• J S Bach's medium of expertise was his precise control of sound.
• Bach composed a book of music that says directly it is about a proper way to tune keyboard instruments.
• The music in the book presents the practical musical problem: to play in all the possible scales, following the examples of this music, we need 27 different notes. (7 natural notes, 7 notes with sharps, 7 notes with flats, 5 more notes with double-sharps, plus B double-flat.) But, this hard requirement must be handled somehow on a keyboard that has only 12 levers per octave, not 27, through some kind of a practical compromise. It is necessary to "temper" the 12 pitches carefully, to make them sound musically pleasing within all these scales.
• At the top of that book's title page is a spiraling figure whose asymmetry draws attention to itself. For the ease of drawing it accurately, it appears to have been drawn with the book rotated (i.e., working along the "bottom" edge), fitting it in between the words and the edge of the page.
• The papers and demonstrations here explore the following hypothesis. To fit the evidence of all these required 27 notes and their scales, Bach's spiraling line-drawing in its context is to be taken as additional evidence: it provides a direct practical diagram for tuning keyboards by ear, solving the problem. It is purposefully irregular, not haphazard or fanciful. It shows a quick and effective hands-on method of tuning an instrument to play this book, giving a tastefully unequal balance, installed with no calculations. It shows how to control the sound.
• [2005 article] [2022 article]

Let's begin.

It appears that Johann Sebastian Bach notated a specific and unequal method of keyboard tuning. He did not express it in our normally-expected modern formats of theory, or numbers. Rather, he drew a diagram that looks like a practical hands-on sequence to adjust the tuning pins, working entirely by ear. The music constrains any solution to be nearly equal temperament, and then, the diagram's irregularity shows how to shape the desired subtle inequality.

In my simple interpretation, it keeps the six main notes of the C major scale (C, D, E, F, G, A) in evenly-spaced positions, at their normal spots within the context of late 17th century practice. The tuner is then to install the keyboard's six remaining notes (B and the sharps F#, C#, G#, D#, A#) in tastefully raised positions, with adjustments as indicated by the diagram, so they can also serve well as flats.

This process of minimal but necessary compromises makes the keyboard ready to play music in all 24 major and minor scales. Every scale has a subtly different expressive character, as the steps are not all exactly the same size. The harmonies have various tensions and spice, when the notes of the scales are built together into chords.

Bach demanded and exhibited a system of this enharmonic flexibility not only in the diagram, but also through the music in Das Wohltemperirte Clavier. It presents his tuning challenge (and gives the solution!), where most of the preludes and fugues each require the smooth handling of more than 12 notes. For example, his D major prelude and fugue use all of Bb, F, C, G, D, A, E, B, F#, C#, G#, D#, A#, and E# in the same piece: 14 notes. Some of the other pieces require 13, 15, up to 25! The last piece in the book, the B minor fugue, requires 17 (Eb up to Fx), and presents 13 of them as early as the subject: six naturals C, G, D, A, E, B, and seven sharps F#, C#, G#, D#, A#, E#, and B#.

Bach obviously knew how to set up his keyboards appropriately before writing his music for them. The aim is to restore the specific intonation scheme of his everyday keyboard tuning, the sound relationships he expected to hear in his melodies and harmonies, as they may have influenced his creative imagination. By hearing how these musical elements work through composition and improvisation, we gain new clues into the interpretation of Bach's music: affecting at least the areas of articulation, phrasing, dynamics, timing, intensity, and drama. The tensions and resolutions within the music suggest fresh ideas in performance, both through intuitive reactions and through close analysis.

The resulting temperament has been absent from our history books. It was lost under layers of assumptions and habits that have led away from it.

I believe that the particular sound of this temperament, as an integrated part of musical practice, has profound implications for all of Bach's instrumental and vocal music that uses keyboards: either with written-out parts or in the basso continuo. Since every scale has a different Affekt or mood, from the different musical tension within the intervals, the music sounds colorful and "alive" as it moves.

Bach's music itself is a large body of primary evidence for temperament: sounding convincingly brilliant and expressive when the intonation is right, or rough and ugly in temperaments that don't properly solve the enharmonic problems.

This "LaripS.com" web site clarifies and explains the published papers, both through theory and practice. It provides various introductions to this work, for different levels of readers' interest. It serves as an archive of ideas as this temperament is used and discussed among musicians, researchers, and enthusiasts.

The necessary background: understanding "ordinary" tempering methods

In these regular temperaments, however, the sharps and flats are not interchangeable. The only available notes are Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#, or occasionally substituting D# for Eb. Where the keyboard has been tuned with a regular Bb, that pitch will sound wrong when played in scales, melodies, or harmonies as if it were A#. The Bb is tuned too high to be A#. Similarly, F is too high to be played as E#, B is too low to serve as Cb, and this problem continues into all other enharmonic spellings of the notes.

Furthermore, the sharps and flats do not connect with one another by 5ths. Somewhere, typically at G#-Eb or at D#-Bb, there is a diminished 6th that is considerably wider than the regular 5ths. Any intervals played across this enharmonic gap sound rough and out-of-tune, both harmonically and melodically, because some of the notes are being misinterpreted with wrong enharmonic spelling. Those notes are missing from the scales.

To be able to handle the enharmonic requirements of the music, the musical spelling of the notes for their scales and harmonic intervals, there are several ways to get through those problems with regular temperaments:

1. Provide separate levers on the keyboard for sharps and flats (although this still does not address the problems of notes such as E# and Cb, serving also as F natural and B natural).
2. Let the sound be wrong for the misspelled notes, and hope that it does not bother the listeners or the other participating musicians too much.
3. Retune some of the notes to be able to play different compositions, assuming that we do not need both names for the same lever within one composition. (For example: if you don't need Eb or Bb, retune them lower in pitch be D# and A#.) The layout can remain regular, moving the gap to a different place.
4. Reduce the regular amount of tempering in the 5ths by so much that the layout evolves into 12-note equal temperament, discarding all enharmonic distinctions and becoming atonal. Unfortunately, this method removes any acoustic advantages that came from having the most frequently-used scales and harmonies the best in tune.
5. Change the regular layout to be tastefully irregular, moving some of the pitches up or down a bit so they can serve more smoothly with two different names. This is "ordinary" or temperament ordinaire practice, setting up compromises so the keyboard can be played in a wider range of music.

Already by the 17th century, much of the extant music calls for the exotic notes such as D#, A#, E#, B#, Ab, Db, Gb, Cb, and/or Fb. It is also common to find enharmonic pairs (such as A# and Bb) both needed in the same composition. The problem gets more extreme with Bach's music, even his earliest, and that of the musicians around him: so frequently needing more than 12 named notes within a single composition that any old-style regular temperament was not feasible. The music itself tells us which notes it needs, in finding a satisfactory tempering scheme to play it.

Where the choice is to derive an ordinary or irregular layout, the careful listening and adjusting at a keyboard can be summarized into process flowcharts.

• For more on this, see my introductory and detailed flowcharts. These two three-page flowcharts are best studied by working through them step-by-step at a harpsichord, as I have done in 30 years of tuning by ear. They go through practical questions about the performance situation and the musical scores, and then through the listening and testing process for each note.
• This practical background in ordinary tempering is necessary, working directly at the instrument, if one is to understand fully the material presented in these articles and web-based resources. My thesis about Bach's tuning method comes from musical practice as a performing harpsichordist and tuner; it is not graphology, as some critics have mistakenly assumed from a too-thin reading of parts of the articles, or from listening to circular arguments from one another's hearsay.
• The music determines within a narrow boundary what the tuning solution must be, even before any arguments about Bach's drawing as further evidence of his tuning practice.
• To put the notes at points where they can serve double duty in scale relationships with all the other notes around them, sharps are raised from their default positions in a regular temperament, and flats are lowered similarly. We work in turn with G#, Eb, C#, Bb, F#, F, and B, and their enharmonic respellings, testing everything carefully with all the competing scales and harmonies that contain them.

After the hands-on adjustments of pitch are made, and tested thoroughly in the music to be played, the resulting irregular temperament can be written down. This makes it easier to reproduce accurately on other occasions, so one does not have to solve the same problem of tasteful adjustments again every time.

That, I believe, is what Bach did here: through a hands-on "ordinary" process of listening to and adjusting the scales, he (or someone around him) found a very good irregular layout that solves all the musical problems. It allows one to play beautifully in all 24 major and minor scales, while it also keeps the most acoustically-favorable sounds in the most commonly-needed scales. It provides enough contrast that every scale sounds subtly different, which can make the music more exciting and interesting as it modulates. He then wrote down enough details, a picture of the tempering result, to show how to do it again.

The recipe

• We set up the notes of the C major scale first, and then we fit the remaining notes into carefully compromised spots: not like the old style, where you were forced to choose either a sharp or a flat, and have it sound bad as the other one.
• Tune your natural 5ths F-C-G-D-A-E with your everyday process of making all the 5ths (or 4ths) waver with gentle equal quality, and checking that F-A and C-E each end up "a little sharp"; your checkpoint here is that F-A is wide at about 3 per second in the tenor. Here's the little jot on one side, showing the 3 beats.
• You normally would have finished everything else with that same type of 5ths: E-B-F#-C#-G#, and F-Bb-Eb, leaving a gap there between the Eb and G# where it doesn't connect. Instead, we are going to put all those remaining notes into tastefully compromised positions so we can play music that has either sharps or flats, all the way around.
• From E, put the next three notes higher than usual, simply doing pure 5ths E-B-F#-C#. You'll notice that C# gets up fairly high and bright against A, but when you check it as Db-F, the A-C# and Db-F have the same quality as each other. That's how it works well playing all music.
• Continuing on from C#, fit the last three notes G#-D#-A# into place with only half as much tempering as you used on the natural 5ths: only a very slight waver, each. That is, when you do each one, put each one into its pure-5th spot but then take it flat the tiniest bit. Check that you're making your Ab-C a little better than E-G#, the Eb-G better than B-D#, and the Bb-D better than F#-A#.
• As your final checkpoint from A# back to F, that leftover spot is a slightly wide 5th, but not troublesome (nobody will hear it, a few paces away from the instrument): it wavers the same amount as the C#-G#-D#-A# you just finished, but in the opposite direction. Here's the jot on the other side, showing the quality of the A#-F leftover, the diminished 6th.
• Finish the instrument by octaves in both directions. This diagram shows you the quality of all the 5ths, if you want to check your octaves with them as you go.
• Now, play and improvise music in all possible scales; it all works! You'll notice that it matches the physical layout of the keyboard: whenever you have to stretch a finger to play a sharp or flat, the music sounds a little more spicy than when you're playing on only the natural notes.

Articles

A supplementary article "The 'Bach temperament' and the clavichord" is available in the November 2005 issue of Clavichord International. It contains further discussion of practical issues: some specifically for clavichord, some more generally in analysis of Bach's keyboard music, scale structure, enharmonic considerations, and by-ear tuning instructions. The compositions presented include BWV 772-801, 802, 808, 849, 887, 988, 1079, and 1080. [Outline] [Full text]

A November 2005 essay "The Tuning" gives a two-page summary of the temperament and its musical character. [Full text] It is printed in the booklets of Peter Watchorn's CDs.

The 2006 article "Bach's Art of Temperament" for BBC Music Magazine further explains this temperament from several additional practical angles, focusing especially on the blend of the C major and B major scales. (C, D, E, F, G, A, B; B, C#, D#, E, F#, G#, A#) [Full text] The BBC's version was shortened and given the title "In Good Temper".

Other articles about this temperament are listed here. There is also a page of my responses to other people's articles and books where they present their agreements or disagreements with this temperament.

My October 2008 lecture notes for James Madison University turn out to be almost a complete article, in themselves.

In Early Music November 2009 I have a letter to the editor, calling for fair argumentation on this topic...especially from Mark Lindley, who has mis-represented and disdained my work three times in that journal.

In "Unequal Temperaments", The Viola da Gamba Society Journal vol 3 part 2 (2009), 137-163, I review a 2009 book by Claudio di Veroli. I also address some recent argumentation about Bach keyboard temperament, and debunk the 1979 analytical methodology of John Barnes. [PDF]

See also the lecture notes from October 2010, a presentation at the University of Colorado: "Recovering Bach's tuning from the Well-Tempered Clavier". These give a deeper perspective on the technique of examining which notes are used in various compositions, and the way that that process informs the inferences about temperament.

My Autumn 2022 article, "The Notes Tell Us How to Tune", revisits the scale theory and the enharmonic measurements from the 2005 article. It explains everything again after the intervening 18 years since 2004, written from a musical scale perspective. I set for myself the challenging task of describing all the principles using as little math as possible.

Quick start!

What might a beginner to this material read first, most productively? Open the streaming audio page and start up some music in the background. Then, start reading either the "Bach's art of temperament" article or the informal lecture. I have prepared a more formal set of notes (slides in a PowerPoint presentation) for a public lecture at James Madison University, 22 October 2008. Also, take a look at the videos showing a harpsichord being tuned and played. Enjoy!

Easy way to tune it accurately and mostly electronically

• On your electronic device, select "Vallotti". It must be an accurate Vallotti, not a weird hybrid of "Vallotti/Young".
• Set all of the following notes exactly where they are in Vallotti: C, D, Eb, E, F, G, and A.
• Turn it off.
• Check the accuracy of the naturals, by ear: F-C, C-G, G-D, D-A, A-E. These should all have an identical quality (slightly narrow 5ths or wide 4ths), each wavering gently, making a smooth cycle of 5ths/4ths. Listen for quality, not for counting any beat rates.
• From E, tune B as a pure 5th or 4th.
• From B, tune F# as a pure 5th or 4th.
• From F#, tune C# as a pure 5th or 4th.
• From C#, tune G# as a pure 5th, but then compare it with D#. Lower G# slightly, testing both these intervals (C#-G#, and G#-D#), until you find the average spot where it is equally tempered from both: a very gentle impurity.
• Finally, tune Bb pure from both Eb and F. From there, lower the Bb slightly, the same amount that you lowered G#, until both of Eb-Bb and Bb-F show the same impurity that you gave to C#-G#.
• That's all! Finish the instrument with pure octaves and pure unisons everywhere.
• When setting the octaves, check all the 5ths and 4ths by ear for appropriate quality. This habit is especially useful when working on the treble.
• You end up with: F-C-G-D-A-E as they are in Vallotti ("1/6 comma" each); E-B-F#-C# pure; C#-G#-D#-A# very gently tempered ("1/12 comma" each); A#-F leftover is also gently tempered, but happens to be wide.

Offsets for ClearTune, PitchLab, and other electronic tuners

C (+5.9), C# (+3.9), D (+2), Eb (+3.9), E (-2), F (+7.8), F# (+2), G (+3.9), G# (+3.9), A (0), Bb (+3.9), B (0).

Check it all by ear, as described in the above section. You should have an even and smooth F-C-G-D-A-E (all with the same quality as one another, 1/6 comma narrow), a pure E-B-F#-C#, and a very gently wavering C#-G#-D#-A#. The Bb/A# should be placed at a spot very slightly lower than the spot where it would have been pure from both D# and F.

When setting the octaves, check all the 5ths and 4ths by ear for appropriate quality. This habit is especially useful when working on the treble.

For the PitchLab tuner, which is newer (2014) and better than ClearTune, do the following:

• Get the "unlock everything" paid version of the app, to get full control over what you are doing.
• Create "Bach/Lehman 1722" as a custom temperament.
• Enter the offsets from C, not A. The numbers to use are:
  C (0.0), C# (-2.0), D (-3.9), Eb (-2.0), E (-7.8), F (2.0), F# (-3.9), G (-2.0), G# (-2.0), A (-5.9), Bb (-2.0), B (-5.9).
• When tuning with it, be sure that the "root note" is set to C, not A.
• Check all your work by ear, when finished, to be sure that no errors or transpositions have happened. You should have an even and smooth F-C-G-D-A-E (all with the same quality as one another, 1/6 comma narrow), a pure E-B-F#-C#, and a very gently wavering C#-G#-D#-A#. The Bb/A# should be placed at a spot very slightly lower than the spot where it would have been pure from both D# and F.

See also this review of PitchLab by Paul Poletti.

Audio samples and video demonstrations

More than a dozen recordings by other musicians using this "Bach/Lehman 1722" temperament: on harpsichords, fortepianos, pipe organs, digital organs, synthesizers, and more....

A survey of the temperament's use in public performances and recordings by hundreds of musicians, and built into pipe organs and other instruments....

The LaripS Recordings label....

There is a growing collection of video demonstrations, showing how to tune harpsichords by ear in this and several related temperaments.

Other resources

A temperament diagram with remarks about the scales and intervals.... Dozens of Frequently Asked Questions (FAQ), answered.... The historical and practical assumptions within my research.... Several sets of instructions for tuning instruments by ear, or with electronic devices.... Several additional temperaments to set by ear: see especially my ordinary temperament to play 17th century German music ("Bonus 5" on that page).... New in November 2008! Jean-Philippe Rameau's published preference in 1726 was apparently for a system with regular tempering in Bb-F-C-G-D-A-E-B, and the other four notes tastefully arranged to fill the gap. My presentation of this is in section 6 of the "practical instructions" page.

A historical survey of other "Bach" temperaments as hypothetical reconstructions....

Additional resources for music theory, practice, and history....

Features / Site Map....

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Bradley Lehman, A.Mus.D. (harpsichord), info@larips.com [Biography] [Home page]