historical temperaments analysis system designed for musicians with advanced knowledge of intervalic theory. By introducing the concept of positive trajectory reduction in the fifth circle diagram of the temperament to be analyzed, it is possible to do a very fast calculation of the size of any interval inside that temperament, within an error of ±1 cent. The corresponding exact frequency ratio becomes obvious in many cases and it is easy to summarize all the intervalic information in one analysis sheet. In addition to the fifth circle diagram, this sheet includes frequency ratios for the chromatic, diatonic, and hexatonic scales (especially their size in cents), intervals with respect to the tonic, and the deviation in cents of each note from equal temperament. With this system, piano tuning students can make comparisons of different temperaments, classify them, and understand their practicability for different types of keyboard music. Besides, the practical tuning instructions for each temperament can be comprehended much better. Also, with this analysis system, composers and keyboardists are able to program any temperament in an electronic synthesizer with user‐tuning option, and the system is even helpful for creating new temperaments.