Intriguing New Release

Donald Satz wrote (March 25, 2001):
I just finished listening to some sound samples from a recent recording on Arion of Bach's Musical Offering performed by Arte Resoluta. What I heard sounded very interesting; the strings had that "scrappy" period instrument sound of past times which I always liked. As it happens, I've seen this recording in one of the local stores and will buy it very soon. The Arion catalog number is 68526.

Bach's Musical Offering

Zeev Schor wrote (March 30, 2001):
I have acquired a CD, called "A Musical Offering" by J.S.Bach (performed by members of the Florilegium) which claims to be a complete set of instrumental trio sonatas. I am not a performing musician, only an avid listener of classical music. The term "Musical Offering" has created for me quite a lot of confusion. I would appreciate if a member of the list would explain to me if there are more than one "Musical Offering" by J.S.Bach, and how they are catalogued?

Donald Satz wrote (April 2, 2001):
Zeev Schor asks:
< The term "Musical Offering" has created for me quite a lot of confusion. I would appreciate if a member of the list would explain to me if there are more than one "Musical Offering" by J.S.Bach, and how they ar catalogued? >
There's only one Musical Offering and it's BWV 1079. I think Zeev is getting a little confused because of the 'trio sonata' designation.

The Musical Offering has two Ricercari, many Canons, and one four-movement trio sonata. It is specificially the trio sonata from the Musical Offering. Bach wrote many other trio sonatas including for organ, and some trio sonatas attributed to Bach are likely not composed by him. Regardless, each of these trio sonatas, except for the one from the Musical Offering, has a BWV number different than 1079.

The Florilegium disc from Channel Classics can only add to the confusion. The bigger lettering on the cover states - "A Musical Offering". The smaller letters state - "The complete instrumental trio sonatas". But this disc only has the Ricercar A 6 and the Trio Sonata from the Musical Offering; there are no Canons nor Ricercar A 3. It's definitely either deceptive or dumb presentation on the part of Channel Classics. As for the 'complete instrumental trio sonata' statement, that's pure bull. The six Trio Sonatas for organ alone take up an entire disc, and the last time I investigated the matter, the organ was still considered an instrument to be reckoned with.

As an aside, some of these cd covers defy worthy explanation. On the Channel Classics release, the cover art is a war-horse used picture of a bunch of angels praying to an elevated lamb representing you-know-who. It's the same cover as on Gardiner's Mass in B minor, although in darker shades. Actually, it looks like they just might cut off the lamb's head after praying to it; give it a look. Unless I'm mistaken, there's a guy behind a set of trees on a grassy knoll who appears mighty hungry.

BWV 1079 - Musical Offering - What are the parts?

Juozas Rimas wrote (May 3, 2001):
Hello. I'd be extremely grateful for the FULL list of all parts of the Musical Offering. I thought there were 16 parts before (I have a CD with 16 tracks) but now it seems there are more in fact??

Have you heard CANON A 7 or CANON DUPLEX, for example?

2 questions on the Musical Offering

Juozas Rimas wrote (January 5, 2004):
Are the instruments specified in the canons?

Eg, in the Canon perpetuus super Thema Regium, Marriner uses a flute, Savall does not but he uses more and more strings with each repetition in the canon.

Is the number of repetitions in the canons specified? Is performed explicitly allowed by the score to choose the number of repetitions?

Eg Marriner repeats the Canon perpetuus super Thema Regium 3 times and Savall 6 or 7, with different beginning and ending. Is the score so "loose"?

Tomek wrote (January 5, 2004):
Juozas Rimas wrote:
< Are the instruments specified in the canons?
Is the number of repetitions in the canons specified? Is performed explicitly allowed by the score to choose the number of repetitions? >
No, the instruments aren't specified in Bach's Musikaliches Opfer, except the trio sonata & one fuge which are for transverse flute, violin and continuo. Number of repetitions is also unspecified.

David Glenn Lebut Jr. wrote (January 5, 2004):
[To Juozas Rimas]
1.) In the score I have (Dover Edition reprint of the BGA Edition) many of the Kanonen do have specified instruments (see the solutions part), which for the most part include Keyboards.

2.) The number of repetitions are not specified, but I favor the "3-time rule"-repeat three times. The only exception I feel is the Kanon a2 (Per tonos), which actually traverses through all keys.

Musical Offering excerpts in the Casals festival, 1950

Bradley Lehman wrote (January 23, 2001):
In the set of Brandenburgs released recently as Pearl 200: http://www.pavilionrecords.com/html/pearl/pearl_frma.html
Casals conducted the Prades Festival Orchestra, 1950. I've written about these several times before; anyone else heard this set yet, with other reactions?

& There's a review by Jonathan Woolf at: http://www.musicweb.uk.net/classrev/2003/Aug03/Bach_Festival_casals.htm

Anyway, the second disc has the filler of excerpts from the Musical Offering, from that same festival. Casals doesn't play in them but probably had some role in coaching the performers; after all, they all came to Prades expressly to work with him. And the booklet has some notes about a Pleyel piano they brought in for the festival, and a gift of a refrigerator from the orchestra to Casals.

Leopold Mannes (director and later president of the Mannes College of Music in New York) plays the three-voiced ricercar; he is joined by Alexander Schneider, John Wummer, and Leopold Teraspulsky for the trio sonata; and then a string sextet (plus bass, so seven players) plays the six-voiced ricercar--really a keyboard piece but working OK this way also. Beautifully shaped phrasing, with admirable flexibility both of dynamics and tempo, from everybody here in these MO excerpts.

Anyone else here know these performances yet, either from the ancient Columbia LPs or this Pearl reissue? How about the other volumes by Pearl from these 1950s festivals, which I haven't picked up yet?

And anyone know of other recordings by Leopold Mannes?

Charlie Ervin McCarn wrote (January 23, 2001):
Bradley Lehman wrote:
< Anyone else here know these performances yet, either from the ancient Columbia LPs or this Pearl reissue? How about the other volumes by Pearl from these 1950s festivals, which I haven't picked up yet?
And anyone know of other recordings by Leopold Mannes? >
Leopold Mannes made at least one chamber music LP for American Decca. I think that it has a piano trio by Clara Schumann on it.

The liner notes for the Pearl sets are full of inaccuracies. For instance, in one of them, the annotator says that, by process of elimination, Eugene Istomin was the continuo pianist in the Szigeti recording. There is a session photo on the cover of the American LP release, which I have in my collection, and you can see the pianist. It's Mannes. The annotator also writes that Valenti played the harpsichord for the Fritz Reiner recordings of the Brandenburg Concertos. That's wrong. Sylvia Marlowe did.

It's sad that Pearl did not get Terri Towe to write the annotations and produce the series. I have heard more than once that he is VERY difficult to deal with, but I also have been told that he knew Casals, Horszowski, Schneider, Istomin, Valenti, Wummer, and a number of the other participants, and that he has unpublished interviews with some of them. One of my friends went to college with him and told me that he did radio shows with Horszowski as his guest. If his liner notes to the reissue of the Casals recordings of the Bach Cello Suites are any indication, he really would have provided us with liner notes that no oneelse could match.

The other transfers in the Pearl set are of the same calibre as the set that you have - serviceable but not extraordinary. (I am sure that Seth Winner could have done a significantly better job than Roger Beardsley, had he been chosen as the transfer engineer. His transfers of the Cello Suites for the set that Terri Towe produced and wrote the notes for are better than all of the other ones.) And the Pearl series of 6 CDs is complete, unlike the competing Cascavelle 5 CD box which does not include the Sonatas for Cello and Piano and the B Minor Flute Sonata with Wummer and Mannes and one or two other pieces.

When I was playing some of these recordings the other day, I realized that now that Eugene Istomin has died , the only remaining soloist from these recordings is the flutist Bernard Goldberg.

Musical Offering (was: 4-hpsi)

John Pike wrote (February 7, 2005):
Bradley Lehman wrote:
< What do you think of the published theory that the Musical Offering's theme for improvisation was composed not by the king, but by one (or several) of Bach's former students there at court, namely CPE or Nichelmann or Agricola? And that some or all of the keyboard parts in MO are piano parts instead of harpsichord parts? >
Recordings of the MO were reviewed on BBC Radio3 on Staurday. I missed much of it I'm afraid, but the top recommendation was for the Hänssler Bachakademie recording. Also recommended were Jordi Savall and Davitt Moroney on Harpsichord.

Peter Bright wrote (February 7, 2005):
[To John Pike] I'm really pleased that recording came out on top (I presume its the disc with Michael Behringer on fortepiano...) - this is one of my favourite CDs - and I LOVE their treatment of those 14 canons based on the fundamental bass notes of the Goldbergs aria at the end of the CD. This disc isn't talked about very much, but I do favour it slightly over the Savall (my next favourite...).

Thomas Shepherd wrote (February 7, 2005):
[To John Pike] It is still possible to hear the programme on Radio 3 streamed via the internet for the next few days. There is half an hour of other stuff (Brahms 3^rd mvt.4 LSO/Haitink: Cavalieri etc.) before the MO is reviewed. Be sure to listen BEFORE next Saturday's broadcast of "CD Review". http://www.bbc.co.uk/radio3/index.shtml?logo > "Listen Again" > "CD Review".

John Pike wrote (February 7, 2005):
[To Peter Bright] That's exactly right. There was another version reviewed with Michael Behringer doing the MO by himself on keyboard, but Moroney came out better. I have the Haenssler recording as well and will have another listen to it very soon.

Bradley Lehman wrote (February 8, 2005):
[To Thomas Shepherd] Thanks, gentlemen, for the notice about this programme. I listened on the web today and enjoyed it. (Brahms 4^th, by the way, not 3^rd.)

But I do have to take exception with the host's remark that "The harpsichord is an austere kind of instrument"... in his setup of one of the recordings. My own musicianship is predicated on the observation and practice that it's not an austere instrument, but rather a sensual one and capable of profound beauty.

Yes, in the 6-part Ricercar too. That piece takes strong advantage of harpsichord sonority. It's supremely difficult, like a huge puzzle of fingering to be solved, but Bach kept all of it playable by two hands and in a way that does not sound particularly difficult...and that's part of the difficulty, to bring off the piece with suave control!

Neil Halliday wrote (February 8, 2005):
Bradley Lehmanwrote:
<"Yes, in the 6-part Ricercar too. That piece takes strong advantage of harpsichord sonority".>
Certainly, in Moroney's rendition, near the end of the program. This is one of the more powerful harpsichord sonorities I have heard, very impressive, reminding me of the sound of the forte-piano that opened the M.O. program (indeed, this forte-piano sounded more like a harpsichord than a piano - the awesome power of the modern instrument being impressively and fully realised in that Rossini transcription in the "Tribute to Horowitz" example we heard earlier in the program).

But did you notice what happened to the sound of the harpsichord in Savall's immediately following ensemble version of the Ricercar? It turned into the pitchless little rattling sound that we so often hear with harpsichord in ensemble. I would have enjoyed that Savall more without that 'noise' going on in the background. (I realize it's obviously a problem of capturing the sound of many instruments onto a CD). It's interesting that the final Haenssler example gave us just such an example of an ensemble version without this 'rattling' noise; if the harpsichord was there, it was inaudible, and in no way missed.

Richard Bradbury wrote (February 8, 2005):
On the BBC website this week it is possible to hear the sublime Bill Patterson reading James Gaines's "Evening in the palace of reason", a review of which Brad posted a few weeks ago: http://www.bbc.co.uk/radio4/arts/book_of_the_week.shtml
elsewhere on the BBC website, to go completely OT, it's also possible at the moment to listen to a dramatisation of "a la recherche de la temps perdu"!

Canons of the MO. (was: Spiritual Illumination)

Neil Halliday wrote (March 19, 2006):
Raymond Joly wrote:
>"The canons in the MUSICAL OFFERING are a feat indeed, but I confess I do not care to listen to them; as music, I feel they are quite uninteresting".<
Might I suggest a performance recommended by Eric Bergerud some time ago: Karl Muenchinger and the Stuttgart Chamber Orchestra (knowing Eric is partial to HIP, I decided to investigate this version for modern orchestra).

The arrangement of these canons for strings and woodwinds indeed raises them to the status of sometimes quite beautiful abstract music. Only two (of the ten) canon movements with only two parts (lines) sound a bit dry in this recording (tracks 9 and 10; these are also the shortest, allowing for the virtual repeat that occurs in track 10), but all the other movements with three or more parts are at least quite attractive as music.

The 4-voice canon (track 13) is a stupendous piece of music (over 5 mins.) in which the entire canon melody (in this case based on the royal theme) is heard at least twice in all four of the SATB lines, with the full orchestra coming in at the end.

The apparent atonality of the `canon through the tones' is especially remarkable and effective as music (track 11); the royal theme on the oboe eerily drifts higher and higher, while a 2-part canon is heard on independent material in the strings.

Several other canons are also very attractive as music in this recording (this is not the place to elaborate).

Here is an excellent site for study, while listening to these canons: http://www2.nau.edu/~tas3/musoffcanons.html

Listeners will need to identify which canon on their own recording corresponds to those listed as 1 to 10 on this site. The full score, with pointers to the canon leader, canon follower, voice with free counterpoint, etc. ie, the fascinating, if not to say mind-blowing, structure of these canons, can be obtained by clicking on the score
diagram under the title of each canon.

Juozas Rimas wrote (March 19, 2006):
Neil Halliday wrote:
< Several other canons are also very attractive as music in this recording (this is not the place to elaborate). >
I find it surprising that the musicality of the MO canons is questioned at all. EVERY single canon of the MO is not only perfectly listenable but provides, at least to me, inexaustible source of beautiful, moving music.

For instance, the "Violini in unisono" canon. I dug it right away which happens seldom to me with Bach's music. The canon is really easy-listening, so sorrowful, reminiscent of the best moments in Bach's solo violin works. I can't imagine someone seriously calling it non-musical -it's simply heart-grippin.

The canon "a 2 per augmentationem, contrario motu". Fantastic sad music.

"A 2 circularis per tonos" - imagine flying higher and higher with every wave (luckily,the ensemble is helping to create this impression).

"Perpetuus, contrario motu" - flute stuff as strong as the 2^nd Orchestral Suite. Not calling this canon musical? With its almost catchy melody and dreamy ending?

Neil has given a great idea for those sceptical about the MO: try listening to the non-HIP versions. I enjoy Marriner's take a lot, not caring much about its authenticity - the lush music is inviting to be listened to. If, after accustomation to the MO, one desires some more tension and conflicts among sounds, HIP ensemble versions are available, Savall's, for instance.

Julian Mincham wrote (March 19, 2006):
Juozas Rimas wrote:
< I find it surprising that the musicality of the MO canons is questioned at all. EVERY single canon of the MO is not only perfectly listenable but provides, at least to me, inexaustible source of beautiful, moving music. >
Well put. I support this view strongly. The one canon which at first hearing may seem stilted and lacking in natural flow is the fourth of the canones diversi. It is is three parts, a version of the king's theme in the middles and the lower part is a canon with a version of itself inverted and augmented in the upper part. Technically this means that the upper part is the lower part turned upside down in notes twice as long-----something which it is difficult to follow by ear after the first couple of bars.

But, as with much great music it requires repeated listening and familiarity. It then becomes apparent that this is the most profound of the canons, deeply moving and personal like a couple of the minor key variations from the Goldberg variations BWV 988 (which have similar technical constraints which, inhibiting as they might be for some composers, always seemed to act as a stimulus for JSB)

Can't recall the actual reference at the moment but I remember reading Donald Tovey some years ago where he was highly critical of the musical value of a couple of the Art of Fugue (BWV 1080) canons. (he was similarly critical of JSB's habit of making a fugue subject invert when it didn't always flow convincingly, giving as examples WTC Book one (BWV 846)the G major and A minor fugues) For myself I have always enjoyed the MO canons more than those from the AOF. The retrogade canon is delicious (and barely a minute long) and the one described above, profoundly moving. Expressively, the others lie in between these extremes.

And, for me the trio sonata is simply the greatest example of its genre.

Those interested in the MO should read James Gaine's fascinating book 'Evening in the Palace of Reason' (Harper Perennial 2005) which I should think must have been recommended here by subscribers already.

Yang Jingfeng wrote (March 19, 2006):
[To Juozas Rimas] I wonder whether it is inevitable to cost some musicality for structure perfection.

I really think the cantatas sound much more musical than the strict perfect cannons in MO.

Alain Bruguieres wrote (March 19, 2006):
Juozas Rimas wrote
< I find it surprising that the musicality of the MO canons is questioned at all. EVERY single canon of the MO is not only perfectly listenable but provides, at least to me, inexaustible source of beautiful, moving music.
For instance, the "Violini in unisono" canon. I dug it right away which happens seldom to me with Bach's music. The canon is really easy-listening, so sorrowful, reminiscent of the best moments in Bach's solo violin works. I can't imagine someone seriously calling it non-musical - it's simply heart-gripping.
The canon "a 2 per augmentationem, contrario motu". Fantastic sad music.
"A 2 circularis per tonos" - imagine flying higher and higher with every wave (luckily,the ensemble is helping to create this impression).
"Perpetuus, contrario motu" - flute stuff as strong as the 2^nd Orchestral Suite. Not calling this canon musical? With its almost catchy melody and dreamy ending? >
I am one of the culprits. I mentioned explicitly the canon 'cancrizans' as a piece I find musically hard to appreciate (and it is the only piece I mentioned). I notice that you talk about other canons, which I find musically very satisfactory, but you omit 'cancrizans'. Is this omission intentional? Would you be as lyrical about cancrizans as you are about other canons?

Juozas Rimas wrote (March 19, 2006):
[To Alain Bruguieres] I regard the canrizans as the introduction to the canons, simple but musical enough to listen to (1^st of prelude of WTC1 and BWV 924 are also very simple but they are very musical nonetheless). I strongly prefer the two-violins version, as Marriner does, over the harpsichord version. In a less than one minute we are given with an outline of what awaits in the rest of the canons. Marriner presents the canrizans in very clearly audible two voices: one violin is scuttling swiftly (it's reminiscent to me a bit of the prelude in the 1 mvt of the BWV 1023 sonata) and another one is presenting the royal theme in the background in a quite jarred fashion which still seems constrained compared to the first voice.

I'm not sure about authenticity of this approach, but Marriner clearly did his best to present the cancrizans not as a tedious exercise but as a sort of an energetic warm-up.

Have you listened to this or another ensemble version of the cancrizans?

Juozas Rimas wrote (March 19, 2006):
[To Yang Jingfeng] I have passed the line where ALL cannons of the MO appear very listenable to me, without any exceptions. I still have problems with parts of AoF, but I firmly believe that every bar of it is listenable and will sound rewarding to me, when I find enough time to listen to it through several times. I'm sure, though, that BWV 1072 canon (and some other canons from the same lot) is a pure trick and no one will ever convince me that it is listenable.

Speaking of cantatas, I'm afraid I'm not able to appreciate the chorus of the BWV 14 cantata musically. It's so complex that my ear cannot catch any tune or any other "foothold" to start appreciating it somehow.

Alain Bruguieres wrote (March 19, 2006):
[To Juozas Rimas] We're all different. I thouroughly enjoy the whole of The Art of the Fuge. In fact I found it very pleasant to listen to from the start. In particular I like the canons of The Art of the Fuge, even if somehow I find them a bit abstract, remote, of a mineral beauty so to speak...

I greatly enjoyed the opening chorus of BWV from the start, too. However I still find cancrizans lacking in musicality. My problem is not that it sounds simple, as you suggest. Quite the contrary. It sounds rather 'alien' to me. Perhaps it is too short a piece for me to begin to grasp its inner beauty before it is over; I don't think I will come to appreciate it better with time (an impression I often have when listening to a Bach piece for the 1st time). The fact that it is an introductory piece is no excuse to me, since Bach has accustomed us to highly musical and endearing introducing pieces, notably the opening prelude of the WTC which you cared to mention. Perhaps my difficulty comes from the fact that, while I can relatively easily grasp the link between a melody and its accelerated or even inverted forms, and grasping this link is a pleasure in itself, I don't sense any relation between a melody and its retrograde form; and in particular the retrograde form of Thema Regium sounds downright queer not to say unpleasant to me.

Julian Mincham wrote (March 19, 2006):
Yang Jingfeng wrote:
< I really think the cantatas sound much more musical than the strict perfect cannons in MO. >
Fair enough as an expression of personal taste. But they cannot really be compared. Leaving aside the fact that Bach uses some very strict' fugal and canonic devices in some cantata movements, comparing the cantatas with the MO canons is a bit like comparing the first Beethoven sonata with the late quartets. i.e. diffent genres, different purposes and coming from very periods of the composers' creative life. The MO canons are very late works; most of the cantatas which most people enjoy are early or 'middle' (i.e. written when Bach was around 40) works.

It is interesting, as an aside, how many great composers developed stylistically in their later years in ways that produced extreme contrasts of their composing styles. The JSB of the early keyboard toccatas is not the JSB of the MO, AOF or Canonic variations (BWV 769) One needs to listen to them differently and from different perspectives.

I find it rather interesting that it is not too difficult to differentiate early and late Bach, Beethoven, Mozart--or even, despite his early death, Schubert. But with Mendelssohn (a composer I like very much) I find it very difficult to differentiate between the last works and the early masterpieces (e.g. the octet).

Julian Mincham wrote (March 19, 2006):
Juozas Rimas wrote:
< Speaking of cantatas, I'm afraid I'm not able to appreciate the chorus of the BWV 14 cantata musically >
I'm not surprised. This is a very dense chromatic chorus, dourly minor and generally un-lightened by major keys. It employs a dense texture of counterpoint and the kind of intense chromatic harmony which was becoming part of JSB's harmonic language pervading those late works already being discussed by members of the group. Hopefully the denseness of this movement is alleviated somewhat by the much more accessable arias which follow (for sop and bass).

It might (or might not) help to be aware that the opening rising theme, announced immediately by the tenors, is accompanied by the same theme one bar later by the basses--but turned upside down so it becomes a falling melody. tenors rise--basses fall.

The same process is immediately followed by the altos (same rising theme) and the sops a bar later (with the falling theme of the basses). What might be helpful is to play through the first dozen or so bars a few times trying to follow this process aurally--it might then click and the rest of the movement may begin to make more sense.

Alternatively just try to follow through one line at a time starting, perhaps with the basses and afterwards, the sopranos, concentrating on that line and that alone, following the twists and turn of the single melody before trying to make sense of the complicated texture and inter-relatedness of the various lines.

Anyhow, good luck--or as Aryeh would say---enjoy! I hope you do.

Alain Bruguieres wrote (March 19, 2006):
Alain Bruguieres wrotet :
< I greatly enjoyed the opening chorus of BWV from the start, too. >
Oops! I meant the opening chorus of BWV 14, of course.

It is true that such a chorus is very complex. But the first time I heard it, I was conquered. I felt immediately that there were tons of things there which I would discover in time, and probably megatons I would never even suspect... but the thrill was there the first instant. Not so with 'cancrizans'! Although I may be greatly mistaken of course.

The point is not whether I like cancrizans or not, really. The point I was trying to make is that in many of Bach's works (most of them?) there is a 'mathematical' streak. That is, not mathematics in a literal sense, but rejoicing in formal combinations, the challenge of arranging a limited number of elements in the presence of strong constraints and produce complex and unexpected patterns in a dramatic progression which arouses a sense of beauty. Often, the music depicts affects such as sadness, joy, expectation, and what not, which touch me in a vivid way so that the 'mathematical streak' remains in the background. This is often the case with the cantatas! Sometimes, it is the 'mathematical streak' which dominates. I would then be very much at pains to describe the affects depicted by the music, and yet I am very deeply affected! Most important perhaps, the same work, - in the same recording - will sometimes touch me on the affective level, and on other occasions on the 'mathematical level'. And sometimes on both at once! So that, paradoxically, the same work may seem very sad (taken on a sentimental level so to speak) and jubilant (the perfect workmanship being a source of jubilation).
What's wonderful is that we are free to choose!

The aim of this long speech was to explain that, to me, it makes absolutely no sense to say that Bach's music is too mathematical, or to claim that it is not mathematical at all.

Stephen Benson wrote (March 20, 2006):
Julian Mincham wrote:
< Alternatively just try to follow through one line at a time starting, perhaps with the basses and afterwards, the sopranos, concentrating on that line and that alone, following the twists and turn of the single melody before trying to make sense of the complicated texture and inter-relatedness of the various lines. >
Trying to take in at once a multiplicity of contrapuntal lines can be extremely challenging. I find the technique suggested here to be extremely useful whenever I'm having trouble sorting out polyphonic lines. While focusing on one line, I incidentally, and perhaps not surprisingly, hear the other lines more clearly. The very term "counterpoint" suggests such a relativity. Having command of one line enables the other lines to acquire clarity and individual character as contrasting voices.

Raymond Joly wrote (March 20, 2006):
Alain Bruguieres wrote:
«The point is not whether I like cancrizans or not, really. The point I was trying to make is that in many of Bach's works (most of them?) there is a 'mathematical' streak. That is, not mathematics in a literal sense, but rejoicing in formal combinations, the challenge of arranging a limited number of elements in the presence of strong constraints and produce complex and unexpected patterns in a dramatic progression which arouses a sense of beauty. Often, the music depicts affects such as sadness, joy, expectation, and what not, which touch me in a vivid way so that the 'mathematical streak' remains in the background. This is often the case with the cantatas! Sometimes, it is the 'mathematical streak' which dominates. I would then be very much at pains to describe the affects depicted by the music, and yet I am very deeply affected! Most important perhaps, the same work, - in the same recording - will sometimes touch me on the affective level, and on other occasions on the 'mathematical level'. And sometimes on both at once! So that, paradoxically, the same work may seem very sad (taken on a sentimental level so to speak) and jubilant (the perfect workmanship being a source of jubilation). What's wonderful is that we are free to choose!
«The aim of this long speech was to explain that, to me, it makes absolutely no sense to say that Bach's music is too mathematical, or to claim that it is not mathematical at all.»
This is a very valuable contribution indeed. In the 18^th century, some writers, Jean-Jacques Rousseau among them, upheld the theory that music was an imitative art as well as the others, but reinterpreted it to an essential extent. True, music is rather limited in its ability to imitate occurrences in the outside world, when one is done with running streams, rising and falling objets, sighing and so on. On the other hand it can powerfully imitate the movements of the human soul. One of them I think is what Alain Bruguières describes: the joy at working towards a solution and affirming the freedom and triumph of the mind, which it so seldom achieves in real life. That is where «mathematics»-paradoxically for donkeys like myself-can become a source of elation. The rest of Bruguières' observations on the two kinds of elements causing emotion when hearing music (is this the same as musical emotion?) would be sufficient stuff for a week-end seminar on aesthetics.

Tom Hens wrote (March 23, 2006):
Julian Mincham wrote:
<snip>
< The MO canons are very late works; most of the cantatas which most people enjoy are early or 'middle' (i.e. written when Bach was around 40) works. >
<snip>
< I find it rather interesting that it is not too difficult to differentiate early and late Bach, Beethoven, Mozar-or even, despite his early death, Schubert. But with Mendelssohn (a composer I like very much) I find it very difficult to differentiate between the last works and the early masterpieces (e.g. the octet). >
If Bach had died at the age Mozart or Mendelssohn did, late Bach would have been what he wrote in Köthen. Perhaps people these days would then be discussing why the early Bach wrote church cantatas, but the late Bach concentrated on purely instrumental music. Had he lost his faith, perhaps? If Bach had died at the age Schubert did, he'd never even have made it to Köthen, and late Bach would be Weimar works.

Of course there are cases of people who reach a certain age, or a certain state of health, who are aware they're producing their last work(s). But on the whole, I find this pervasive habit of trying to impose retroactive periodisation in this way slightly amusing, whether applied to the output of an individual or to whole periods. In another place, I recently saw a discussion about whether Monteverdi was a "late renaissance" or "early baroque" composer. Yes, I'm sure that's a dilemma the poor man spent sleepless nights worrying about.

Bradley Lehman wrote (March 23, 2006):
Tom Hens wrote:
< Of course there are cases of people who reach a certain age, or a certain state of health, who are aware they're producing their last work(s). But on the whole, I find this pervasive habit of trying to impose retroactive periodisation in this way slightly amusing, whether applied to the output of an individual or to whole periods. In another place, I recently saw a discussion about whether Monteverdi was a "late renaissance" or "early baroque" composer. Yes, I'm sure that's a dilemma the poor man spent sleepless nights worrying about. >
Having already spearheaded the "second practice" he probably wracked his brain trying to decide what to do third or fourth: to stay ahead of all his would-be imitators. Like maybe having something else trump both the music and the words (e.g., inventing the genre of commercial jingles for the local vintners, or taking even greater care to specify his orchestrations, or writing out all his costuming/staging instructions in full, or larding his compositions with arbitrary feats of numerology, or laying down guidelines for the style of Ornette Coleman).

Of course, that would have screwed up the plot of Richard Strauss's "Capriccio" (or at least added more suitors to the cast), but that needn't have concerned Monteverdi.

Julian Mincham wrote (March 23, 2006):
Tom Hens wrote:
< Of course there are cases of people who reach a certain age, or a certain state of health, who are aware they're producing their last work(s). But on the whole, I find this pervasive habit of trying to impose retroactive periodisation in this way slightly amusing, whether applied to the output of an individual or to whole periods. >
This was not quite my point. Whilst I agree about the dangers of retroactive periodisation, I was musing on (amusing myself with??) the thought, which I think is largely self evident, that some composer's styles changed much more throughout their lifetimes than others. One amusing aside to this relates to a lady, now dead, who, some years ago gained considerable publicity in GB by claiming that she was getting messages from dead composers who were dictating their posthumus pieces to her to copy down. The pieces claimed to be dictated by Beethoven all shared clear stylistic similarities to his early works e.g. the Pathetique piano sonata.

It seems somewhat absurd that, from his grave and after the 9th symphony and the late quartets and piano sonatas he would have returned to a style of writing associated with his earliest works.

The point is that these differences in style are clearly apparent and discernable without any retrospective imposition of 'periodisation'.

Regarding when composers died Mozart, Schubert and Mendelsohn all died in their 30s, but (to my ear at least) their stylistic development was quite different in each case.

Musical Offering for the New Year

Aryeh Oron wrote (December 25, 2006):
Discography

Following previous discographies of Bach's non-vocal works, I have added now a comprehensive discography of the Musical Offering BWV 1079 (MO), which, AFAIK, is the first ever web-discography of this work.

I have used every possible source I could find, including web-catalogues as J.S. Bach Home Page and All Music Guide, web-stores as Amazon, JPC, CD Universe and eBay, web-magazines as Gramophone and MusicWeb, and other websites I have been able to find with Google search engine, as well as various printed catalogues and my private collection.

As with previous discographies at the BCW, the complete recordings of the MO are split into several pages, a page for a decade. Since there are many individual recordings of the Trio Sonata from the MO, as well as the 2 Ricercars, which are not part of complete recordings of this work, I have also created pages for them. In 1935 Anton Webern orchestrated the 2^nd Ricercar [Fuga (Ricercata) a 6 voci], the recordings of which were put into a separate page. Previous discussions of the MO have also been compiled into Discussions page. You can find them all through the main page of the MO at the BCW:
http://www.bach-cantatas.com/NVD/BWV1079.htm

This initial version of the discography includes 57 complete (or near complete) recordings of the MO, 16 of the Trio Sonata, 36 of the Ricercars and 10 of the Webern's Ricercar. As in previous discographies in the BCW, each recording is listed only once. All the issues of each recording are presented together. If a performer has recorded the MO more than once, the info includes also the recording number. Nevertheless, I am quite certain that there are somewhere recordings I have missed, and for many recordings, especially from the 1960's and 1970's, the info presented is only partial. Please help me making this discography more comprehensive and more accurate. If you are aware of a recording of the MO not listed in these pages, or if you find an error or missing information, please inform me, either through the BRML or to my personal e-mail address.

Background

Taka Kidokoro wrote a good short description of the MO and the background to its composition in the liner notes of DVD, which contains a live recording of the chamber music concert given in July 2000 as part of the Leipzig Bach Festival. It was performed by the Kuijken Brothers exactly on the 250^th anniversary of J.S. Bach's death.

Bach's late masterpiece The Musical Offering (1747) is music of homage written on the occasion of Bach's visit to King Frederick the Great of Prussia. By that time, the choirmaster and organist at St. Thomas's Church in Leipzig was already enjoying considerable recognition in Saxony and Thuringia, and his circle of admirers and, in some cases, patrons was beginning to grow. One of these admirers was the Russian ambassador to Prussia, Hermann Carl von Keyserlingk. The Count, who is regarded as having commissioned the Goldberg Variations, referred the king to the Leipzig musician. As a result of this referral, Bach received an official invitation from the Prussian court.

Bach gladly accepted the offer, and the reason for this decision must have been the honour and fame accorded him by the audience. But a personal motive certainly played a part in it too: his second son Carl Philipp Emanuel had been active as royal harpsichordist at the Prussian court since 1740, and in 1745 a son was born to him, Bach's first grandchild. The grandfather probably used the audience as an opportunity for a family reunion as well. Thus it was on May 7, 1747, accompanied by his first son Wilhelm Friedemann, that he arrived in Potsdam. It is conjectured that the meeting took place not in the newly built Sanssouci, but the city castle. On learning of Bach's arrival, the king is said to have announced to those present "with a kind of disquiet": "Gentlemen, old Bach has arrived!"

The king led him to one of the Silbermann fortepianos spread out among various rooms of the castle and asked for an improvisation on a theme set by him. Bach performed this with an artistic fugue à 3, eliciting all-round admiration in the process. The next day, the king again demanded a six-part prelude on the same theme. Bach felt less than sure about the task and played a six-part fugue on a different (more suitable) theme. Although the playing was of an "equally magnificent and learned nature" (Forkel/Wilhelm Friedemann Bach), the composer felt that he had offered a less than satisfactory service. There and then he resolved "to elaborate this right regal theme more perfectly, and thereafter to make it known to the world". That was the basis of his Musical Offering.

Straight after the trip Bach began work on the composition. Work went really swiftly, the oeuvre being completed by 7 July and then sent to Potsdam in the form of a special volume destined for the king. Following that, in September of the same year, Bach had the music printed. The first edition consisted of 100 copies. The entire labour, from composition to printing, took only a matter of months, which testifies to Bach's special ambition to do justice to the self-appointed task. The Musical Offering is a rare blend of different chamber music pieces with two ricercares (three or six-part fugues), a trio sonata and ten elaborate canons. The "thema regium" forms the basis for all these numbers, and the first ricercare à 3 is a transcription of the extemporisation performed in Potsdam. The renowned Bach researcher Christoph Wolff claims that the score dedicated to the king originally contained only this piece and seven additional canons. However, this would mean that Bach decided relatively late on to write the fugue not performed in Potsdam.

Yet it is hard to imagine that he refrained from sending the desired six-part fugue and only penned it after the event to coincide with the printing of the score.

At any rate, the score of The Musical Offering presents us with various problems of performance technique. The printed music - possibly owing to lack of time - is singly bound, and the exact sequence intended by the composer is therefore not known. Uncertainty also surrounds the orchestration. Of the entire composition, only a handful of parts are provided with directions for orchestration: trio sonata and canon perpetuus with transverse flute, violin and basso continuo, canon à 2 violini in unisono with two violins and bass (played in the present recording on violin and harpsichord, however). Above all, the sequence fuelled discussion, and even today there is still no clear-cut insight into the order.

Bach's intentions can be recognised at every turn of the orchestration, however. 1n this work he is trying to accommodate the practical facilities available to the musical king. The four-movement trio sonata is chamber music with flute, as mentioned; the king himself played this instrument and frequently made music at the court. With a contemplative andante Bach takes on the then fashionable "sentimental style" of Carl Philipp Emanuel or Graun and thereby does justice to the rococo-like taste of the king. 1n the present recording, the ricercare a 6 is also played in the orchestration of the trio sonata, lucidly conjuring up the picture of the musical soiree at the Prussian court.

Musical Offering theme

Continue of discussion from: Evening in the Palace of Reason (By. James R. Gaines] [Books]

Julian Mincham wrote (January 3, 2007):
Julian Mincham wrote:
< There are few similar fugue subjects in the Bach repertoire, the one with most similarities being that of the B minor fugue from the WTC (no 24) This too contains all the notes of the scale but one, but is more focussed and precise in its repeated motivic structure i.e the six falling motivic minor seconds enclosed by three note arpeggios on B and F sharp minor. >
A correction to myself. I wrote the above when away from my books and music from memory, always a dangerous thing to do!

In fact the WTC fugue subject (book 1) contains ALL the 12 notes of the chromatic scale in its 20 note theme---mea culpa. It remains true that the Musical Offering theme contains all but one (Bb) in its 21 notes.

However, when I came to look at these two pieces again another thought struck me; and apologies for becoming somewhat technical. Whilst it is the case that it is not unusual for baroque and Bachian themes and fugue subjects to be highly chromatic (see, as examples, that in D minor from WTC Book 2 and the last movement of the E minor toccata) the two pieces mentioned above are different.

Most highly chromatic fugue subjects (generally they are in minor rather than major keys) make extensive use of the chromatic notes in the upper part of the octave e.g. taking Cminor as a reference key, these are the notes between G and the upper octave C (C, B, Bb, A. Ab, G). This is a popular sequence because it is easily and powerfully harmonised by a simple sequence of chords based upon roots a 4th apart.

What distiguishes the Bm and Musical Offering themes is their unusual employment of chromatic notes in the LOWER section of the octave----F sharp, E natural and Db in the key of C minor. Whilst it is not difficult to harmonise these notes singly, when all three are used within the same phrase (the same bar in the case of the Bm subject) they present problems which can produce a harmonic awkwardness.

My guess is that this may lie at the root of the desire to either test or humiliate Bach. CPE, if not the King, would certainly have been aware that the use of such notes as a fundamental part of a fugue subject was highly untypical of his father's output.

And if Bach was correctly reported as indicating that the 'King's theme' was not suitable as a basis for improvising a 6 part fugue, this may well have been the reason why.

Ed Myskowski wrote (January 3, 2007):
Julian Mincham wrote:
< A correction to myself. I wrote the above when away from my books and music from memory, always a dangerous thing to do! >
Good to see that you are surviving the '12 days of Xmas'. It is a variant of Murphy's Law: the opportunity for error guarantees the error. Something like that.

< In fact the WTC fugue subject (book 1) contains ALL the 12 notes of the chromatic scale in its 20 note theme---mea culpa. It remains true that the Musical Offering theme contains all but one (Bb) in its 21 notes. >
So much for novelty of the twelve-tone school.

< However, when I came to look at these two pieces again another thought struck me; and apologies for becoming somewhat technical. >
No apologies needed. That is the true beauty of this forum. We can have a laugh to ease the bitter pills down!

Bradley Lehman wrote (January 4, 2007):
< In fact the WTC fugue subject (book 1) contains ALL the 12 notes of the chromatic scale in its 20 note theme---mea culpa. It remains true that the Musical Offering theme contains all but one (Bb) in its 21 notes. >
If we're gonna be really technical and picky, that WTC B minor subject actually has 13 differently named pitch classes in it, not only 12. C natural and B# are both in there, serving different melodic functions.

Carry on.... (or check your luggage....)

Julian Mincham wrote (January 5, 2007):
[To Bradley Lehman] Well, if you are going down that road it's actually 14 because there is an E natural and an E sharp as well. However I don't think this is particualrly helpful to the general listener (or even musician) where the basic triadic harmony theory is predicated upon 12 notes to the octave:--which is why I didn't mention it or feel it to be appropriate for this list. IT only muddies the waters for the majority of people.

Peter Smaill wrote (January 5, 2007):
[To Julian Mincham] Far from muddying the waters, has not Julian just discovered another crystal-clear example of the gematric 14 (BACH in the number alphabet )?

Julian Mincham wrote (January 5, 2007):
Peter Smaill wrote:
< Far from muddying the waters, has not Julian just discovered another crystal-clear example of thegematric 14 (BACH in the number alphabet )? >
Interesting observation, Peter--I hadn't thought of that.

Nevertheless I still think that traditional C18 harmonic theory (predicated upon 12 notes to the octave) is the simplest and most meaningful way of analysing such pieces. What I actually should have said is that counting enharmonically altered notes as 'additional notes' tends to be confusing. Sure, they might have different Functions as Brad suggests but that seems to me not to be relevant------ it happens all the time. The note of B in the major scale of C has different functions if it

a) is a part of a dominant chord rising to the C above orb) is a suspended note falling to an A as a part of chord 1V

Still the same note though for harmonic purposes though; a string player might instinctively play it slighly sharper or flatter in pitch but is is still the 7th note of the scale, functioning in different ways.

Bradley Lehman wrote (January 5, 2007):
[To Julian Mincham]
13:
F#, D, B, G, (F#), (B), A#, E, D#, C, (B), (F#), E#, (D,) C#, B#, (C#), A, (F#), G#, (F#).

That's touching all the 12 chromatic notes on a keyboard at least once each, plus using both C natural and B# (sharing the same physical key) in two different functions. Total of 13 differently named pitch classes.

As for muddying the waters, maybe so, but anyway it's 13. :)

Ed Myskowski wrote (January 5, 2007):
[To Julian Mincham] Nicely stated, so that even the non-musicians can grasp the discussion!

BTW, I also find Peter's gematric note interesting. I have often expressed skepticism over some of the more extreme gematric or numerologic interpretations. This should not be misunderstood as skepticism about the entire topic.

In fact, just the opposite. One of my objections to extremes of interpretation is that they tend to discredit an entire field or topic, throwing out the baby with the bathwater, as it were (ACE).

Bradley Lehman wrote (January 5, 2007):
Peter Smaill wrote:
< Far from muddying the waters, has not Julian just discovered another crystal-clear example of the gematric 14 (BACH in the number alphabet )? >
The book 1 fugue with a hidden set of 14 notes in it (for what that's worth, which I suspect is merely a coincidence) is the C# minor, not the B minor.

Like this: http://www-personal.umich.edu/~bpl/larips/clavichord.html
"But let us look more closely at the extant music. The C# minor fugue of book 1 negates this notion of retuning: it uses 14 different notes, G, D, A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx, and Cx. The G, D, Fx, and Cx (among others) occur in accented thematic positions in this piece. This extraordinary fugue has real entrances of its subject in six different keys, and it gives us all of the following diminished 4ths in melodic contexts: B#-E, Fx-B, E#-A, Cx-F#, D#-G, A#-D." Blah blah blah.

The B minor fugue, as a whole, happens to have a total of 17 differently named notes in it.
- Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#, D#, A#, E#, B#, Fx
And the subject alone has 13 of those, already: being not only a "really chromatic" piece but a "really really chromatic piece". :)

The book of inventions/sinfonias, as a whole, happens to visit 25 differently named notes.......

Julian Mincham wrote (January 5, 2007):
Bradley Lehman wrote:
< F#, D, B, G, (F#), (B), A#, E, D#, C, (B), (F#), E#, (D,) C#, B#, (C#), A, (F#), G#, (F#). >
That's touching all the 12 chromatic notes on a keyboard at least once each, plus using both C natural and B# (sharing the same physical key) in two different functions. Total of 13 differently named pitch classes.

As for muddying the waters, maybe so, but anyway it's 13. :)

It seems quite bizarre to me , and against all triadic harmonic theory I know to count the octave C as a different note from the lower C simply because it can be written enharmonically as a B sharp. How does this relate to the fact that you can also count D as a C double sharp, or a E double flat? or F as an E sharp as in the Bach theme (ot G double flat for that matter).

Yeah sure, you touch 13 notes if you begin on C and go up to include the top C. That is simply where the process begins all over again. It's the beginning of a new sequence of 12 more notes---- C up to B (i'e' before you repeat the sequence ) is 12 notes not 13.

Maybe Schoenberg was at fault in calling serialism a '12 note system'----should be the 13 note system! And John Mehegan should have had a 65 chord system rather than the 60 chord one he produced.

Bradley Lehman wrote (January 5, 2007):
I wrote:
< F#, D, B, G, (F#), (B), A#, E, D#, C, (B), (F#), E#, (D,) C#, B#, (C#), A, (F#), G#, (F#).
That's touching all the 12 chromatic notes on a keyboard at least once each, plus using both C natural and B# (sharing the same physical key) in two different functions. Total of 13 differently named pitch classes. >
Julian replied (in part):
< It seems quite bizarre to me , and against all triadic harmonic theory I know to count the octave C as a different note from the lower C simply because it can be written enharmonically as a B sharp. How does this relate to the fact that you can also count D as a C double sharp, or a E double flat? or F as an E sharp as in the Bach theme (ot G double flat for that matter).
Yeah sure, you touch 13 notes if you begin on C and go up to include the top C. That is simply where the process begins all over again. It's the beginning of a new sequence of 12 more notes---- C up to B (i'e' before you repeat the sequence ) is 12 notes not 13. >
It's not about going up them chromatically, or from any particular endpoints. And it's not about merely academic spelling exercises on enharmonic notes, but it's about the way they sound.

Irrespective of any specific temperament choice, enharmonic pairs such as C natural and B# (or G#/Ab, or B/Cb, or whatever) simply are different notes for their various melodic and harmonic functions. Whenever both of them get used within the same composition, with an enharmonic swap having occurred somewhere in between, we somehow have to burn off a Pythagorean comma between one and the other. If that's done gracefully and with moderation, both of them might sound decently in-tune enough within their contexts. Or, if it's done roughly, one or the other will sound rotten.

For example, a note that's been tuned "too purely" as a C simply cannot be played as a B#, or as a Dbb, without sticking out as grossly wrong for its context.

The "circle of fifths" is really a spiral that doesn't meet itself: by the time we've come around twelve notes we're either a Pythagorean comma higher or lower than we started, unless we've been diddling off little fragments of it as we go along.

...Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B,
...F#, C#, G#, D#, A#, E#, B#, Fx, Cx, Gx, Dx, Ax, ....

Attempting not to belabor this too much, here are some resources:

Ross Duffin's new book released in the past couple of weeks:
http://www2.wwnorton.com/catalog/fall06/006227.htm
http://www.amazon.com/Equal-Temperament-Ruined-Harmony-Should/dp/0393062279

The "Figure 4" graph and explanation in my article here:
http://www-personal.umich.edu/~bpl/larips/clavichord.html

My mapping of enharmonic notes onto the model of a Rubik's Cube, which is a fun exercise!:
http://www-personal.umich.edu/~bpl/larips/cube-layout.html

Julian Mincham wrote (January 5, 2007):
Brad I suspect we are approaching this from somewhat different viewpoints.

I come at it principally from a harmonic point of view and the implications of the 12 notes of the chromatic scale to form chords, consonances, dissonances etc.

My reactions to you main points inserted below.

Bradley Lehman wrote:
< It's not about going up them chromatically, or from any particular endpoints. And it's not about merely academic spelling exerciseon enharmonic notes, but it's about the way they sound. >
so far agreed. It is about sound---and the different contexts which alter our perceptions of those sounds.

< Irrespective of any specific temperament choice, enharmonic pairs such as C natural and B# (or G#/Ab, or B/Cb, or whatever) simply are different notes for their various melodic and harmonic functions. >
again mainly agreed. But why only pinpoint the b sharp/C natural relation particularly as in the original postings (to make 13 notes)? all notes may have variant enharmonic spellings according to context. By this logic we could end up with 24-30 scale notes within an octave and this, to me, is a harmonic absurdity when looked at from a point of view of harmonic practice (and theory) and the building of 'chords' in series of thirds.

< Whenever both of them get used within the same composition, with an enharmonic swap having occurred somewhere in between, we somehow have to burn off a Pythagorean comma between one and the other. If that's done gracefully and with moderation, both of them might sound decently in-tune enough within their contexts. Or, if it's done roughly, one or the other will sound rotten.
For example, a note that's been tuned "too purely" as a C simply cannot be played as a B#, or as a Dbb, without sticking out as grossly wrong for its context.
The "circle of fifths" is really a spiral that doesn't meet itself: by the time we've come around twelve notes we're either a Pythagorean comma higher or lower than we started, unless we've been diddling off little fragments of it as we go along. >
Absolutely, this (last parar above) is the veryfact of nature which presents us with these particular problems.

< ...Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B,
...F#, C#, G#, D#, A#, E#, B#, Fx, Cx, Gx, Dx, Ax, .... >
I don't want to hang this thread out either particularly as its probably of a minority interest tp many on this list. If I put it in pedagogical terms, however, I forsee problems of teaching students triadic harmony theory (the basis of Western European musical tradition for some time) on the basis of the fact that there might be 12, 13 or more actual notes in the octave.

Question ---does a composer consider the Fsharp to be an actual different note when it occurs as a part of V7 in the key of G as against when (as written G flat) it appears as part of the dom 7th chord in the key of Db? Sure, they sound very different because of the context---just as the very same triad of G sounds very different in the established keys of C, G--or to really stress the point, as a Neapoliton in F sharp minor, do thy not? They are all repititions of precisely the same chord--but their differing contexts makes them SOUND most radically different.

That to me is one of the fundamental characteristics of tonal music and one of the reasons why it has so much potential variety, interest and expressivity.

PS on a personal level, it's a pleasure to have an interesting argument or point of different which can be explored without rancour-------roll on!

Bradley Lehman wrote (January 5, 2007):
< For example, Duffin includes on page 47 a diagram of a violin fingerboard drawn in Peter Prelleur's book The Modern Musick-Master, 1730-31, clearly illustrating that the sharps are to be fingered at different places on the string next to the enharmonic flats. He has put this same illustration here: http://music.cwru.edu/duffin/Vallotti/T1/Prelleur.html >
p.s. On that particular illustration, what I find especially startling and instructive is the recommendation that A# and Bb, on the E string, are not even supposed to be played by the same finger! And similarly for the other three strings, when the hand is playing in first position. The third finger of the left hand is supposed to reach up to play the sharp, while the fourth finger is supposed to stretch down to play the enharmonically paired flat.

Any string-playing professionals or keen amateurs here care to comment: have you encountered this before, in years of taking or teaching lessons on your instrument?

And this is by no means Duffin's only presented evidence along this line. I especially like the section where he cites a Haydn quartet (with autograph score 1799, published score 1802 both presented in facsimile), where Haydn wrote in a special warning for the cellist during an enharmonic shift.

Bradley Lehman wrote (January 5, 2007):
Julian Mincham wrote:
< Brad I suspect we are approaching this from somewhat different viewpoints. >
Agreed!

<< Irrespective of any specific temperament choice, enharmonic pairs such as C natural and B# (or G#/Ab, or B/Cb, or whatever) simply are different notes for their various melodic and harmonic functions. >>
< again mainly agreed. But why only pinpoint the b sharp/C natural relation particularly as in the original postings (to make 13 notes)? all notes may have variant enharmonic spellings according to context. >
Because those are the 13 specific pitch classes in the B minor fugue subject (WTC1), which was the original question....... Here they are:C, G, D, A, E, B, F#, C#, G#, D#, A#, E#, and B#.

Another great book about this, by the way, is Easley Blackwood Jr's The Structure of Recognizable Diatonic Tunings....

< By this logic we could end up with 24-30 scale notes within an octave and this, to me, is a harmonic absurdity when looked at from a point of view of harmonic practice (and theory) and the building of 'chords' in series of thirds. >
But, there are "25 scale notes within an octave" notated in Bach's set of 15 sinfonias and 15 inventions......

And if we consider both books of WTC together, there are even more than 25 differently named notes there. Book 2 especially goes way up to the double-sharps and double-flats, stretching both ends. Book 1 doesn't venture quite so deeply into the double-flats. It does use as high as Cx and Gx, but I don't remember offhand if it also gets to Dx.


<< The "circle of fifths" is really a spiral that doesn't meet itself: by the time we've come around twelve notes we're either a Pythagorean comma higher or lower than we started, unless we've been diddling off little fragments of it as we go along. >>
< Absolutely, this (last parar above) is the veryfact of nature which presents us with these particular problems. >
Yep.

<< ...Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B,
...F#, C#, G#, D#, A#, E#, B#, Fx, Cx, Gx, Dx, Ax, .... >>
< I don't want to hang this thread out either particularly as its probably of a minority interest tp many on this list. If I put it in pedagogical terms, however, I forsee problems of teaching students triadic harmony theory (the basis of Western European musical tradition for some time) on the basis of the fact that there might be 12, 13 or more actual notes in the octave. >
Hence, in part, the need for Duffin's book and others like it! It calls for consideration of the way Leopold and Wolfgang Mozart (among others) taught such things, as ordinary practice, where there definitely were more than 12 in the octave.

For example, Duffin includes on page 47 a diagram of a violin fingerboard drawn in Peter Prelleur's book The Modern Musick-Master, 1730-31, clearly illustrating that the sharps are to be fingered at different places on the string next to the enharmonic flats. He has put this same illustration here: http://music.cwru.edu/duffin/Vallotti/T1/Prelleur.html

< Question ---does a composer consider the Fsharp to be an actual different note when it occurs as a part of V7 in the key of G as against when (as written G flat) it appears as part of the dom 7th chord in the key of Db? >
That depends at least somewhat on the tuning system(s). In everything BUT equal temperament, when the string of notes is constructed by a series of regularly spaced fifths, they (pairs such as F# vs Gb etc) are absolutely different notes.

< Sure, they sound very different because of the context---just as the very same triad of G sounds very different in the established keys of C, G--or to really stress the point, as a Neapoliton in F sharp minor, do thy not? They are all repititions of precisely the same chord--but their differing contexts makes them SOUND most radically different. >
I agree, they sound radically different because of context, even if the intonation (in some forced system such as equal temperament) happens to be exactly the same. The difference of function still contributes difference of effects, yes.

< That to me is one of the fundamental characteristics of tonal music and one of the reasons why it has so much potential variety, interest and expressivity. >
Agreed. At the same time, those tonal rules weren't generated by equal temperament in the first place, and they get washed out considerably if the music is constrained to be played only in equal. Equal gives the "variety, interest, and expressivity" of (an analogy here) a black-and-white movie, even though the real-life takes with the actors occurred in colour.

< PS on a personal level, it's a pleasure to have an interesting argument or point of different which can be explored without rancour-------roll on! >
Amen to that, too!

Cara Emily Thornton wrote (January 5, 2007):
Bradley Lehman wrote:
<< For example, Duffin includes on page 47 a diagram of a violin fingerboard drawn in Peter Prelleur's book The Modern Musick-Master, 1730-31, clearly illustrating that the sharps are to be fingered at different places on the string next to the enharmonic flats. He has put this same illustration here: http://music.cwru.edu/duffin/Vallotti/T1/Prelleur.html >>
< p.s. On that particular illustration, what I find especially startling and instructive is the recommendation that A# and Bb, on the E string, are not even supposed to be played by the same finger! And similarly for the other three strings, when the hand is playing in first position. The third finger of the left hand is supposed to reach up to play the sharp, while the fourth finger is supposed to stretch down to play the enharmonically paired flat.
Any string-playing professionals or keen amateurs here care to comment: have you encountered this before, in years of taking or teaching lessons on your instrument? >
Yes, for a violinist this fingering appears perfectly normal (4^th for b-flat'', 3rd for a#'' - for those who are not familiar with this notation, the double prime after the note signifies that it is in the second octave above 'middle C', which is called that, no doubt, because it is in the middle of the piano keyboard). The exception to this fingering rule would be if the third finger is already occupied with, for example, the note a tritone higher (note: violin is tuned in fifths, so the perfect fifth would involve stopping both strings with the same finger, so the tritone, being half a step smaller, would involve use of adjacent fingers).

This obviously would not happen on the E string, but for example: if the third finger in first position on the E string is occupied with the note a'', then you would use the fourth finger on the A for d#'' (a tritone lower). But again, in a normal two-octave B major scale in first position, you definitely would use the third finger for g#', d#", a#" (on the D, A and E strings respectively).

And yes, the a#'' being a leading tone to b'', we will definitely place the a#'' close to the b''. Whereas, if we have a progression C7-F, the b-flat'' is the '7' in that C7, and it resolves down a half step to the note a'', which is part of the F chord. And we will definitely place the b-flat'' close to the a'', with the result that b-flat'' ends up, oh, maybe one cycle per second (Hz) or so lower than the a#'', given which octave we are in.

Thomas Braatz wrote (January 6, 2007):
Bradley Lehman wrote:
>>For example, Duffin includes on page 47 a diagram of a violin fingerboard drawn in Peter Prelleur's book "The Modern Musick-Master, 1730-31, clearly illustrating that the sharps are to be fingered at different places on the string next to the enharmonic flats. He has put this same illustration here: http://music.cwru.edu/duffin/Vallotti/T1/Prelleur.html
p.s. On that particular illustration, what I find especially startling and instructive is the recommendation that A# and Bb, on the E string, are not even supposed to be played by the same finger! And similarly for the other three strings, when the hand is playing in first position. The third finger of the left hand is supposed to reach up to play the sharp, while the fourth finger is supposed to stretch down to play the enharmonically paired flat. Any string-playing professionals or keen amateurs here care to comment: have you encountered this before, in years of taking or teaching lessons on your instrument? And this is by no means Duffin's only presented evidence along this line. I especially like the section where he cites a Haydn quartet (with autograph score 1799, published score 1802 both presented in facsimile), where Haydn wrote in a special warning for the cellist during an enharmonic shift.<<
All of this has little if anything to do with Bach's music and his performance practices. In citing sources from different countries and cultures as well as sources a generation or more after Bach's death, the chances that anything meaningful regarding the focus which here on these lists should directed more specifically to a certain time and place rather than jumping far afield are truly very slim and the information presented not very significant to the matter at hand.

Johann Friedrich Agricola, who performed under Bach's direction for a few years while still residing in Leipzig, comments on Pier Franceso Tosi's "Opinioni de' cantori antichi e moderni....", 1723 who still argues against the newest temperament ["die neumodische Intonation" ("the newly fashionable temperament") or (equal temperament)which was beginning to take hold at that time as translated in Agricola's translation and commentary, "Anleitung zur Singkunst", Berlin, 1757. Agricola has a footnote on p. 19 stating that in earlier times keyboard instruments had split keys, two to every octave between the G and A and the D and E so as to distinguish between the G# and Ab and the D# and Eb. This practice, Agricola explains, had been completely abandoned and replaced by the effort to make the temperament provide a better distribution of notes without resorting to these differences. From his use of "Temperatur" or "schwebende Stimmung" it is not clear whether this refers to any number of well-tempered, non-mean-tone temperaments or to Equal Temperament. Taken literally, however, "schwebende Stimmung" would imply that other than the octaves, all intervals would demonstrate noticeable beating, some intervals less than others. That definition could apply to Equal temperament. Tosi comments that more and more opera composers (1723) were introducing quite a number of arias with string accompaniment only. What this could imply is that the string players could still play G# or Ab (or D3 or Eb) differently and the singer would have to match these distinctions. The use of other instruments, particularly keyboard instruments without broken/split keys would begin to cause noticeable clashes unless the string players adjusted their tuning of these notes accordingly and gave up their old methods. Singers encountered a real problem faced with the prospect of performing with an organ or harpsichord which could not account for these finer distinctions and then trying to adjust the intonation differently when the strings predominated. Other instruments like the flute have greater difficulty in adjusting quickly to these 'split-key' notes. When playing notes slowly it is easier to adjust by using a different fingering or by adjusting the embouchure, etc. than when the notes are played at a or fast tempo.

It would appear that the string instruments held on to the older split distinctions longer than any other type of instrument while the keyboard instruments were on the vanguard of doing away with these distinctions as fast as possible, one main reason being the cumbersome additional difficulty of playing these split keys the faster the tempo was taken.

Ed Myskowski wrote (January 6, 2007):
Bradley Lehman wrote:
< The book of inventions/sinfonias, as a whole, happens to visit 25 differently named notes....... >
Which is 13 (the number of the Last Supper) x 2, minus 1 (Jesus? Judas?).

Witch is also, on my Wiccan block, the Pentangle squared.

Take your pick.

Ed Myskowski wrote (January 6, 2007):
Julian Mincham wrote:
<< PS on a personal level, it's a pleasure to have an interesting argument or point of different which can be explored without rancour-------roll on! >>
Bradley Lehman wrote:
< Amen to that, too! >
Even enjoyable as a spectator!

Thomas Braatz wrote (January 6, 2007):
Bradley Lehman wrote:
>>That depends at least somewhat on the tuning systems). In everything BUT equal temperament, when the string of notes is constructed by a series of regularly spaced fifths, they (pairs such as F# vs Gb etc) are absolutely different notes....
Agreed. At the same time, those tonal rules weren't generated by equal temperament in the first place, and they get washed out considerably if the music is constrained to be played only in equal. Equal gives the "variety, interest, and expressivity" of (an analogy here) a black-and-white movie, even though the real-life takes with the actors occurred in colour.<<
The following quotation is extremely valuable in assessing properly the movement away from making audible distinctions between notes such as G# or Ab or another pair D# or Eb. Agricola's commentary looked back at the advances made in the preceding decades. Mattheson, picking up on the changes toward adopting Equal Temperament that were already taking place, writes persuasively early on how to achieve the goal of eliminating antiquated distinctions caused by less successful temperaments and by older instrumentalists who were not amenable to change. The split keys on an organ or harpsichord keyboard had been proven unnecessary as soon as it could be demonstrated that tuning organs according to Equal Temperament was feasible and practical. The next effort that immediately had to follow was to change/modify wind instruments so that they would comply with the requirements of Equal Temperament. A new generation of players would now need to be taught to play on these newer instruments, thus discarding the old way of making distinctions where now these distinctions were no longer necessary.

Note the time, June 1722, and place, Hamburg, in a musical journal that was widely read by musicians and composers throughout German-speaking countries. It is highly likely that Bach also would have read these lines:

Johann Mattheson "Critica musica" Hamburg, June, 1722, p. 53
[After having presented on the preceding page the precise measurements according to which anyone could tune an octave in Equal Temperament using a monochord, Mattheson continues:]

"Und weil keine bessere noch vollkommenere 'Temperatur' in der Natur zu finden / so fehlet es anitzo (nachdem deren Gebrauch auf den Orgeln auch ,practicable' befunden worden) an nichts mehr / als deren ,application' auch an die Blase=Instrumenten / als ,Hautbois, Bassons, Flutes &c.' wozu man denn auch zulängliche Nachricht zu geben sich getraute / wenn ein dergleichen Instrumentmacher hier ,in loco' wäre. Wenigstens hat man schon auf einer ordinairen Flöte / jedoch mit andrer ,application' der Finger / alle 12. ,Intervalla' rein / nach dem ,Monochordo', und dieser Temperatur / herausgebracht. Wäre es also auf den andern Blase=Instrumenten / zweifelsohne / auch müglich : falls nur die Herrn ,Virtuosi' gedachter Instrumenten sich die Mühe nehmen wollten / eine andere ,application' der Finger sich anzugewöhnen. Dafern es aber die Alten nicht thun wollten / so könnte es doch bey den Jungen / dergleichen Blase=Instrument erst=lernenden / geschehen / und also / nach und nach / die Music in mehre ,Perfection' und Vollkommenheit /'ratione intervallorum,' mit der Zeit kommen."

("And since nowhere in all of Nature can anyone find a better or more perfect temperament [than Equal Temperament], all that is lacking now (after the introduction and use of Equal Temperament on organs has also been determined to be practicable) is to apply the same temperament to wind instruments (oboes, bassoons, flutes, etc.) as well for which you [must] trust yourself to be able to pass on sufficiently specific information to a wind instrument maker who might happen to live in your area. At least some people have already succeeded in playing all 12 intervals/notes of this Equal Temperament in tune according to the [sounds of the] monochord on a regular flute; however, this was accomplished by applying different fingerings. Without a doubt it would then also be possible to do the same on other wind instruments [trumpets, horns, etc.], if only the virtuosi playing such instruments would make the effort to become accustomed to different fingerings. But if the older instrumentalists do not want to do this, then let this happen with the young instrumentalists who are still learning to play wind instruments of this type. Thus gradually, little by little, in time [the performance of] music will achieve greater perfection in regard to the theory of intervals [as they relate to each other].")

Where is there concern expressed here about the 'washing out' of distinctions and about having less variety, interest, and expressivity when performing in Equal Temperament?

Julian Mincham wrote (January 7, 2007):
12 notes---or not?

Just to sum up with a final comment on where I feel I am at on this one

1 HARMONICALLY I still maintain its a 12 note scale. All tonal practice in Western Civilisation from Bach to Hindemith and Mehegan allied to my own composition arrangement, study of scores, performance and teaching confirm this for me

2 Note s(and whole chords) will SOUND very different according to context----and the context is very often a tonal one conditioned by the forces of tonality brought about by triadic combinations of these 12 notes.

3 Individual Notes cannot be raised and lowered on fixed keyboards (even if they are written differently and enharmonically) ALTHOUGH i WOULD MAINTAIN THAT ON SOME INSTRUMENTS---E.G. PIANO--- WITH SUBTLE BALANCING AN ILLUSION OF THIS KIND CAN BE CREATED.

4 Voices and stringed instruments particularly CAN AND DO alter the pitch of notes subtely according to context and certain tuning approaches may well be predicated upon such practice.

5 The fact that notes (and chords) may sound different in different contexts is an immensely complicated issue and there is no one simple conditioning factor.

Brad---this is where I stand---possibly alone, but undaunted!

Ed Myskowski wrote (January 8, 2007):
Bradley Lehman wrote:
<< For example, Duffin includes on page 47 a diagram of a violin fingerboard drawn in Peter Prelleur's book The Modern Musick-Master, 1730-31, clearly illustrating that the sharps are to be fingered at different places on the string next to the enharmonic flats. He has put this same illustration here: http://music.cwru.edu/duffin/Vallotti/T1/Prelleur.html
p.s. On that particular illustration, what I find especially startling and instructive is the recommendation that A# and Bb, on the E string, are not even supposed to be played by the same finger! And similarly for the other three strings, when the hand is playing in first position. The third finger of the left hand is supposed to reach up to play the sharp, while the fourth finger is supposed to stretch dowto play the enharmonically paired flat.
Any string-playing professionals or keen amateurs here care to comment:
have you encountered this before, in years of taking or teaching lessons on your instrument? >>
Cara Emily Thornton wrote:
< Yes, for a violinist this fingering appears perfectly normal (4^th for b-flat'', 3rd for a#'' - for those who are not familiar with this notation, the double prime after the note signifies that it is in the second octave above 'middle C', which is called that, no doubt, because it is in the middle of the piano keyboard). The exception to this fingering rule would be if the third finger is already occupied with, for example, the note a tritone higher (note: violin is tuned in fifths, so the perfect fifth would involve stopping both strings with the same finger, so the tritone, being half a step smaller, would involve use of adjacent fingers).
This obviously would not happen on the E string, but for example: if the third finger in first position on the E string is occupied with the note a'', then you would use the fourth finger on the A for d#'' (a tritone lower). But again, in a normal two-octave B major scale in first position, you definitely would use the third finger for g#', d#", a#" (on the D, A and E strings respectively).
And yes, the a#'' being a leading tone to b'', we will definitely place the a#'' close to the b''. Whereas, if we have a progression C7-F, the b-flat'' is the '7' in that C7, and it resolves down a half step to the note a'', which is part of the F chord. And we will definitely place the b-flat'' close to the a'', with the result that b-flat'' ends up, oh, maybe one cycle per second (Hz) or so lower than the a#'', given which octave we are in. >
This is all a bit technical for a lapsed clarinetist, which my violinist high-school orchestra conductor called a machine. Nevertheless, it certainly strikes me as a very illuminating response to the question. We could use more of that on BCML. We could also use more participation from the ladies, thanks for not giving up!

Julian Mincham wrote (January 8, 2007):
Bradley Lehman wrote:
<< p.s. On that particular illustration, what I find especially startling and instructive is the recommendation that A# and Bb, on the E string, are not even supposed to be played by the same finger! >>
To put this in simple terms the fingering chosen is usually because of the context of the note. Same on the keyboard. If the note D, E and F follow the note of C (played in the right hand) the note of C is not best served by playing it with the little finger. You would play the C with the thumb or first finger leaving other convenient fingers available to continue the upward scale.It's a very different situation if the notes immediately following the C are lower in pitch.

Thus the same notes, even written the same way will be differently fingered according to context. On the keyboard of course, there is no alteration of the pitch of the note (although as I have already indicated in some circumstances this illusion may be created ,especially on the piano)

In the specific example given above the fact that the note is written first as an A sharp then as a Bb would indicate that they probably occur in very different contexts technically and different fingerings would seem inevitable. But because of the matter of tempering that have formed a recent thread, the two notes may also deviate very slightly in pitch from each other in a way that cannot happen on the keyboard.

Bradley Lehman wrote (January 8, 2007):
Cara Emily Thornton offered remarks about violin playing:
<< And yes, the a#'' being a leading tone to b'', we will definitely place the a#'' close to the b''. Whereas, if we have a progression C7-F, the b-flat'' is the '7' in that C7, and it resolves down a half step to the note a'', which is part of the F chord. And we will definitely place the b-flat'' close to the a'', with the result that b-flat'' ends up, oh, maybe one cycle per second (Hz) or so lower than the a#'', given which octave we are in. >>
But, this is the opposite of what Prelleur said on that fingering chart, 1730: that the a#'' (even if it's a leading tone) should be lower than bb''.
>> http://music.cwru.edu/duffin/Vallotti/T1/Prelleur.html

And a point of Duffin's book: that the 19^th century practices eventually flip-flopped the 18th century practices, such that the sharps became higher than flats instead of vice versa. The high-leading-tone stuff is a late 19^th and early 20^th century practice, not 18^th; and it's anachronistic to use it in 18^th century music.

Again, Duffin explains these things far beyond this single example from Prelleur, and the discussion here is no substitute for actually reading his book. And it's enlivened with a couple dozen little biographical sketches of the people discussed in the text...plus some commissioned cartoon drawings. Mozart drives his carriage up to Sternbach's Interval Cafe and orders a "mezzo tuono grande mit schlag" to go. (And instead of a microphone there at the drive-through window, it says: "Please yell into the speaking tube".)

=====

That said, apart from the typical practices of any century, there's also a good case to be made for playing bb'' low when it's the minor 7^th above C major or C minor as in your example here...but that's for harmonic reasons, not melodic (i.e. to resolve down to a''). A low bb'' gets it closer to the minor 7th that is found in the harmonic series.

Bradley Lehman wrote (January 8, 2007):
Thomas Braatz wrote::
< Note the time, June 1722, and place, Hamburg, in a musical journal that was widely read by musicians and composers throughout German-speaking countries. It is highly likely that Bach also would have read these lines:
Johann Mattheson „Critica musica“Hamburg, June, 1722, p. 53
[After having presented on the preceding page the precise measurements according to which anyone could tune an octave in Equal Temperament using a monochord, Mattheson continues:]
“Und weil keine bessere noch vollkommenere ‘Temperatur’ in der Natur zu finden (....) >
...Which offers no case that Johann Sebastian Bach would have agreed with any or all of that journal article, or necessarily would have done any of that in his own practice as an expert musician, already being more than 20 years into his own career by 1722. Bach had already been tuning harpsichords and his violin for more than half his lifetime, to that point. Without a monochord, and presumably well enough to suit his fine musical ear. So, why should Bach bother with this except perhaps to be amused by it, if he bothered to read it?

Furthermore: Mattheson was notably a polemicist (one of those guys to agitate discussions by publishing whatever), and Mattheson changed his own mind across his various publications, as to issues of key character and tuning. It simply doesn't do to grab one isolated publication of his, hold it up triumphantly for exhibit, and assert that nothing else but that one article could possibly have been relevant to Bach's practices, just because it was allegedly "widely read by musicians and composers throughout German-speaking countries". (Which process is the same old argumentation-by-authority batch of red herrings that we see here, regularly, and again here. Pick one authority, hold it up as the only thing that could possibly matter to JS Bach's current/future practice, and demand a defense against it. All to suit a smugly foregone conclusion, in this case for Equal Temperament, but the overriding problem of fallacy here is the process of self-serving selectivity!)

How do we "know" that Bach did such-and-such? Because one polemicist, Mattheson, can be located and his writings selected to have said so on such-and-such a date, case closed. This is absurd.

Consider for the moment that Bach's WTC (1722-3) perhaps was some manner of response to Mattheson's article. Bach as bandwagon-jumper? Really? (Which fits almost zero with what we know of his personality and reputation....) Or rather, Bach as expert practical musician retorting: eff this Equal Temperament waste-of-time basura for math-heads and speculative dillwads! Here's my system that's much easier to do and it makes the music sound better through all 24 keys, and it takes absolutely no calculations of anything, so there, ye speculators and monochord-weenies.

Tom Dent wrote (January 8, 2007):
[To Bradley Lehman] Some comments on enharmonic differences.

Clearly, in a (usually irregular) circulating temperament such as would be needed to play the WTC or Inventions/Sinfonias, there is in terms of pitch no such thing as an enharmonic difference. B# and C are the same pitch class, regardless of name.

Then the concept of enharmonic difference must be defined through the harmonic function of a note. This is quite difficult to give in theoretically watertight fashion. However, I guess one way to do it is that major thirds and major sevenths above the root of a chord move upwards by (semitone) step as a regular resolution, whereas minor thirds and minor sixths move downwards by (semitone) step. Then, in the context of a circulating temperament, a note serves as more than one enharmonic function if it resolves upwards by a semitone in one place but downwards by a semitone in another.

So for example C resolves downwards to B as a minor 6^th above E, but B# resolves upwards as a major 3rd above G#.

However, the use of chromatic melody such as C-B-Bb-A-Ab-G blurs the distinction, since depending on how it is harmonised one can hear it as C-B-Bb-A-G#-G, etc etc. The supposed difference is that B-Bb is not a 'usual' melodic progression. But if the music contains enough such chromaticism, it becomes harder to tell which progressions are 'usual' and which not. Case in point is the Bb minor fugue of WTC bk.II in which some passages, if heard in isolation, would be ambiguous as to which were the 'diatonic' and which were the 'chromatic' progressions. Only within a context of several chords in succession does it become clearer. (Or, if there is an enharmonic modulation, remains ambiguous!!)

About the violin fingering on the Prelleur chart, by which one should use different fingers for notes with different letter-names, but the same finger for notes with the same letter name (e.g. Db - D - D#). I think this was absolutely standard until the mid 20^th century. This results in some 'slides' between notes when fingering chromatic scales, which can still be heard on some old recordings (e.g. Busch quartet, 1930s). But more on Prelleur later.

With respect to regular temperaments (where every fifth is tuned the same, and B# and C are different pitches unless we have ET) I disagree with Brad's saying the enharmonic difference is 'always a Pythagorean comma'. The difference between B# and C in quarter-comma meantone is nearly half a semitone (nearly twice as much as a comma); the difference B#-C in 1/6-comma meantone is slightly less than a comma; the difference B#-C in equal temperament is zero ... etc. The Pythagorean comma is, by definition, the enharmonic difference in Pythagorean intonation - where fifths are not tempered at all.

With respect to Prelleur, one can't draw any exact conclusions from the diagram. Certainly it shows enharmonic sharps being played lower in pitch than flats. But beyond that it is inaccurate, chaotic and exaggerated. The size of these enharmonic differences is varied up and down the fingerboard in a rather random fashion... as are the tones and semitones. Look at the G-string for instance (far left column): the B-Cb difference looks much larger than the C#-Db. Further down, the E-F semitone is scarcely smaller than the D-E tone!

Some of the enharmonic differences look to be about 1/5 of a tone, some of them don't... A warning perhaps that any attempt to fit violin tuning into a regular system is unrealistic. Anyway, the slight unevenness of gut strings means that any attempt to calculate finger positions mathematically is doomed.

Returning to possible irregular Bach keyboard temperaments, the thing is that there can be no connection between harmonic function and size of interval: because in an irregular temperament each interval comes in many different sizes!

For example, the C-B diatonic semitone may be fairly large. But the C#-B# diatonic semitone may be small. And conversely these intervals can be rewritten as B-B# and C-C# chromatic semitones. On the keyboard, every semitone, of whatever size, has to fulfil two different functions.

Haydn is another kettle of fish. I believe his intention in that quartet (the last he completed) was to have Eb and D# be the same pitch: the warning "l'istesso tuono" is just "same note" written in Italian. This could be taken as negative evidence that a cellist would have played Eb and D# as different notes unless warned. Of course, it doesn't tell us which would be higher. Also, at some point the theory switched over to having sharps sharper than flats ... as Cara said.

Anyway, there is not even any good reason why one Eb on a cello must be exactly the same pitch as another, or why any string player should follow a theory if it conflicts with their ears.

As to what string players would have done playing in concert with keyboard accompaniment, that is yet another question.

Ed Myskowski wrote (January 8, 2007):
Brad Lehman wrote:
< Bach as bandwagon-jumper?
Really? (Which fits almost zero with what we know of his personality and reputation....) Or rather, Bach as expert practical musician retorting: eff this Equal Temperament waste-of-time basura for math-heads and speculative dillwads! Here's my system that's much easier to do and it makes the music sound better through all 24 keys, and it takes absolutely no calculations of anything, so there, ye speculators and monochord-weenies. <
I agree with the principle of what you say, but I wonder if basura, dillwad, and especially monochord-weenies shouldn't have some sort of indication that they are not yet standard words, even in American English?

Ed Myskowski wrote (January 8, 2007):
Tom Dent wrote:
< Also, at some point the theory switched over to having sharps sharper than flats ... as Cara said. >
I know what you are saying, because I have been following the thread. But <sharps sharper than flats> is not the best choice of words, and not exactly what Cara said:

< And we will definitely place the b-flat'' close to the a'', with the result that b-flat'' ends up, oh, maybe one cycle per second (Hz) or so lower than the a#'', given which octave we are in. >
Much better!

Bradley Lehman wrote (January 8, 2007):
Tom Dent wrote:
< (...) With respect to regular temperaments (where every fifth is tuned the same, and B# and C are different pitches unless we have ET) I disagree with Brad's saying the enharmonic difference is 'always a Pythagorean comma'. (...) >
OK, but I didn't say any such thing. Let's not get off with such a misunderstanding, where what I said (and meant) is getting misquoted.

Let me try again, for clarity. Let's do it using examples, and examining a common enharmonic pair such as G# and Ab. (And of course assuming we can pop back down by a pure octave whenever things get out of hand, so we stay in the same general region.)

I know that Tom Dent already understands what I'm saying below, but this is for the benefit of whatever few other interested parties remain, listening in. I actually do this illustration in lectures using several paper plate rims taped together at one slit, and note-names written onto all 12 clock positions on each plate. The thing makes a helix in three dimensions, like a Slinky.

- 1. Starting from Ab, tune 12 pure 5ths:
Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#
The ending G# turns out to be one Pythagorean comma higher than the starting Ab.

- 2. Starting from Ab, tune 12 5ths modified by on average 1/12 Pythagorean comma each:
Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#
The ending G# turns out to be the same as starting Ab, because we've wasted off that single Pythagorean comma in bits and pieces along the way. And this might be equal temperament, or any number of "well temperaments" aka "circulating temperaments"...saying only that on average it's 1/12 PC per 5th being burned off, so it doesn't overshoot.

- 3. Starting from Ab, tune 12 5ths modified by 1/6 Pythagorean comma each:
Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#
The ending G# turns out to be one Pythagorean comma lower than the starting Ab. Now we're overshooting in the other direction. This one happens to be "extended 1/6 comma", of the type described in Duffin's book and elsewhere. The circle doesn't meet itself.

In example 2 the set of paper plate rims all sits together flat on a table. The helix has been compressed down into two dimensions (ignoring for the moment the fact that it might be bumpy in one or several places; the starting and ending points meet one another at a flat spot). In example 1, the helix opens out in one direction. In example 3, it opens out in the other direction.

In each of these three examples, from #1 to #3, the resulting major 3rds are getting better and better, in terms of coming down in size. They're still larger than pure, of course. They only get to be pure if we do one of these similar cycles having the 5ths all modified by 1/4 syntonic comma. In that case, Ab and G# end up even more than one Pythagorean comma apart from each other. The helix is stretched out even farther, vertically, than in example 3.

One can also construct 13-note cycles like this, an infinite batch of them, based on other sizes of intervening 5ths (either absolutely or on average)...and the difference in resulting pitch between the Ab and the G# is going to be determined by whatever that average amount is. Stretch the helix out as much as you want to, in either direction, making different distances between the layers each time the thing goes around.

It doesn't stop at 13 notes, either, but it keeps going in both directions: into the sharps and double-sharps (and beyond) on one side, and the flats and double-flats (and beyond) on the other.

This spiral-of-5ths business is only going to meet itself (coming round the mountain) if we picked the cases where the average amount in the 5ths is 1/12 Pythagorean comma; burning off all the excess from example 1. One Pythagorean comma exactly, which is what I was trying to explain the first time. Just by virtue of tuning twelve consecutive 5ths of whatever size, we have to burn off one Pythagorean comma in total if we plan to meet ourselves. The enharmonic renaming of the note is a function that either contributes or loses one Pythagorean comma: getting there by visiting all the intervening notes chained by 5ths.

Bradley Lehman wrote (January 8, 2007):
<< Bach as bandwagon-jumper?
Really? (Which fits almost zero with what we know of his personality and reputation....) Or rather, Bach as expert practical musician retorting: eff this Equal Temperament waste-of-time basura for math-heads and speculative dillwads! Here's my system that's much easier to do and it makes the music sound better through all 24 keys, and it takes absolutely no calculations of anything, so there, ye speculators and monochord-weenies. >>
< I agree with the principle of what you say, but I wonder if basura, dillwad, and especially monochord-weenies shouldn't have some sort of indication that they are not yet standard words, even in American English? >
Good point!

Bach as bandwagon-jumper?
Really? (Which fits almost zero with what we know of his personality and reputation....) Or rather, Bach as expert practical musician retorting: eff
[VULGAR ABBREVIATION OF AN IMPERATIVE VERB IN EITHER GERMAN OR ENGLISH]
this Equal Temperament waste-of-time basura
[SPANISH]
for math-heads
[NEOLOGISM, INCLUDING A PLAY ON WORDS EVOKING "METH{amphetamine}-HEADS"]
and speculative dillwads!
[NEOLOGISM AS EUPHEMISM FOR A VULGAR WORD HEARD IN A "BILL AND TED" MOVIE, CHANGING ONE CONSONANT SOUND AND ONE VOWEL SOUND]
Here's my system that's much easier to do and it makes the music sound better through all 24 keys, and it takes absolutely no calculations of anything, so there, ye
[ARCHAIC ENGLISH]
speculators and monochord-weenies.
[NEOLOGISM: SUGGESTING BOTH A SAUSAGE-SHAPED OBJECT, AND WEANING FROM RELIANCE ON MECHANICAL TUNING DEVICES INSTEAD OF LISTENING]

Macaronically yours,

Tom Dent wrote (January 9, 2007):
Brad Lehman Lehman wrote:
<>
< (cut personal abuse, both real and put into the mouth of Bach, in an extraordinary feat of ventriloquism-as-musicology)... >
What is sauce for one is sauce for the other. Both TB and BL have exactly one named source: Prelleur versus Mattheson. Neither has said why this or that source is particularly credible, why Bach would bother with them (Prelleur being a violinist in London and Mattheson a gossipy ex-composer with hearing loss), or what the relevance is to Bach's cantatas.

Brad has asserted that there is a mound of other sources which corroborate Prelleur in some respect, but it is not clear exactly what respect, and what this might mean for Bach. He has also sought to minimize the importance of the Mattheson quotation. Well, two can play at that game.

I can assert that there are other references to equal temperament being used on keyboards in the first half of the 18^th century in Germany, of which Mattheson is only one representative. (He reported that a Cantor in southern Germany had been using ET for many years already...) So Mattheson might have been influenced by equal-tempered musical innovators - of which Bach might, or might not have been one.

Now to Prelleur. His chart has the advantage that it might allow a violinist to produce purer intonation - if that musician ignored the exact marked finger positions and took away only the notion that D# can be lower than Eb. It has the disadvantage of being quite inaccurate if one tries to apply the exact finger positions. And, which is the main thing, it does not give a tuning compatible with circulating keyboard tunings, of the type probably used by Bach in his continuo instruments. For example it would give a very low G#, whereas Brad's proposed keyboard tuning has a high G#. So Prelleur and other sources indicating meantone-like intonation on instruments would seem to be irrelevant to the question of what happens when combining violins and voices with a circulating keyboard continuo tuning.

The question seemed to be: equal- or unequal-temperament for Bach? But this is a false choice, due to conflating keyboard temperament with the tuning of free (e.g. string) instruments and voices. String instruments and voices do not have to use tempered intervals.

On one side, the meantone which one might deduce from Prelleur's text is very unequal and colourful, but quite unsuitable for use with keyboard continuo instruments tuned with circulating temperament a la Bach (whatever that means).

On the other, equal temperament lacks variety as a tuning for keyboard solos, but is extremely practical as a continuo tuning; and there is no reason why violins and voices must reproduce exactly the pitches of the continuo keyboard, rather than making tiny adjustments towards purer or more colourful harmony or melody. To borrow Duffin's phrase, equal-tempered continuo need not ruin the cantatas' harmony.

I don't see how Brad can argue that the supposed use of regular meantone systems with enharmonic B#-C etc. differences on violins (or any other instrument) would be a point in favour of his proposal of an irregular keyboard temperament without enharmonic difference. It is apples and frankfurters.

Bradley Lehman wrote (January 9, 2007):
Tom Dent wrote:
< (...) What is sauce for one is sauce for the other. Both TB and BL have exactly one named source: Prelleur versus Mattheson. Neither has said why this or that source is particularly credible, why Bach wouldbother with them (Prelleur being a violinist in London and Mattheson a gossipy ex-composer with hearing loss), or what the relevance is to Bach's cantatas. >
Hold on a moment, there. I haven't asserted that Prelleur has any relevance to Bach's cantatas, or that it/he doesn't. I simply mentioned that Prelleur's diagram is discussed in an interesting and current book about tuning and harmony, which I feel is a worthwhile book that Bach-enthusiasts might want to read and learn from (in general areas of intonation, musicianship, and scale structure).

This book: http://www2.wwnorton.com/catalog/fall06/006227.htm

< The question seemed to be: equal- or unequal-temperament for Bach? But this is a false choice, due to conflating keyboard temperament with the tuning of free (e.g. string) instruments and voices. String instruments and voices do not have to use tempered intervals.
On one side, the meantone which one might deduce from Prelleur's text is very unequal and colourful, but quite unsuitable for use with keyboard continuo instruments tuned with circulating temperament a la Bach (whatever that means).
On the other, equal temperament lacks variety as a tuning for keyboard solos, but is extremely practical as a continuo tuning; and there is no reason why violins and voices must reproduce exactly the pitches of the continuo keyboard, rather than making tiny adjustments towards purer or more colourful harmony or melody. To borrow Duffin's phrase, equal-tempered continuo need not ruin the cantatas' harmony. >
But, whoa again: I haven't asserted (and neither has Duffin in this book) that the instrumentalists and singers should be trying to match whatever temperament happens to be on the keyboard. Nor do I believe they should, or that it's especially feasible at any tempo other than an extremely static adagio.

Look again: the person asserting that players and singers must reproduce the pitches of the continuo keyboard happens to be...Thomas Braatz, asserting that they should be playing/singing in equal temperament to match an equal-temperament keyboard. Please don't confuse/conflate me with him or his ideas about this, as if they were mine.

Thank you.

=====

I do believe (from years of practice doing this, both as ensemble singer and as keyboardist) that well-tuned continuo keyboards can help to center and focus the intonation of the whole ensemble, as well as to set appropriate Affekt associated with different keys. All of which is not the same thing as trying to match any of its pitches exactly, either.

In fact, I wrote directly against matching keyboard pitches, here and thus:
"The goal is not necessarily to match every pitch on the keyboard exactly, but rather to play as naturally and comfortably as possible, being free to explore expressive nuances and not worry about intonation at all, consciously." (page 17, February 2005, Early Music)

It's pretty hard to mis-construe my meaning there! That sentence also leads to endnote #88, which cites both an article by Duffin (web) and another by Bruce Haynes (Early Music, 1991), for more about ensemble playing and not-matching keyboard pitches.

Thomas Braatz wrote (January 9, 2007):
Thomas Braatz had previously stated:
>>Note the time, June 1722, and place, Hamburg, in a musical journal that was widely read by musicians and composers throughout German-speaking countries. It is highly likely that Bach also would have read these lines:
.. Johann Mattheson "Critica musica" Hamburg, June, 1722, p. 53
>>[After having presented on the preceding page the precise measurements according to which anyone could tune an octave in Equal Temperament using a monochord, Mattheson continues:] "Und weil keine bessere noch vollkommenere'Temperatur' in der Natur zu finden (and because there simply is no better/more perfect temperament to be found in all of nature.)"<<
Bradley Lehman responded with:
>>...Which offers no case that Johann Sebastian Bach would have agreed with any or all of that journal article, or necessarily would have done any of that in his own practice as an expert musician, already being more than 20 years into his own career by 1722 Bach had already been tuning harpsichords and his violin for more than half his lifetime, to that point. Without a monochord, and presumably well enough to suit his fine musical ear. So, why should Bach bother with this except perhaps to be amused by it, if he bothered to read it?<<
Just because you do not use a monochord, does not mean that Bach did not at some point earlier in his lifetime do so as he was acquainting himself with the various temperaments that were being used and propagated at that time. So you know that Bach would be amused by this article? This definitely demonstrates your prejudicial viewpoint in this matter which you now project on to Bach as if he would think and feel likewise about this extremely important issue which affected profoundly his life's work. Likewise in regard to the monochord which was the supreme tool used by Werckmeister to define, explain and tune temperaments precisely.

BL: >>Furthermore: Mattheson was notably a polemicist (one of those guys to agitate discussions by publishing whatever), and Mattheson changed his own mind across his various publications, as to issues of key character and tuning. It simply doesn't do to grab one isolated publication of his, hold it up triumphantly for exhibit, and assert that nothing else but that one article could possibly have been relevant to Bach's practices, just because it was