ARTICLES_ Bach's Art of Temperament (2006)

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LaripS.com , Bradley Lehman, 2005-10, 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.

Bach's Art of Temperament Bradley Lehman, April 2006 (with several improvements added June 2006)

Introduction

Tuning historian J Murray Barbour wrote in a 1947 article: "Explanations of the tuning of themusical scale are usually so full of figures that the non-mathematician shies away from themin terror. This is unfortunate, for the subject is not only of interest to the musicologist andtheorist, but of immediate and practical concern to the performer." (Barbour, "Bach and the Artof Temperament", Musical Quarterly 33/1, January 1947)

Tuning also has direct impact on listeners: affecting the expressive powers of the music, and

the resonance and melodic character of the instruments. Let us visit some of the basicpractical matters, without calculations.

We will work our way toward a method of keyboard tuning I believe was notated by JohannSebastian Bach. My 2004 hypothesis in that regard is based on both his music and a drawingby him, in his textbook about musicianship in all keys.

Perspective

There are hundreds of reasonable ways to tune keyboards for the classical repertoire. Theirappropriateness depends on several questions. Do they clarify the music, aiding its beauty andstrengths? Is the tuning historically plausible for the composer's context or known practices?Does the tuning allow the music, the instruments, and the performers to sound competent - orbetter yet, inspired? Is the tuning for all-purpose use (a whole concert or a whole season), or

for some more restricted set of compositions?

Obviously, all this quickly becomes a minefield of historical, practical, and aesthetic issues.What is a proper balance of musical, antiquarian, and practical priorities as we choose amethod to set up the keyboard for any specific day, or any specific repertoire?

Here is the familiar appearance of a keyboard. See Figure 1.

Figure 1:

As we zigzag through the white and black keys, the notes might look equally spaced, or nearlyso. However, the pitches produced need not be evenly spaced (as most of us are accustomedto hearing from modern equal temperament on pianos and organs). A keyboard is merely aset of levers to activate whatever pitches have been programmed into their opposite ends. The

layout can offer additional musical choices, with much music sounding aesthetically "better", if not all the pitches are required to be equally spaced.

v Introduction> Articles

- Outline- Errata- Clavichord* ArtOfTemp- TheTuning

v FAQv Practicev Theoryv Historyv Etcv Recordings

ARTICLES: Bach's Art of Temperament (2006) - www.larips.com http://www-personal.umich.edu/~bpl/larips/art.html

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died with him - if there ever was one. But, we can make reasonable guesses at an appropriateharmonic/melodic balance: using as corroborative evidence the historical context plus thesound and expressive range of Bach's music.

Any meaningful interpretation of the drawing must fit into musical taste and the behavior of the appropriate instruments, in practice. We must deliver performances, improvisations, andcompositions of satisfying music. With various arguments presented to interpret Bach's clues,or to decide what constitutes evidence at all, we must never lose sight of the main goal: which

is to catalyze the production of beautiful music.

Tuning Process

Let us now work out a specific tuning process from Bach's drawing. My temperament layoutderived from this is a very simple sequence, working hands-on at a harpsichord. This methodtakes only a few minutes, by ear. The drawing suggests both the strategy of working on thenotes, and the relative amounts of tempering necessary along the way. The crucial listeningprocess is to sense how much of a deliberate spice or wobble is being added to an otherwise

pure 5 th or 4 th , by knocking one of the interval's two notes slightly out of tune.

A general principle: when we tune seven notes in sequence by 5 ths (or by going the opposite

direction by a 4 th ), those seven notes give us a complete major scale. Fa-Do-Sol-Re-

La-Mi-Ti gives us all the notes connected by 5ths

, and in conventional scale order this isDo-Re-Mi-Fa-Sol-La-Ti[-Do]. This basic structure of diatonic scales is explained fully inEasley Blackwood's book; see the "Further resources" below.

For all seven notes of the diatonic scale to be generated by 5ths(and not diminished 5ths or augmented 5ths!), we must start on Fa=0, Do=1, Sol=2, Re=3, La=4, Mi=5, Ti=6.

In C major, this sequence is F-C-G-D-A-E-B as the0-1-2-3-4-5-6, or Fa-Do-Sol-Re-La-Mi-Ti. The notes may goup by a 5th and/or down by a 4th, if we want to stay within asingle octave. Also notice that Ti (6=end) does not connect back to Fa (0=beginning), as that would form a diminished 5th or augmented 4th.

The character of that major scale, in intonation, is determined by how nearly we make those

5 ths pure as we go along. They might be tuned exactly pure, or slightly tempered eithernarrow or (less commonly) wide. The notes La, Mi , and Ti are especially open to tastefuladjustment, occurring near the end of the sequence: by the time they are generated, we canhear how they fit into the scale both melodically and harmonically, and make any slightalterations that seem prudent.

Most harpsichord temperaments are generated by some simple set of rules for this process:

describing the tasteful character of those pure or tempered 5 ths . The easiest rule is to make all

those 5 ths the same size as one another (see the Reference discussion of "meantone", below).In the case of Bach's tuning style, I believe he has drawn a simple picture of those rules to

adjust the 5 ths/ 4 ths , using a careful variety of three sizes.

With regard to Bach's drawing, I take another of my conceptual models from the physicalprocess of tuning a harpsichord: from the tuning-pin's position where an interval is soundingas pure, how many units of slightest nudge on the tuning lever does it take to rotate the pin,to deliver the appropriate impurity? One micro-nudge, two, or none (leaving the intervalpure)? This practical business of nudging the rotation is all there is to it, once one knows the

appropriate pattern of which notes to nudge and by how much. Each nudge causes anincreased "twang" in the interval's quality, on very close listening.


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Figure 3:

We establish all of the keyboard's natural notes first (the white notes on pianos or most organs

- obtaining the C major scale of C-D-E-F-G-A-B-C; Do-Re-Mi-Fa-Sol-La-Ti-Do ), and thenproceed to the sharps/flats to finish. The C major scale uses the common practice of medium

tempering on the 5 ths F-C-G-D-A-E, with a consistent size: two nudges from pure. Each 5 th in

turn (F-C, C-G, etc.) is made slightly too small, and this helps the quality of major 3 rds (F-A,

C-E, etc.) when they result from the several tempered 5 ths in succession ( F-C-G-D- A).

The exceptional step is the last one, E-B. The loop between them has no nudges in it, and Ibelieve this indicates a pure 5 th . We have tuned Fa-Do-Sol-Re-La-Mi-Ti (F-C-G-D-A-E-B);and that last step of the pure E-B (generating Ti ) makes that note B a slight bit higher inpitch, which helps it to serve well as a "leading tone" upward to Do . The C major notes werederived with consistent tempering on all the intervals F-C-G-D-A-E. This is true not only for

this Bach layout, but for all the common "meantone" schemes back into the 17 th century andearlier: regularity and consistency as a base, at least within the natural notes.

Instead of continuing into the sharps/flats with that same "two nudges" consistency off bothends, our aim here is to make them tastefully irregular in a way that improves their utility.This is "modified meantone" (see the Reference section, below) or "ordinary" practice, from

the 17 th century forward: to preserve as much illusion of regularity as possible, while makingadditional scales sound beautiful beyond those with two sharps or flats in the key signature.Such temperaments allow keyboards to be played in all or most keys, without extremedissonance, and providing interesting variety among the scales and chords. The farther these

systems evolved in both practice and theory, into the 18 th century with dozens of examples,adjustments began to affect the naturals as well. Some of these increasingly irregular systemsare occasionally described today as "well-temperament," but that is a modern (andungrammatical!) term, turning Bach's adjective " Wohltemperirte" into a noun.

With the notes of C major already installed, it remains for us to find a source for the fiveaccidentals. These are the keyboard's raised notes, usually black by contrasting color - andthese notes can be found most conveniently in the B major scale, B-C#-D#-E-F#-G#-A#-B.The notes when arranged by 5 ths (Fa-Do-Sol -etc.) are the sequence E-B-F#-C#-G#-D#-A#.

The same way that we took the first half of the diagram for the seven notes of C major, wetake the other half to generate B major. These two scales have the notes E and B in common.E is " Mi " of C major and " Fa " of B major; B is " Ti " of C major and " Do " of B major.

C major:

Again in the tuning sequence, by 5ths and/or 4ths: Fa=0,Do=1, Sol=2, Re=3, La=4, Mi=5, Ti=6.

B major:


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In B major, this sequence is E-B-F#-C#-G#-D#-A# as the0-1-2-3-4-5-6, or Fa-Do-Sol-Re-La-Mi-Ti.

The Bach diagram, continuing where we stopped before, gives us three plain loops (pure 5 ths )for E-B-F#-C#, and then three single-nudge loops for the remaining C#-G#-D#-A#. Our startwith several pure 5 ths gives the B major a "Pythagorean" character (see the Reference section,below), but then the notes G# ( La ), D# ( Mi ), and A# ( Ti ) each get a tasteful downwardadjustment that removes some of their bright edginess. E-B-F#-C#-G#-D#-A#; Fa-Do-Sol-Re-La-Mi-Ti ...where the concluding La-Mi-Ti have been slightly altered.

We now have generated all twelve notes that we need to play music on this "well-tempered"keyboard. Like the whole book of Das Wohltemperirte Clavier , the music starts with therudiments in the C major scale and it ends in B, giving us a play through everything else inbetween. Both the C major scale and the B major scale have been generated by a series of 5ths/4ths: Fa-Do-Sol-Re-La-Mi-Ti , adjusted carefully to taste so everything works out. Withthe overlapping of E and B common to both, these two seven-note scales together give us thecomplete chromatic scale of all twelve.

In review, our 5 ths are tempered as follows: F-C-G-D-A-E with the medium amount in commonpractice; then E-B-F#-C# pure; then C#-G#-D#-A# tempered with only a light amount, as acompromising average. The whole line is therefore: F-C-G-D-A-E-B-F#-C#-G#-D#-A#,typically taking either the C or the A from a tuning fork to start setting up the naturals. Bach'sdiagram has given us a map of the appropriately tasteful adjustments. The leftover interval of

A# back to F - technically a "diminished 6 th " - is not tuned directly; it is very nearly pureanyway, as a checkpoint that we have not ruined ourselves with cumulative errors. Thecomplete mapping is shown in Figure 3 (or we could read Figure 2 from right to left).

Figure 3, repeated:

In practice, sitting at the keyboard with our tuning lever and referring to Bach's diagram, thisis much simpler to do than it might appear in all the theoretical explanation above. Just dothis: wherever those "knots" appear inside the loops, make that note slightly lower than itwould sound at the pure position, by nudging the tuning lever counterclockwise to "impurify"that 5th or 4th deliberately. The picture shows the whole sequence. It takes less than fourminutes, with experience. Some doubly-nudged 5ths, then some pure 5ths, then some singly-

nudged 5ths, and we're done.

All 24 Major/Minor Scales

As a result, all 24 major and minor scales become usable and distinctive, with a smoothvariety passing through all modulations. Each key signature uses some different blend of thenotes borrowed from B major and C major, the two halves of the diagram, sharing the centralloop. If our music needs flats instead of sharps, we simply rename the entire B major scale onthe keyboard: Cb, Db, Eb, Fb, Gb, Ab, Bb, Cb. All twelve notes of the chromatic scale dodouble duty, triple duty, and more as they are used within many more scales than merely Cand B majors.

Our scale alloy makes the entire layout thoroughly flexible, and just barely irregular enoughthat it holds and renews a listener's attentions. The adjustments are all so subtle that nothingsounds unduly odd: yet there is enough spice that we avoid the immobilizing blandness of equal temperament. Risking a metaphysical comment: this temperament makes keyboard


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instruments seem magically alive and alert, in the handling of tonal music.

How does this temperament affect musical practice? Intonation can influence a sensitivekeyboard player's phrasing, dynamics, timing, articulation, and analysis of a composition.Because all the scales are differently patterned from one another, the ear can pick up clueseach time the music shifts from one scale to another. The irregularities, sensed more ashardenings or softenings than heard consciously, offer a surprise against expectations. The earis drawn to any notes that are coming into the composition from outside the current major or

minor scale of a passage, bringing attention to harmonic modulations as they happen.

This resembles CPE Bach's advice, in his essay about playing keyboard instruments, that theplayer should give such foreign tones a bit of extra weight or emphasis, showing off thesurprise. The pattern of dynamic emphasis, due to subtleties of this temperament, matchesmusical examples given by both CPE Bach's essay and the flute tutorial by his colleague atFrederick's court, Johann Joachim Quantz, instructing basso continuo players in tastefuldynamics. Notes and chords are weighted by the relative amounts of dissonance they arebringing into the musical texture, the surprise against expectations.

What musical effects emerge from this temperament, in tonal music by the Bachs and others?C major and F major make the gentlest music, and E major the brightest. The overall sound ismoderate and euphonious, gently melodic and harmonic, while drawing no attention to itself.It does not make an overwhelming presence, but rather it serves to catalyze beautiful and

natural expression. Different characters are projected with little or no expenditure of effort bythe performer.

At increasing distance from the instrument, anything beyond a meter or two, it simply soundsalmost like equal temperament: always smooth. It would have sounded especially so to peoplewho grew up in a meantone-based musical environment. It keeps meantone's relaxed,resonant effect of purity, while eliminating all the remarkably dissonant dead-ends of theexotic keys. Music seems in tune, and magically unproblematic, all the time.

Exploration

For those who wish to explore these ideas further: the full scholarly version of this is availablein Early Music (Oxford University Press) February-May 2005, and online at < www.larips.com >with the more recent news items.

This temperament's usage today

Five new CD releases during the first months of 2006 use this special keyboard tuning system,which may have been Johann Sebastian Bach's own.

"A Joy Forever: Opus 41 at Goshen College" - Bradley Lehman, organ, Taylor & Boodyinstrument in northern Indiana, USA, completed 2005. Various repertoire by Bach,Fischer, Brahms, Bhm, Walther, et al., including the complete book Ariadne musica(1715) by Fischer. LaripS 1002: Three-disc set. Available through< www.gcmusiccenter.org > and < www.larips.com >.

"Playing from Bach's Fancy" - Bradley Lehman, harpsichord and organ. Selected

compositions by Johann Sebastian Bach and Wilhelm Friedemann Bach: preludes,fugues, sinfonias, polonaises, chorale preludes, and more. LaripS 1003:< www.gcmusiccenter.org > and < www.larips.com >.

Bach: Goldberg Variations BWV 988 and 14 Goldberg Canons BWV 1087 - Richard Egarr,harpsichord. Harmonia Mundi France 907425-26: Two-disc set.< www.harmoniamundi.com >.

Anna Magdalena Bach's Book - Elizabeth Anderson, harpsichord and organ; JacobLawrence, boy soprano. Move MD 3304: < www.move.com.au >.

Bach: Well-Tempered Clavier Book 1 - Peter Watchorn, harpsichord and pedalharpsichord. Musica Omnia MO 0201: Two-disc set. < www.musicaomnia.org >.

Since spring 2004 many other musicians have been exploring this temperament in solo andensemble music, and in instrument construction. The first broadcast was a fortepiano recital


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by Robert Hill on Swiss radio, May 2004. The BBC has broadcast ensemble performances byThe English Concert and The Purcell Quartet using this in the basso continuo , and parts of thenew solo CDs mentioned above. In April 2006, Netherlands Radio presented a performance of Bach's "St Matthew Passion" using this, from a concert series by the Netherlands Bach Society.

Alongside this temperament's use in hundreds of harpsichords, fortepianos, continuo organs,modern pianos, and synthesizers, it is a lso installed in instruments that require more hardwarecommitment: at least eight larger organs, several fretted clavichords, guitars with microtonal

fingerboards, a bandoneon, and a carillon(!).

Additional adventures of this temperament are listed at < www.larips.com >.

Reference: Other keyboard tuning strategies, historically

Keyboard tuning methods fall historically into several competing styles, as to the size(s) of 5 ths

used to generate the notes. All of these styles have been brought back into use in the 20 th and

21 st centuries, for reasons that are variously historical, aesthetic, or practical.


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Pythagorean (before the 17th

century): Eleven pure 5 ths or 4 ths are easily tuned in

succession. The resulting major 3 rds are very wide. The cycle of 5 ths does not meet itself exactly; a dissonance results between the twelfth note and the first note.

Just intonation (15th

century forward): All the notes are related to one another

variously by pure 5 ths or pure major 3 rds . The available intervals and chords areoutstandingly resonant within the single home key, but strong dissonances arise in otherkeys (thereby restricting modulations). Melodies are bumpy, as the steps within thescale are of vastly different sizes.

Equal temperament (16th

century forward, but not a universal "standard" until

the 20th): The same amount of tempering is given to each of the eleven 5 ths , so the

beginning and end meet with a twelfth 5 th of the same size. All twelve notes are equallyspaced, and all scales have the same character. The tempering of the 5 th is so slight thatit is difficult to control with precise equality, in practice.

Quasi-equal (18th

century forward): These variously subtle methods make a neutraleffect similar to equal temperament's, without a s trongly recognizable character to anykey. Some of these are easier to tune than equal temperament, as they have several

pure 5 ths spaced symmetrically among their tempered 5 ths .

Meantone or "regular" systems (16th

-19th

centuries): Each tone (whole step) isplaced at an exact mean (geometric average) position within the major 3 rd , whatever itssize. For example, C-D and D-E are equally spaced within C-E. All of these meantone

systems are generated by tuning eleven identical ( regular ) 5 ths of some selected size,

until there are twelve different notes. There is a leftover gap, or "wolf" diminished 6 th ,from the twelfth back to the first note; this rift is usually placed at G#-Eb, D#-Bb, orC#-Ab. Notes such as Db, A#, and E# usually do not exist in this scheme; and theysound rough if they occur in the music. The most common meantone system had astrongly tempered 5 th so that the usable major 3 rds worked out to be pure; but from the

early 17 th century forward this was gradually relaxed toward slightly sharper major 3 rds

and gentler 5 ths , as practical compromises.

Modified meantone, or "circulating" or "irregular" or "ordinary" (17th

-19th

centuries): The series of 5 ths is regular in the midsection (on the natural notes of the Cmajor scale, ...C-G-D-A-E...), but increasingly wide toward the outsides. This sharpensthe sharps and/or flattens the flats gradually, so they can serve passably as one another,and it reduces or eliminates the "wolf" intervals. Some of these schemes also have aflattened F or a raised B, as transition into the flats or sharps.

Split keys (15th

century forward): Extra key-levers are added within the octave, e.g.,having two separate keys to play G# and Ab. Such keyboards are usually tuned in a

meantone style, taking advantage of excellent major 3 rds .

Further resources

Ledbetter, David: Bach's Well-Tempered Clavier: The 48 Preludes and Fugues. YaleUniversity Press, 2002, 352 pages. See especially chapter 2, "Well-tempered", pp.35-50. An excellent 16-page presentation of the historical and practical issues in Bach'smilieu.

Devie, Dominique: Le Temprament musical: Philosophie, histoire, thorie et pratique.Socit de musicologie du Languedoc, Bziers, 1990, 540 pages. [in French] Acomprehensive history of temperament styles, well-illustrated with hundreds of examples.

Lindley, Mark: "Stimmung und Temperatur", Hren, messen und rechnen in der frhenNeuzeit, Frieder Zaminer (ed.), vol. 6 of Geschichte der Musiktheorie, WissenschaftlicheBuchgesellschaft, Darmstadt, 1987, pp. 109-331. [in German] Another comprehensive

history.Lindley, Mark: "A Quest for Bach's Ideal Style of Organ Temperament", Stimmungen in


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17. und 18. Jahrhunderts: Blankenburg, Harz, 1994, pp. 45-67. Algebraic theory toderive Bach's presumed preferences, by averaging and modifying other systems in hismilieu.

Lindley, Mark: "J.S. Bachs Klavierstimmung", Alte Musik als sthetische Gegenwart:Bach, Hndel, Schtz: Stuttgart 1985 vol. 1, Brenreiter, Kassel, 1987, pp. 409-421.English translation "J.S. Bach's Tunings", The Musical Times vol. 126 no. 1714,December 1985, pp. 721-726.

Blackwood, Easley: The Structure of Recognizable Diatonic Tunings. Princeton UniversityPress, Princeton NJ, 1985, 318 pages. A thorough set of mathematical theoremsconcerning scale structure.

Barbour, J Murray: Tuning and Temperament: A Historical Survey. Michigan StateCollege Press, East Lansing, 1951. Reprint Da Capo Press, New York, 1973, 228 pages.Reprint Dover, New York, 2004. A classic and widely-used reference, written as Barbour'sdissertation in 1932; watch out for a number of typographical errors in his tables.

Barbour, J Murray: "Bach and the 'Art of Temperament'", Musical Quarterly vol. 33 no. 1,Nauray, 1947, pp. 64-89. Also in Garland Library of HWM vol. 6, 1985, pp. 2-27.

Steblin, Rita: A history of key characteristics in the eighteenth and early nineteenthcenturies. 1981. Reprint University of Rochester Press, Rochester NY, 1996, 396 pages.See especially the Second Edition, 2002. ISBN: 1-58046-041-0. Comprehensive study of literature and theory, about the reported effects of different musical keys.

Norrback, Johan: 'A Passable and Good Temperament' - A New Methodology for Studying Tuning and Temperament in Organ Music. PhD diss., Gteborg University,September 2002, 166 pages. GOArt Publications, Gteborg, 2002.

Bach, Carl Philipp Emanuel: Versuch ber die wahre Art das Clavier zu spielen , Berlin,1753 and several later editions. An inexpensive reprint is available from Breitkopf,1958/1992, edited by Lothar Hoffmann-Erbrecht. The standard English translation of thisbook is by William J Mitchell, Essay on the True Art of Playing Keyboard Instruments .New York: Norton, 1949. I believe that CPE Bach had the same career temperament ashis father's, or very close to it, in his ordinary use: details of that hypothesis are here .

Lehman, Bradley: articles "Bach's extraordinary temperament: Our Rosetta Stone"

(2004-5) and "The 'Bach Temperament' and the Clavichord" (2005), and the additionalcitations within both of those

Frequently Asked Questions answered at LaripS.com

Informal speech about these tuning concepts

Summary of the most common "meantone" (regular) temperaments

Remarks about the strategy of "ordinary" (or "irregular" or "modified meantone" or"circulating") temperaments

Other recordings in this temperament, by various people

Video demonstrations

Lecture notes from October 2008

Lecture notes from October 2010, University of Colorado

This is the full-length original version of this article; a shortened feature drawnfrom it appears in the BBC Music Magazine , August 2006 (Vol 14, #13), pages42-44.

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