The Well-tempered Clavier

Below is the com­plete set of Preludes and Fugues by J.S. Bach known as The Well-Tempered Clavier, Books II and II, pub­lished around 1722 and 1742 respectively.

What was the com­poser’s inten­tion when he used the term “well tempered”?

All the scores in this cor­pus have been con­vert­ed from MusicXML to Bol Processor syn­tax — see Importing MusicXML scores. This paved the way for tonal analy­sis using Bol Processor’s tonal batch pro­cess­ing tool, described in detail on the Bach Well-tempered tonal analy­sis page.

Each musi­cal work has been com­pared with a set of tun­ing schemes imple­ment­ed on the Bol Processor. These include tem­pera­ments doc­u­ment­ed by Pierre-Yves Asselin ([1985], 2000) and “nat­ur­al” scales sys­tem­at­i­cal­ly con­struct­ed — see pages Microtonality and Creation of just-intonation scales.

The match­ing algo­rithm select­ed the tun­ing scheme(s) that best matched the def­i­n­i­tions of ‘con­so­nant’ and ‘dis­so­nant’ melod­ic and har­mon­ic inter­vals. Two sets of def­i­n­i­tions were used: “stan­dard” and ” alter­na­tive”. Obviously, under any hypoth­e­sis, some tun­ing schemes are more suit­able than oth­ers for achiev­ing the com­poser’s pre­sumed per­cep­tion of ‘con­so­nance’. Therefore, the fol­low­ing sound pro­duc­tions of the Preludes and Fugues, with their “best” tun­ing schemes, should not be tak­en as a defin­i­tive answer to the ques­tion of tem­pera­ment dis­cussed by Bach’s stu­dents and fol­low­ers. They may, how­ev­er, come clos­est to what the com­pos­er intend­ed, with­in the lim­its of the ear’s abil­i­ty to dis­crim­i­nate between intervals.

Note that if sev­er­al tun­ing schemes ranked first for match­ing a piece, only one of them was used for the demo. It is pos­si­ble that anoth­er may sound better.

Settings of an audio unit for the post-processing

All the pieces were played and record­ed on a Csound instru­ment, sim­i­lar to a harp­si­chord, allow­ing a clear appre­ci­a­tion of the tonal inter­vals. This kind of “mag­ni­fy­ing glass” view of the tonal inter­vals pro­duced harsh sound­ing ver­sions, avail­able in the Standard (raw) and Alternate (raw) fold­ers. These have been post-processed with a bit of reverb to pro­duce soft­er attacks. The post-processed sound files are the ones accessed in the tables below. Readers famil­iar with sound edit­ing are invit­ed to down­load the raw files and sug­gest bet­ter post-processing options.

The last two columns of each table con­tain record­ings of human inter­pre­ta­tions of the same works by out­stand­ing harp­si­chordists. These explore dimen­sions of musi­cal­i­ty that the mechan­i­cal inter­pre­ta­tion of the score with per­fect tonal inter­vals could not reach. It remains that the chal­lenge of accu­rate tonal­i­ty was a pri­or­i­ty for this cor­pus, as evi­denced by the title “well-tempered” giv­en 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, to Csound and to the author/editor of its MusicXML score (list­ed on the Bach Well-tempered tonal analy­sis page).

Listen with head­phones or a very good sound system!

The record­ings of Wanda Landowska’s (1879-1959) are from YouTube. Other inter­pre­ta­tions are cour­tesy of Wikimedia Commons.

As explained on the Bach Well-tempered tonal analy­sis page, the D’Alambert-Rousseau tem­pera­ment was found to be equiv­a­lent to H.A. Kellner’s BACH in terms of scale intervals.

(favourite: Sauveur)
(favourite: D’Alembert-Rousseau)
1BWV 846CmajSauveurSauveurRameau en sibMarpurgMartha GoldsteinMartha Goldstein
2BWV 847CminSauveurSauveurD’Alambert-RousseauD’Alambert-RousseauWanda LandowskaMartha Goldstein
3BWV 848C#majAbmajZarlino nat­ur­alD’Alambert-RousseauD’Alambert-RousseauWanda LandowskaWanda Landowska
4BWV 849C#minWerckmeister 4Werckmeister 4CminCminWanda LandowskaWanda Landowska
5BWV 850DmajSauveurSauveurD’Alambert-RousseauRameau en doMartha GoldsteinMartha Goldstein
6BWV 851DminSauveurSauveurD’Alambert-RousseauD’Alambert-RousseauMartha GoldsteinMartha Goldstein
7BWV 852E♭majRameau en sibRameau en sibRameau en sibD’Alambert-RousseauWanda LandowskaWanda Landowska
8BWV 853E♭min/D#minEbminMarpurgEbminDminWanda LandowskaWanda Landowska
9BWV 854EmajSauveurWerckmeister 4EmajD’Alambert-RousseauWanda LandowskaWanda Landowska
10BWV 855EminSauveurSauveurD’Alambert-RousseauD’Alambert-RousseauWanda LandowskaWanda Landowska
11BWV 856FmajSauveurSauveurD’Alambert-RousseauD’Alambert-RousseauWanda LandowskaWanda Landowska
12BWV 857FminZarlino nat­ur­alSauveurD’Alambert-RousseauD’Alambert-RousseauWanda LandowskaWanda Landowska
13BWV 858F#majMarpurgF#majD’Alambert-RousseauD’Alambert-RousseauWanda LandowskaWanda Landowska
14BWV 859F#minSauveurSauveurFminD’Alambert-RousseauWanda LandowskaWanda Landowska
15BWV 860GmajSauveurSauveurD’Alambert-RousseauD’Alambert-RousseauWanda LandowskaWanda Landowska
16BWV 861GminSauveurSauveurRameau en sibRameau en sibWanda LandowskaWanda Landowska
17BWV 862A♭majAbmajZarlino nat­ur­alD’Alambert-RousseauD’Alambert-RousseauWanda LandowskaWanda Landowska
18BWV 863G#minGminAbminGminDminWanda LandowskaWanda Landowska
19BWV 864AmajSauveurSauveurD’Alambert-RousseauD’Alambert-RousseauWanda LandowskaWanda Landowska
20BWV 865AminSauveurSauveurD’Alambert-RousseauD’Alambert-RousseauWanda LandowskaWanda Landowska
21BWV 866B♭majSauveurSauveurRameau en sibMarpurgMartha GoldsteinWanda Landowska
22BWV 867B♭minSauveurMarpurgAminAminWanda LandowskaWanda Landowska
23BWV 868BmajBmajMarpurgD’Alambert-RousseauEbmajWanda LandowskaWanda Landowska
24BWV 869BminSauveurSauveurD’Alambert-RousseauD’Alambert-RousseauWanda LandowskaWanda Landowska
➡ Click a scale or a per­former’s name to lis­ten to the recording

Book II sound examples

The record­ings of Ottavio Dantone are from YouTube.

As explained above, D’Alambert-Rousseau tem­pera­ment was found to be equiv­a­lent to H.A. Kellner’s BACH.

(favourite: Sauveur)
(favourite: D’Alembert-Rousseau)
1BWV 870CmajSauveurSauveurMarpurgD’Alambert-RousseauOttavio DantoneOttavio Dantone
2BWV 871CminRameau en sibSauveurD’Alambert-RousseauD’Alambert-RousseauOttavio DantoneOttavio Dantone
3BWV 872C#majMarpurgDbmajD’Alambert-RousseauD’Alambert-RousseauOttavio DantoneOttavio Dantone
4BWV 873C#minSauveurWerckmeister 4D’Alambert-RousseauD’Alambert-RousseauOttavio DantoneOttavio Dantone
5BWV 874DmajSauveurSauveurD’Alambert-RousseauD’Alambert-RousseauOttavio DantoneOttavio Dantone
6BWV 875DminSauveurSauveurD’Alambert-RousseauD’Alambert-RousseauOttavio DantoneOttavio Dantone
7BWV 876E♭majRameau en sibRameau en sibRameau en sibRameau en sibOttavio DantoneOttavio Dantone
8BWV 877D#minMarpurgMarpurgD’Alambert-RousseauDminOttavio DantoneOttavio Dantone
9BWV 878EmajSauveurWerckmeister 4Werckmeister 4Werckmeister 4Ottavio DantoneOttavio Dantone
10BWV 879EminSauveurSauveurD’Alambert-RousseauD’Alambert-RousseauOttavio DantoneOttavio Dantone
11BWV 880FmajSauveurSauveurMarpurgD’Alambert-RousseauOttavio DantoneOttavio Dantone
12BWV 881FminZarlino nat­ur­alZarlino nat­ur­alD’Alambert-RousseauD’Alambert-RousseauOttavio DantoneOttavio Dantone
13BWV 882F#majF#majBmajD’Alambert-RousseauD’Alambert-RousseauOttavio DantoneOttavio Dantone
14BWV 883F#minSauveurSauveurD’Alambert-RousseauD’Alambert-RousseauOttavio DantoneOttavio Dantone
15BWV 884GmajSauveurSauveurD’Alambert-RousseauD’Alambert-RousseauOttavio DantoneOttavio Dantone
16BWV 885GminSauveurSauveurRameau en sibD’Alambert-RousseauOttavio DantoneOttavio Dantone
17BWV 886A♭majZarlino nat­ur­alAbmajD’Alambert-RousseauD’Alambert-RousseauOttavio DantoneOttavio Dantone
18BWV 887G#minAbminAbminD’Alambert-RousseauD’Alambert-RousseauOttavio DantoneOttavio Dantone
19BWV 888AmajSauveurSauveurRameau en doD’Alambert-RousseauOttavio DantoneOttavio Dantone
20BWV 889AminSauveurSauveurD’Alambert-RousseauD’Alambert-RousseauOttavio DantoneOttavio Dantone
21BWV 890B♭majSauveurSauveurMarpurgMarpurgOttavio DantoneOttavio Dantone
22BWV 891B♭minBbminMarpurgAminAminOttavio DantoneOttavio Dantone
23BWV 892BmajBmajBmajD’Alambert-RousseauD’Alambert-RousseauOttavio DantoneOttavio Dantone
24BWV 893BminSauveurSauveurD’Alambert-RousseauBbminOttavio DantoneOttavio Dantone
➡ Click a scale or a per­former’s name to lis­ten to the recording

More examples?

Interestingly, sim­i­lar clas­si­fi­ca­tions of tun­ing sys­tems apply to anoth­er famous cor­pus by J.S. Bach: the Goldberg Variations (1741). Read the Bach well-tempered tonal analy­sis page.

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.

In the same peri­od (1730), the French musi­cian François Couperin com­posed Les Ombres Errantes, for which our tonal analy­sis sug­gests a Rameau en sib temperament:

François Couperin’s “Les Ombres Errantes” inter­pret­ed by the Bol Processor + Csound
with a “Rameau en sib” tem­pera­ment ➡ Image
Source: MusicXML score by Vinckenbosch in the MuseScore com­mu­ni­ty

Conclusive remarks

The title of this cor­pus, The Well-Tempered Clavier, sug­gests that its com­pos­er intend­ed to demon­strate the suit­abil­i­ty of one or more tem­pera­ments for the per­for­mance of musi­cal works in any tonal­i­ty. As pre­vi­ous­ly stat­ed, this does not imply that they were all intend­ed to con­form to the same unique solu­tion; how­ev­er, it is tempt­ing to hypoth­e­size that the same instru­ment and tun­ing scheme were uti­lized for the entire­ty of the set. This has led to spec­u­la­tion by J.S. Bach’s pupils, who were not instruct­ed on how to pro­ceed. Part of the rep­u­ta­tion of great artists in those days was based on the with­hold­ing of information.

It would be illog­i­cal in the con­text of human musi­cians and phys­i­cal instru­ments to play a pre­lude in one tun­ing and then retune the instru­ment sole­ly for the pur­pose of play­ing the fugue. Consequently, these audio exam­ples are not intend­ed to emu­late a gen­uine per­for­mance. They can, how­ev­er, assist in eval­u­at­ing the suit­abil­i­ty of a pre­ferred tun­ing scheme for each musi­cal work.

A “deaf musi­col­o­gist” employs a method of tonal­i­ty assess­ment that involves mea­sur­ing melod­ic and har­mon­ic inter­vals in terms of fre­quen­cy ratios. The results of this assess­ment depend on the val­ues (weights) assigned to cer­tain ratios in advance. Our find­ings demon­strate that equal­ly mean­ing­ful sets of hypothe­ses can lead to com­plete­ly dif­fer­ent results, which only trained ears can dis­tin­guish. The com­bi­na­tion of hypothe­ses may not elu­ci­date the sit­u­a­tion. An appar­ent pref­er­ence for a tun­ing scheme may be the result of a numer­i­cal arte­fact rather than proof of its validity.

A metic­u­lous lis­ten­ing to these record­ings, with the exclu­sion of the some­what inel­e­gant ren­der­ing of fast trills in the low­er octave, reveals a musi­cal dimen­sion that can­not be reduced to the con­cept of ‘inter­vals’. Each piece can be likened to a pre­cious stone, exhibit­ing an aston­ish­ing reg­u­lar­i­ty in its struc­ture. The lis­ten­er is invit­ed by the artist to explore all sides of the crys­tal and to appre­ci­ate its puri­ty: a ‘tonal land­scape’. In this approach, the slight­est imper­fec­tion, such as a few cents up or down, is ampli­fied by the struc­ture. In short, the most impor­tant fea­ture may be less the choice of a struc­ture than its con­sis­ten­cy in ren­der­ing each musi­cal phrase.

It is uncer­tain whether J.S. Bach had a spe­cif­ic, unique musi­cal tem­pera­ment in mind when he com­posed The Well-Tempered Clavier. This is because the high­est rat­ing in terms of inter­vals may not be the most appro­pri­ate. This ques­tion remains open to art his­to­ri­ans and musi­col­o­gists. However, from the sound exam­ples, it is evi­dent that play­ing this reper­toire on improp­er­ly tuned instru­ments — in terms of con­so­nance — is tan­ta­mount to expos­ing plas­tic imi­ta­tions of diamonds.


Asselin, P.-Y. Musique et tem­péra­ment. Paris, 1985, repub­lished in 2000: Jobert. Soon avail­able in English.

Musicians inter­est­ed in con­tin­u­ing this research and relat­ed devel­op­ment can use the Bol Processor BP3 to process musi­cal works and imple­ment fur­ther tun­ing pro­ce­dures. Follow the instruc­tions on the Bol Processor ‘BP3’ and its PHP inter­face page to install BP3 and learn its basic oper­a­tion. Download and install Csound from its dis­tri­b­u­tion page.

Bernard Bel — January 2022

Please join the BP users help forum , BP open dis­cus­sion forum and/or the BP devel­op­ers list to stay in touch with the progress of work and dis­cuss relat­ed the­o­ret­i­cal issues.

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