The Well-tempered Clavier

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 ([1985], 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.

Settings of an audio unit for the post-processing

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!

Wanda Landowska’s (1879-1959) record­ings are bor­rowed from Youtube. Other inter­pre­ta­tions belong to Wikimedia Commons.

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.

(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

Ottavio Dantone’s record­ings are bor­rowed from Youtube.

As explained above, D’Alambert-Rousseau tem­pera­ment rat­ed 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: Goldberg Variations (1741). Read page Bach well-tempered tonal analy­sis.

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:

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 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

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

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