------------------------------------------------------------- Lehman's Bach tuning and Neidhardt's "Third-Circle No. 4" A quick comparison with open questions ------------------------------------------------------------- In Bradley Lehman's article "Bach's extraordinary temperament: Our Rosetta Stone - 1," (_Early Music_, Vol. 33 No. 1, February 2005), available at his Web site , an ornamental design on the title page of Bach's _Well-Tempered Klavier_ (1722) is interpreted as a diagram showing the great composer's preferred keyboard temperament. As musicians and scholars take up Lehman's invitation to weigh both the historical plausibility of this proposal for Bach's tuning and its artistic effect when applied to Bach's music, I would like to call attention to a temperament from this same epoch with some striking similarities and contrasts: Johann Georg Neidhardt's Third Circle No. 4. My source for Neidhardt's tuning is James Murray Barbour's _Tuning and Temperament: A Historical Survey_ (East Lansing: Michigan State College Press, 1953), p. 169 (Table 157), where a previous footnote (p. 167 n. 38) apparently applying to this tuning also cites Neidhardt's _Sectio canonis harmonici_ of 1724 (only two years after Bach's WTK). ------------------------------- 1. A tale of two tuning circles ------------------------------- In his article, Lehman reads the different types of loops in Bach's design on the WTK title page as showing the degree of tempering for each fifth n a 12-note circle. Thus a doubly-ornamented loop shows a fifth narrowed by 1/6 of a Pythagorean comma; a singly-ornamented loop, by 1/12 comma; and a simple loop, an untempered fifth at a pure ratio of 3:2 (about 701.955 cents). Reading the artwork upside down, and taking one symbol to show the location of the note C in the chain or circle, he arrives at this temperament: [fixed-width or nonproportional font recommended] F C G D A E B F# C# G# D# A# (E#/F) -1/6 -1/6 -1/6 -1/6 -1/6 0 0 0 -1/12 -1/12 -1/12 +1/12 In this scheme the five nearest fifths of the circle F-C-G-D-A-E are each tempered narrow by 1/6 Pythagorean comma; the three fifths E-B-F#-C# are left pure; the three fifths C#-G#-D#-A# are tempered by 1/12 comma; and the fifth joining the ends of the chain, A#/Bb-F, is consequently wide by 1/12 comma. As Lehman notes, this scheme is like a typical well-temperament of the variety defined by a modern theorist such as Owen Jorgensen[1] and exemplified by a tuning such as Werckmeister III insofar as it most heavily tempers the nearest fifths of the circle, thus achieving in this portion of the circle the major thirds closest to a pure ratio of 5:4 (about 386.31 cents). However, it differs in two important regards. First, unlike in the Jorgensen paradigm where all fifths are either pure or tempered narrow, here the fifth A#/Bb-F joining the two ends of the chain and closing the circle is 1/12 Pythagorean comma _wider_ than pure. This is an interesting case of a tuning with a wide fifth where all major thirds are smaller than the Pythagorean 81:64 (about 407.82 cents) -- unlike a typical French _temperament ordinaire_ of this era, where some major thirds exceed this size. Secondly, while a well-temperament of the Werckmeister/Jorgensen type places the largest and most tense or active major thirds in the most remote positions, where they would be spelled as diminished fourths in a traditional Eb-G# meantone (B-Eb, F#-Bb, C#-F, G#-C), here the largest major third is E-G# at 1/12 comma narrower than Pythagorean (i.e. about 405.865 cents). Lehman argues that this different key-color scheme or "shape" better fits Bach's tonality than the familiar "modified meantone" pattern. Neidhardt's Third-Circle No. 4 strikingly uses identical fifth sizes and numbers of fifths in each size, but differs in their arrangement, at once generally adhering to a familiar "modified meantone" shape and gently moderating that shape. Here, following Lehman's ordering of the fifths based on his reading of the WTK design, I show the two temperaments together: Lehman reading of WTK F C G D A E B F# C# G# D# A# (E#/F) -1/6 -1/6 -1/6 -1/6 -1/6 0 0 0 -1/12 -1/12 -1/12 +1/12 Neidhardt's Third-Circle No. 4 F C G D A E B F# C# G# D# A# (E#/F) 0 -1/6 -1/6 -1/6 0 -1/6 0 -1/12 -1/12 -1/12 +1/12 -1/6 In Neidhart's circle, as in Lehman's, the most heavily tempered fifths and the major thirds nearest to a pure 5:4 occur in the nearer portion of the circle, here including the fifths Bb-F and C-G-D-A and the thirds Bb-D, F-A, C-E, and G-B. Neidhart's arrangement, however, has the familiar attribute that the largest major thirds (F#/Gb-Bb and G#/Ab-C) are located in the remote portion of the circle where diminished fourth spellings would apply in Eb-G# meantone; these thirds are 1/6 comma narrower than Pythagorean (about 403.91 cents). While following this convention, Neidhardt both moderates and adds interesting nuances to the traditional contrast between nearer or "regular meantone" and remote regions: as in Lehman's reading of the WTK design, the thirds A-C# and C#/Db-F are identically tempered (here 1/4 comma narrow in, or about 401.955 cents, and 1/6 comma narrow of Pythagorean scheme, or about 403.91 cents, in Lehman's Bach scheme). Thus both schemes share a 12-note circle based on five fifths narrowed by 1/6 comma, three narrowed by 1/12 comma, three left pure, and one fifth wide by 1/12 comma. However, the two schemes arrange these fifths differently, producing similarities and contrasts which I will here briefly attempt to discuss in a bit more detail. Before launching into this discussion, however, I should provide definitions of both temperaments in the format for Scala, an excellent and free scale generation and analysis program by Manuel Op de Coul available on the Internet at ! Lehman_WTK.scl ! Bradley Lehman's reading of Bach's WTK title page spiral design (1722) 12 ! 98.04500 196.09000 298.04500 392.18000 501.95500 596.09000 698.04500 798.04500 894.13500 998.04500 1094.13500 2/1 ! Neidhardt_Third-CircleNo4.scl ! Johann Georg Neidhardt's "Third-Circle No. 4" 5 -1/6P, 3 -1/12, 3 J, 1 +1/12 12 ! 96.09000 196.09000 296.09000 396.09000 4/3 596.09000 698.04500 796.09000 894.13500 1000.00000 1094.13500 2/1 ---------------------------------------------------------------- 2. Some comparisons in Brombaugh's "temperament units" and cents ---------------------------------------------------------------- In 18th-century circulating temperaments, as Lehman nicely explains, we have two paramount considerations: the quality of the fifths and the need to have them form a closed circle where all 12 are "playable"; and the quality of the thirds, with the nearer ones ideally at or not too far from a pure 5:4. These two considerations bring into play respectively the Pythagorean and syntonic commas. While only musical performance and experience can show how apt a tuning scheme such as Lehman's reading of Bach or Neidhardt's Third-Circle No. 4 is to Bach's music, a consideration of these commas helps us in analyzing the "vital statistics" of the two temperaments. The Pythagorean comma, a natural unit to use in describing a 12-note circulating temperament, is the amount by which 12 fifths at 3:2 exceed seven pure octaves at 2:1, a difference of 531441:524288 or about 23.46 cents. Thus in a 12-note circle, the net narrowing of all the fifths must add up to this amount in order to balance the "temperament budget." Lehman by the way, has developed an educational spreadsheet to assist in this process, . Thus in either the Lehman WTK or Neidhardt tuning, we have three pure fifths plus eight narrowed by a total of 13/12 Pythagorean comma (1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/12 + 1/12 + 1/12), so that the remaining fifth must be 1/12 comma _wide_ to balance out the circle. The syntonic comma, a typical unit for measuring temperament in regular meantone tunings, is the difference between the rather active and complex Pythgorean major third formed from four pure 3:2 fifths (e.g. C-G-D-A-E), 81:64 or about 407.82 cents, and a pure 5:4 third at about 386.31 cents. The difference is 81:80, about 21.51 cents. While active Pythagorean thirds are nicely suited to many 13th-14th century Gothic styles where these intervals are regarded as relatively blending but unstable, often expanding for example to stable fifths, thirds at or near a pure 5:4 ratio are the ideal for late 15th-17th meantone temperaments, and remain the norm for the nearest keys of an 18th-century "modified meantone" or well-temperament. For more background see, e.g., . In order to achieve near-pure thirds in the nearer portion of the circle while balancing out the fifths, the general approach in a well-temperament is to temper the nearer fifths (or many of them) rather heavily, so that a chain of four will remove a large portion of the syntonic comma, a procedure typically using up most (as here) or all of the allotted one Pythagorean comma net narrowing for the whole circle. In many schemes such as Werckmeister III, most of the remaining fifths remain pure, producing Pythagorean or near-Pythagorean thirds in the remote part of the circle. While a necessary compromise of the budgeting process, these tense and active thirds are often seen as a coloristic resource of the furthest keys, giving them a special "character." In the Lehman WTK reading or Neidhardt's scheme, most of the temperament budget is expended in the nearer part of the circle by tempering five fifths at 1/6 Pythagorean comma, or 5/6 comma in all, but an additional 3/12 comma is spent in ameliorating further portions of the circle by narrowing three fifths at 1/12 comma each -- thus slightly overrunning the budget, with the remaing fifth at 1/12 comma wide redressing this balance. While the 23.46-cent Pythagorean comma is our basic "budgeting" unit, the 21.51-cent syntonic comma is the unit for measuring the quality of the thirds. Lehman (e.g. "Bach's extraordinary temperament," p. 5, and in his temperament spreadsheet) ably adopts the approach of organ builder John Brombaugh, who has ingeniously devised a most intuitive unit for understanding the relationship between these two relevant quantities: the "temperament unit" or TU. In this system, a Pythagorean comma is taken as having 720 TU, and a syntonic comma as having 660 TU, a rounding which reflects their actual sizes with the latter almost exactly 11/12 of the former. As Lehman points out, these handy units are not only intuitive and conveniently divisible by various factors but incredibly accurate. Here we'll abbreviate Pythagorean comma as PC, and syntonic comma as SC. For example, in Lehman's WTK reading the thirds F-A and C-E are formed from four fifths each tempered by 1/6 PC or 120 TU, for a total of 2/3 PC or 480 TU. This is 480/660 or 8/11 SC, the amount that these thirds are narrower than the Pythagorean 81:64. Since the difference between 81:64 and a pure 5:4 is one SC, this means that the thirds are only 3/11 SC wider than 5:4, thus rather nicely fulfilling the meantone ideal for the nearer keys. In Neidhardt's tuning, many of the nearer fifths are also tempered by 1/6 PC or 120 TU, but with some pure fifths interspersed so that F-A and C-E get three 1/6 PC fifths plus a pure fifth (pure F-C plus tempered C-G-D-A, or C-G-D-A plus pure A-E). Thus the overall tempering is 1/2 PC or 360 TU, which means 360/660 or 6/11 SC, the amount by which these thirds are narrower than Pythagorean -- leaving them 5/11 SC larger than 5:4, a bit less "ideal" than in Lehman's tuning. For people accustomed to the cents measurement, giving the size of these thirds at respectively about 392.18 cents and 396.09 cents conveys similar information: the first size in within what might be called a charcteristic meantone range, while the second is rather larger, in the region about midway between 5:4 and 81:64. While Neidhart's scheme requires greater compromises for the nearest thirds, it mitigates the tension of the largest or most impure ones, here F#/Gb-Bb and G#/Ab-C. For the first, the fifths F#-C#-G#-D#/Eb are each tempered by 1/12 PC, while D#/Eb-A#/Bb is wide by 1/12 PC, for a net narrowing of 1/6 PC or 120 TU; for the second, G#-D#/Eb is narrow by 1/12 PC, D#/Eb-A#/Bb wide by the same amount, Bb-F narrow by 1/6 PC, and F-C pure, again a net narrowing of 1/6 PC or 120 TU. These thirds are thus narrowed by 120/660 or 2/11 SC, remaining wide of 5:4 by 9/11 SC -- fairly near Pythagorean, but somewhat milder. In Lehman's WTK reading, the widest major third E-G# has a chain with the fifths E-B-F#-C# pure and C#-G# at 1/12 PC or 60 TU narrow, and is thus 60/660 or 1/11 SC narrower than Pythagorean, and 10/11 SC wider than 5/4, getting close to Pythagorean although still a bit milder. Modern composer and theorist Easley Blackwood regards this size as around the limit for acceptable major thirds in an 18th-19th century tonal style, in contrast to the more active thirds of a medieval context, although some theorists of Bach;s era are ready to accept and indeed relish the color of full Pythagorean thirds in the most remote keys. Again, people accustomed to cents would find the respective sizes of around 403.91 cents and 405.865 cents helpful in comparing these third sizes; I find both the cent and TU measurements useful and communicative. On this basis, we can survey the "shapes" of the two systems by comparing the temperings and transitions of the major thirds, listed side by side in TU and cents, here starting at C-E and moving flatward until we pass through the traditional meantone region of diminished fourth spellings and then into the region with traditional regular spellings involving sharps, eventually arriving at G-B to complete our tour. Here "L" is Lehman's WTK reading and "N" is Neidhardt's Third-Circle No. 4: Third Tempering TU Size Cents Impurity TU Impurity Cents L N L N L N L N ---------------------------------------------------------------------- C-E -480 -360 392.180 396.090 180 300 5.866 9.776 F-A -480 -360 392.180 396.090 180 300 5.866 9.776 Bb-D -300 -360 398.045 396.090 360 300 11.731 9.776 Eb-G -240 -180 400.000 401.955 420 480 13.686 15.641 G#/Ab-C -180 -120 401.955 403.910 480 540 15.641 17.596 C#/Db-F -120 -180 403.910 401.955 540 480 17.596 15.641 F#/Gb-Bb -180 -120 401.955 403.910 480 540 15.641 17.596 B-D#/Eb -120 -180 403.910 401.955 540 480 17.596 15.641 E-G# - 60 -240 405.865 400.000 600 420 19.551 13.686 A-C# -120 -180 403.910 401.955 540 480 17.596 15.641 D-F# -240 -240 400.000 400.000 420 420 13.686 13.686 G-B -360 -360 396.090 396.090 300 300 9.776 9.776 The "Tempering TU" column shows how much each third is narrowed from a Pythgorean 81:64, while the "Impurity TU" column shows how wide it is with respect to a pure 5:4, am amount equal to 660 TU or 1 SC less the amount of narrowing; the size and impurity of each third is also shown in cents. In this table we have in traditional meantone terms four regularly spelled major thirds, followed by four diminished fourth spellings (reflected in the dual names for certain degrees), followed by four more regular spellings: thus the four most "remote" thirds are graphically in the middle of the table. The Lehman WTK reading and Neidhardt scheme handle these four thirds almost exactly alike, with alternating sizes of 480 and 540 TU, or about 15.64 and 17.60 cents, wide of 5:4 (about 401.955 and 403.910 cents). Two notable differences occur: the greater impurity of the best "meantone-like" fifths in Neidhardt, at 300 TU or about 9.776 cents wide of 5:4, in contrast to 180 TU or about 5.866 cents in the Lehman WTK tuning; and the decidedly gentler treatment of E-G# at 420 TU or about 13.686 cents from 5:4, in contrast to 600 TU or about 19.551 cents in Lehman's reading of Bach. This latter point, as Lehman emphasizes, is a significant distinction in color or shape. As he also emphasizes, his reading provides two best thirds, F-A and C-E, tempered as in 1/6 (Pythagorean) comma meantone, often espoused by theorists and likely familiar to many musicians in practice.[2] Thus his tuning actualizes one aspect of the meantone ideal as adapted to the tastes of the 18th century (see Lehman, "Bach's extraordinary temperament," pp. 7 and 22 n. 20, noting that his forthcoming Part 2 will address some relevant theorists in more detail). Neidhardt's circle, while its smallest major thirds are rather larger than in a traditional meantone (somewhere between those found in 1/7-SC and 1/8-SC regular temperaments), might be seen as fulfilling a different aspect of the meantone ideal by providing four major thirds closer to 5:4 than to the Pythagorean 81:64, or in other words impure by less than 1/2 SC (330 TU or about 10.75 cents). These thirds, Bb-D, F-A, C-E, and G-B, all have the size noted above of about 396.090 cents, 300 TU or about 9.776 cents wide of 5:4. Lehman's Bach tuning, in addition to the more nearly pure F-A and C-E, has G-B at the same size as these best thirds of Neidhardt's circle -- or three major thirds in all within 1/2 SC of 5:4. To sum up, both tunings show a very similar treatment of the traditionally most "remote" transpositions or keys; but the two schemes differ in their "coloring," with Neidhardt more ready substantially to compromise the nearest thirds, and notably gentler in his treatment of E-G#. The system for the most part maintains the general "meantone shape" (or Jorgensen-type key color gradation), while evening out both ends of the spectrum. Thus major thirds range from 300 to 540 TU or 9.8-17.6 cents wide (or sizes of 396.0-402.0 cents). Compare the Lehman WTK range from 180 to 600 TU or 5.9-19.6 cents of impurity (or sizes of 392.2 to 405.9 cents). My purpose here is only to sketch a quick comparison of the two temperaments, and to raise some questions which might relate to the issues both of historical likelihood and of appraisal through musical performance. --------------------------------------------- 3. Some open questions of practice and theory --------------------------------------------- In presenting his intonational interpretation of the WTK title page, Lehman argues that this scheme provides a viable and indeed most attractive alternative to the more familiar paradigms either of a 12-note equal temperament (indeed questionable as a standard period practice for keyboards on both technical and aesthetic grounds) or of a "modified meantone" variety of well-temperament like Werckmeister III. Without attempting to judge the historical and calligraphic issues regarding this reading of the WTK design, I do find Neidhardt's Third-Circle No. 4 of interest as a tuning from this very epoch which I might describe as an "equable temperament,"[3] maintaining the general outline of a meantone-like shape while moderating the contrasts to achieve a smooth graduation. This is not to say that such an approach is more or less musical than one such as Werckmeister III, nor that it is more or less suited to Bach's music, only that it is a noteworthy one to consider in these dialogues on Bach and tuning. In his article (_Early Music_, February 2005, p. 15), Lehman tells the story of a contest to tune an organ in "absolutely equal temperament" in which Neidhardt, using a mathematically prepared monochord, was bested by his young colleague Nikolaus Bach "setting his set of 8' flutes entirely by ear." He suspects that Neidhardt "did achieve equal temperament or something indistinguishably close to it," while Bach sought and attained a temperament that "'_seems_ equal'" (Lehman's emphasis) to "all but the most finicky keyboard geeks, but is more musically pleasing." He accordingly concludes that the temperament of his WTK reading "seems equal _in effect_," as judged by many "friends and colleagues": it is "so suave that it _is_ equal, in musical context... unless one listens very closely with the technical skills of a keyboard tuner by ear." Here I am again tempted to substitute the term "equable temperament" in order to emphasize the idea of an artistic as opposed to strictly mathematical "equability" or "evenness." A possible analogy might be to the typographical practice of pair kerning, in which the regular spacings between certain pairs of characters are adjusted to obtain an effect of _visually even_ spacing (for example, often involving a reduction in the spacing of pairs such as "AW" or "Aw" in a proportional font). While his WTK reading is one temperament that might satisfy this ideal of "equability," and indeed he suggests that Nikolaus Bach may have won his contest with Neidhardt using this system, which was thus a "family heirloom" by the time of Johann Sebastian's great publication of 1722, the Neidhardt temperament of 1724 seems yet more equable and subtle in its gradations, while still maintaining some key color and indeed following a traditional meantone shape for the most part. Lehman (p. 13) contrasts his Bach temperament with oft-cited period schemes where "the traditional mean-tone extremities are harsher than E major is." Neidhardt's temperament presents an interesting case where these traditional "extremities" indeed include the widest thirds (at F#/Gb-Bb and G#/Ab-C), but are at the same degree of temperament as in Lehman's WTK reading, and indeed milder than his E-G#. Of course, to borrow Lehman's apt term, a judgment of equability or musical evenness may also involve a "suave" quality: not only the theoretically smooth gradation but also the harmoniousness of the thirds. This raises the question: "Are Neidhardt's nearest thirds narrowed enough to fit the ideals of an era still steeped in the practice and theory of meantone, regular or `modified' in various ways?" Neidhardt's scheme seems in some ways a kind of middle road between the kind of meantone shape with dramatically polarized key colors exemplified by Werckmeister III, and the ingenious irregular tunings of Neidhardt and others that closely approximate precise equal temperament (e.g. evenly alternating six fifths narrowed by 1/6 PC and six pure fifths), as surveyed by Barbour, pp. 170-174, as well as Jorgensen (1991) in an impressive catalogue of such quasi-equal methods. Whatever the judgment of scholars as to the WTK design, Lehman has done a great service by inviting us to consider the variety of Bach-era tunings which _might_ have been used to play his music -- likely, then as now, with a range of aesthetic appraisals. A scheme with the nearer keys comparable to 1/7-(syntonic)-comma, and the most remote keys rather milder than Pythagorean, is possibly a scenario not so often explored. Neidhardt's Third-Circle No.4, using fifth sizes and numbers of fifths in each size identical to Lehman's proposed Bach tuning, but differently arranged, shows that this kind of scenario was known and in print in 1724, and invites trial through performance and discerning appraisal of the kind which Lehman laudibly both practices and encourages. ------------ Notes ------------ 1. See, e.g. Owen Jorgensen, _Tuning the Historical Temperaments By Ear: a manual of eighty-nine methods for tuning fifty-one scales on the harpsichord, piano, and other keyboard instruments_ (Northern Michigan University Press: Marquette, 1977); and _Tuning. Containing: The perfection of eighteenth- century temperament, The lost art of ninteenth-century temperament and the science of equal temperament. Complete with instructions for aural and electronic tuning_ (Michigan State University Press: East Lansing, 1991). 2. As was recently discussed in rec.music.early, the common label "1/6-comma meantone" for a regular temperament tends to draw the inference of 1/6-syntonic comma, since this is a usual unit for measuring such temperaments where the quality of the thirds is the main concern. Thus it is good practice to specify that the relevant comma is in fact Pythagorean, or possibly unspecified (as Barbour, p. 42, reports for Sorge's 1748 description of Silbermann's meantone). The 55-note equal division of the octave favored by some leading Bach-era theorists has a fifth tempered about 0.14 cents larger than in a 1/6-PC temperament, but 0.19 cents smaller than a 1/6-SC temperament. Jorgensen suggests a "1/6-comma" tuning taking the comma as the intermediate step of 1/53 octave (about 22.64 cents or a rounded 695 TU), and thus tempering the fifths by 1/318 octave, which produces a fifth differing according to the Scala program by only about 0,00040 cents from 32/55 octave. 3. I borrow this term from the theory of just intonation based on integer ratios, where Ptolemy's Equable Diatonic with its tetrachord of 12:11:10:9 exemplifies the idea of a division with steps equably or gently graduated in size. Most appreciatively, Margo Schulter mschulter@calweb.com mschulter@www.bestii.com