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

The basic, minimal, hands-on instructions

Background

This tuning method is based on straightforward principles (among the additional historical material as presented in the Oxford paper):
  • Axiomatically (& historically), no major 3rd should be smaller than C:E. Some others might be similar or the same, but none smaller.
  • Axiomatically (& historically), at least the natural 5ths from C-G-D-A-E should all be of consistent size geometrically...i.e. from the core of "regular" (aka meantone) tempering practice, with ordinaire types of adjustments outside that core. With those regular 5ths, what's good for violins/violas/cellos/violas da gamba on the open strings is good for music: gentle and consistent tempering of the naturals. (And Bach himself was a string player; what might he do as normal practice on those instruments?)
  • Axiomatically (& historically), the C major scale is the natural center of harmony, and the one that should be most regular melodically...again from meantone practice.
  • Axiomatically (& historically), there cannot be any noticeably bad 5ths/4ths anywhere; all major and minor triads have to be usable.
  • Axiomatically (& historically), if the major 3rds in a temperament are changing sizes, it has to be gradual and sound steady when we modulate normally around the circle of 5ths. The easiest test is to play major triads all the way around in both directions, like dominant or subdominant progressions. No major 3rd should be grossly different from the ones immediately on either side of it, in root motions by 5ths.
  • Premise: the whole WTC is playable (and to be played) in a single temperament without stopping to retune any notes between pieces. A good solution makes everything playable and sufficiently interesting as well. On fretted clavichords and organs, retempering between pieces is out of the question anyway.
  • Practical observation (from experimentation and from historical models): it works well to have E:G# smaller than or equal to Ab:C, not vice versa, because Ab is closer to C than G# is (C-F-Bb-Eb-Ab, vs C-G-D-A-E-B-F#-C#-G#). The closer we are to the home key of C major, the less tempered out of regularity we should be.
  • Practical observation: it works well to have F to Bb slightly wider than a pure 5th (as in Italian/French ordinaire practice), yielding a decent major 3rd on F#:A#; the cost to Bb situations is much less than the gain in A# situations, both melodically and in dominant harmonies.
  • Practical observation: the major 3rd C#:E# must be rather good, as Bach audaciously started the C# major prelude with an open exposed occurrence; likewise plenty of G#:B#. Also on the subject of Db:F, this interval is very important to music in the frequently used keys of C minor, F minor, and Eb major, among others; it just doesn't do to have this interval be nasty or obtrusive. Music (such as Bach's F minor prelude/fugue of WTC 1, or the Eb major or the Bb minor p/f, or the much later F major Duetto BWV 803) develops suddenly intrusive bumpiness on the occurrences of Ab:C and Db:F, in temperaments like Werckmeister 3 where those major 3rds are the widest.
  • Practical observation: the major 3rd B:D# must also be very good, for straightforward use in E minor and A minor.
  • Practical observation: if an organ is tempered with the WTC's temperament (in at least one or more accompanimental ranks, if not the whole instrument), it also has to handle the Chorton/Cammerton transposing continuo parts for the compositions that were written that way, playing the continuo in its originally notated keys; and this affects the overall sound of the ensemble.
  • Premise: Bach was clever enough (and musically enterprising enough) to have understood all this and made full use of it before writing his music, treating temperament issues as a musical virtue rather than an unwelcome liability. The tuning style perhaps affected his creative imagination, symbiotically, as to the types of themes and harmonic adventures that made their way into his music; and they only pop back out most clearly if we can re-create the same or similar tuning balance to hear those effects directly. Set up the same conditions he likely had at home or in his office, during the compositional process, to hear how its sound can influence improvisation and composition.
  • Practical observation: the best way to test all this is to play the music directly. The compositions themselves tell us more than any paper argumentation does, in their sound.
A version of this list was cross-posted to HPSCHD-L on 31 May 2006.

Bach's schematic, as it appears on the page

Simple presentation, with no math

  • Tune the home key, the C major scale, first. We do that by putting the Ut-Re-Mi-Fa-Sol-La into order by 5ths or 4ths. Fa-Ut-Sol-Re-La-Mi. That is F-C-G-D-A-E. Tune all of those in the normal way, which is to make the 5th or 4th pure and then give the tuning lever the slightest couple of nudges to make the 5th narrow (or the 4th wide). When you get to E-B, just leave it as a pure 5th. Play a few things in C major to check that it all sounds good. Yes, the major 3rds are a little wide.
  • All our natural notes are done. We just need the accidentals now, doing the B major scale for convenience. Fa-Ut-Sol-Re-La-Mi is E-B-F#-C#-G#-D#. We already have E-B pure. Make B-F#-C# pure also. This gets the sharps up good and high so they can also be used as flats. For the C#-G#-D#-A#, when you do the G#, D#, A# in turn give them a single nudge a little flatter, each, so they don't all get too high where they would sound bad. Test the B major scale with some simple music, and improvise around to some other keys if you feel like it. Check the C# major, F# major, and other spots that would sound rough in other methods, and make sure they sound decent to you here.
  • That's the whole setup. Now play through this book that demonstrates all the scales sounding marvelous, in some worked-out music. All the major scales Ut, Re, Mi, and all the minor scales Re, Mi, Fa.
  • Here's a simple little picture showing that same nudging process as a reminder which notes get what, but you've probably already memorized it. The C major scale is this half with the double nudges, and the B major scale is that other half. They overlap with the E-B, there in the middle.
Bach's schematic, rotated for use

Instructions

Here is the diagram, with explanation of an easy sequence to get the work done, tuning a harpsichord by ear. Electronic-aided instructions are below.

With the help of a decent teacher (or one's own previous experience as tuner) demonstrating the rudimentary handling of the tuning lever on the harpsichord's tuning pins: the following sequence of events shows what is important to listen for, to get the correct results onto the keyboard. Bach's schematic, rotated for use

  • Copy the note A from some reliable source (tuning fork, woodwind instrument, whatever). It also works easily from C, D, E, F, or G if we have to do that instead...but for the first time to try it, A is simplest.
  • Our home pitch is the A immediately nearest middle C on the keyboard, below middle C.
  • Tune the F below that as a pure major 3rd.
  • Flatten F slightly until it beats (wobbles) 3 times per second against our home A. That speed of wobbling is memorizable either through musical experience (feeling that tempo), or check it with a reliable indication of seconds.
  • Do all the F-C-G-D-A 5ths slightly narrow: or if using 4ths for some of those, the 4ths are slightly wide. The important point is that all of these have the same musical quality as one another, the same amount of deliberate queasiness to them. (There are also methods available to do all this with exactness by ear, but concentrate first on understanding what "same quality" is.)
  • Do the A-E 5th with that same queasiness. Check that the C-E major 3rd has the same quality as the F-A 3rd we started with.
  • Our ground plan of all the notes C, D, E, F, G, and A is now in place.
  • Tune pure 5ths or 4ths E-B-F#-C#. If you care to test them: the major 3rds C-E, then G-B, then D-F#, then A-C# will each sound a little "brighter" than the previous one.
  • Tune G# pure from C#, and then back it down flatward ever so slightly: only half as much queasiness as our original main 5ths/4ths. Just barely enough to notice on super-close listening. This takes a little bit of the bright edge off the major 3rd E-G# (test it), where it was getting too high for comfortable taste.
  • Likewise tune D# pure from G# and back it down the slightest bit. This takes a little bit of the bright edge off the major 3rd B-D# (test it).
  • Likewise tune A# pure from D# and back it down the slightest bit. This takes a little bit of the bright edge off the major 3rd F#-A# (test it).
  • Check that our A# to F is not terrible; it should simply sound like an almost-pure 5th (or 4th) even though it really is a leftover diminished 6th. If it doesn't sound decent, back up several steps in the instructions to find the mistake, and redo it better.
  • Tune all the rest of the instrument with pure octaves and unisons, from this central region outward in both directions.
  • That's all.
Bach's schematic, rotated for use

That's the picture. At the left side of the diagram: our indication that our starting F-A is 3 wobbles per second, as the only speed we really need to know well.

Then a standard consistent tempering of F-C-G-D-A-E. All those are narrowed the same as one another, and enough to notice.

Then E-B-F#-C# pure. Those are easiest.

Then C#-G#-D#-A# each backed off flat a little bit, only half as much each as our starting 5ths. Barely enough to notice, and only at closest listening.

At the right side of the diagram: the flourish is our checkpoint that A# to F is not terrible. It's just a single bit of queasiness like these other 5ths we just ended with.

And we're done, that's the whole thing. With practice and experience we can do it more carefully and accurately, or work out some shortcuts that deliver the same result with a couple of additional test points. But this is the basic thing, this picture of the finished results.

If we started from C, D, E, F, or G instead of A, we still have to get exactly the same finished results as this. We just have to work more carefully doing that basic batch of F-C-G-D-A-E 5ths, going in both directions from wherever we started...so it results in the F-A major 3rd having 3 wobbles per second, our important speed. We can't go anywhere unless we have laid down the regular set of F-C-G-D-A-E reasonably accurately first.

The circle of 5ths, in Bach's tuning

Major 3rds in scale context

F-A and C-E must have the same quality as one another, and be our nicest (i.e. closest to pure) major 3rds anywhere. They are slightly sharp of a pure harmonic interval (5:4 ratio). Those are our main two major 3rds in the rising scale of C major, and used in the simplest music we'd expect to play.

The book tells us--in its title page's words--that we are going to use the "ut re mi" scales in all the possible keys. The first piece in here is in C major. Home base for the easiest prelude. And the first fugue's subject uses all six of these same notes, C-D-E-F-G-A. The first composition in the book nicely demonstrates C major's harmonic (prelude) and melodic features (fugue).

  • C, D, E, "ut re mi".
  • F, G, A, "fa sol la" (or "ut re mi" again).
The other major 3rds in the other scales are somewhat different from these, and most are different from one another: giving a pleasing variety as our music moves from one scale or harmony to another. The other major 3rds are all larger (wider) than C-E and F-A. As Bach's pupil Kirnberger reported, a tuning lesson shows how to tune "all the major 3rds sharp".

The thing to learn about the taste of all this is: they vary smoothly up toward the brightest ones, and then those are backed off slightly during the tuning instructions, taking the roughest edge off them. These: E-G#, B-D#, and F#-A#. It's done in each case by tuning a pure 5th up to the top note of these, and then tastefully knocking it down the slightest bit so the major 3rd sounds a little bit better than it would if the 5th were pure.

Other perspectives of the same sequential method:


Video demonstration of the basic principles on YouTube, August 2007: explaining the drawing and the tuning sequence


Video demonstration on YouTube, August 2007: setting the Bach temperament


Video demonstration on YouTube, September 2007: setting late 17th century temperament ordinaire, and playing Purcell


Video demonstration on YouTube, August 2007: setting meantone accurately [Transcript of the captions]


Video demonstration on YouTube, August 2007: 1/6 comma meantone morphed into Bach

Electronic device instructions

This is the main version, for solo or instrumental music. (If setting up a harpsichord, organ, or rehearsal piano for Bach's vocal music there is a separate set of instructions, as the layout for that is different.)
  • Establish the tempered C, D, E, F, G, A: copy them exactly from the "Vallotti" setting on an electronic device.
  • Turn off the device.
  • Tune B pure to E.
  • Tune F# pure to B.
  • Tune C# pure to F#.
  • Tune Bb pure to F (this is temporary).
  • Tune Eb pure to Bb.
  • Set G# very slightly impure from both C# and Eb. This is a midpoint between the places where it would be pure to either one of them, and getting only a vaguely slow wobble from each.
  • Go back to Bb and give it this same vaguely impure quality against Eb, by lowering it slightly from pure.
  • Check everything by ear: E-B-F#-C# pure 5ths, C#-G#-D#-A# 5ths with only the slightest wobble, and F-C-G-D-A-E 5ths with a bit more wobble.
If your electronic device doesn't have a "Vallotti" setting, but only an equal-temperament meter:
  • Tune A exactly to its equal-temperament position.
  • Tune B exactly to its equal-temperament position.
  • Tune E pure to B, and check that it is 2 cents low (from the prescribed equal-temperament position) on your device.
  • Tune D 2 cents higher than its equal-temperament position.
  • Tune G 4 cents higher than its equal-temperament position.
  • Tune C 6 cents higher than its equal-temperament position.
  • Tune F 8 cents higher than its equal-temperament position.
  • Tune C#, G#, Eb, and Bb each 4 cents higher than its equal-temperament position.
  • Tune F# pure to both C# and B, i.e. 2 cents higher than its equal-temperament position.
  • Check everything by ear: E-B-F#-C# pure 5ths, C#-G#-D#-A# 5ths with only the slightest wobble, and F-C-G-D-A-E 5ths with a bit more wobble.

See also: Instructions for phone apps.


Return... Bach's schematic, rotated for use