MILKY WAY TEMPERAMENTS

a proposal of five well-tempered tuning systems

Abstract

Research Question/Problem

Tuning systems in Western music can be organized into four major categories: Pythagorean tuning, meantone temperaments, well temperaments, and equal temperament. Temperament can be defined as “the process of altering fifths and thirds in order to balance these differences while maintaining the octave intact” (Sturm, 2010, our translation). While Pythagorean tuning, with pure[1] perfect fifths, generated the Pythagorean comma[2], later systems involved different ways of tempering both this and the syntonic comma[3], seeking a balance between intervallic purity and tonal mobility. During the Renaissance, meantone temperaments favored certain tonalities, limiting harmonic circulation (Zumpano & Goldemberg, n.d.). From the Baroque period onward, well temperaments made it possible to explore all tonalities through the unequal tempering of the Pythagorean comma, preserving the identity of each key. In the twentieth century, equal temperament became consolidated, which “met the needs of composers since the Classical period” (True, 2018, p. 72, our translation) and standardized intervallic relationships, but reduced the timbral differences among harmonies. In light of this context, the following question arises: would it be possible to resume the use of well temperaments in contemporary pianistic practice, especially in concert halls, in order to expand the expressive palette without compromising the tonal mobility required by today’s repertoire?

Objectives

To demonstrate the creation of five tuning systems, belonging to the category of well temperaments, inspired by the Milky Way: Norma-Cygnus, Scutum-Crux, Carina-Sagittarius, Orion, and Perseus.

Method

The development of these temperaments was grounded in an empirical-theoretical method, in which theory guides practice and practice validates theory. Concepts related to the Pythagorean and syntonic commas were applied and investigated through the practice of piano tuning, with emphasis on the auditory perception of intervallic deviations. At the same time, “mathematical steps were developed to obtain optimized well temperaments” (Broekaert, 2019, our translation), through the creation of a musical temperament calculator, making it possible to generate graphs on the behavior of fifths and the dilution of the Pythagorean comma.

Results

This research resulted in five well-tempered tuning systems in which tonal mobility proves empirically broad. With the exception of the Norma-Cygnus system, all the others partially temper the Pythagorean comma (PC), making their perfect fifths (P5) widened (+), narrowed (−), or pure (p), according to the desired aesthetic. However, from an auditory perspective, all function as well temperaments. Some aesthetic strategies were applied, such as heartbeat pulsation in Scutum-Crux (PC: 109.1%; seven -P5s, three pP5s, two +P5s). In Perseus (PC: 79.7%; six -P5s, two pP5s, four +P5s), the Fibonacci sequence was applied to beating relationships between certain major thirds and P5. Carina-Sagittarius (PC: 66.0%; eight -P5s, four +P5s) presents some meantone-like dilutions. Orion explores some pure thirds and fifths, alluding to the purity of creation.

Conclusions

It can be concluded that the developed systems demonstrate acoustic and musical viability, allowing performance in all keys and expanding expressive possibilities in contemporary pianistic practice without compromising tonal mobility.

Keywords

well temperaments, musical tuning, musical temperament, piano

[1] Without beats

[2] 23,46 cents (100 cents=1 semitone)

[3] 21,51 cents