The following is the complete set of preludes and fugues by J.S. Bach known as The Well-tempered Clavier, books II and II published circa 1722 and 1742 respectively.
What did he mean by “well tempered”?
All musical scores of this corpus have been converted from MusicXML to Bol Processor syntax — read Importing MusicXML scores. This paved the way to a tonal analysis by Bol Processor’s tonal batch-processing tool discussed in detail on page Bach Well-tempered tonal analysis.
Each musical work was matched against a set of tuning schemes implemented on the Bol Processor. These comprise temperaments documented by Pierre-Yves Asselin (, 2000) and “natural” scales constructed systematically — read pages Microtonality and Creation of just-intonation scales.
The matching algorithm selected the tuning scheme(s) most compliant with definitions of “consonant” and “dissonant” melodic and harmonic intervals. Two sets of definitions have been enlisted: “standard” and “alternate”. Evidently, each hypothesis renders some tuning schemes more eligible than others for achieving the composer’s presumed perception of “consonance”. Therefore, the following sound productions of preludes and fugues with their “best” tuning schemes should not be taken as a definitive answer to the issue of temperament discussed by Bach’s pupils and followers. Nonetheless it might be closest to what the composer had in mind, within the limits of ear’s discrimination of tonal intervals.
Note that when several tuning schemes ranked first for their compliance with a piece, only one of them was used for the demo. It is possible that a different one sounds better.
All pieces have been played and recorded on a Csound instrument resembling a harpsichord, thereby allowing a clear appreciation of tonal intervals. This kind of “magnifying glass” of tonal intervals produced harsh sounding versions available in folders Standard (raw) and Alternate (raw). These have been post-processed with a little bit of reverberation yielding softer attacks. Post-processed sound files are the ones accessed in tables below. Readers conversant with sound processing are invited to download the raw files and suggest better options of post-processing.
The last two columns of each table contain the recordings of human interpretations of the same musical works by outstanding harpsichord players. These explore dimensions of musicality which the mechanical interpretation of the score with perfect tonal intervals could not reach. It remains that the challenge of accurate tonality was of prior importance for this corpus, as evidenced by the title “well-tempered” assigned by its composer.
Book I sound examples
These Bol Processor + Csound recordings may be reused under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) licence. Attribution includes links to the present page, Csound and the author/editor of its MusicXML score (listed on page Bach Well-tempered tonal analysis).
➡ Listen with earphones or a very good sound system!
As explained on page Bach Well-tempered tonal analysis, D’Alambert-Rousseau temperament rated equivalent to H.A. Kellner’s BACH in terms of scale intervals.
Book II sound examples
As explained above, D’Alambert-Rousseau temperament rated equivalent to H.A. Kellner’s BACH.
➡ Listen to the synthesis of Goldberg Variations with Sauveur’s meantone temperament.
➡ Listen to the synthesis of Goldberg Variations with D’Alembert-Rousseau temperament.
➡ Listen to the Aria on a harpsichord tuned with Werckmeister III meantone temperament.
At the same epoch (1730), French musician François Couperin composed Les Ombres Errantes for which our tonal analysis suggests a Rameau en sib temperament:
The title of this corpus, The Well-tempered Clavier, suggests that its composer intended to demonstrate the adequacy of temperament(s) for the performance of musical works in every tonality. As earlier suggested, this does not imply that all of them should match the same unique solution, although one is tempted to believe that the same instrument and the same tuning scheme have been used for the whole set. This has led to speculations by J.S. Bach’s disciples who had not been instructed how to proceed. Part of the reputation of great artists, in those days, relied on things kept secret…
It would not make sense in “real life” (human musicians and physical instruments) to play a prelude on a certain tuning and retune the instrument just to play the fugue… Therefore, these sound examples do not aim at mimicking a real performance. They may only help evaluating the tunefulness of a presumably favourite tuning scheme for each musical work.
A “deaf musicologist’s” way of appreciating tonality lies on measuring melodic and harmonic intervals in terms of frequency ratios. Results depend on values (weights) assigned a priori to certain ratios. We have shown that equally meaningful sets of hypotheses lead to utterly different findings which only trained ears might differentiate. Piling up hypotheses may not clarify the situation: an apparent “preference” for a tuning scheme might be the outcome of a numeric artefact rather than a proof of its validity.
Carefully listening to the set of recordings — and ignoring inelegant renderings of fast trills in the lower octave — highlights a musical dimension that may not be reduced to “intervals”. Each piece is like a precious stone displaying an amazing regularity in its structure. The listener is driven by the artist to exploring all sides of the crystal and appreciate its purity: a “tonal landscape”. In this approach, the slightest defect — maybe a few cents up/down — is amplified by the structure. In short, the most relevant feature may be less the choice of a structure than its consistency for the rendering of each musical phrase.
If J.S. Bach had a specific unique musical temperament in mind when composing The Well-tempered Clavier, this might not even be one rating highest in terms of intervals. This question remains open (to art historians and music experts). The only point made clear by sound examples is that playing this repertoire on improperly tuned instruments amounts — in terms of consonance — to exposing plastic imitations of diamonds!
Musicians interested in continuing this research and related development may use Bol Processor BP3’s beta version to process musical works and implement more tuning procedures. Follow instructions on page Bol Processor ‘BP3’ and its PHP interface to install BP3 and learn its basic operation. Download and install Csound from its distribution page.
Bernard Bel — January 2022