Musical Offering BWV 1079
General Discussions - Part 2

Continue from Part 1

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 the gematric 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 n.......

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 exercises on 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 tobe 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 moderate 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 thprecise 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 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? >>
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 differecontexts 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. 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 the 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 monweenies. >>
< 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 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. >
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 pitc.

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 allegedly "widely read by musicians and composers throughout German-speaking countries".
Let's turn this around properly to read: this article by Mattheson may have been the catalyst to bring to a head the discussion of Equal temperament which had been quickly spreading throughout Germany since its "re-discovery" "re-calculation" by Hänfling in 1703 and subsequently by its promotion through Bümler for over 18 years before Mattheson printed it. It was primarily through Bümler that other German cantors and organists discovered the advantages of Equal temperament, long before 1722. The latter is also the ultimate solution that Werckmeister was working towards when he died. In the decades before his death, he had published various 'well-tempered' (but not Equal) temperaments. These are known as Werkmeister I, II, III etc. which he continued to modify and improve. He regretted that he was unable to provide in an easily readable form the final temperament he had been working on. If there is any reasonable assumption that can be made about Bach's acquaintance with Werckmeister's ideas about (and definitions of) temperament, it is that Bach either read or owned his books at one time or received all the pertinent information from them through his cousin, Johann Gottfried Walther, who, history records, had personal contact with Werckmeister. Bach may simply have taken the final step which Werckmeister was unable to achieve, and Bach being Bach, he did not publish it (after having used the monochord once or twice he could tune without a monochord), or talk about it, he simply used it continually wherever he could and wrote music which could make the most of all the possibilities which Equal temperament offered.

BL: >>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.<<
George J. Buelow (Grove Music Online, Oxford University Press, 2006, acc. 1/8/07):

>>Mattheson's books are written in a difficult, exceedingly prolix style requiring considerable expertise in the German language. Very little from these texts is available in English and the definitive study of his treatises remains to be written. For the student of German Baroque music, however, they are a source of inestimable value, musical documents of unique importance to the history of 18^th-century music in Germany.<<

I suggest that you reread the last sentence slowly three times until it finally sinks in. I have quoted this passage before, but then I have no way to determine whether such information is overlooked or simply not understood properly.

Ed Myskowski wrote (January 9, 2007):
Thomas Braatz wrote:
< For the student of German Baroque music, however, they are a source of inestimable value, musical documents of unique importance to the history of 18^th century music in Germany.<<
I suggest that you reread the last sentence slowly three times until it finally sinks in. I have quoted this passage before, but then I have no way to determine whether such information is overlooked or simply not understood properly. >
Actually, it is a well written sentence, easy to understand in a single reading. The phrases <inestimable value> and <unique importance> in no way suggest that Mathesson is always correct. More important, it makes no comment one way or the other on Brad's observation that Mathesson is not internally consistent, i.e., that he changes his mind from one publication to another.

I do not have the time or resources to form an independent opinion, so I hope this particular point can be resolved.

Bradley Lehman wrote (January 9, 2007):
Mattheson…

Thomas Braatz wrote:
<< For the student of German Baroque music, however, they are a source of inestimable value, musical documents of unique importance to the history of 18th-century music in Germany.<<
I suggest that you reread the last sentence slowly three times until it finally sinks in. I have quoted this passage before, but then I have no way to determine whether such information is overlooked or simply not understood properly. >>
Ed Myskowski wrote:
< Actually, it is a well written sentence, easy to understand in a single reading. The phrases <inestimable value> and <unique importance> in no way suggest that Mathesson is always correct. More important, it makes no comment one way or the other on Brad's observation that Mathesson is not internally consistent, i.e., that he changes his mind from one publication to another. >
And I have no dispute with the general value of Mattheson's stuff, either (other than the overrating as "of unique importance"--since every scrap of info by everybody is "of unique importance"!).

My dispute, as already stated, is simply that we can't assume Bach agreed 100% with anything Mattheson wrote, or that he considered it in any way binding on his own music even if he did speculatively agree with any of it.

Mattheson's stuff is just one set of clues, among many others. We can't hold them up triumphantly as the one single source that could possibly matter, and fluff off everything else just because it's inconvenient or contradictory! It's the same old fallacy of argumentation-by-author, of grabbing one thing that can be forced to deliver the desired outcome, and asserting that everything else is irrelevant.

Thomas Braatz wrote (January 9, 2007):
Bradley Lehman wrote:
>>And I have no dispute with the general value of Mattheson's stuff, either (other than the overrating as "of unique importance"--since every scrap of info by everybody is "of unique importance"!).<<
But Mattheson had a much better and more reliable overview than anyone else of what was happening in the field of music in the first quarter of the 18^th century. This is why true scholars at least make an attempt to understand everything that he had written and reported before making the accusation that he has been overrated (and this without even having read most of what is available in the original, but which still has not yet been translated into English!) I am assuming here correctly that you depend on the English translations and have not yet read any of the untranslated works.

BL: >>My dispute, as already stated, is simply that we can't assume Bach agreed 100% with anything Mattheson wrote, or that he considered it in any way binding on his own music even if he did speculatively agree with any of it.<<
No, we cannot assume that Bach would easily agree with anything that Mattheson or anyone else would have said. If anything contradicted in Bach's mind what he had learned through experience as an autodidact, he would reject it forthwith. But, coming back to Equal temperament for a moment here, Bach most likely would have learned about Equal temperament through other earlier sources than Mattheson, whose newspaper-like report was simply the culmination of a much longer period (almost two decades) of investigation and trial. As an astute musician and organist interested in all aspects of the organ, he would most likely have encountered Equal temperament even before Mattheson's article was published in 1722.

>>Mattheson's stuff is just one set of clues, among many others. We can't hold them up triumphantly as
the one single source that could possibly matter, and fluff off everything else just because it's inconvenient or contradictory!<<
Mattheson's reports and views are extremely important if only in this instance (discussion of Equal temperament) to give official proof and historical documentation. Otherwise there would be demands for historical evidence that Equal temperament even existed before Bach's WTC1 by those who want to offer their own favorite non-Equal-temperament solutions. These latter individuals would receive great comfort from the fact that no record of any German musician ever using Equal temperament (precise mathematical description required) before Neidhardt's inclusion of it ever existed.

BL: >>It's the same old fallacy of argumentation-by-authority, of grabbing one thing that can be forced to deliver the desired outcome, and asserting that everything else is irrelevant.<<
The precise evidence for Equal temperament and reports of its use as it spread through various parts of Germany between 1703 and 1722 is factual. It has nothing to do with a logical fallacy (argumentation by authority) since Mattheson, in this instance, is only a facilitator. Here he is a reporter who does not fully understand (nor has he personally tried out this temperament) Equal temperament except that he has his finger on the pulse of things that are going on in all areas of music. He recognizes the importance of something because there has been an ongoing movement toward adopting this temperament in various parts of Germany.

There are, to be sure, other roles that Mattheson with his vast experience in musical matters assumes. He can express strong personal beliefs but he can also play the role of the devil's advocate. Over the course of at least a quarter century of publications by him, he is inconsistent, but I see this as a human quality and one which demonstrates that he is willing to try to understand and adopt new ideas while letting old notions die. Unfortunately for a writer who has written so much and one who has outgrown old ways of thinking, these inconsistencies viewed side-by-side with possibly decades intervening can become a stumbling block for those who seek only consistency. Consider Bach's oeuvre. Would you cast out Bach's verified early works because they do not represent the 'mature' Bach (which one might happen to prefer) and appear to be inconsistent when viewed strictly according to the theory that the latest ideas are always better than the earlier ones?