Bach's schematic, as it appears on the page
LaripS.com, © Bradley Lehman, 2005-13, 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 the musical scale are usually so full of figures that the non-mathematician shies away from them in terror. This is unfortunate, for the subject is not only of interest to the musicologist and theorist, but of immediate and practical concern to the performer." (Barbour, "Bach and the Art of 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 basic practical matters, without calculations.

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

Perspective

There are hundreds of reasonable ways to tune keyboards for the classical repertoire. Their appropriateness depends on several questions. Do they clarify the music, aiding its beauty and strengths? 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 - or better 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 a method 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 nearly so. However, the pitches produced need not be evenly spaced (as most of us are accustomed to hearing from modern equal temperament on pianos and organs). A keyboard is merely a set 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.

"Tempering" refers to a bit of impurity that is introduced deliberately into an otherwise perfectly harmonious interval, such as a 5th or a 4th on the keyboard. The pitches are put slightly off their expected positions, giving a vibrato-like wobble when both notes are played together. There is impurity, asymmetry, and subtle variety: if used carefully, all of these features can strengthen the musical effects and enliven the sound.

It is a principle similar to fashioning objects with metal alloys, deliberately introducing impurities: the synthesized blend can be designed with a desired hardness, elasticity, strength, weight, cost, spring, lustre, colour, to be waterproof, etc. - qualities not available simultaneously in a single pure metal. A careful alloy helps the resulting tool (in our case, a musical scale instead of a bolt or a spanner) to be more durable and reliable than it would be with absolutely natural and pure elements.

Why are clever tuning schemes necessary at all, on keyboards? The short answer is: nature has not provided any definitively obvious way in which all the available material (within a musical octave) fits most neatly into twelve small packages, to serve all possible musical needs. If we try to tune everything directly as well as possible, we quickly run into dead-ends where other note combinations do not sound as good, since they were not given such direct attention. We must add some careful refinements and deliberate impurities, along the way, for the final result to work out well.

As Johann Sebastian Bach instructed his son Carl Philipp Emanuel on the topic of combining musical influences: "Though one style may on the whole be better than another, there is nevertheless something of particular value in each, and no style is so perfect that it will not suffer any additions. Through these additions and refinements we have progressed up to this point, and we shall progress even farther; but this will never come about if we pursue and, so to say, worship only one style; on the contrary, we must utilize all that is good, no matter where we may find it." I believe this applies to the structure of musical intonation, as well: blended styles of scales.

All the notes, and groups of notes, must work together reasonably. The requirements of harmony and melody are already in some conflict with one another, as to balancing resonant stasis vs forward motion through the time of a composition. Each individual pitch has relationships with (at least) the semitone, tone, minor 3rd, major 3rd, and 5th above it...and below it. All is delicate balance: the resulting set of pitches must sound musically satisfying (and, one hopes, also historically plausible) for whatever music is to be played on the instrument.

Bach's Drawing

When a musician has a nicely workable layout, it is convenient if it can be reproduced again on other occasions. I have hypothesized that Johann Sebastian Bach himself had at least one practical solution that was regularly usable...and that he wrote it down on the title page of his music most concerned with issues of tuning: his main personal copy of the Well-Tempered Clavier, containing exemplary music in all 24 keys. As a textbook for teaching Bach's Leipzig students and his own children, the WTC probes the musical effects and playing techniques of all the major and minor scales.

Figure 2:

The oddly asymmetric line drawing at the top of that page looks suspiciously like a carefully-proportioned diagram. Its irregularity suggests that it is meaningful, and not merely an unbalanced decoration. See Figure 2. Some others before me, and since, have suggested that this Bach drawing indicates something else (decorative, mathematical, musical, or otherwise). Perhaps it conveys no measurable information that is reliable enough for some people's own epistemology; or perhaps it was only fanciful for Bach's own amusement.

Or perhaps, as I believe, the drawing is a direct depiction of Bach's keyboard-tuning method. It supplies the minimum amount of sufficient information to set twelve appropriately adjusted notes to "well-tempered" taste...if one knows how to read it properly! That secret may have died with him - if there ever was one. But, we can make reasonable guesses at an appropriate harmonic/melodic balance: using as corroborative evidence the historical context plus the sound 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, and compositions 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 layout derived from this is a very simple sequence, working hands-on at a harpsichord. This method takes only a few minutes, by ear. The drawing suggests both the strategy of working on the notes, and the relative amounts of tempering necessary along the way. The crucial listening process is to sense how much of a deliberate spice or wobble is being added to an otherwise pure 5th or 4th, by knocking one of the interval's two notes slightly out of tune.

A general principle: when we tune seven notes in sequence by 5ths (or by going the opposite direction by a 4th), 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 is Do-Re-Mi-Fa-Sol-La-Ti[-Do]. This basic structure of diatonic scales is explained fully in Easley 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.

Tuning sequence of C major

In C major, this sequence is F-C-G-D-A-E-B as the 0-1-2-3-4-5-6, or Fa-Do-Sol-Re-La-Mi-Ti. The notes may go up by a 5th and/or down by a 4th, if we want to stay within a single 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 5ths pure as we go along. They might be tuned exactly pure, or slightly tempered either narrow or (less commonly) wide. The notes La, Mi, and Ti are especially open to tasteful adjustment, occurring near the end of the sequence: by the time they are generated, we can hear how they fit into the scale both melodically and harmonically, and make any slight alterations 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 5ths. The easiest rule is to make all those 5ths 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 5ths/4ths, using a careful variety of three sizes.

With regard to Bach's drawing, I take another of my conceptual models from the physical process of tuning a harpsichord: from the tuning-pin's position where an interval is sounding as 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 interval pure)? 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 an increased "twang" in the interval's quality, on very close listening.

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 then proceed to the sharps/flats to finish. The C major scale uses the common practice of medium tempering on the 5ths F-C-G-D-A-E, with a consistent size: two nudges from pure. Each 5th in turn (F-C, C-G, etc.) is made slightly too small, and this helps the quality of major 3rds (F-A, C-E, etc.) when they result from the several tempered 5ths 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 I believe this indicates a pure 5th. 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 in pitch, which helps it to serve well as a "leading tone" upward to Do. The C major notes were derived 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 17th century and earlier: 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 both ends, 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 17th century forward: to preserve as much illusion of regularity as possible, while making additional 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 extreme dissonance, and providing interesting variety among the scales and chords. The farther these systems evolved in both practice and theory, into the 18th century with dozens of examples, adjustments began to affect the naturals as well. Some of these increasingly irregular systems are occasionally described today as "well-temperament," but that is a modern (and ungrammatical!) 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 five accidentals. These are the keyboard's raised notes, usually black by contrasting color - and these notes can be found most conveniently in the B major scale, B-C#-D#-E-F#-G#-A#-B. The notes when arranged by 5ths (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, we take 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:

Tuning sequence of 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:

Tuning sequence of B major

In B major, this sequence is E-B-F#-C#-G#-D#-A# as the 0-1-2-3-4-5-6, or Fa-Do-Sol-Re-La-Mi-Ti.

The Bach diagram, continuing where we stopped before, gives us three plain loops (pure 5ths) for E-B-F#-C#, and then three single-nudge loops for the remaining C#-G#-D#-A#. Our start with several pure 5ths 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 downward adjustment 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 the rudiments in the C major scale and it ends in B, giving us a play through everything else in between. 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. With the overlapping of E and B common to both, these two seven-note scales together give us the complete chromatic scale of all twelve.

In review, our 5ths are tempered as follows: F-C-G-D-A-E with the medium amount in common practice; then E-B-F#-C# pure; then C#-G#-D#-A# tempered with only a light amount, as a compromising 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's diagram has given us a map of the appropriately tasteful adjustments. The leftover interval of A# back to F - technically a "diminished 6th" - is not tuned directly; it is very nearly pure anyway, as a checkpoint that we have not ruined ourselves with cumulative errors. The complete 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, this is much simpler to do than it might appear in all the theoretical explanation above. Just do this: wherever those "knots" appear inside the loops, make that note slightly lower than it would 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 four minutes, 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 smooth variety passing through all modulations. Each key signature uses some different blend of the notes borrowed from B major and C major, the two halves of the diagram, sharing the central loop. If our music needs flats instead of sharps, we simply rename the entire B major scale on the keyboard: Cb, Db, Eb, Fb, Gb, Ab, Bb, Cb. All twelve notes of the chromatic scale do double duty, triple duty, and more as they are used within many more scales than merely C and B majors.

Our scale alloy makes the entire layout thoroughly flexible, and just barely irregular enough that it holds and renews a listener's attentions. The adjustments are all so subtle that nothing sounds unduly odd: yet there is enough spice that we avoid the immobilizing blandness of equal temperament. Risking a metaphysical comment: this temperament makes keyboard instruments seem magically alive and alert, in the handling of tonal music.

How does this temperament affect musical practice? Intonation can influence a sensitive keyboard 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 clues each time the music shifts from one scale to another. The irregularities, sensed more as hardenings or softenings than heard consciously, offer a surprise against expectations. The ear is 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 the player should give such foreign tones a bit of extra weight or emphasis, showing off the surprise. The pattern of dynamic emphasis, due to subtleties of this temperament, matches musical examples given by both CPE Bach's essay and the flute tutorial by his colleague at Frederick's court, Johann Joachim Quantz, instructing basso continuo players in tasteful dynamics. Notes and chords are weighted by the relative amounts of dissonance they are bringing 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 is moderate 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 by the performer.

At increasing distance from the instrument, anything beyond a meter or two, it simply sounds almost like equal temperament: always smooth. It would have sounded especially so to people who 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 the exotic 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 available in 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 & Boody instrument in northern Indiana, USA, completed 2005. Various repertoire by Bach, Fischer, Brahms, Böhm, 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; Jacob Lawrence, boy soprano. Move MD 3304: <www.move.com.au>.
  • Bach: Well-Tempered Clavier Book 1 - Peter Watchorn, harpsichord and pedal harpsichord. Musica Omnia MO 0201: Two-disc set. <www.musicaomnia.org>.

Since spring 2004 many other musicians have been exploring this temperament in solo and ensemble music, and in instrument construction. The first broadcast was a fortepiano recital by Robert Hill on Swiss radio, May 2004. The BBC has broadcast ensemble performances by The English Concert and The Purcell Quartet using this in the basso continuo, and parts of the new 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 also installed in instruments that require more hardware commitment: 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 5ths used to generate the notes. All of these styles have been brought back into use in the 20th and 21st centuries, for reasons that are variously historical, aesthetic, or practical.

  • Pythagorean (before the 17th century): Eleven pure 5ths or 4ths are easily tuned in succession. The resulting major 3rds are very wide. The cycle of 5ths 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 5ths or pure major 3rds. The available intervals and chords are outstandingly resonant within the single home key, but strong dissonances arise in other keys (thereby restricting modulations). Melodies are bumpy, as the steps within the scale 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 5ths, so the beginning and end meet with a twelfth 5th of the same size. All twelve notes are equally spaced, and all scales have the same character. The tempering of the 5th is so slight that it is difficult to control with precise equality, in practice.
  • Quasi-equal (18th century forward): These variously subtle methods make a neutral effect similar to equal temperament's, without a strongly recognizable character to any key. Some of these are easier to tune than equal temperament, as they have several pure 5ths spaced symmetrically among their tempered 5ths.
  • Meantone or "regular" systems (16th-19th centuries): Each tone (whole step) is placed at an exact mean (geometric average) position within the major 3rd, whatever its size. 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) 5ths of some selected size, until there are twelve different notes. There is a leftover gap, or "wolf" diminished 6th, from the twelfth back to the first note; this rift is usually placed at G#-Eb, D#-Bb, or C#-Ab. Notes such as Db, A#, and E# usually do not exist in this scheme; and they sound rough if they occur in the music. The most common meantone system had a strongly tempered 5th so that the usable major 3rds worked out to be pure; but from the early 17th century forward this was gradually relaxed toward slightly sharper major 3rds and gentler 5ths, as practical compromises.
  • Modified meantone, or "circulating" or "irregular" or "ordinary" (17th-19th centuries): The series of 5ths is regular in the midsection (on the natural notes of the C major scale, ...C-G-D-A-E...), but increasingly wide toward the outsides. This sharpens the 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 a flattened 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 3rds.


Further resources

  • Ledbetter, David: Bach's Well-Tempered Clavier: The 48 Preludes and Fugues. Yale University 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's milieu.
  • Devie, Dominique: Le Tempérament musical: Philosophie, histoire, théorie et pratique. Société de musicologie du Languedoc, Béziers, 1990, 540 pages. [in French] A comprehensive history of temperament styles, well-illustrated with hundreds of examples.
  • Lindley, Mark: "Stimmung und Temperatur", Hören, messen und rechnen in der frühen Neuzeit, Frieder Zaminer (ed.), vol. 6 of Geschichte der Musiktheorie, Wissenschaftliche Buchgesellschaft, Darmstadt, 1987, pp. 109-331. [in German] Another comprehensive history.
  • Lindley, Mark: "A Quest for Bach's Ideal Style of Organ Temperament", Stimmungen in 17. und 18. Jahrhunderts: Blankenburg, Harz, 1994, pp. 45-67. Algebraic theory to derive Bach's presumed preferences, by averaging and modifying other systems in his milieu.
  • Lindley, Mark: "J.S. Bachs Klavierstimmung", Alte Musik als ästhetische Gegenwart: Bach, Händel, Schütz: Stuttgart 1985 vol. 1, Bärenreiter, 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 University Press, Princeton NJ, 1985, 318 pages. A thorough set of mathematical theorems concerning scale structure.
  • Barbour, J Murray: Tuning and Temperament: A Historical Survey. Michigan State College 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's dissertation 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 nineteenth centuries. 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., Göteborg University, September 2002, 166 pages. GOArt Publications, Göteborg, 2002.
  • Bach, Carl Philipp Emanuel: Versuch über 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 this book 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 as his 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 additional citations 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 drawn from it appears in the BBC Music Magazine, August 2006 (Vol 14, #13), pages 42-44.


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