Number Symbolism in Bach's Vocal Works
Discussions - Part 3

Continue from Part 2

BWV 29 -Bach & Bar Numbers

Continue of discussion from: Cantata BWV 29 - Discussions Part 2

Peter Smaill wrote (August 15, 2008):
BWV 29 -Bach & Bar Numbers

Douglas Cowling wrote:
<< "Gratias/Dona," which is adapted from "Wir Danken," is perhaps the most dramatic of them all. The first trumpet does not enter until Bar 16 and then merely doubles the soprano line. At bar 31 (with only 16 bars until the end), Bach brings the brass and timpani with new independent contrapuntal lines which take the first trumpet up to a high D. >>
Stephen Benson wrote:
< Note that, although taken from BWV 29, the Gratias and Dona choruses in the BMM (BWV 232) differ in measure numbers from BWV 29 (see the introduction to BWV 29 and Doug's recent post) because of the difference in time signatures. Same measures; different numbers. >
Stephen raises the question of time signature variance in the transposition of BWV 29/2 by parody to the BMM (BWV 232). Why did Bach bother to do this?

It is curious that Bach changes the time notation such that BWV 29/2 has 92 bars of 4 crotchets, whereas the Gratias and Dona Nobis Pacem in the BMM (BWV 232) each have the same music expressed in 46 bars of 8 crotchets. This suggests that there may be a structural cause, following the amazing discoveries by Ruth Tatlow regarding the B Minor Mass (BWV 232).The arguments are complex but also compelling from a number-sceptical source and I touch on some of the most striking analyses.

In the BMM (BWV 232) the Kyrie/Christus/Kyrie has 270 bars in stile antico notation, a Trinitarian number.

The Symbolum Nicenum in the original setting through to the Osanna has exactly 1100 bars; the remainder (including the BWV 29 transpositions) precisely 300, bringing up 1400 bars.So if Bach had left the transpositions in the original notation, his overall bar symmettry proportions would have been disturbed.

By comparison,there are precisely 2800 bars in the SMP (BWV 244).

Now, is there any significance for BWV 29? This late work is directly related to the BMM (BWV 232); could Bach have been in it also working out bar number proportions?

The answer is (work in progress!) up to a point. BWV 29/1 has 138 bars; BWV 29/2 as stated has 92, or 46 in the BMM (BWV 232) setting. This is the exact ratio of 3 to 2 as set, or of course 3 to 1 in the BMM (BWV 232) setting (i.e., 46x3=138).

As for the rest of BWV 129, the complications (also in the BMM (BWV 232) research) are how to count da capos and part bars. I have'nt figured anything more yet but there could be a link....note that in the lute transcription of the related BWV 1006/1 Bach extends by a further bar to 139, also a number of Trinitarian appearance. (1-3-9) Or is the last extra bar just a final chordal strum?

Douglas Cowling wrote (August 16, 2008):
Peter Smaill wrote:
< The Symbolum Nicenum in the original setting through to the Osanna has exactly 1100 bars >
The Credo ends with the Amen of "Et exspecto". The Osanna forms a four movement ABCB "cantata": Sanctus - Osanna - Benedictus - Osanna.

Peter Smaill wrote (August 16, 2008):
[To Douglas Cowling] Yes indeed the break in bar structures in the BMM (BWV 232) which creates 1100 bars then 300 is not in an obvious place, since the Osanna is generally classed as Part 1 of section IV; but the numeric split is nonetheless there and I feel beyond coincidence taken together with all the other numeric patterns. The overall total of 1400 in the original casting of the BMM (BWV 232), and its relation to the SMP (BWV 244) at 2800 bars, plus the B-A-C-H number alphabet value of 14, are there to see; but as to the implications, if any.........? Tatlow herself is quite reticent in the absence, as often with Bach, of direct evidence. Her full dissertation is considerably more complex than the surface points which I've led in this discussion of the adaptation of BWV 29/2.

If one accepts that Bach was conscious of bar totals as an aid to a sort of divine,or simply aesthetic, proportionality, then the parody method (by which nearly all the content of the BMM (BWV 232) is from preexistent Cantatas (and those mainly with precise theological resonance I argue), demonstrates to an even greater degree than imagined the compositional skill of Bach. Add to this the chiastic patterns in the BMM (BWV 232).

The source or more likely reflection of the technique is in Mattheson:

"What is required first and foremost for disposition is a clever ordering of all the parts of the melody or complete musical work, almost in the way that one constructs a building and designs a ground plan, making a sketch or draft, to show where for example a lounge, a study, a chamber and so on should be." (Kern Melodischer Wissenschaft, (Hamburg, 1737)).

"In my opinion, to get the very best results,with every part measured,the composer should set out each part with the others in an exact proportion and correspondence...there is nothing more satisfying in the world to the ear than this" (Der Vollkommene Capellmeister (Hamburg, 1739)).

Douglas Cowling wrote (August 16, 2008):
Peter Smaill wrote:
< "In my opinion, to get the very best results,with every part measured, the composer should set out each part with the others in an exact proportion and correspondence...there is nothing more satisfying in the world to the ear than this" (Der Vollkommene Capellmeister (Hamburg, 1739)). >
Mattheson must be talking in generalities here, or referring to structures such as a da capo aria. I find hard to accept that any listener can consciously detect bar numbers. I can tell when a da capo has been shortened, but I would have to get out scores of Cantata BWV 29 and BMM (BWV 232) to tell you where there are difference between "Wir Danken", "Gratias" and "Dona Nobis".

Peter Smaill wrote (August 16, 2008):
[To Douglas Cowling] I think Doug has put his finger on the pulse of the numerological issue; despite what Mattheson says, which is vaguely true in that some balance of movement length and structure to dramatic balance?is perceptible, the deployment of numerology does not in the detail that emerges from these types of study actually aid the hearing of the works. It is really only in the baroque mind with its emphasis on celestial unity and mental love of hermeneutics that these analyses have a point.

Ever since Smend's "J S Bach bei seinen Namen begrufen" in 1950 the numerological work has been characterised by hostility from an aesthetic point of view because it lies in the background to the compositional process, and not really in the auditory impact analysis which dominates discussion of music.By contrast,?other hermeneutical features, such as dissonance, abnormal drops in pitch,instrumentation ,walking bass etc. do have an auditory impact. But?bar numerology does begin to answer compositional questions such as , "Why did Bach change the time signature in the transpofrom BWV 29?".

Julian Mincham wrote (August 16, 2008):
[To Peter Smaill] Back to the age old debate of whether it is necessary to 'hear' or detect a musical process for it to be valid. Do we hear--i.e.detect aurally?inverted crab canons without the aid of a score? or the permutations of the row in serial compositions? Or Bartok's use of the golden section?

Do we not detect and respond to the effect the music has rather than the processes from which it is generated? These may often be private and personal methods by which a composer stimulates his/her own creative thinking. I would hazard a guess that Bach would not have expected his congregations to have absorbed much of the musical processs from one hearing of a cantata (possibly repeated every five years!) and without the aid of a score.

It would follow from this that the musicological detection of such processes may tell us something of considerable interest?about the composer's methods but do very little to enhance the emotional effect of the music for the listeners be they amatuer or professional????

Douglas Cowling wrote (August 16, 2008):
Julian Mincham wrote:
< Back to the age old debate of whether it is necessary to 'hear' or detect a musical process for it to be valid. Do we hear--i.e.detect aurally inverted crab canons without the aid of a score? or the permutations of the row in serial compositions? Or Bartok's use of the golden section? >
I don't doubt for a moment that the expereince of a piece of music is greater than its aural component. "Eye music" has been around since the Middle Ages. An obvious example is Machaut's "Dans ma fin est mon commencement" in which the circularity of the text and music was expressed by notating the music in a circle: an effect which was apparent only to the performer. Closer to home, in Handel's "O Thou That Tellest" in "Messiah", the alto sings "Get thee up into the high mountain", and the violins have a little rising and falling figure which literally draws a picture of a mountain range on the musical staff.

Then there are the effects which are only apparent in the full score. In the Sanctus of the B Minor, it is a revelatory moment when you first look at the full score and see all those Trinitarian threes: three groups of three instruments (oboes, trumpets, strings) and the 3x2 six-part choir. That indicates to me that the conductors, students and copyists who worked with the full score were given an even deeper expereince of the music through this numerical symbolism.

But I rarely count measures in a piece of music, and so I have always been rather disinterested in the numerology of Bach's music. However, I have copied out parts from a full score many times, and I can attest that that is a very special experience of the music full of insights. That's the activity where you DO count bars and you do discover delightful things. I imagine Bach's copyists counting bars and notes and smiling in delight at the message which Bach was sending them and which was imperceptible to performers and audiences.

Vivat205 wrote (August 16, 2008):
Douglas Cowling wrote:
< the Sanctus of the B Minor, it is a revelatory moment when you first look at the full score and see all those Trinitarian threes: three groups of three instruments (oboes, trumpets, strings) and the 3x2 six-part choir. That indicates to me that the conductors, students and copyists who worked with the full score were given an even deeper expereince of the music through this numerical symbolism. >
...and two groups of 3 x 3 x 3 = 666...but it's possible to read lots more into that than is warranted. Granted, Bach was "into" numerology and tricks, and was certainly religious, but I'd be skeptical of overinterpreting. Did Bach's letters or other written legacy say anything about his intent?

Peter Smaill wrote (August 16, 2008):
[To Vivat205] The issue of whether Bach admits numerology and other hermeneutical devices in writing is a vexed question.

Firstly, all scholars agree that our written sources for Bach in direct terms are scarce; most everythng is in the great "Bach Reader" by Hans Davis and Arthur Mendel.Numerology in general has only a weak link to music and Bach gives no direct reference; all of this is covered in Ruth Tatlow's definitive?"Bach and the Riddle of the Number Alphabet". But to prove the negative, that there is no link at all, is harder. Sometimes the connection is undeniably striking.

To discount the number alphabet then the appearance of 14, 27, and the resultant 41 have to be entirely coincidental. (BACH, Trinity 3x3x3, J S BACH in terms of the alphabet/number code)

Earliest Cantata: BWV 150: 41 bars in the trio section. B-A-C-H acrostic in final four lines of text. Acrostic also in recently discovered "Alles mit Gott" strophic aria.

Next; BWV 4/1, "Christ lag in Todesbanden"; 14 bar sinfonia; 27 entries of the "hallelujahs" in BWV 4/2

Sanctus in BMM (BWV 232); 3x3x3 = 27 as per Doug Cowling's explanation

Earliest organ works : BWV 1113 "Ich hab mein sach Gott heimgestellt" : extended to 41 bars by three additional cadences (also central chorale in "St Luke Passion" (BWV 247) and quoted in BWV 106, "Gottes Zeit")

St John Passion : Johann Sebastian Bach numerogically evident according to analysis of Bach scholar?William Scheide

Advent Cantata BWV 61, Nun Komm der Heiden Heiland": 14 sharps in the chorale.

Cantata for 14^th Sunday in Trinity, BWV 78 "Jesu der du Meine Seele": 27 entries of the subject (and at least one other Cantata discussed here has this level of repetition, noted by Duerr?purely due to persistence)

Klavierubung III: 27 component pieces

Last works: B-A-C-H? tonally in Art of Fugue, undisputed since its publication

Cantus firmus of "Vor deinen thron tret' ich allhier", final work, (BWV 668):14 notes in its first lines and 41 in all.

There are many more instances in Smend?which no doubt in the context of 1100 or so known works could be dismissed statistically. But the context in which they appear, and the clusters (I argue particularly in the early cantatas) make a complete denial of numerology implausible. That is why the debate and discoveries after a half century of scepticism remain a lively topic!

Julian Mincham wrote (August 17, 2008):
[To Douglas Cowling] There is no doubt that the perusal of scores and parts can effect aesthetic pleasures not the least of which is the recognition and discovery of the techniques and processes the composer employed.

The issue I was addressing is slightly different in that I was wondering how much of this is consciously recognised and detected by hearing alone and, furthermore, whether it actually matters or not.

Rd Myskowski wrote (August 18, 2008):
Bach & Bar Numbers

>...and two groups of 3 x 3 x 3 = 666...<
Ed Myskowski responds:
No disrespect intended to <number-challenged> correspondents, but I cannot allow this example to pass unchallenged.

Two groups of 3x3x3 = 27 + 27 = 54. Numerologically (not Bachs method, as I understand it), 54 can be reduced to 9, or Trinity squared. Note that 27 = Trinity cubed (3x3x3), which also reduces to Trinity squared. QED? More likely not.

Peter Smaill wrote:
>If one accepts that Bach was conscious of bar totals as an aid to a sort of divine, or simply aesthetic, proportionality,<
Ed Myskowski adds:
Bachs interest in the divine aspect of numbers is evidenced in the Calov Bible marginalia and underlining. An objective reader (me?) might even suggest this interest is the single most noticeable subject. I certainly enjoy his emphasis on the twelve sacred stones of the Earth, among many examples. I note a local favorite here in New England USA, Jasper, is number 12. go figure.

For further interesting detail related to Bach and numbers, see the reference (I first noticed it from from Brad Lehman, I believe, with reference to Bach/Lehman tuning. Gaining traction? Richter, Dudes.):

Numbers at Work: A CPerspective, Rudolf Taschner, German original c. 2005 (English translation c. 2007 by AK Peters Wellesley, MA, USA). I believe the English translation may also include updates; I note that it reflects the demotion of Pluto from planetary to minor planet status. No impact on Bach. His Astrology (ouch?) did not even include Uranus or Neptune , now clearly major planets. In Bachs day they were as yet unknown (to humans), but they would have increased the total to 9! Mercury, Venus, Mars, Jupiter, Saturn, Sun, Moon (his accepted seven), plus Uranus and Neptune = 9. Clearly Trinitarian? The plot thickens? Or thins?

Ed Myskowski wrote (August 18, 2008):
Juliam Mincham wrote:
>The issue I was addressing is slightly different in that?? I was wondering how much of this is consciously recognised and detected by hearing alone and, furthermore, whether it actually matters or not. <
My sentiments, precisely. Nice to see you back, mate!

Bach and gemiatria

Ed Myskowski wrote (April 7, 2010):
I have just noticed that my familiar name, Ed, reduces to 9 in Bach's gemiatric formula. That is the largest irreducible number, numerologically.

I have also noticed that my full name, Edziu Myszkowski, is a pretty big number. Perhaps a resident expert will be kind enough to cacluclate it for me?

OT: Bach use of caballism and gematria

Bruce Simonson wrote (May 10, 2011):
Ed Myskowski wrote:
< Not to overlook contributions from Peter Smaill:
If BACH = 14, and JSBACH = 41 (gemiatric), and 14 is 41 inverted, can we discount the significance of those numbers, when they occur? >
Notwithstanding Ruth Tatlow's book "Bach and the Riddle of the Number Alphabet" that calls into question the proposition that Bach intentionally employed gematria in his compositions, I still find the possible "14" reference in his autograph manuscript of the St Matthew Passion BWV 244:63 compelling.

The bass line in the chorus sings, "Wahrlich, dieser ist Gottesohn gewesen", in 14 notes.

In addition, the layout of this page in the autograph does several things that are of note:

a) red ink, for words of Christ
b) the recitatives on either side of the ensemble treatment of "Wahrlich ... gewesen" seem purposely designed to create a visual cross in the manuscript (recit-TUTTI-recit).

(FYI: It was Helmuth Rilling who pointed (b) out to me, years ago, for me to consider).

Tatlow's book: (http://www.amazon.com/Bach-Riddle-Number-Alphabet-Tatlow/dp/0521361915 ),

Wiki image of gematria: (http://en.wikipedia.org/wiki/File:Cabalistic-Gematria.png)

JPG of facsimile of SMP BWV 244:63. (http://tinyurl.com/BWV-244-63-facsimile)

Bruce Simonson wrote (May 10, 2011):
Bruce Simonson wrote:
< a) red ink, for words of Christ >
Oops, scratch that! I think red ink in this manuscript might indicate scripture verbatum, or maybe it's just the evangelist recits. Actually, I'm not sure! I stand totally educatable on what the red ink signifies. :)

Ed Myskowski wrote (May 11, 2011):
Bruce Simonson wrote:
<< a) red ink, for words of Christ >>
< Oops, scratch that! I think red ink in this manuscript might indicate scripture verbatum, or maybe it's just the evangelist recits. Actually, I'm not sure! I stand totally educatable on what the red ink signifies. :) >
I did not take the time to confirm, but I recall a reference (cited in BCW archives?) that the red ink is in fact a later addition (CPEBach?), not original with JSB.

To reiterate: BACH = 14, JSBACH = 41, gemiatrics. Numbers, plain and simple. The significance is a matter of interpretaion.

Aloha, Edziu Myszkowski (I once did the gemiatricd calculation, quite a large number, as I recall)

Peter Smaill wrote (May 11, 2011):
[To Ed Myskowski] The appearance of 41 in the St Luke recalls a talk by the veteran scholar William Scheide at Princeton five years ago where he demonstrated that, based on the layout in one of the sources, the St John Passion has 41 movements.

Bach's Numbers

Charles Francis wrote (September 7, 2015):
Ruth Tatlow has a new book that may be of interest:

Bach's Numbers - Compositional Proportion and Significance, AUTHOR: Ruth Tatlow (Cambridge University Press)

Note, there’s a promotional video and the opportunity to view excerpts.

This hardback is presumably priced for the limited academic market (lending libraries etc.), but perhaps if enough people click the box requesting availability as an ebook (you don’t have to order to do that), then that option will eventually be provided.

Linda Gingrich wrote (September 7, 2015):
I just finished reading Tatlow's article on proportional parallelism in Understanding Bach 2, 2007. It outlines her research into and development of this idea, more fully developed, I assume, in her book. Really interesting reading. You can find it at: http://www.bachnetwork.co.uk/understanding-bach/ub2/

Look for Collections, bars and numbers: Analytical coincidence or Bach's Design?

Quaesivi wrote (September 8, 2015):
Unfortunately, the academic presses tend to make their electronic editions e.g. $120 versus $150 for the book. And that e. e. cost is still too great for many of us.

Luke Dahn wrote (September 8, 2015):
[To Quaesivi] Both the last comment and the unfortunate book cover artwork put me in mind of this little piece from last Friday: Academics are being hoodwinked into writing books nobody can buy An editor called me up to ask me if I’d like to write a book (The Guardian)

Julian Mincham wrote (September 10, 2015):
[To Luke Dahn] Interesting point. It is why I, and others, are putting our work out on the internet rather than publishing (which had been the original intention). It has the advantages that it reaches far more people, is able to be edited and updated continuously and can incorporate live examples of music, performances etc.

It doesn't make you money--but then nor would the published book. My website on the cantatas has been accessed by well over 100,000 people in over 70 countries. Published, it would have reached, at most, the shelves of a few hundred universities and libraries.It does not get formally peer reviewed but it does get reviews--and comments from a wide range of people in different cultures. (Many think that the peer review process today is suspect anyway)

The only way to attack the throat hold of the publishing cartell is to self publish or put material out on websites. As I say, no money--but you reach your audiences. You also avoid having people who know so much less about the subject telling you how it should be presented!

Charles Francis wrote (September 10, 2015):
[To Linda Gingrich] In that paper, she suggests that bar-count served as a useful proxy for duration. Comparison of the respective Praetorius / Mizler heuristics pp. 40-41, intimate a 33% increase in performance tempo over time, plausibly a consequence of common practice moving away from learned counterpoint towards the galant.

Julian Mincham wrote (September 10, 2015):
[To Charles Francis] I have not had the chance to read Ruth's book as yet but I was at the international Bach conference in Warsaw in 2013 when she delivered a paper outlining her main themes. The thrust of her work relates to the philosophical concepts of 'Perfect' proportions, in Bach's case particularly related to bar numbers. She quotes a wide variety of compelling examples where Bach is known to have added bars to previously completed movements and sets of works in order to achieve, for example, a 2:1 relationship. In fact the only convincing reason as to whyBach added bars to already seemingly 'perfectly formed' pieces was to conform to this philosophical and religious notion of perfect proportion.

It raises the fascinating speculation that Bach may have been more interested in ensuring that his 'perfectly proportioned' music would be immortalised by performances in heaven rather than on earth.

Linda Gingrich wrote (September 10, 2015):
On a slightly different note, in the article I was fascinated by what might be glimpses of Tatlow's personal struggle to deal with numbers and symbolism. On p.45, as part of a discussion of the golden section, she refers to the definition of such in Zedler's Lexicon as "almost dismissive" in its brevity. Does brevity make something dismissive? Or is that her interpretation of its shortness? And on p. 56, as part of a discussion of Bach's birthdate--321, which coincides with the numbers assigned to C, B and A in the number alphabet--she says she is forced to ask whether, because of Bach's love of permutations, he might have occasionally used these numbers to represent his signature. The words "force", "might" and "occasionally" are intriguing in this context. Does this reveal some reluctance? She makes it clear that she is striving for scientific evidence (see p. 57), which is measurable and therefore valuable to the 21st century, and that proportional parallelism has nothing to do with symbolism (see p. 58), which seems clear. But is it necessary to make that point? I don't know the answers to these questions, but I wonder...18th and 21st century viewpoints in collision, perhaps?

William L. Hoffman wrote (September 11, 2015):
IMHO, numerology is found in Bach, particularly in Cantata 77 on the 10 commandments, see the recent BCML Discussion Part 4, Cantata BWV 77 - Discussions Part 4, which got numerology in Bach really going. While this symbolism is important, particularly in the Age of Enlightement, and Ruth Tatlow has done some important work, just as Eric Chafe does marvelous work with tonal allegory and musico-theology, I believe Sigmund Freud once reminded his colleague that sometimes a cigar is just a cigar, which he cherished.

Luke Dahn wrote (September 11, 2015):
[To William L. Hoffman] I honestly have a difficult time understanding what significance numerological study such as is being done in Tatlow's article has on engaging with Bach's music, whether as performer or as listener.

At the end of the article, she states: "The usefulness of proportional parallelism to the editor and interpreter of Bach's music is limited, however, unless one fundamental question can be answered: is proportional parallelism analytical coincidence or Bach's design?" But even if we are able to affirm Bach's intentional design, how do the kinds of parallelism she brings up inform performative interpretation? How does knowing that the number of bars in Violin/Harpsichord Sonatas 3 & 5 create a 1:2 proportion with Sonatas 1, 2, 4, & 6 help guide a persuasive performance or a more rewarding listening experience?

At the end of her piece, she gracefully leaves it to the reader to decide ("to weigh up the evidence"). In my humble opinion, the question is irrelevant, so long as we are dealing with largely (entirely?) inaudible numerological correspondences and so long as our interest remains on engaging with the music itself. (I don't deny that there are interesting audible correspondences and symbolism closer to the music's surface.) In general, I don't find barline-counting ("number hunting" as Tatlow herself puts it) particularly interesting because it's not what makes Bach's music so incredibly marvelous in experience.

In other words, I'll enjoy the subtle flavors of my cigar without measuring the proportion of its circumference against its length.

Humbly, ...and I would very much welcome push back from these thoughts. My questions aren't merely rhetorical.

George Bromley wrote (September 11, 2015):
[To Luke Dahn] I agree, I see it as total nonsense, when I lived in Capetown I did not walk up Tablemountain with a tapemeasure, I now live in London, I dont stroll through the streets with a theodolite.

John Hendron wrote (September 11, 2015):
[To Luke Dahn] I'd wager that the driving interest one would have with this type of study would come from an interest more in Bach the man than today's performer's needs in performing Bach's music.

That said, what performer wouldn't be more deeply educated about the music than through more deeply understanding the composer's habits, practices, and ideas about composition?

I realize the utility this brings to questions of tempo, phrasing, and instrumentation may be weak, however I've long wondered how Bach was able to come up with such interesting melodic subjects. To know that the length of music he wrote was not only pragmatic by its intended purpose for performance, but by some grand design, may have bearing on my understanding of his process that begins with some of these interesting new findings.

Formal structure in Bach is an important topic of study and it's been an important attribute to other composers and their music too. I think we've yet to see of Bach's simple counting of bars, note alphabets, etc. didn't also extend vertically within the score. I'd remain open minded about these kinds of studies and what they do to further clarify our understanding of Bach, and perhaps later, performance of his music.

Julian Mincham wrote (September 11, 2015):
John Hendron wrote:
< I've long wondered how Bach was able to come up with such interesting melodic subjects >
Responding to the above comment first, a study of the cantatas is invaluable when attempting to address this question. Bach continually derived his musical ideas from the textual imagery--a puff of smoke, cutting off the serpent's head, flowing water, bustling clouds and hundreds more images, physical and metaphysical, were ones that he translated into melodic, harmonic and textural ideas (Schweitzer knew this but his development of the idea was somethat suspect). This helps to explain the range of non-repetitive ideas Bach created, though not necessarily the quality. For that we might turn to Glenn Gould's comments to Humphrey Burton in an interview around 50 years ago. Bach was not, he claimed, a natural contrapuntalist (although be sheer application he made himself into one) but he was born with a natural sense of expressive melody which he managed to infuse into every contrapuntal line. Andre Previn wrote something similar when he claimed that expressive melodic gifts were not infrequently to be found in people with virtually no other musical skills (e.g. Irving Berlin and Lionel Bart ) and absent in people with a high degree of musical training. We have all benefitted from the fact that Bach had both qualities--and made the most of them.

Taking up Luke's invitation to comment, and again rising principally from my study of the cantatas, I often wondered why, when time must have been so pressing Bach took so much trouble with the abstruse imagery and complex number patterns. It would have been impossible for even an educated congregation with the text in front of them, to have picked up every nuance---and it is surely right, as you say, that such knowledge is not necessarily essential for the appreciation and performance of the music. The explanation which satisfies me (and which relates back to Ruth's work on perfect proportions) must be that he was writing for God and Heaven first and mankind second. I don't think it really mattered to him if the hoi poloi did not pick up all the nuances--presumably God would and that was what mattered.

Melanie aka Picotsnkeys wrote (September 11, 2015):
[To Luke Dahn] I would humbly share that I also have conflicted thoughts regarding the significance of analyzing music through numerology.

I'm one of those lurkers that seldom are seen or heard by this group. I have little right to speak up as my Bach quotient is quite lacking and my education limited. As an organist, my musical experience is wrapped up in Bach constantly. But, life has been full and my knowisn't as specific or extensive as those who post here. I became interested in this group about the same time I gained experience tatting lace with a shuttle. Veteran tatters encouraged me to create new designs in thread drawing inspiration from how music is organised. At first, I shied away as numerology always seems to miss the point of the aural experience. But then, I was trying to translate the aural into visual. Kinda of reversing the process of "program music" so popular in previous times.

Well, I'm still not sure if designing lace using this method really is important. There's so much more to my life. But, the more people bump up against this idea, the more people like it. I've yet to have someone say it completely bonkers. I've yet to figure out the method and process. Each time I hear a piece, study a score, pick up the shuttles, I seem to be engaging in something important. I'm enjoying the ride at any rate.

I must put into this message a thank you for this group. The courage it takes to express thoughts about music is valued. All of us are gathered here because of the music of Bach. That's something. All of us respond as individuals. That's something. Reading what members write gives me courage to express in thread. Please continue to post.

Thank you,

Julian Mincham wrote (September 11, 2015):
Melanie C picotsnkeys wrote:
< 'I have little right to speak up as my Bach quotient is quite lacking and my education limited. As an organist, my musical experience is wrapped up in Bach constantly.' >
Melanie I was struck by the contradiction of these sentences. I would suggest that if you are an organist and wrapped up in Bach you have every right to speak (and write) about it.

Linda Gingrich wrote (September 11, 2015):
I also think that part of the appeal of studying Bach's symbolism is understanding the man, particularly his penchant for puzzles. This is purely subjective and speculative, but in my own studies I often have the sense that Bach enjoyed creating these hidden hints. But that may be because of my own enjoyment in ferreting them out. But I think musicologists like Robert Marshall and theologians like Jaroslav Pelikan said it best: Bach's ultimate audience may have been God, from whom nothing is hidden. The thought that, if we can't see it it is useless strikes me as a product of our secular, 21st-century, Enlightenment influenced age. I suspect residents of the 18th century saw things much differently. In other words, the people in the seats, including us, may have been a little less important to Bach than we like to believe!

Julian Mincham wrote (September 11, 2015):
Linda Gingrich wrote:
< But that may be because of my own enjoyment in ferreting them out. >

But this sort of enjoyment is not, I would suggest, limited to ferreting out number puzzles. There is a great aesthetic enjoyment in working out what (say) Beethoven of Brahms did to solve their musical, harmonic and structural problems. In other words an intelligent questioning analysis of the scores brings forth a similar sort of pleasure---which is quite different from the pleasures of listening.

Linda Gingrich wrote (September 11, 2015):
[To Julian Mincham] Well put!

Luke Dahn wrote (September 11, 2015):
[To Julian Mincham] I appreciate the thoughtful feedback from everyone (John, Julian, Melanie, Linda). If the only thing my post did was generate conversation, then that's a good thing.

I can certainly appreciate the idea that Bach put into his music hidden constructs that the average person will not notice, that perhaps only God would notice. I can also appreciate the beauty of an elaborate math equation.

I also understand that the listening experience is an extremely complicated exercise and that we hear many things that we do not realize we hear -- i.e. large-scale proportion created by symmetrical structures. To this point, I do understand how some of the proportional symmetries could inform a performer's interpretation.

One could also make a point that, in some cases, forcing an interpretation in order to create temporal symmetry can lead to less persuasive performances. I don't particularly care for the 1981 Gould recording of the Goldberg Variations, a performance in which he explicitly emphasized temporal proportions between variations. I much prefer his 1955 studio recording and his 1957 live recording. I don't know how much that has to do with the approach he takes regarding proportion, but I do find this later recording a bit stilted and lacking the vitality of the earlier recordings.

(I once did a fun study of tempos taken in recordings of the Goldberg. In most of my favorite recordings (e.g. Schiff's 2001 recording) you find little to no exact proportional correspondences in tempo choices. https://lukedahn.wordpress.com/2013/06/09/goldberg-tempo-variations/)

I also have to push back against a couple other thoughts. Julian refers to the solving of "musical, harmonic and structural problems." But this gets at my precise point. The solving of such problems is almost always perfectly audible, and it's precisely what my composer's/theorist's ears are trained to listen for and what my ears appreciate. (I suppose we could talk about what is meant by "structural".) That is absolutely where much of the "aesthetic enjoyment" rests. But to what extent are we able to hear that there are 410 bars of tenor, bass and choir arias in both the first and second halves of the SMP?

My only point is this: Even if measure-number proportions were intentional on Bach's part, proportions of this kind fall extremely low on the list of reasons I love his music (if they don't fall completely off the list). There is plenty of music by other composers that is far more symmetrically structured from a mathematical standpoint than Bach's. Most of such music ends up being uninteresting. Speaking very generally, I find asymmetry more aesthetically pleasing and interesting than symmetry. But to each his own.

I know that I am (and you are) in many ways a product of post-Enlightenment secularism. But I will say that as a Christian, I worship through Bach's music. I just don't think my ability to do that is dependent upon large scale proportional symmetry. That ability to worship through Bach's music has been greatly enhanced by people like Pelikan. But most of what Pelikan points about about Bach's music in Bach Among the Theologians, at least in the parts I remember, have to do with things that can be easily heard in the music. But perhaps my memory is only failing me.

Stephen Clarke wrote (September 11, 2015):
[To Luke Dahn] Dear Mr.Dahn:

So very Well Said! Listening to Bach is worship(ing with Bach) for me, also. Many lovely small things fall into place if the Big Picture things are taken well care of, and a professional artist will invariably play little games within his work. But never the cart before the horse!

Linda Gingrich wrote (September 12, 2015):
[To Luke Dahn] I hear all you are saying Luke, and I agree, these proportions, numbers, symbols, are not hearable, and don't affect listening to and marveling at Bach's works, and even worshiping if one is so inclined. Discovering these hidden things simply gives me joy. I vividly remember what spurred me towards my eventual dissertation. In class one day a professor pointed out that the bass line in the final 2 1/2 measures or so of one chorale cantata contained the chorale tune on which the cantata for the next day was based. I was absolutely captivated. They are the third and fourth cantatas in Bach's second cycle, and it turns out they are the only two cantatas in the Trinity season performed on a Saturday and a Sunday. It seems that Bach may have built that tiny, hidden connection into the end of the first cantata. Who would notice such a thing? Maybe, possibly, perhaps, a bass student of Bach's, but certainly no one else unless it was pointed out. I found many other hidden connections of all types between the cantatas in that cycle's Trinity season as I did my research, and they still captivate me. Ithink it might be Robert Marshall who said something about the best way to study Bach was to spend time with his scores on your own music stand, one on one with the man and his music. That sums it up, for me anyway.

Claudio Di Veroli wrote (September 12, 2015):
My thanks go to the outstanding musicians that have provided interesting insights into this topic.

Needless to say, I fully agree that Bach's numerical proportions are mostly HIDDEN marvels.

However in a few cases, as we all know, the proportions are in full view.

Perhaps the most ostensible one is the beginning of the 3rd Cantata from the Christmas Oratorio, "Herrscher des Himmels". It can be clearly divided into the following sections:

1. Instrumental Question (16 bars), clearly divided in 4 groups of 4 bars each.
2. First Fugato (16 bars): Tenor (4 bars), Soprano (4 bars), Alto (8 bars).
3. Question repeated but now complete with the singers.
4. Instrumental answer (16 bars), again divided 4x4.
5. Second Fugato (16 bars): Tenor (4 bars), Soprano (4 bars), Alto (8 bars).
6. Answer repeated but now complete with the singers.

Certainly one of the most impressive and exhilarating short pieces JSB ever composed. Rhythmically at all times you know what is coming, yet the piece has its good dose of surprises. If only conductors would refrain from unstylish features such as performing it blazingly fast, or finishing it with a few bars of never-ending ritardando (which I accept only at the end of the penultimate bar), it would be perfect!

Ed Myskowski wrote (September 12, 2015):
Julian Mincham wrote:
<snip>
< The explanation which satisfies me (and which relates back to Ruth's work on perfect proportions) must be that he was writing for God and Heaven first and mankind second. I don't think it really mattered to him if the hoi poloi did not pick up all the nuances--presumably God would and that was what mattered. >
I find Julians interpretation satisfying, but likely beyond proof. Further, Bach striving for details and perfection of proportion becomes an essential part of his creative process, of interest in its own right, regardless of the motivation..

Julian Mincham wrote (September 13, 2015):
[To Luke Dahn] I think we are conflating and possibly confusing two issues. One is why did Bach go to so much trouble to create complications, not audible to most listeners, in his musical structures. As I have conjectured, a simple (and possible simplistic) explanation is that he wrote for God not man (but I do belive it to be true) . Another is that he was the sort of artist who needed challenges to stimulate his artistic imagination. He seemed to go out of his way to create them if they were not readily apparent.

The second issue is how we, as listeners, respond to the abtruse images and number proportions. The answer to this must be particularly subjective as we all listen differently and aspects of music which are not overtly recognised may well have an impact on how we react emotionally. But the further point is that there is a different kind of delight and real aesthetic pleasure to be gained from studying scores and finding aspects of the musical structures that one does not, or maybe cannot recognise aurally. I remember well the excitement when, as a student, I found structural aspects of movements that I had not recognised simply through listening (although maybe I should have). Inversions and retrogrades of themes offer very good examples, even without going down the numerology and structural proportions road (even more so in later twelve-tone music but plenty of examples abide in Bach as well)

Our responses to the first issue mentioned above must be speculative, to the second they are personal and subjective.

Still fun to debate, though!

Peter Smaill wrote (September 13, 2015):
Bach's Numbers- Ruth Tatlow's discoveries

Once it is generally acknowledged that Bach's quest for mathematical order is rarely consciously audible, the we can move to examine the fascinating if distant world in which Baroque conceptions of divine pattern existed, alongside musical considerations such as melody, tonality, fugue and harmony/modality.

The suggestion of order in the "Christmas Oratorio" is most striking, in the context of the doctrinal view that the incarnation is the crux of divine intervention so as , in time, to perfect creation.

There is a certain mathematical beauty at work at a meta-level, as Tatlow observed. All six Cantatas were performed at the Nikolaikirche; only four (1,2,4,6) at the Thomaskirche.Four of the Cantatas have 2310 bars; two have exactly half, 1155. This happens to produce sum-of-digits for the two groupings of 6 and 12.

Furthermore (may I add) 2310 is noticeable as having a prime number on either side (2309 and 2311.) 2310 is also the product of the first seven primes (1 x2 x3 x5 x7 x11). However- caution required here- there seems to be no reference to 2310 in contemporary mathematicians engaged in number theory, such as Ferdinand Euler, Gottfried Leibniz or in Fermat, alluding to this special number. In modern number theory it is I think called a primorial.

The Tatlow discovery and her overall structural thesis is strong enough on its own without such further speculations by me! She setavlished beyonf doubt that In Bach's last decade (but earlier too) there is powerful evidence of forming and revisiting works and groups of works to create 1:2 and 2:3 relationships, and often the "numerus perfectus", 6, is also an end-point.

Suffice to say that number theory and universal order were both theological and academic pursuits at the time of J S Bach, the "Learned Musician", as Christoph Wolff describes him. This latest publication may well lead on to further analysis and discovery.......

......... and I'd encourage all interested with access to an academic library to borrow the new book, a breakthrough in Bach scholarship regarding compositional methodology. Meanwhile there is a clip available: https://www.youtube.com/watch?v=TtjHD9zB6bg&feature=y

Linda Gingrich wrote (September 13, 2015):
[To Peter Smaill] Thank you for the link to the clip, Peter! Really fascinating, and it's wonderful to see Tatlow's own passion for this discovery.

Jane Newble wrote (September 15, 2015):
Luke Dahn wrote::
< In general, I don't find barline-counting ("number hunting" as Tatlow herself puts it) particularly interesting because it's not what makes Bach's music so incredibly marvelous in experience. >
But it could just help to explain why "Bach's music so incredibly marvelous in experience".

Julian Mincham wrote (September 16, 2015):
George Bromley wrote to Luke Dahn:
< I agree, I see it as total nonsense, when I lived in Capetown I did not walk up Tablemountain with a tapemeasure, I now live in London, I dont stroll through the streets with a theodolite. >
With respect I disagree. There are many aspects of musical structures which may not be overtly apparent to the listener who nevertheless adores the music e.g. techniques of inversion, retrograde, diminution, invertable counterpoint etc etc. The fact that one does not 'measure' or even take particular note of them on a conscious level does not mean that they should be dismissed as 'total nonsense'. They are part of the structure of the music that the composer has conceived, like it or not.

Peter Smaill wrote (September 16, 2015):
No-one contends that the numerology issue contributes to acoustic pleasure to any significant degree, although it is possible that elegant proportion, such as the "Herzstuck" of the St John Passion, for example, may have an almost subconscious effect.

Once this concession to the argument has been made it is difficult to understand why the anti-numerology camp is so hostile since the Baroque world itself did not create divisions in its contemplation of the created world between mathematics and music. In Bach's cousin's dictionary of musicians,Johann Walther's Musicalische Lexikon, there is a single entry for a British individ(other than the naturalised Handel).

It is for Alexander Malcolm, Professor of Mathematics at Edinburgh University; whose work "A Treatise of Music", is suffused with the mathematical content in music. Coupling this with the recent discovery that eight of Bach's Cantata texts were written by the mathematician-theologian Christoph Birkmann it seems unwise in terms of wider scholarship to attempt to close off this line of enquiry, On the other hand if any individual is subjectively uninterested in it there are many aspects of Bach still awaiting research and elaboration!

Numbers, as a result of Ruth Tatlow's research and prior art in this field, undoubtedly played a part in Bach's compositional methodology. The wonder is that the result does not sound mechanistic (generally speaking), but, as the musicologist Alec Robertson observed, reveals an almost romantic effect in the ability of the output to move the listener. The tension between the rigour of form in Bach and the emotional appeal is a most amazing feature of his compositions.

Julian Mincham wrote (September 16, 2015):
Just to add to my earlier comments in response to George, if people are attempting to locate and invent numbers, proportions and ratios which do not exist in the music (which, I suspect, some do) it then may be called a nonsense, with justification. But if they do exist they cannot be denied and it is a legitimate question to ask why the composer did it.

Luke Dahn wrote (September 16, 2015):
Peter Smaill wrote:
< No-one contends that the numerology issue contributes to acoustic pleasure to any significant degree, although it is possible that elegant proportion, such as the "Herzstuck" of the St John Passion, for example, may have an almost subconscious effect.
Once this concession to the argument has been made it is difficult to understand why the anti-numerology camp is so hostile since the Baroque world itself did not create divisions in its contemplation of the created world between mathematics and music. >
I'm not sure it's true that "no one" contends that numerology contributes to acoustic pleasure to any significant degree. It seems that someone on this thread just suggested that it does or at least might contribute. So I'm not sure this concession is made by everyone, and I think this may explain part of the strong reaction against it.

I have acknowledged that the listening experience is a complex thing, and that we may subconsciously hear things. To put it another way, audibility is not an either/or question but a "to what extent" question. For me personally, when answers of the "to what extent" question fall so far to the right (or is it left?) of the spectrum, I lose interest, especially when there is little or no good evidence that the proportion or symmetry was consciously constructed. Personally, I won't say it's pointless (at least, I won't anymore), I just simply lose interest.

Julian's mention of retrograde struck me, and if we tease out Bach's use of retrograde, at least as I think through it, it may actually lend support to both camps simultaneously. A retrograde version of a theme is quite a difficult thing to hear as being what it is -- a backwards version of the theme. We may have some vague awareness of an affinity, but the fact is that we don't listen backwards well. Retrograde, therefore, falls quite a bit lower on the inaudibility scale than melodic inversion. The fact that Bach used melodic inversion far more than he used retrograde may give evidence that Bach was more concerned with the consciously audible (supporting the anti-numerology camp). But the fact that we do have pieces that consciously make use of retrograde (e.g. musical offering), albeit in pieces that are more in the vein of intellectual exercises, prevents us from dismissing less audible structures entirely (support for numerology).

Having conceded that, there is a huge difference, in my opinion, between a retrogradable piece as in the musical offering on the one hand and symmetry of measure numbers on the other. Yes, they both come from the same composer's mind, but the former shows unparalleled genius, especially when it engages the heart as well as the head ("rigour of form and the emotional appeal" in Peter's words). Anyone, and I mean anyone, can match up measure numbers. That's no evidence of genius whatsoever.

Claudio Di Veroli wrote (September 16, 2015):
[To Julian Mincham] Fully agree. Without any criticism to the interesting work by Tatlow, we should always remember that there is authoritative recent musicology about the use of numerology to produce quite a few modern erroneous inferences and conclusions about Bach and his music. Unfortunately, some of these beliefs are still found in musical practice today, in spite of sound published proof on the contrary. A well known case is in the field of musical temperament.

Ed Myskowski wrote (September 16, 2015):
[To Julian Mincham] I do not see this as an either/or choice, I expect that neither does Julian, but perhaps he can confirm that (or not!).

Perhaps all folks who have stared at a blank page of paper, canvas, or block of stone can appreciate the value of artificial constraints on creativity.

Jane Newble wrote (September 17, 2015):
Personally I see it as architecture. We can admire the beauty of a building, but behind it are lots of designs and calculations.

Some people appreciate the beauty even more if they can look at the design behind it.

Others might prefer just to look at the end result without worrying about all the work behind it, but that does not mean that it isn't there.

Julian Mincham wrote (September 17, 2015):
Ed Myskowski wrote:
< I do not see this as an either/or choice, I expect that neither does Julian, but perhaps he can confirm that (or not!). >
True

Perhaps all folks who have stared at a blank page of paper, canvas, or block of stone can appreciate the value of artificial constraints on creativity.

I think this gets to the heart of the topic (I nearly wrote 'problem there but then realised that I don't see this as a problem but as a perfectly logical process). I am always fascinated by the ways in which artistis deal with (self or otherwise) imposed technical problems, often solving them in the most imaginative and elegant ways. Examples abound---the 'missing' notes on early trumpets and horns, the rewriting of a classical sonata form second subject theme when it returns in a range inappropiate or impossible to execute (See Beethoven's D minor sonata op 31 where the rising second subject in the first movement 'runs out of keyboard space' so he completely rethinks it under an inverted pedal). Most listeners don't notice such details as they listen--but this does not impair their enjoyment--and nerds like me get pleasure out of discovering such aspects of the composer's thinking.

Bach, perhaps more than most, seems to have required challenges and restrictions in order to bring forth his most creative works. Examples--the constant experimentation with the combinations of traditional musical structures, the setting of texts with seemingly contradictory texts within the same movements---and the imposition of number proportions which, doubtless, led to the creative manipulations of phrase and section lengths and thus a seemingly endless parade of musical variety. How often did the man repeat himself one asks?

I repeat what I said . I think that Bach wrote for God primarily and that he deliberately sought out and created his own challenges which stimulated his imagination, leading to an incredible variety of musical outputs. I see no contradiction between the self imposed restrictions and the marvels of the music. In fact the latter almost certainly resulted, at least to some degree, from the stimulus of the former.

Linda Gingrich wrote (September 17, 2015):
Just to add to the "restrictions" thought, in Haydn's case, in the mid- to late 1790s Esterhazy fired his entire wind section due to financial considerations. Haydn had only strings, trumpet and timpani, but he created masterpieces in his six late Masses, the first three a much reduced orchestra. Plus Napoleon was breathing down Austria's neck! Hence Haydn's Mass in time of war and the so-called Nelson Mass, really titled Missa in angustiis--Mass in time of straitened circumstances, or anxiety!

I agree with Julian; Bach responded to challenges, and he wrote primarily for God. In general, the baroque view of the cosmos differed from ours. It's always good to remember that Enlightenment "rationality" didn't always hold sway. In fact, in many of the cantatas Bach warned against the dangers of "rationality." But not "reason", which was a gift from God.

Julian Mincham wrote (September 18, 2015):
[To Linda Gingrich] Well, quite by chance I came across a quotation this week that seems particularly apposite to the discussion we've bee having recently.

It was stated by one of the C20 leading composers. If you don't know who it is, try and guess.he speaking from the artistst's viewpoint of course, not that of the state).

'The more art is controlled, limited, worked over, the more it is free.

'My freedom will be so much the greater and more meaningful the more narrowly I limit my field of action and the more I surround myself with obstacles. Whatever diminishes constraint, diminishes strength.The more constraints one imposes, the more one frees oneself of the chains that shackle the spirit,'

QED!

Ed Myskowski wrote (September 18, 2015):
[To Julian Mincham] I had to look it up, and naturally, I had it incorrect. Right state however! My instinct was Shostakovich.

Julian Mincham wrote (September 18, 2015):
[To Ed Myskowski] good guess--but no coconut!

Claudio Di Veroli wrote (September 18, 2015):
[To Julian Mincham] Marvellous Julian!

And, allow me an out-of-topic rant, it applies to freedom of performance itself.

Sooo many performers telling me like "But if you play, say, Baroque music follow all those rules about playing the ornaments, playing the inègales, avoiding wide rubato, playing on period instruments, playing with period technique, you lose your freedom of expression!". To which I retort "I lose freedom of MODERN expression, and gain instead freedom of BAROQUE expression, as all those rules still give me lots of freedom, to be used WITHIN the Baroque style uses, not outside them!". And when I show them early treatises and play them comparative examples on the harpsichord, they are not completely convinced: they are too much used to the modern fad of doing rubato everywhere even grossly affecting the beat, and find my micro-rubato "too stiff"! Needless to say, they also find stiff most of what K.Gilbert, Hogwood, Christie, Pinnock, Rousset and others have played and conducted, so at least I am in excellent company!. They should be reading Mozart when criticising a pianist for his bad taste because he "did rubato with his left hand!"

Jane Newble wrote (September 18, 2015):
[To Julian Mincham] Stravinsky?

Luke Dahn wrote (September 18, 2015):
[To Jane Newble] Yes, Stravinsky. If he didn't say this exact quote, he said something very similar.

Of course, Stravinsky also said: "Of all the musicians of his age Haydn was the most aware, I think, that to be perfectly symmetrical is to be perfectly dead." :o)

Julian Mincham wrote (September 19, 2015):
[To : Luke Dahn] Yep---from Poetics of Music.

But one can almost imagine Bach saying it too!

Number Symbolism: Part 1 | Part 2 | Part 3