Article URL: https://xenharmlib.readthedocs.io/en/latest/whats_new_0_4_0.html Comments URL: https://news.ycombinator.com/item?id=49086059 Points: 10 # Comments: 1

As far as new features are concerned, version 0.4.0 is not a step but a leap forward. Xenharmlib’s core has been refactored to allow not only equal-division temperaments but also tunings with multiple generator intervals. This means that it now supports the entire range of regular temperaments, including Just Intonation. With the sole exception of irregular temperaments (such as the “Werckmeister temperaments”), xenharmlib can finally represent the entire cosmos of musical traditions in mathematical form. It can now be used for the accurate analysis of music from the Byzantine Empire, the European Renaissance, Indian classical music, the Indonesian gamelan tradition, Arabic Maqam and 20th-century American experimental music (such as the compositions of Harry Partch or Ben Johnston) In keeping with its holistic approach, xenharmlib allows the use of existing analytical tools for all regular temperaments, provided they are mathematically compatible with them. For example, Prime Form and Normal Form calculations can also be performed on (multi-dimensional) regular temperaments. In addition, the documentation has been completely revamped: feature examples are organized by tuning and notation, and comprehensive, dedicated documentation is provided for all harmonic primitives. We will give a brief overview of the aforementioned (and other) features hereinafter. (For the full Changelog see here) Just Intonation refers to tunings that are based on “pure intervals”. A pure interval is characterized by a fraction of two integers, e.g. \(\frac{3}{2}\). Historically, most tuning systems were based on Just Intonation. It was not until the 18th and 19th centuries that equal temperament systems began to gain widespread acceptance in practice. Xenharmlib now introduces support for Just Intonation with the Prime Limit Tuning construct: A Pythagorean 12-tone chromatic scale can, for example, now be generated by stacking 5 ascending pure fifths and 6 descending pure fifths from the root pitch: Most software programs implement functions for Just Intonation not as a complete arithmetic system, but as a system of scalar transposition. In the case of a transposition by a fifth, for example, a definition of the chromatic scale (such as the one mentioned above) is typically used as a basis, and the transposition is understood as a shift of 7 scale degrees: As you can see with the last pitch of the output (\(\frac{243}{128}\)), transposition by scale degree in a Pythagorean chromatic scale does not always result in an actual transposition of the interval in question. This is a common problem when tuning a keyboard instrument (such as a piano or harpsichord) that has only a limited number of notes per octave. Mathematically speaking, this definition of transposition by a fifth is irregular: the size of the fifth changes depending on which note is chosen as the starting point.