Bach's schematic, as it appears on the page, © Bradley Lehman, 2005-14, all rights reserved.
All musical/historical analysis here on the web site is the personal opinion of the author,
as a researcher of historical temperaments and a performer of Bach's music.

Why John Barnes was wrong to invent a 'Bach' temperament using fake statistical methods

John Barnes wrote a 1979 article where he analyzed part of Bach's Well-Tempered Clavier by counting up the use of major 3rds. He then used these data (putatively) to concoct several hypothetical keyboard temperaments. Barnes's premise was that Bach would have made most use of the "best" sounding 3rds, and that we can guess at his tuning practices by running that process backwards from statistical tabulation of the resulting music.

Let me ask: who composes systematically in that manner, constrained to use the "best" intervals more times than other intervals? Bach? Wouldn't that severely limit his musical creativity against every other musical element that could be important? It's absurd.... Going beyond that: when an absurd procedure from an absurd premise is performed again and again by other people, it does not suddenly become credible.

Here is my published rebuttal from 2009, with simple re-formatting. I reproduce here two of my sections that have broader importance in the theory of assessing temperaments, beyond merely saying why Barnes was wrong.

This assessment of Barnes's method was part of a book review for The Viola da Gamba Society Journal: vol 3 part 2 (2009), 137-163. [PDF of that issue] The reviewed book was: Claudio Di Veroli, Unequal Temperaments: Theory, History and Practice (Scales, Tuning and Intonation in Musical Performance). Second revised edition, eBook, Bray Baroque, Bray, Ireland 2009.

These theoretical points were a substantial part of my review because Di Veroli had used the flawed foundation of Barnes to build his own methods of analyzing and deriving temperaments.

Why does this matter?

My broader point is: such an analysis of major 3rds is both too misleading and too superficial to be of much musical or theoretical value. The task must go far beyond looking at key signatures, counting the usage of individual intervals, or judging the quality of simple triads as static "chords".

To assess historical or conjectural (pseudo-historical) keyboard temperaments for use in real music, there are at least four other directions of investigation that are all more important than statistical crunching of selected data:

  1. Study all the available historical clues from pedagogy, instrument construction, performance practices, and music theory (all contemporary with the compositions) to know what intonation practices were normal in given places and times. What did the composer expect people to know and do when learning and performing the music? Did theorists write down what people were actually doing? Were theoretical treatises used to prescribe practices? How can we reconstruct a reasonable description of practices?

  2. Study the individual correctly-spelled notes required within the compositions, and the way those notes are used within scales, to determine if a tuning scheme delivers those correctly-spelled notes within reasonable tolerances. (N.B.: Enharmonic pairs such as Eb and D# are absolutely different notes, because they belong to completely different scales.)

    These notes in scores are hard evidence, going far beyond looking at key signatures. Axiomatically, all of these notes in the compositions had to sound reasonably acceptable to the people using and hearing them, else those notes wouldn't be there (or the composition would be considered unusable).

  3. Study historical records of aesthetics contemporary with the music and locations, to know what errors of pitch would have been considered acceptable in practice. (How much could those people tolerate playing or singing out of tune, along with a given keyboard scheme where misspelled notes are being played? What did listeners, collaborative musicians, or students consider to sound correct and normal?)

  4. Directly test the music on instruments that the composer knew and played: real harpsichords, clavichords, and organs, not merely computer simulations with electronically-generated tone.

    Preferably, tune those instruments by ear using only the physical devices and theoretical constructions available at the time and place the music was composed. This procedure tests the plausibility of the composer (and colleagues) having done it without any computers or other electronic devices.

The pertinent excerpts from that book review:

A closer look at Barnes's analytical method,
and Di Veroli's own 'WTC Optimal+'
and 'Couperin' temperaments
derived by Barnes's procedures

[Omitting the Couperin part here]


Statistics are meaningful and reliable only if they are based on a truly representative data set, and if they are designed to measure all the features that are important. Barnes's statistical method [82] fails both these criteria.

Any of the following defects individually make the Barnes experiment seriously flawed, but taken all together, the Barnes measurement emerges as being nearly (perhaps entirely) meaningless.

  • Major 3rds are given too much importance, above musical features such as smoothness of diatonic scale steps, audible beating in 5ths and 4ths, and any aesthetic judgement in favour of variety. Those criteria are all treated as negligible, being outside the experiment.
  • Major 10ths and 17ths are given too little importance, and in this experiment none, while in musical contexts they show tuning problems more clearly than major 3rds do. They are simpler consonant intervals (5:2 and 5:1, as compared with 5:4), and their overtones beat more prominently, at least on harpsichords. In a major 17th, the upper note's fundamental coincides with the fifth overtone of the lower note, producing beats directly at that frequency if there is any tempering.
  • Tuning problems show up more clearly in thin contrapuntal textures than in thicker chordal situations. However, Barnes's method discounts music of thin texture, sometimes to the point of having that music disappear from the data set altogether. It heavily favours pieces where Bach happened to write four-part music, or punctuating chords.
  • Analysis of major 3rds is a superficial way to assess temperaments for real music. It is more important to look at the sizes of steps within diatonic scales (consistency, or nearly so), so no individual notes stick out as being too high or low for all the contexts in which they are used. It is also important that no notes be as much as a comma away from the spot where they would be in a regular system (generated by the spacing of the naturals).
  • Barnes's data were not necessarily accurate. In a musical example printed in his article, showing part of the C major prelude of WTC book 2, one of the major 3rds is not circled. [83] How many other errors of tabulation or analysis are there?
  • The Barnes temperament was then derived by trial and error, rather than systematically from the data set (even though the data set itself was flawed); this is not properly scientific procedure, for an experiment that proposes to be objective. [84]
  • The weighting of relative 'prominence' values is subjective, and not defined clearly by Barnes. (Di Veroli is more systematic and forthright about this, with regard to deriving his 'Couperin' temperament, but the method remains subjective.)

The Barnes method omits from the data set:

  • All the fugues (which bring up some of the worst offenders in ill-chosen temperaments: B major, D# minor, Bb minor, G# minor, Ab major, C# major, F# major, and F minor of both books; B minor of book 1; C minor, G minor, E minor, A major, and G major of book 2). The fugues comprise more than half the music of the WTC!
  • All the minor-key music. It is more difficult to tune well for music in minor than for music in major keys, because it generally requires a more uncommon set of accidentals (venturing higher into the sharps). None of this music comes to Barnes's tables.
  • All 10ths and 17ths, i.e. all music with a rather thin texture, having the hands spaced farther apart than an octave. In book 1, the F# major prelude contributes zero major 3rds (is wholly invisible in this experiment's analysis), and the C# major only two isolated major 3rds at its end! The D, F, G, and Ab major preludes do not get much say, either. In the C, Eb, E, G, Ab, and A major preludes of this book, the long final chords don't count, just because Bach did not insert one more note to double the bass with the right-hand thumb.

Coupled with all of that is the flawed assumption that Bach should have used his major 3rds systematically, favouring the best ones the most often, without much (any?) influence from other rules or ideas of composition. I am surprised that Di Veroli still champions Barnes's method, in light of its devastating defects. The criticism against it is not new, either; Rudolf Rasch already presented about half of these problems in his 1985 article [85] -- not in UT's bibliography.

Di Veroli does address the general problem of 10ths and 17ths, briefly, but not in the context of deducing any Bach temperament. It is only in his presentation of his 'Couperin' temperament, [86] where he made a decision that gets it backwards: 10ths and 17ths in the music are signals to reduce the 'prominence' factor, to him, rather than increasing it (as listening experiments and my theoretical point about 5:2 and 5:1 both make clear).

Di Veroli's 'Optimal' and 'Optimal+' temperaments are then chosen on their fit to Barnes's first graph in Figure 1 of his article. C, F, Bb, Eb, Ab, and A (taken as major 3rds: C-E, F-A, etc.) are pressed to be the smallest intervals in the temperament. This comes up most obviously in UT's section called 'Smoothing out the outsider', [87] where Di Veroli tries to explain away that anomalous spike in Barnes's data on A-C#. That Barnes graph also informs the remarks in several places about the way Bach's music favours flats ahead of sharps.

So: that profoundly biased Barnes experiment in statistics leads to all this theoretical conjecture about both Couperin and Bach. The time could be better spent in playing the music on suitably clear-toned harpsichords, clavichords, and organs, where the intonation flaws become more immediately obvious than they are in spreadsheets.


The notion of Bach 'avoiding' the bad major 3rds of F#-A# and Db-F is asserted, [89] but never demonstrated, and a play-through of the WTC in 'Optimal+' shows it to be false. Apart from the WTC, which obviously uses all tonalities and 3rds, the F#-A# and the Db-F (or C#-E#) come up in hundreds of other pieces, while playing in keys as simple as E minor, B minor, D major, A major, F# minor, G minor, or C minor; not to mention their importance in music with signatures of four or more sharps or flats.

An especially problematic piece in this regard is the opening of WTC 1's C# major prelude, with its open 17ths on downbeats. (If Bach wanted to show off the importance of appropriately moderated major 3rds/10ths/17ths, how much more blatant could he be, beyond writing music in this two-voiced texture with such a spacing, and putting those intervals on strong beats? There is no place for problems to hide. He did it again in the four Duetti of Clavierübung III, where the two-voiced texture and the modulatory adventurousness make perfect test pieces for a temperament.) Another obvious example is the last bar of WTC 2's F# major fugue: F#, a#', and f#'' played together, with that unconcealed open 17th between the bass and the alto.

UT gives an exceedingly complicated method to set 'WTC Optimal+' by ear, requiring the user to count more than ten different beat rates. I have worked out a much simpler method, giving the same temperament through easier steps oriented toward musical listening (instead of exhaustive beat counting). It is available on my web site. [90] I invite any reader of UT to try out those grimace-inducing spots I have mentioned here, not only in 'WTC Optimal+' but in a variety of other temperaments as well, to understand the magnitude of the musical problem which any 'Bach' temperament must solve. [91]


Getting beyond UT's insufficient analytical methods

Let us go through several case studies that show why the analytical methods in UT are insufficient, where it merely measures 5ths and major and minor 3rds. I hope that this may inspire a new path forward, as musicians and scholars grapple with the practical and historical decisions regarding temperaments.

Case 1

The first case study is the accompaniment of a violinist in two sonatas by Jean-Marie Leclair, in 'easy' keys of zero or one sharp. The violinist chooses the sonatas from book 1 (1725), #1, A minor, and book 2 (n.d.), #1, E minor.

Taking UT as it stands, we have no option but to choose the 'Standard French' (1/4 comma) or 'Homogeneous French' (1/5), because 1/6 is allegedly at least 25 years too early for mention in France, and because Di Veroli has told us what was 'Standard': a weighty word implying contemporary consensus. All of the major 3rds based on B, F#, C#, Ab, and Eb are much worse than in equal temperament, as the notes D#, A#, and E# are not designed to be any good in these temperaments. Those pitches serve their functions very roughly.

Now, let's take a closer look at the primary evidence of the music. That A minor sonata requires all the notes from Bb to E# (14 different notes), inclusive, and the E minor sonata requires Bb to A# (13). Despite the simple key signatures of zero or one sharp, the compositions support the argument that we need a much better circulating temperament than UT has been able to provide for that period in France.

The broader point here is: when selecting a keyboard temperament to participate in ensemble music, it does not suffice simply to observe the date and place of origin, and then to apply some generalist solution based only on that, assuming all will be well. Wherever a piece of music goes beyond 12 differently-named notes, or exceeds the usual set of 12 notes (Eb to G#), one needs some way to handle them: whether that is making adjustments to get the 'right' accidentals, or having some split keys, or using a circulating temperament that handles all the enharmonic exchanges smoothly enough (i.e., a temperament that is close enough to equal, abandoning 1/4 comma or 1/5 comma schemes).

I take it as axiomatic that no single note should be more than a comma 'off-spot' for its spelling, vis-a-vis all the other notes around it, and especially the naturals--or else it will never sound smooth in its musical contexts, but merely like a 'wrong' note. So, for example, if a composition requires both A# and Bb, all of the 3rds/4ths/5ths/6ths on either side of those two notes ought to sound 'good' within one comma of the point where they would make a pure interval. The pitch must be moderate enough to work with C#/Db, D, Eb/D#, F/E#, F#/Gb, and G both above and below it. It should also sound good with the minor 7ths: C/B# below and Ab/G# above. Every note has important intervallic relationships with almost every other note.

If that harmonic consideration were not enough, we must also consider how a pitch fits into scales melodically, proceeding by either semitones or tones on each side. If we place our Bb to fit well into a diatonic scale fragment such as G-A-Bb-C-D, does it also function well as A# within a competing fragment such as F#-G#-A#- B, as we have some cadence into B minor? Are F#-G# and G#-A# nearly the same size?

Case 2

The second case study is a closer look at some earlier repertoire, along a similar line of investigation: Arcangelo Corelli's book of violin sonatas, Op. 5, published 1700, with a title page that calls for accompaniment by harpsichord or cello. Playing it with harpsichord, what notes do we require of the temperament?
  • 1: D major, F to E# (13)
  • 2: Bb major, Ab to G# (13)
  • 3: C major, Db to D# (15)
  • 4: F major, Eb to G# (12)
  • 5: G minor, Ab to C# (12)
  • 6: A major, C to B# (13) 161
  • 7: D minor, Eb to G# (12)
  • 8: E minor, F to A# (12)
  • 9: A major, D to B# (11)
  • 10: F major, Eb to G# (12)
  • 11: E major, A to Fx (11)
  • 12: D minor, Bb to C# (10 - 'La Follia' variations, no modulation)
As noted here in this set of 12 Corelli sonatas, regarding intonation, different pieces in a collection might have radically different needs.
  • 'Should' the D minor and F major pieces be done in a typical regular system because they can be? If we do that, what should guide our choice of tempering in the 5ths, anywhere in the continuum from 1/4 to 1/6 comma?
  • 'Should' we retune the notes Eb, Bb, F, C, and G to D#, A#, E#, B#, and Fx to be able to play #9 in A or #11 in the process smoothing out whatever exotic character the composer may have intended in choosing the scale he did?
  • 'Should' we use some mostly-regular system but with a moderated Ab/G# serving roughly, to play #2 in Bb? If we do that, how should we handle #6 in A where the compromised note is C/B#, or #1 in D where it is F/E#?
  • Does the presence of #3 in C major, needing four flats and four sharps (and all in simple-triad situations, presumably to sound consonant), argue that the other pieces in the book 'should' be accompanied by a nearly-equal keyboard, even though they would not require it technically on their own?

Case 3

The third case study brings in the enharmonic requirements of some solo keyboard music. Readers may draw their own conclusions from these data, thinking through the questions I raised above.

Henry Purcell (1659-95), eight suites for harpsichord, published 1696 by his widow:

  • G major, Z. 660: C to A# (11)
  • G minor, Z. 661: Ab to C# (12)
  • G major, Z. 662: F to D# (11)
  • A minor, Z. 663: F to D# (11)
  • C major, Z. 666: Bb to G# (11)
  • D major, Z. 667: C to A# (11)
  • D minor, Z. 668: Eb to G# (12)
  • F major, Z. 669: Eb to C# (11)
Georg Böhm (1661-1733):
  • Capriccio in D major, Bb to E# (14)
  • Präludium, Fuge und Postludium in G minor, Ab to C# (13)
  • Suites:
    • #1 in C minor, Ab to C# (12)
    • #2 in D major, Bb to A# (13)
    • #3 in D minor, Ab to G# (13)
    • #4 in D minor, Bb to G# (11)
    • #5 in Eb major (now attributed to Froberger, with Allemande about crossing the Rhine), Ab to F# (11)
    • #6 in Eb major, Gb to F# (13)
    • #7 in F major, Db and Eb to G# (no Ab) (13)
    • #8 in F minor, Db to B (11)
    • #9 in F minor, Db to B (11)
    • #10 in G major, F to A# (12)
    • #11 in A minor, F to D# (11)

Johann Sebastian Bach (1685-1750),

  • Toccatas, c1707-c1715:
    • F# minor BWV 910, G to Cx (14)
    • C minor BWV 911, Gb to C# (14)
    • D major BWV 912, Bb to Fx (16)
    • D minor BWV 913, Cb to D# (17)
    • E minor BWV 914, F to E# (15)
    • G minor BWV 915, Cb to C# (15)
    • G major BWV 916, F to A# (12)
  • Capriccio in Bb BWV 992, c1704, Gb to G# (15)
  • Capriccio in E BWV 993, c1725, C to Cx (15)

Further remarks

I present all these questions here as a practical challenge, because they arise out of hard musical evidence in these published compositions. These data are from core repertoire, and from an investigative method that I believe is much more meaningful than the Barnes/Di Veroli tallying of 3rds.

UT (unfortunately) leaves the student at a superficial level, where these types of question do not enter the reasoning process. Instead, it advises the reader to maintain only a few 'one size fits all' temperaments in one's practical repertoire, make a choice from that tiny menu, and live with it. A keen musician who does that will get a mistaken impression: that unequal temperaments in general will always have 'wrong' spots that we can't do anything about, or it's too complex to deal with, or too difficult to set up without modern beat-rate techniques.

Did all the seventeenth and eighteenth century experts accept crude temperaments and wrong enharmonics, just because they didn't know any better or have any refined skills? I refuse to accept that. Why did they write and improvise music [96] in keys that require these exotic notes such as E# and B#, unless they expected those notes to sound pleasing and colourful, worth bothering with as a positive feature of the music? To me, the presence of those notes shows that those musicians cared about key differentiations, and had satisfying temperament methods that didn't make it into the realm of their contemporary speculators who wrote treatises. Furthermore, any speculators and researchers today who focus on only the treatises and on related historical scraps of information are profoundly missing the point: in the sets of enharmonic notes it asks for by name, to fill out the scales it employs, the music itself largely tells us what it needs.



[82] J. Barnes, 'Bach's Keyboard Temperament: Internal Evidence from the Well-tempered clavier', Early Music, vol. 7 no. 2, April 1979, 236-249.

[83] It is the right hand's f'-a' in the penultimate bar, 33.

[84] Di Veroli's derivation of 'WTC Optimal' and 'WTC Optimal+' falls into that same trap, preferring visual inspection ahead of any mathematically-based outcomes; see pp. 423-425. This is poor science: throwing out any unwanted data that don't fit a preconceived model, and going only with visual inspection as the ultimate arbiter of truth. Furthermore, visual inspection and calculations are not sufficient, either, since the ultimate point is to play music on appropriate instruments, and to listen to it.

[85] R. Rasch, 'Does "Well-tempered" mean "Equal-tempered"?', in the book Bach, Händel, Scarlatti Tercentenary Essays (Cambridge University Press: Cambridge, 1985) 293-310.

[86] Section 21.2, 395.

[87] Section 21.9, 424.


[89] Section 9.7, 131.


[91] The practical gist of mine is as follows: start by setting the naturals F-C-G-D-A-E (i.e. the home base of the C major hexachord, C-D-E-F-G-A) in their ordinary regular positions, 1/6 comma each. The remaining six notes, B and the five sharps, are carefully raised as compromises so they can also serve as flats. E-B-F#-C# are pure 5ths, and C#-G#-D#-A# a gentle 1/12 comma each, taking Bach's drawing as a diagram indicating these tasteful adjustments in turn. See


[96] Including basso continuo parts....


HPSCHD-L discussions

Here are the related postings from early January 2009: before I had decided to write the review, but after I had already analyzed the Barnes data some years earlier. This public newsgroup was a place to discuss harpsichord-related topics with colleagues.

From:		Brad Lehman <[log in to unmask]>
Reply To:	Harpsichords and Related Topics <[log in to unmask]>
Date:		Wed, 7 Jan 2009 10:18:30 -0500
Content-Type:	text/plain
Parts/Attachments:	text/plain (88 lines)

I've been reading Claudio's new book, and am about 1/5 of the way 
through it so far (but skipping around).  I'll reserve review until I've 
read the whole thing.

But, apropos of that, I've gone back and reread John Barnes's 1979 
article.  I also put his temperament onto my harpsichord and played some 
WTC for a few hours.  Frankly...I have to say that that Barnes article 
(while historically important in the 20th C) is such a flawed experiment 
that its results are meaningless, both musically and historically.  It 
starts off enticingly well, but when it gets into the statistical 
assessment of the WTC, it falls dead.  Here's why:

- Barnes disregarded all the minor-key music, and all the fugues.  His 
sample (i.e. only major-key preludes of both books of WTC) is too small 
to say anything meaningful about Bach's compositional achievement in 
those two books of music.  The F-minor fugues of both books bring up 
some of the worst behavior of Barnes's own temperament, with respect to 
obviously raunchy major 3rds, but these compositions are not in the data 

- As his printed excerpt from C major book 2 shows, alongside his tables 
of prelude data: Barnes disregarded major 10ths and 17ths.  He counted 
ONLY major 3rds in their close position, and not in inversion as minor 
6ths, either.  The F# major prelude of book 1 therefore has 0 (zero!!) 
data points, the C# major has only two, book 2's G major only six, and 
book two's F# major only ten.  Book 1's G and D major bring up only five 
and seven data points, respectively.  Barnes rationalized this away by 
observing that the musical texture in those pieces happens not to fit 
his choice of methods.  Well, yeah.  What about those listeners who find 
some far-out-of-tune major 10ths or 17ths to be MORE objectionable than 
the corresponding 3rds?  As these thinly-textured preludes show, along 
with the Inventions and the four Duetti, wide spacing of two voices can 
lay out any tuning problems naked to the world.  (A 15-minute play 
through the Duetti is a great way to find infelicities in 
temperaments....)  But Barnes's experimental method reveals only the 
truism that Bach happened to compose thicker-texture music in the WTC 
preludes that generated lots of data points.  Those thicker pieces 
overwhelm the experiment, at the expense of pieces in wide-open texture, 
just because they have closer spacings, more notes, and therefore more 
major 3rds.

- Barnes's classification of major 3rds into five "prominence 
categories" is understandable, but it's also subjective...and it will 
shift somewhat, when the music is played on instruments of different 
tonal qualities.  Other factors such as tempo, articulation, phrasing, 
and registration would also come into play, here.

- Why should we expect Bach, or any other systematically-oriented 
composer (Chopin's or Shostakovich's 24 preludes as a gold mine of 
temperament data, anyone?), to have featured any specific major 3rds 
ahead of any others...ONLY because of their relative qualities of 
consonance?  This assumption reduces the compositional process to a 
numbers game.  Taking this to its absurd extreme, the preponderance of 
C-E in Terry Riley's "In C" doesn't prove that it should be played in 
any specific temperament.

- IF we were to take Barnes's experiment on the major-key preludes as 
meaningful, then he and di Veroli have both missed a temperament that 
fits Barnes's data even more simply than his own.  I reproduced his 
experiment several years ago, inputting all his data from this 1979 
article, and noticed that the most elegant fit is 1/5 PC F-C-G-D-A-E, 
with pure 5ths everywhere else.  It doesn't sound especially good in the 
music (pardon the value judgment), but it fits Barnes's thin data better 
than his own layout does, and by Occam's Razor it should win.  Still, 
all this is an aside...because the experiment itself delivers 
meaningless results about tuning.

- IF we must be fixated on the relative qualities of tempered major 
3rds, then minor 6ths also deserve more attention than they get.  Other 
intervals come into play beyond the 8:5.  For example, try playing in 
regular 1/6 comma (which, according to Claudio's book, has no 
advantages: "In a nutshell, this is a temperament that manages to put 
together all the worst features of both Standard Meantone and Equal 
Temperament, but none of their virtues."...).  In this temperament, the 
augmented 5ths C-G#, Eb-B, F-C#, and Bb-F# are each very close to a pure 
11:7.  They sound startlingly consonant, given an expectation from the 
misspelling that they should be awful as minor 6ths.  That is: playing 
continuo or other heavily triadic music in regular 1/6, one can get away 
with playing B major, C# major, and F# major in first inversion, even 
though they "shouldn't" work according to superficial analysis of major 
3rds.  Barnes's article doesn't go there...or into any of the other 
interesting intervals that populate real music.

- Music does more than sit idly on 5ths and major 3rds, anyway....

Brad Lehman

From:		Brad Lehman <[log in to unmask]>
Reply To:	Harpsichords and Related Topics <[log in to unmask]>
Date:		Thu, 8 Jan 2009 12:09:22 -0500
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Claudio wrote:

 > Scores of pages have been written and published on the temperament 
for > Bach, especially in recent years...
 > - by yourself supporting your own Bach temperament proposal based on
 > the WTC title-page
 > - by Lindley and Ortgies strongly arguing that your proposal is
 > fundamentally flawed

Claudio, I hope you've read my detailed rebuttals to the latter, and to 
O'Donnell's.  They're here:

And the notes of my one-hour lecture from October 2008:

To me, on the topic of Bach temperament, any argument worth its salt has 
to go into the way the temperament handles the notes actually called for 
in his compositions.  Not just superficial observation of major 3rds, 
but the way all the named notes fit into scales and music.  Especially 
important is the notion of enharmonic equivalence.  18 of the 
prelude/fugue units from WTC book 1, and all 24 of book 2, call for more 
than 12 notes.  (And of the six in book 1 that have only 12, only one of 
them sticks to the central Eb-Bb...C#-G#.  The other five use a 
different 12 notes.)  That's using the implicit assumption that one is 
supposed to play both a prelude and fugue together without retuning 
anything between them; and probably also to play any and all of the book 
without retuning.  (Bach played the whole thing straight through for 
student Gerber....)

Here's a list of the WTC requirements for enharmonic equivalence, across 
both books.

Ebb and D: 2
Bbb and A: 3
Fb and E: 5
Cb and B: 8
Gb and F#: 7
Db and C#: 8
Ab and G#: 5
Eb and D#: 8
Bb and A#: 9
F and E#: 7
C and B#: 8
G and Fx: 7
D and Cx: 9
A and Gx: 9
E and Dx: 4
B and Ax: 1

That is: 8 of the prelude/fugue pairs need a Db/C# at the same time, 
because the music modulates so far that it needs both.  8 of them need a 
C/B#.  Etc....  The whole collection needs an A at such a well-balanced 
place that it can also serve as either Bbb or Gx.  B has to sound like a 
decent Cb or Ax.  F has to sound like a decent E#.  Etc.  The fugue 
subject of the B minor book 1 requires both B# and C natural, and only 
one bar apart, within the melody!  Bach's music lays out the problem 
clearly: tuned pitches needing to be balanced very carefully to serve 
all their multiple functions, melodically and harmonically.

And then, in my hypothesis, Bach's drawing solves the problem he has 
laid out: it's a diagram showing exactly the type of hands-on 
adjustments needed, listening closely and working by 5ths, proceeding 
from each note to the next.  Put down all the C major scale first 
(F-C-G-D-A-E-B), continue into the remaining five notes (F#-C#-G#-D#-A#) 
with the appropriate adjustments, and then proceed to play music using 
every possible major or minor scale.  It is straightforward hands-on 
work, not calculation.

Enharmonic reckoning is important for other contemporary music, too, in 
choosing a workable temperament.  For example, to play all eight of the 
preludes in Couperin's "L'art de toucher le clavecin", we need all the 
notes Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#, D#, A#, and E#. 
That's 16 notes, and if we assume no retuning between pieces, we have to 
have an Ab/G#, Eb/D#, Bb/A#, and F/E# all at somewhat intermediate 
positions so they can serve both ways.  We also have to have the basic 
tempering unit considerably lighter than 1/4 comma, and closer to 1/6 
comma on average, to be able to connect the circle at all for these 
enharmonic requirements.  The music itself demands it.

My basic method, for all this, is to try to keep the notes of the C 
major scale as close to regularity as possible...and to adjust the 
others minimally until they're good in their dual functions.

- In music that requires extended flats such as Fb/Cb/Gb/Db, we might 
have to break the regularity of the C major scale at its last note (B), 
sharpward, to get the chain of sharps high enough to serve as flats.

- In music of extended sharps, requiring A#/E#/B#, we might have to 
break the regularity of the C major scale at its first note (F), 
flatward, to get F-Bb-Eb low enough to serve as sharps for all their 
musical contexts.

It's a process of adjustment by irregular 5ths, stretching them wider 
than the "downtown" regular 5ths: listening to and checking scales 
hands-on at the harpsichord, NOT doing calculations.  Every note has to 
work well in all its enharmonically-required contexts.  Ye got a 
keyboard that needs to play a bunch of notes outside the old-fashioned 
set?  Here's how you adjust it until it all works out smoothly and you 
can play anything.  The first fugue of WTC book 1 lays out all six of 
the regularly-spaced notes (F-C-G-D-A-E as the C-D-E-F-G-A hexachord). 
Everything else is tastefully adjusted off that regularity so you can 
finish that prelude/fugue (which happens to need both Ab and G#), and 
the rest of the book.

That first fugue subject is pretty amazing, all by itself.  It walks up 
from Ut to hit Fa strongly on the first downbeat.  It then decorates the 
expressive descent from Fa to Mi.  (Witness Bach's later canon that 
asserts in its description: "Fa-Mi et Mi-Fa est tota Musica", playing 
with the way Fa and Mi interact during modulations.)  Having established 
Ut, Fa, and Mi as the three fixed points in the chain, the subject then 
shows us how to zigzag by 4ths and 5ths to generate La, Re, and Sol.  Fa 
and Mi are the endpoints of the hexachord, generated Fa-Ut-Sol-Re-La-Mi 
by 5ths/4ths, and laid out as the scale Ut-Re-Mi-Fa-Sol-La.  The longest 
note of the subject is Sol, before falling with decoration down to Mi.

And this fugue (about melody?) is coupled with a prelude (about 
harmony?) that showed us how to handle triads and modulations.  Bar 1 of 
the prelude gave us Ut, Mi, and Sol.  Bar 2 adds Re, Fa, and La. 
There's the complete hexachord already.  Bar 3 gives us the seventh 
note, Ti (Si), completing the diatonic scale.  The first foreign note, 
F#, shows up a bit later and we diddle around with that for a while, 
having modulated via the F# (replacing F) to the G major scale.  The 
introduction of both Bb and C# together lurches us into the scale that 
contains them both: D minor.  The same two bars then happen again, a 
step lower, juxtaposing Ab against B, and taking us back to C major. 
When the next Bb appears (replacing B), it sends us to the nearest scale 
that contains it: F major.  The extraordinary motion in the bass, with a 
melodic diminished 3rd of F# to Ab, decorates our move into a long pedal 
point on dominant G.  When that thing finally gets resolved to C, a Bb 
intrudes immediately: really surprising.  On our way back to home C 
major, we have to deal with the subdominant F major and dominant G major 
once more, each.

And, if my hypothesis is correct: once we get off the regularity of the 
C major hexachord and into the irregular notes (B/Ti, and the sharps and 
flats), each of them by their careful irregularity signals the ear that 
we're heading into a modulation.  Whenever a flat or a sharp intrude 
into the plainness of C major, they sound like a relatively large 
surprise, because they're off-spot.  (They break the regularity of the 
55-note division of the octave, the one Sauveur wrote in 1707 was the 
common practice of musicians.)  The off-spot-ness emphasizes the new 
scale that we're going to.  Any intruding accidental is an intense 
event, an irritant, a break of expectations.  More things will happen 
before the music can relax back into plainness and regularity.  And the 
really weird stuff, such as the F# to Ab motion in the bass, comes 
through as doubly intense.

That's what I believe Bach was illustrating in this prelude and fugue. 
The first piece in the collection shows us how to handle harmony and 
melody in this home key, before we set off to music that requires more 
exotic notes: a 27-note enharmonic gamut, all the way from Bbb to Ax, 
inclusive.  The title page in front of that gives a diagram, showing how 
to start from the notes of the C major hexachord as Fa-Ut-Sol-Re-La-Mi, 
and then adjust the other six properly with less tempering.  The whole 
book demonstrates the system in action, by musical example (Bach's 
preferred manner of instruction) rather than in words or any numbers. 
No calculation is required; just DO THIS and then play anything, in any 
and every scale.  It goes through every "Ut Re Mi" and every "Re Mi Fa" 
possible on a 12-note keyboard.

Brilliant, brilliant music, whether I'm right about the tuning scheme or 

Book 2 has a different 27 notes: it loses the Ax off the top, but gains 
the Ebb at the bottom.  And, as I mentioned above, every one of its 24 
preludes/fugues requires more than 12 notes.  These pieces all force the 
player/tuner to grapple with enharmonic compromises.


Claudio also wrote:

 > Brad clearly stated that he would "review" not just Barnes but my
 > WHOLE book here.

To be clear: I didn't promise to review the whole book here.  I only 
said that "I'll reserve review until I've read the whole thing."  That 
is, I won't write evaluative comments about it UNLESS I've invested the 
time to read the whole thing FIRST...which is the way I wish people 
would treat my work, too.  Read the whole argument, and listen to any 
proposed solutions hands-on and ears-on in real music, before 
formulating an evaluation.

Now, please, let's discuss those reasons why Barnes's method yields 
skewed and ultimately meaningless results.  If Barnes's article is (at 
best) a shaky foundation, anything built upon it is not going to be 
solidly convincing least, not to me.  If we're going to tune 
for, say, Couperin, we can look at all the enharmonic notes required; 
or, we can look at only the major 3rds in only one organ mass.  The 
latter approach seems superficial, to me.  Major 3rds don't tell us 
enough, by themselves, about what music actually does.  If I'm playing 
straight through the fairly short Ordre 25 by Couperin, with its pieces 
in Eb major, C major, and C minor, I have to have a temperament that 
makes the instrument sound non-silly through all the notes Db, Ab, Eb, 
Bb, F, C, G, D, A, E, B, F#, C#, G#, and D#.  Statistical methods 
applied to small samples aren't going to tell me what I need to know, to 
accomplish that.

I'm curious about the reaction of PP's alter ego to this Barnes stuff, too.

Brad Lehman

From:		Brad Lehman <[log in to unmask]>
Reply To:	Harpsichords and Related Topics <[log in to unmask]>
Date:		Thu, 8 Jan 2009 13:29:52 -0500
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Dennis asked:

 >>That is: 8 of the prelude/fugue pairs need a Db/C# at the same time,
 >>because the music modulates so far that it needs both.
 >But do the pieces in question really need a well-tuned Db and C#
 >(though obviously not "at the same time")? If the C# is a long value 
in > a consonant tonic chord, and the Db a short passing dissonance, for
 > instance, then the music doesn't really need both? A good C# is not
 > only enough but preferable. Could you give a few examples of pieces
 > that need consonant enharmonic pairs?

Book 2, F major fugue.  Db major triad on the downbeat of bar 75, and 
big Bb minor chord on the downbeat of 87; A major triads in bar 21 of 
the prelude, and at several other places in both prelude and fugue.

Book 2, Bb major prelude.  Db-F 10th struck together on a strong beat of 
bar 63, and plenty of other Dbs elsewhere (and in the fugue); sustained 
C# in bar 42 where it's tonicizing D major on the next beat.

Book 2, G minor fugue.  A major in bar 31; Dbs in the decoration of Ab 
major, bar 62.  A stepwise alternation of Eb with a too-low Db (as C#) 
would sound odd there, where the Eb is the tonic of bars 61-63.  The 
prelude also uses both notes: C# at several places leading to D minor, 
and Db at one (admittedly short) spot going into F minor.

Brad Lehman

Subject:	Re: Bach temperament: Lehman vs Barnes
From:		Brad Lehman <[log in to unmask]>
Reply To:	Harpsichords and Related Topics <[log in to unmask]>
Date:		Thu, 8 Jan 2009 15:59:14 -0500
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Claudio wrote:

 > (...) My main objection to your Bach temperament is a small, simple
 > thing, but with far-reaching implications: your system makes E major
 > the worst tonality, even if Bach wrote quite a few important works 
for > it (as detailed in my book). Conversely, you make more consonant 
other > tonalities (F# major, C#/Db major) for which Bach hardly wrote 
at all.
 > (...)

Thanks for the cordial and complimentary reply!

The E-G# in my system is wide-ish, yes; but as you know, Vallotti's has 
THREE major 3rds larger than that, and Barnes's has two.  So, anyone 
favoring either Barnes or Vallotti can't really object to the mere size 
of E-G#, but only to its placement.  It breaks the *expectation* that 
the widest should be on F#-A#, B-D#, Ab-C, or Db-F (Lindley 
perpetually!).  Those four are classically the wolves, coming down from 
meantone.  Well, as you also know, there are also some of Neidhardt's 
and Sorge's published systems that do have E-G# as their widest major 
3rd.  If one accepts E-G# as slightly wider than all four of F#-A#, 
B-D#, Ab-C, and Db-F, all the old meantone-inherited problems are wiped 
out.  It looked ridiculous to me, too, until I actually tried it.

One might also think of it this way: the note G#/Ab is closer to C (the 
home key of tonality) by moving four 5ths flatward than by moving eight 
5ths sharpward; so, why *shouldn't* E-G# be at least as wide as Ab-C?

It's not merely a matter of looking at what Bach used as tonic triads. 
It's a look at where he went inside compositions.  To me, playing in 
Barnes, I can't stand the way it makes the music sound: whenever F minor 
or C minor music modulates to the related major keys, or whenever the 
hideous dominant F# major comes up within music in B minor and E minor.

For that matter, in the Bach E major harpsichord music (WTC, Invention, 
Sinfonia, early Capriccio, French suite, concerto...)...why "should" we 
expect things to go farther and farther out of tune whenever the music 
does ordinary things with dominants, secondary dominants, or into the 
relative minor (C# minor)?  Why "should" there be a lousy B# and E# when 
it gets near C# minor, or a hotly high A# anywhere, since these are 
perfectly normal notes (for Bach) within E major?

We can't just point in isolation at a tonic triad such as E and assert 
that things are "worst" there, as if that's all that matters.  The 
primary reason why meantone-inherited temperaments such as Barnes's 
don't work -- at least for me -- is that all these extreme sharps or 
flats aren't treated smoothly enough.  They sound more exotic than they 
necessarily would need to.

Apropos of my temperament getting a good workout and use: this week I'm 
enjoying Julia Brown's new Naxos disc of WF Bach fantasias and fugues. 
Those things go to some of the wildest spots, and apparently for shock 
value sometimes.  That's on a Kingston harpsichord.  There are at least 
three complete sets of WTC 1 out now (Watchorn, Egarr, Beausejour), plus 
the excerpts disc by me, giving good opportunity to hear how it 
interacts with players of different personal temperament in the music. 
I get complaints from some people that the tuning sounds too moderate 
for their tastes, and complaints that it's too spicy for others, or 
peaks in the "wrong" place.  There are also plenty who don't even know 
it's in there, like on Beausejour's and Brown's, because the program 
notes say nothing; where the reviewers simply say things like "that 
instrument sounds especially good", I recognize that the temperament is 
drawing no undue attention to itself, but rather highlighting the music 
and the instrument's tone appropriately.

Brad Lehman

Subject:	Re: Bach temperament: Lehman vs Barnes
From:		Brad Lehman <[log in to unmask]>
Reply To:	Harpsichords and Related Topics <[log in to unmask]>
Date:		Thu, 8 Jan 2009 18:23:21 -0500
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I wrote:
> It breaks the *expectation* that 
> the widest should be on F#-A#, B-D#, Ab-C, or Db-F (Lindley 
> perpetually!).  Those four are classically the wolves, coming down from 
> meantone.  Well, as you also know, there are also some of Neidhardt's 
> and Sorge's published systems that do have E-G# as their widest major 
> 3rd.  

Clarification on this: I meant to say "Neidhardt's and Sorge's published 
systems that do have E-G# wider than Ab-C."

That's where a crux of that matter is.  Having established the points 
where C and E are, is the G#/Ab going to be higher than, exactly at, or 
lower than the midpoint of the remaining distance?  There are some 
Neidhardt examples of each type.

Carry on.

Brad Lehman

Subject:	Re: Bach temperament: Lehman vs Barnes
From:		Brad Lehman <[log in to unmask]>
Reply To:	Harpsichords and Related Topics <[log in to unmask]>
Date:		Fri, 9 Jan 2009 08:40:42 -0500
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To Tom's possibly disingenuous questions:

 > Well, for the n'th time:
 > Why are there such strong verbal and graphical similarities (not to
 > mention dates and locations) between the first two images and the
 > third?

For my opinion of it, perhaps you should read what I wrote about the 
first two images (Suppig's), on the lower left corner of page 226 (in 
_Early Music_ 2005).

Tell us honestly, Tom: had you heard of or seen those two Suppig pages 
before I mentioned them there in the article?  The things I've seen 
about them are Rasch's facsimile edition (of course), and Craig Wright's 
book _The Maze and the Warrior_ (Harvard, 2001).  Wright draws in 
possible connections with Marais and Heinichen, as well.

 > What ideas, musical or otherwise, are expounded in the documents that
 > were behind the first two images?

Mathematically calculated tuning through all keys; but, I wouldn't call 
Suppig's enclosed composition especially "musical".  I've played it. 
It's formulaic and incredibly dull.  So are Suppig's pages upon pages 
upon pages of ratios, handwritten.

 > Is anyone qualified to comment on the 3rd image unless they know
 > something about the first two?

Glad I brought it up in that first article: so people can go look at the 
same things I looked at, and formulate their own conclusions.

 > (Bonus question: What graphical signs were generally used by German
 > organmakers, so far as we know, to denote tempering of thirds and
 > fifths?)

^ , v , and blank space.  Some of them can be seen at:

 > Brad himself has said that he developed at least two different
 > readings, both of which he finds musically satisfactory (whatever
 > other people might happen to think), and anointed one of them only for
 > ease of presentation.

Those are both in the first article and in its PDF "supplementary files" 
(the web portion of the article), as well.  They both have the same 
shape as one another, and differ only in the choice of syntonic vs 
Pythagorean comma as the unit.  That is, they sound slightly more 
intense or less intense in different keys, within the same overall 
pattern of relationships.

More recently, I came up with some others that are duly documented at:

 > What is really 'modern' is the attitude that there are Tuning Experts
 > who know what you want to hear better than you do yourself, who are
 > only too pleased to tell you their answer to your tuning problems (and
 > any old composer's problems, too). This is a hangover from the days
 > when people needed to bring in the piano tuner to maintain their big
 > clunking double-manual.

Was Bach a Tuning Expert that anybody ought to come to for advice? 
According to Sorge, he was.

Were Bendeler and Werckmeister considered Tuning Experts?  According to 
the fact that Bendeler's 1690 work was republished in Leipzig and 
Frankfurt in 1739, that's probably yes, as well.

Glad to help clarify these things.

Brad Lehman

Other "Bach" Temperaments

Back to the assessment of other "Bach" temperaments....
Bach's schematic, rotated for use