Bach well-tempered tonal analysis

Harpsichord jacks in a com­plet­ed harp­si­chord
Source: Material Matters

The fol­low­ing is a “com­pu­ta­tion­al” tonal analy­sis of musi­cal works by J.S. Bach known as The Well-tempered Clavier, books II and II, pub­lished around 1722 and 1742 respec­tive­ly, and the Goldberg Variations (1741).

All musi­cal scores have been con­vert­ed from MusicXML to Bol Processor syn­tax — see Importing MusicXML scores. This tonal analy­sis is gen­er­at­ed by Bol Processor’s tonal batch pro­cess­ing tool.

The aim of this exer­cise was to match each musi­cal work to a set of tun­ing schemes described and imple­ment­ed on the Bol Processor. These include all the tem­pera­ments doc­u­ment­ed by Pierre-Yves Asselin ([1985], 2000) and “nat­ur­al” scales sys­tem­at­i­cal­ly con­struct­ed — see Creation of just into­na­tion scales.

It has been sug­gest­ed that the best match for a scale is the tun­ing scheme that is appro­pri­ate for the inter­pre­ta­tion of a musi­cal work. This assump­tion is based on the hypothe­ses that (1) musi­cians and com­posers of the Baroque peri­od aimed to achieve opti­mal ‘con­so­nance’, and that (2) this notion implied a pref­er­ence for cer­tain inter­vals expressed as inte­ger ratios. These claims are dis­cussed on this page. Accurately tuned sound exam­ples are sug­gest­ed for audi­tive eval­u­a­tion of the results.

The inter­est of this tonal analy­sis goes beyond the under­stand­ing of music the­o­ry and prac­tice. Its epis­te­mo­log­i­cal dimen­sion is the trust­wor­thi­ness of today’s fash­ion­able math­e­mat­i­cal “pre­dic­tive mod­els”. We show that, giv­en a set of hypothe­ses, the solu­tion to an opti­mi­sa­tion prob­lem — find­ing the best tun­ing scheme for all musi­cal works in a reper­toire — is not unique, as it depends on ini­tial con­di­tions. Furthermore, the same ini­tial con­di­tions can pro­duce a cloud of seem­ing­ly iden­ti­cal solu­tions, even though each of them points to com­plete­ly dif­fer­ent pro­ce­dures for its real­i­sa­tion in the “real world” — here, the tun­ing of a harpsichord.

The take-home mes­sage is that sci­en­tists should not be impressed by the accu­ra­cy and appar­ent con­sis­ten­cy of machine-generated solu­tions. They must crit­i­cal­ly exam­ine the ini­tial con­di­tions and the cal­cu­la­tion process itself.

Ultimately, the only accept­able way to (in)validate a com­po­si­tion­al mod­el is to lis­ten to the audio ren­der­ing of the results.

“Standard” analysis

The Well-Tempered Clavier con­sists of two books, each con­tain­ing 24 pre­ludes and 24 fugues in all the usu­al key sig­na­tures. In total, this analy­sis cov­ered 96 musi­cal works (pre­sum­ably) writ­ten by the same com­pos­er under (pre­sum­ably) sim­i­lar conditions.

Our first analy­sis is based on the fol­low­ing set­tings of inter­vals esti­mat­ed to be con­so­nant or dis­so­nant:

Settings for a “stan­dard” analysis

The analy­sis of ascend­ing and descend­ing melod­ic inter­vals looks for com­mon fre­quen­cy ratios close to 3/2 (Pythagorean fifths) and 5/4 (har­mon­ic major thirds), which are wide­ly regard­ed as ‘con­so­nant’. It also includes the ratios 6/5 (har­mon­ic minor thirds) and 9/8 (Pythagorean major sec­onds), which can be con­sid­ered opti­mal. Other ratios are often con­sid­ered ‘dis­so­nant’: 40/27 (wolf fifth), 320/243 (wolf fourth) and 81/64 (Pythagorean major third). These dis­so­nant inter­vals are 1 com­ma (ratio 81/80) high­er or low­er than their “con­so­nant” neigh­bours — see page Just into­na­tion: a gen­er­al frame­work.

Consonant inter­vals are giv­en pos­i­tive weights, for exam­ple ‘1’ for a har­mon­ic major third and ‘2′ for a Pythagorean fifth. Dissonant inter­vals are giv­en neg­a­tive weights, for exam­ple ‘-2′ for wolfish inter­vals and ‘-1’ for Pythagorean major thirds. These weights can be mod­i­fied; indeed, the mod­i­fi­ca­tion will in turn change the rat­ings of the tun­ing schemes.

Each melod­ic inter­val found in the musi­cal work will be sized accord­ing to the same inter­val in the scale being test­ed for com­pat­i­bil­i­ty. For exam­ple, when try­ing to match the D’Alambert-Rousseau tun­ing scheme (see image), a note sequence ‘C’ - ‘Eb’ will be sized 290 cents, which is close to 294 cents or the ratio 32/27 (Pythagorean minor third). When this ratio appears in the set­tings, the scale val­ue is increased by the weight of the ratio mul­ti­plied by the (sym­bol­ic) dura­tion of the inter­val — see Tonal Analysis of Musical Works for details of this procedure.

The same method is applied to har­mon­ic inter­vals, which are giv­en the same weights as melod­ic inter­vals, except for the 9/8 ratio, which is ignored.

The scores for ascend­ing and descend­ing melod­ic inter­vals are then added to the score for har­mon­ic inter­vals, with weights of 1, 1 and 2 respec­tive­ly. This weight­ing may be mod­i­fied if “con­so­nance” is expect­ed to be greater for melod­ic than for har­mon­ic inter­vals, or if ascend­ing and descend­ing melod­ic inter­vals are not con­sid­ered equal­ly important.

Each scale is giv­en a mark if it is found to be the best match for a piece of music. Counting these marks over the entire reper­toire gives the best tun­ing scheme(s) for that repertoire.

Results are stored in tables that can be down­loaded in both HTML and CSV for­mats. The ini­tial set­tings are recalled at the bot­tom of the “All Results” HTML page.

Each cell in the “all results” table indi­cates the rank of a giv­en tun­ing scheme (scale) that match­es a giv­en musi­cal work. For exam­ple, in the fugues of book I, Corette’s tem­pera­ment (col­umn cor­rette) was ranked 6th for the 5th fugue, and the best match for this piece was the Sauveur’s tem­pera­ment (col­umn sauveur).

The line labelled Ranked first (times) shows the num­ber of times each tun­ing scheme ranked first in the clas­si­fi­ca­tion of this cor­pus. The line labelled Average score shows the aver­age glob­al (melod­ic + har­mon­ic) score com­put­ed for this tun­ing scheme, as explained on our Tonal analy­sis page.

Abstract tables show the list of first ranked tun­ing schemes for each musi­cal work.

The full set of scale images is avail­able on this page.

Discussion of the standard analysis

The scale of Sauveur’s temperament

Of of these 96 musi­cal works, 56 chose ‘sauveur’ as their favourite tun­ing scheme, plus 9 as their sec­ond favourite. This tem­pera­ment strik­ing­ly dom­i­nates the clas­si­fi­ca­tion because of its abil­i­ty to pro­duce almost per­fect Pythagorean fifths (ratio 3/2), har­mon­ic major thirds (ratio 5/4) and har­mon­ic minor thirds (ratio 6/5).

Note that it also con­tains a wolf fourth ‘Eb’ - ‘G#’ close to 477 cents (or ratio 320/243) which is per­ceived as a dis­so­nant inter­val. It is assumed that these two notes are nev­er (or rarely) found in melod­ic or har­mon­ic inter­vals in this reper­toire. This illus­trates the fact that there is no one-size-fits-all solu­tion to the prob­lem of tun­ing an instru­ment for this type of music. In a kind of “reverse engi­neer­ing”, we can say that the com­pos­er explored melod­ic and har­mon­ic pleas­ing effects in order to build this reper­toire: play­ing on the instru­ment before notat­ing it on sheets of music sheets.

As sug­gest­ed in our tuto­r­i­al, there is no evi­dence that J.S. Bach was aware of the the­o­ret­i­cal work of the French physi­cian Joseph Sauveur, but the the­o­ret­i­cal frame­work of this tem­pera­ment — a sin­gle sequence of fifths dimin­ished by 1/5 com­ma (see image and read Asselin, 2000 p. 80) — sug­gests that any com­pos­er could work it out inde­pen­dent­ly. This process has been record­ed on the Bol Processor’s Scale page as follows:

Created meantone downward notes “do,fa,sib,mib” fraction 3/2 adjusted -1/5 comma
Created meantone upward notes “do,sol,re,la,mi,si,fa#,do#,sol#” fraction 3/2 adjusted -1/5 comma

Interestingly, “nat­ur­al scales” with names cor­re­spond­ing to the key — for exam­ple, Abmin (i.e. G#min) for Fugue 18 in G♯ minor, Book II (BWV 887) — were often at the top of the pop­u­lar tun­ing schemes, but in most cas­es were over­tak­en by sev­er­al tem­pera­ments. For more details on these scales, see our page on the Creation of just-intonation scales.

In all cas­es, the equal tem­pera­ment (see image) was among the low­est, due to its use of major and minor thirds close to Pythagorean. This con­tra­dicts the pop­u­lar belief that Bach’s series of Preludes and Fugues was intend­ed to equate ‘well-tempered’ with ‘equal-tempered’

This first result also sug­gests that tem­pera­ments often pro­vide a bet­ter tonal struc­ture for achiev­ing max­i­mum con­so­nance than the so-called just into­na­tion scales.

Temperaments are based on empir­i­cal tun­ing pro­ce­dures guid­ed by per­ceived inter­vals (see Asselin, 2000) where­as “just into­na­tion” is the result of spec­u­la­tion about numer­i­cal ratios — a deduc­tive process. This brings us back to a dis­cus­sion of the ancient Indian approach to tonal­i­ty, see the page on The two-vina exper­i­ment.

“Alternate” analysis

At this stage, it is tempt­ing to con­clude that J.S. Bach’s The Well-Tempered Clavier was intend­ed to be played on instru­ments tuned to Sauveur’s tem­pera­ment. However, the result of any analy­sis must always be exam­ined for bias in its hypothe­ses. In the present case, we must revise the choice of cer­tain fre­quen­cy ratios as cri­te­ria for assess­ing the ‘con­so­nance’ of melod­ic and har­mon­ic intervals.

The minor third — either har­mon­ic (6/5) or Pythagorean (32/27) — is in ques­tion because the Pythagorean minor third appears in some tem­pera­ments. For exam­ple, the Cmaj nat­ur­al scale (see image) uses 32/27 for its inter­val ‘C’ - ‘Eb’. Therefore, it makes sense to ignore all minor thirds when eval­u­at­ing har­mon­ic inter­vals and to accept both ratios 6/5 and 32/27 as equal pos­i­tive weights in melod­ic inter­vals. This option is illus­trat­ed by sound exam­ples, read on. A use­ful vari­ant would be dif­fer­ent ratios in ascend­ing and descend­ing har­mon­ic intervals.

The same obser­va­tion applies to major thirds: although 5/4 (har­mon­ic) cer­tain­ly sounds bet­ter than 81/64 (Pythagorean) in har­mon­ic inter­vals, there is no strong rea­son to pre­fer the for­mer in melod­ic inter­vals — again with a pos­si­ble dis­tinc­tion between ascend­ing and descend­ing movements.

Let us start the whole analy­sis again with these changed settings:

Settings for an “alter­nate” analysis

Results are the following:

Discussion of the alternate analysis

The results con­tra­dict the con­clu­sion of the ‘stan­dard’ analy­sis: Sauveur’s tem­pera­ment may not be such a good choice, giv­en the alter­na­tive choice of ratios for consonant/dissonant melod­ic and har­mon­ic intervals.

According to these set­tings, the best tun­ing schemes might be the D’Alambert-Rousseau tem­pera­ment (see pic­ture and read Asselin, 2000 p. 119) and H.A. Kellner’s BACH tem­pera­ment (see pic­ture and read Asselin, 2000 p. 101). Both were designed after J.S. Bach’s death, but sim­i­lar or iden­ti­cal tun­ing pro­ce­dures could have been devised by the composer.

A com­par­i­son of the images and cent posi­tions (iden­ti­cal with­in ± 7 cents) explains why these two tem­pera­ments pro­duced iden­ti­cal match­es, despite their com­plete­ly dif­fer­ent tun­ing pro­ce­dures. Look at the pro­ce­dures (traced by the algo­rithm) and lis­ten to short note sequences pro­duced with these scales:

D'Alembert-Rousseau temperament
Created meantone upward notes “do,sol,re,la,mi” fraction 3/2 adjusted -1/4 comma
Created meantone downward notes “do,fa,sib,mib,sol#” fraction 3/2 adjusted 1/12 comma
Equalized intervals over series “sol#,do#,fa#,si,mi” approx fraction 2/3 adjusted 2.2 cents to ratio = 0.668

Sequence of notes accord­ing to D’Alembert-Rousseau temperament

Kellner's BACH temperament
Created meantone upward notes “do,sol,re,la,mi” fraction 3/2 adjusted -1/5 comma
Added fifths down: “do,fa,sib,mib,lab,reb,solb” starting fraction 1/1
Created meantone upward notes “mi,si” fraction 3/2

Sequence of notes accord­ing to Kellner’s BACH temperament

As a reminder, the same sequence of notes with an equal-tempered scale:

Sequence of notes accord­ing to equal temperament
D’Alembert-Rousseau tun­ing scheme (Asselin, 2000 p. 119)

These tun­ing pro­ce­dures are not exact­ly the same as those described by Asselin (2000, p. 120 and 102), but they pro­duce the same tonal positions.

In these tem­pera­ments, inter­vals such as ‘C’ - ‘Eb’ are ren­dered as Pythagorean minor thirds (32/27), and many Pythagorean major thirds (ratio 81/64) are encoun­tered. This jus­ti­fies their choice, giv­en the new con­di­tions of analysis.

Again, these tem­pera­ments dom­i­nate the clas­si­fi­ca­tion, tak­ing first place 65 times and sec­ond place 15 times, while the equal tem­pera­ment, despite its mas­tery of Pythagorean major thirds, took first place only 21 times. Compared to Sauveur’s tem­pera­ment in the stan­dard analy­sis (56 first posi­tions and 9 sec­ond posi­tions), these tem­pera­ments look ‘bet­ter’, but this com­par­i­son is irrel­e­vant as the two analy­ses focused on dif­fer­ent ratios.

The 33 Preludes and Fugues that do not con­form to these tem­pera­ments often pre­fer a just into­na­tion scale in the same key; for exam­ple, Prelude 8 in E♭ minor of Book I (BWV 853) choos­es the Ebmin scale, and Prelude 9 in E major of Book I (BWV 854) choos­es the Emaj scale. However, this match­ing is less com­mon in the “dis­si­dent” fugues.

More advanced analy­sis is required. Note that chang­ing the weight­ing of inter­vals or the weight­ing in the sum­ma­tion of melod­ic and har­mon­ic scores can rad­i­cal­ly change the classification.

In this dis­cus­sion, we have only exam­ined tun­ing schemes at the top of the clas­si­fi­ca­tion. Other schemes may be prefer­able when look­ing at melod­ic or har­mon­ic scores sep­a­rate­ly — see our tuto­r­i­al Tonal analy­sis.

Sound examples

The auto­mat­ic tonal analy­sis of a large reper­toire, com­pared with the whole set of tun­ing schemes imple­ment­ed in the Bol proces­sor, did not solve the prob­lem of find­ing “the best tun­ing scheme” for this reper­toire, since it depends on the ini­tial con­di­tions: fre­quen­cy ratios esti­mat­ed as “con­so­nant” or “dis­so­nant”, plus the com­poser’s pre­sumed focus on opti­mal con­so­nance. Nevertheless, two analy­ses select­ed 2 (or 3) tun­ing schemes as dom­i­nant in the clas­si­fi­ca­tion. Further analy­sis would be required to refine this result, if it is significant.

All sound exam­ples are com­pared with human inter­pre­ta­tions on (not so well-tempered?) phys­i­cal instru­ments on the page The Well-tempered clavier.

These sound exam­ples are use­ful to hear the dif­fer­ence between tun­ing schemes select­ed on the basis of the “stan­dard” and “alter­na­tive” set­tings. For exam­ple, Fugue 8 of book I may sound more melo­di­ous with a Dmin tun­ing (see illus­tra­tion) than with a Marpurg tun­ing (see illus­tra­tion). The dif­fer­ence may lie in the choice of the most con­ve­nient ratios for minor thirds.

Is this method reliable?

As the results shown in the 4 tables for each book (see above) sug­gest, some pre­ludes and fugues ranked sev­er­al tun­ing schemes as their favourite: num­ber ‘1′ is coloured red in the ‘all results’ tables. However, we only record­ed one of the win­ners. What does this mean?

Take for exam­ple Prelude 12 of book I. In the “alter­nate” set­tings, five scales are ranked first: Emin, Cmaj, BACH, d_alembert_rousseau, bethisy. We have already shown that BACH and d_alembert_rousseau are almost iden­ti­cal despite the dif­fer­ences in their tun­ing pro­ce­dures. Emin and Cmaj are exact­ly the same. This leaves us with the fol­low­ing choice:

Three scales rank­ing 1st for Prelude 12 of book 1 as per “alter­nate” settings

Tonal posi­tions dif­fer by a only few cents, which may not be notice­able in melod­ic and har­mon­ic inter­vals. Below are record­ings using these three scales:

Prelude 12 of book I, Emin tun­ing scheme
Prelude 12 of book I, Bethisy temperament
Prelude 12 of book I, Kellner’s BACH temperament

This exam­ple sug­gests that if the widths of accept­able melod­ic and har­mon­ic inter­vals have been set small enough to pro­vide a well-focused solu­tion set, dif­fer­ences in the first-ranked scales may be inaudible.

Listen to minor thirds

Judging the sizes of the com­mon minor thirds by ear may make it eas­i­er to decide which is more “con­so­nant”. Lucky users of the Bol Processor BP3 only need to cre­ate the fol­low­ing data file:


// Harmonic minor third
_scale(2_cycles_of_fifths,0) DO3 RE#3 DO3 RE#3 DO3 RE#3 {4,DO3,RE#3}

//Pythagorean minor third
_scale(2_cycles_of_fifths,0) DO3 MIb=RE#-c3 DO3 MIb=RE#-c3 DO3 MIb=RE#-c3 {4,DO3,MIb=RE#-c3}

// Sequence harmonic then pythagorean
_scale(2_cycles_of_fifths,0) DO3 RE#3 DO3 MIb=RE#-c3 DO3 RE#3 DO3 MIb=RE#-c3 -{2,DO3,RE#3} {2,DO3,MIb=RE#-c3} {2,DO3,RE#3} {2,DO3,MIb=RE#-c3} {2,DO3,RE#3} {2,DO3,MIb=RE#-c3}

These items pro­duce sequences of ‘C’ - ‘D#’ melod­ic and har­mon­ic inter­vals using har­mon­ic (6/5) and Pythagorean (32/27) minor thirds:

Harmonic minor thirds in sequence then superposed
Pythagorean minor thirds in sequence then superposed
Alternance of har­mon­ic then Pythagorean minor thirds

Listening to these exam­ples sug­gests that both 6/5 and 32/27 are suit­able ratios for minor thirds as “con­so­nant” melod­ic inter­vals, while 6/5 sounds “soft­er” than 32/27 as a har­mon­ic interval.

The “2_cycles_of_fifths” scale

This demo uses the scale “2_cycles_of_fifths” described by Asselin (2000, p. 62) and imple­ment­ed on a Scale page of the Bol Processor — see pages Microtonality and Just into­na­tion: a gen­er­al frame­work.

The names of the notes (inspired by the book, ibid.) sound bizarre but they make the posi­tions explic­it. For exam­ple, “Mib=RE#-c” indi­cates a posi­tion that is usu­al­ly called mi bémol (E flat), which is iden­ti­cal to ré dièse (D sharp) minus one comma.

This scale — and the even more com­pli­cat­ed “3_cycles_of_fifths” — is not prac­ti­cal for writ­ing music… It is used to visu­alise (and hear) tonal posi­tions pro­duced by dif­fer­ent tun­ing schemes that con­form to the just into­na­tion paradigm.

Listen to tempered fifths

Readers unfa­mil­iar with tun­ing pro­ce­dures may need to appre­ci­ate the tiny dif­fer­ences in inter­vals pro­duced by tem­pera­ments cre­at­ed using the meth­ods intro­duced on the Microtonality page and described in detail in Asselin (2000).

Let us lis­ten to Pythagorean fifths in three forms: pure (fre­quen­cy ratio 3/2 = 702 cents), equal-tempered (700 cents), dimin­ished by 1/5 com­ma (697.3 cents) and dimin­ished by 1/4 com­ma (696.2 cents).

Pure fifth (702 cents)
Equal-tempered fifth (700 cents)
Fifth dimin­ished by 1/5 com­ma (697.3 cents)
Fifth dimin­ished by 1/4 com­ma (696.2 cents)
Sequence of fifths: pure, then equal-tempered, then dimin­ished by 1/5 com­ma, then dimin­ished by 1/4 comma

Below is the Csound score of the last example:

i1 0.000 4.000 261.630 90.000 90.000 0.000 0.000 0.000 0.000 ; do4
i1 0.000 4.000 392.445 90.000 90.000 0.000 0.000 0.000 0.000 ; sol4
i1 4.000 4.000 261.626 90.000 90.000 0.000 0.000 0.000 0.000 ; C4
i1 4.000 4.000 391.996 90.000 90.000 0.000 0.000 0.000 0.000 ; G4
i1 8.000 4.000 261.630 90.000 90.000 0.000 0.000 0.000 0.000 ; do4
i1 8.000 4.000 391.399 90.000 90.000 0.000 0.000 0.000 0.000 ; sol-1|5c4
i1 12.000 4.000 261.630 90.000 90.000 0.000 0.000 0.000 0.000 ; do4
i1 12.000 4.000 391.137 90.000 90.000 0.000 0.000 0.000 0.000 ; sol-1|4c4

Goldberg Variations

The same exer­cise was attempt­ed with J.S. Bach’s Goldberg Variations (1741). The aria and its thir­ty vari­a­tions were per­formed in a sin­gle sequence, appar­ent­ly with the same instrument/tuning. For this rea­son, we checked a unique MusicXML score con­tain­ing all the variations.

With the “stan­dard” hypoth­e­sis of con­so­nance, the result is as follows:

(As expect­ed) Sauveur’s mean­tone tem­pera­ment won the game, fol­lowed by Kellner’s BACH. The equal-tempered scale came 28th in this clas­si­fi­ca­tion… (Note that the cal­cu­la­tion of this table took 3 1/2 hours on an old MacBook Pro…).

Listen to the syn­the­sis of the Goldberg Variations with Sauveur’s mean­tone temperament:

The “alter­na­tive” mod­el of con­so­nance gives the fol­low­ing classification:

Favourite tun­ing schemes, accord­ing to this mod­el, would be D’Alembert-Rousseau (see pic­ture) and Kellner’s BACH (see pic­ture) mean­tone tem­pera­ments, both of which have equal val­ue because their tonal inter­vals are almost identical.

Listen to the syn­the­sis of the Goldberg Variations with the D’Alembert-Rousseau temperament:

Source: Wikipedia

These pre­ferred tun­ings are the same as those best suit­ed to the set of pre­ludes and fugues in The Well-Tempered Clavier.

These sound exam­ples can be com­pared to human per­for­mance, for exam­ple the Aria on a harp­si­chord tuned to the Werckmeister III mean­tone tem­pera­ment — lis­ten to this record­ing. In fact, the musi­cians show a more flex­i­ble tim­ing than the Bol Processor, which sticks to the para­me­ters of the MusicXML score. Nevertheless, a com­par­i­son focus­ing on tonal inter­vals remains possible.

The fact that Werckmeister III (see image) ranked low in the auto­mat­ic tonal analy­sis does not indi­cate a wrong choice. This tun­ing scheme may per­form bet­ter against a par­tic­u­lar mod­el of “con­so­nance”.

Let us use cal­cu­la­tions to work out its main dif­fer­ence from D’Alembert-Rousseau and Kellner’s BACH. We can lim­it the analy­sis to bars #1 to #32 (the Aria), which expos­es most of the melodic/harmonic inter­vals; this Aria func­tions sim­i­lar­ly to the open­ing sec­tion (ālāp) in North Indian clas­si­cal music… We notice that nei­ther D - F# (397 cents) nor G - B (398 cents) in Werckmeister III are exact har­mon­ic major thirds (390 cents), inter­vals with a high fre­quen­cy as shown in the table of inter­val frequencies:

Interval fre­quen­cies in the Aria of Goldberg Variations

Below is a com­par­i­son of the Werckmeister III and D’Alembert-Rousseau scales in terms of match­ing melod­ic inter­vals (in the “alter­nate” mod­el of con­so­nance) over the first 32 bars of the Goldberg Variations:

Matching two scales with the melod­ic inter­vals of the Aria in Goldberg Variations:
Werckmeister III (left) and D’Alembert-Rousseau (right)

The width of the yel­low lines is pro­por­tion­al to the occurrence/duration of melod­ic inter­vals in this part of the cor­pus. The pic­ture con­firms the absence of an exact har­mon­ic major third D - F# in the Werckmeister III scale, and the same mis­match of the major third G - B. Another mis­match is on the minor third E - G, here aim­ing at a ratio of 6/5 (315 cents) or 32/27 (294 cents).

Sources of MusicXML scores

Links point to MusicXML scores used in this analy­sis. These links must be cit­ed in the attri­bu­tion part of Creative Commons licences. Updated ver­sions are welcome.

Our thanks to the edi­tors of these scores in the MuseScore community!

The (pub­lic domain) score of the Goldberg Variations has been edit­ed by MuseScore lead devel­op­er Werner Schweer.

Book I sources

1BWV 846CmajPreludeFugue
2BWV 847CminPreludeFugue
3BWV 848C#majPreludeFugue
4BWV 849C#minPreludeFugue
5BWV 850DmajPreludeFugue
6BWV 851DminPreludeFugue
7BWV 852E♭majPreludeFugue
8BWV 853E♭min/D#minPreludeFugue
9BWV 854EmajPreludeFugue
10BWV 855EminPreludeFugue
11BWV 856FmajPreludeFugue
12BWV 857FminPreludeFugue
13BWV 858F#majPreludeFugue
14BWV 859F#minPreludeFugue
15BWV 860GmajPreludeFugue
16BWV 861GminPreludeFugue
17BWV 862A♭majPreludeFugue
18BWV 863G#minPreludeFugue
19BWV 864AmajPreludeFugue
20BWV 865AminPreludeFugue
21BWV 866B♭majPreludeFugue
22BWV 867B♭minPreludeFugue
23BWV 868BmajPreludeFugue
24BWV 869BminPreludeFugue

Book II sources

1BWV 870CmajPreludeFugue
2BWV 871CminPreludeFugue
3BWV 872C#majPreludeFugue
4BWV 873C#minPreludeFugue
5BWV 874DmajPreludeFugue
6BWV 875DminPreludeFugue
7BWV 876E♭majPreludeFugue
8BWV 877D#minPreludeFugue
9BWV 878EmajPreludeFugue
10BWV 879EminPreludeFugue
11BWV 880FmajPreludeFugue
12BWV 881FminPreludeFugue
13BWV 882F#majPreludeFugue
14BWV 883F#minPreludeFugue
15BWV 884GmajPreludeFugue
16BWV 885GminPreludeFugue
17BWV 886A♭majPreludeFugue
18BWV 887G#minPreludeFugue
19BWV 888AmajPreludeFugue
20BWV 889AminPreludeFugue
21BWV 890B♭majPreludeFugue
22BWV 891B♭minPreludeFugue
23BWV 892BmajPreludeFugue
24BWV 893BminPreludeFugue


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

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