Well Temperaments based on the Werckmeister Definition From Werckmeister, passing Vallotti, Bach and Kirnberger, to Equal Temperament Inspired by Kelletat and Amiot

TL;DR

This paper develops mathematical models to optimize and objectively rank historical Well Temperaments based on Werckmeister's definition, aligning musical appreciation with mathematical analysis and enabling precise recognition of temperaments.

Contribution

It introduces a mathematical framework for optimizing and ranking historical Well Temperaments, bridging musical perception with objective analysis.

Findings

Historical temperaments closely align with optimal mathematical models.

The impurity levels of Well Temperaments are between optimal and equal temperament.

RMS modules enable accurate recognition of historical temperaments.

Abstract

Mathematical steps are developed to obtain optimised well tempered model "temperaments", based on a musical Well Temperament definition by H. Kelletat, that is derived from A. Werckmeister. This supports objective mathematical ranking of historical temperaments. Historical temperaments that fit best with mathematical models are often installed on organs. Musical appreciation and mathematical ranking of historical temperaments are in agreement. The diatonic C-major impurity of Well Temperaments lies between that of an elaborated optimal model and that of Equal Temperament. Historical Well Temperaments join these limits very closely and the list is amply filled in small rising steps. Hence, it achieves little to develop new Well Temperaments. The developed RMS computing modules can also be used for precise and accurate recognition of historical temperaments.