Temperament / Key Character / Tuning
Part 6
Continue from Part 5
Tunings and Bach
Ludwig wrote (June 18, 2004):
Yes Bach had certain tunings in mind when he wrote music--that is if he had anything to do with it. He is attributed to being the inventor of the well tempered scale.. the tunings which Bach seemed to be in favor was in the neighborhood of A=440 based on the Organs which he was a jurist and consultant in their building.
Some of the Organs Bach played (and had no say so in their construction etc) were not so well tempered and in certain reconstructed HIP instruments one can easily hear the Wolf at the door if certain music is played. In such instuments; it was not uncommon to find certain notes of the chomatic series missing. Aside from Organs; one often finds this is surviving Carillons from the period. As a Carillonneur I find
this building practice despicable because when playing each different instrument; one has to try to find out what how the bells are tuned and one has to transpose at sight and just hope that the music comes our correctly as one is often caught between the devil and the deep blue sea---running out of batons/keys to play. This also limits the use of Carillon in combinations with other instruments.
Nowhere is this more evident than in the French Baroque School of such composers as Lully, Couperin etc. Their music played in the tunings that they used sounds very different from the same music played in the well tempered scale. For me in a HIP scale (for such composers other than Bach); it has a more agreeable tenure but that is simply a matter of taste as others might not so agree.
My personal preference given the differences between well tempered and no so well tempered scales is that of Bach---as it seems to be the diplomat of all scales and tunings.
Bradley Lehman wrote (June 18, 2004):
Ludwig wrote: < Yes Bach had certain tunings in mind when he wrote music--that is if he had anything to do with it. He is attributed to being the inventor of the well tempered scale.. the tunings which Bach seemed to be in favor was in the neighborhood of A=440 based on the Organs which he was a jurist and consultant in their building. (...)
My personal preference given the differences between well tempered and no so well tempered scales is that of Bach---as it seems to be the diplomat of all scales and tunings. >
Ludwig, you very likely haven't heard Bach's "well tempered scale" yet. It was only discovered in April 2004 (by me), and it's only been used in two public performances so far: a broadcast of fortepiano music on Swiss radio and a live performance of the St John Passion in the USA. As for private performances, I've played through almost all of Bach's harpsichord music, and the violin sonatas, and about half the organ music in it here in my house; and a few others assisting me in this research project have done likewise.
But, please, hold your assertions about its characteristics until you've heard it, and until you've read my paper (to be published later this year). Its sound does NOT resemble equal temperament, most of the time; and it sounds even less like Werckmeister's or Kellner's.
The remarks above about "A=440" and the presumption of modern equal temperament are both superficial and mistaken; the history books have been wrong, and those legends about Bach supposedly inventing equal temperament have been copied and recopied ad infinitum while still not being true.
John Pike wrote (June 18, 2004):
[To Bradley Lehman] Does this mean you have a definite journal publishing it, Brad? Please tell us as soon as you are able about publication and, if possible, post us a copy of the article at the time of publication. I get more intrigued every day.
Bradley Lehman wrote (June 18, 2004):
[To John Pike] I'm trying to finish the proofreading of all my footnotes this weekend, and send it off. The basic paper's been done already for about a month, other than the small fixes and restructuring I've been doing (from the private suggestions and questions of several informal referees who have been reading my drafts: professional musicians, instrument builders, and university professors...peers qualified to assess portions of it).
There might be an appendix or two explaining the basic and normal techniques of tuning keyboard instruments (necessary background to understand where all this comes from), and the comparison with about 25 other currently used temperaments for Bach's music...if there's space in the journal. Bach's temperament arises absolutely from normal 17th century practices, not from walking out of a cornfield, while it's also an exquisitely different blend of several other methods.
I'll of course post a link to the copyrighted results when publication happens. Obviously I can't say more about those details until the paper is formally accepted for publication.
John Reese wrote (June 18, 2004):
[To Bradley Lehman] Does anyone know of a software utility that can filter MIDI data so that it produces different tunings?
The reason I ask is that it sounds like it would make for a good programming project (although I would have to find a way to account for MIDI's treatment of enharmonic spellings).
F# and Gb being different
Bradley Lehman wrote (June 22, 2004):
Juozas Rimas wrote: << I wonder how that "piquancy" sounds. It could be I have heard it already in a recording but I can't say - perhaps I didn't spot it, without knowing that it was unequal temperament and that I should have excepted the piquancy. >>
Ivea Mark wrote: < Apparently, and of course, I don't really know much about this, but I have heard that in unequal tuning, the F# and Gb is supposed to be different in sound (is that technically? 'Cos they're the same physical key, how do you get two sounds?).
Can someone elaborate? >
F# and Gb, in Bach's time, were approximately 22 cents apart: 1/5 or 1/4 of a semitone (there being two different sizes of semitones) in common usage among musicians. In the century before that, the discrepancy was even wider than that...F# being even more than 22 cents flat from Gb.
That's true of all the enharmonic pairs of notes that (today) are crammed into a single key on the piano keyboard. That's why some of the 16th/17th/18th century keyboards were built with split keys, so the player may select either one as appropriate. Or, lacking that, musicians and theorists had to come up with tuning compromises so the sound isn't violently awful when the wrong one is played on keyboards that don't have both. Basically it involves putting the sharp/flat notes of the keyboard somewhere between the points where they really should be, according to acoustical properties, such that they are useful (but differently useful) either way. The tasteful ways to do that are legion, and everybody had an opinion, and the field is still a minefield in that regard.
That's the basic historical problem of temperament, in a nutshell. For a good historical overview, I recommend Mark Lindley's article "Temperaments" in New Grove (2001) as the place to start understanding the complexities; and its bibliography as well.
As of last night, my paper about Bach's tuning method is submitted: it will be in the hands of the peer reviewers shortly, and (I hope) published later this year yet. And then, New Grove and quite a few other reference books will eventually need revisions, in light of my findings. An exciting project...while I'm now also relieved to have turned it in so I can catch back up with other things in my life that have been on hold! That's after I get all the library books turned back in, and all the articles and drafts filed appropriately.
If I stop answering tuning questions here for a while, it's because (1) I have to be careful about revealing too much pre-publication information, and (2) I simply need a rest from it...like recuperation after giving birth, or something (according to my wife's report of that experience).
To answer Juozas' question above: MOST harpsichord recordings of Bach from the past 20 years or so are in various unequal t, with various levels of piquancy or subtlety...listen to as many as you can find! Ditto for many (but I wouldn't say "most") organ recordings of the past 20 years. But none of those on either instrument are tuned Bach's way, yet: for the simple reason that his way hadn't been discovered yet until April 2004.
Fumitaka Sato wrote (June 23, 2004):
In history there have been many professionals who thought that a new and right temperament had been found out or created by themselves. And in my case, I found out the Sacred Temperament. In this Sacred Temperament all the things will be forgiven. Any will not be discriminated out. Of course the coming new temperament is included in the Sacred Temperament.
Bradley Lehman wrote (June 23, 2004):
[To Fumitaka Sato] Again, that's true. Herbert Kellner comes especially to mind: in the 1970s he invented a temperament that has some pretty good musical qualities, but no direct historical tie to Bach (and is indeed contradicted by historical information). Relentlessly Kellner spent the next 30 years explicating his invention through numerology and semantics, and the marketing of truisms about 1/5 comma temperaments in general: all to make it appear that his invention had more historical validity for Bach (as a secret reading of Werckmeister!) than it really did. The musical results have convinced a lot of people, as his temperament sounds especially good for 17th century music; but, that's not proof that Bach used it or even anything similar. In Kellner's published articles, he said explicitly that he got it from an assumption that Werckmeister published less than his own best, reserving this similar solution as a secret offshoot of it. That's a forcing of the evidence by Herbert Kellner, reading his own goals and expectations into the historical record, along with an assumption that Werckmeister was dishonest! Not an especially good scholarly thing to do. The bylines of his articles also explain that his own (Kellner's) training was in engineering, not music or music history; it shows up additionally in the thinness of his musical/historical arguments as he presented them. From a modern scientific standpoint, he's invented something nice and useful; and he spent his remaining years spiritualizing it, in print. But his ties of it back to Bach were too tenuous, for the simple reason that Bach's tuning is entirely different from his. Bach's way has only two intervals in common, anywhere, with Kellner's!
This Sacred Temperament of which you speak: what are its musical details, and does it sound in any way different from equal temperament in practice? Or, is it really a repackaging of equal temperament itself...or, maybe even something removed from mere sound altogether?
Bach's way sounds as "sacred" to me, now, as your method undoubtedly sounds to you. And, Bach's deployment of his as an expressive device in his sacred music is (I believe) both overwhelming and rich; he used it to make theological points directly in the sound of the music, through the subtle inequalities. The evidence here is the music itself, the way it sounds tuned his way, according to his explicitly handwritten instructions; the sound of it affected the way he composed his pieces, the way he deployed specific sounds to make expressive points. That, to me, is sacred: Bach using the sound itself to express his soul and edify his listeners.
Charles Francis wrote (June 24, 2004):
[To Bradley Lehman] Your real quarrel appears to be with the poor quality of the peer-review process in musicology: Herr Dr. Kellner was, after all, the author of 52 publications on the tuning system of J. S. Bach! Given the alleged thinness of Kellner's musical/historical arguments, it would seem to speak volumes about the peer-review process in this "soft" science. If you are to be believed, good marketing (smoke and mirrors, as you once put it) is all that is needed to be become widely respected.
For myself, I will avoid passing judgement on Kellner. Having corresponded with him both publicly and privately, I was saddened to learn of his passing. His last posting to the tuning group, was to express dismay at the 9/11 attacks, after that silence.
Bradley Lehman wrote (June 24, 2004):
Charles Francis wrote: < Your real quarrel appears to be with the poor quality of the peer-review process in musicology: Herr Dr. Kellner was, after all, the author of 52 publications on the tuning system of J. S. Bach! Given the alleged thinness of Kellner's musical/historical arguments, it would seem to speak volumes about the peer-review process in this "soft" science. If you are to be believed, good marketing (smoke and mirrors, as you once put it) is all that is needed to be become widely respected. >
Translation: all academia is bogus, Kellner and Isacoff were as smart as Lindley and Barbour, Lehman's got nothing, pass the beer nuts.
Anyone who thinks this is a "soft" science is advised to read Lindley's book Lutes, Viols, and Temperaments, and Rita Steblin's dissertation (2nd edition), and Easley Blackwood's Structure of Recognizable Diatonic Tunings.
And as for becoming widely respected, there's a difference between impressing people who have no technical background to make value judgments about the material, and impressing people who really know the musical and historical problems of tuning. "Wide respect" in terms of numbers of papers, and wowing the crowd, really means nothing in a truly scientific field as this is; accuracy to the material does, and that most of all is listening to the sound.
< For myself, I will avoid passing judgement on Kellner. Having corresponded with him both publicly and privately, I was saddened to learn of his passing. His last posting to the tuning group, was to express dismay at the 9/11 attacks, after that silence. >
I was saddened to learn of his departure, too, told to me by a longtime colleague and friend of his who has been an excellent sounding board for drafts of my paper. I don't pass judgment on Kellner either, personally, but his work was smoke and mirrors to promote a good system of his own invention (with heavy reliance on Werckmeister) that just happened to have nothing to do with Bach. Somebody can say the same thing 52 times and still be wrong, because of ignoring inconvenient evidence; that's what happened here with Herr Dr K. He found several mathematical truisms and coincidences in his temperament and promoted them about 7000% beyond their real significance (making them appear mystical), eventually falling back on numerology to keep the ball rolling: numerology where anything can be "proven" through enough coincidence. That's not a science. I read about 6 or 7 of Kellner's papers (variously in German, French, and English) over the past month and found them amusing, but probably not for a reason he intended. They're at least good role models of how not to write a convincing paper: about 2% content (the temperament he made up himself) and 98% fluff....
Which temperament to use?
Fredrik Sandström wrote (July 16, 2004):
As a complete amateur I'd like to ask a simple and practical question regarding temperaments:
On my digital piano (that also has a remarkably pleasant harpsichord sound for an electronic instrument, I think) it is possible to use Equal temperament, Pythagorean, Mean tone, Werckmeister and Kirnberger. Which one of these would you suggest that I use for playing Bach's music? (It also offers pure major and pure minor tunings, but these are of course not very usable...)
Bradley Lehman wrote (July 16, 2004):
[To Fredrik Sandström] From my experience playing Bach's music in all of those, the Kirnberger is the one that sounds best (assuming it's the 1/4 comma variety ascribed rather sloppily to him, and not the 1/2 comma one he published). But, that is a very limited set of choices there and not all that attractive to begin with. Quite a few people will prefer equal temperament over the Kirnberger; others might prefer the Werckmeister. All of those choices run into some serious problems of one kind or another.
That "Kirnberger" solution is not thought of very highly in some scholarly circles, dto some different problems with his work as a theorist; but musically it brings some decent effects to the repertoire. And I suspect that--because it's so extraordinarily easy to set up by ear, by beginners--that same solution or very similar may have been in use already 100 years or more, informally, by musicians before Kirnberger. It is a very easy, and elegant, layout that sounds decent.
To do it from scratch, by ear, simply tune C-E pure and then fudge the other fifths C-G-D-A-E within it so they're all equally tempered (as compared with one another) or close enough not to bother you; and then finish up by tuning pure fifths C-F-Bb-Eb-Ab-Db and from E-B-F#. The schisma discrepancy shows up in F#-C# but is so slight that the impurity of that fifth is almost negligible. (Technically, it's not a fifth; F#-Db is a diminished sixth here at this place where the circle is forced to close, or almost to close.)
Hope that helps,
Chorton / Cammerton & Bach's Tuning
Charles Francis wrote (February 1, 2005):
I noticed the recent discussion on Chorton / Cammerton (Dale Gedcke, Thomas Braatz et al.) in relation to BWV 51, and thought some of you might be interested in an article of mine that touches on this topic. It's an abridged version of a longer document that I shortened for technical reasons. But, even so, the PDF-file is still
over 2 Megabytes in size and will take a while to download. The title is "The Esoteric Keyboard Temperaments of J. S. Bach" and the document can be found in the files section of the group: http://launch.groups.yahoo.com/group/BachCantatas/files/
Bradley Lehman wrote (February 1, 2005):
[To Charles Francis]
Quite entertaining.
Some salient remarks and questions. (Well, I'll keep it to twelve, in honor of twelve semitones on a keyboard) :
1. Dr Rudolf Rasch has debunked John Barnes's method as presented in that article, already some 20 years ago. The premise that Bach would pick key signatures with a statistically valid correlation to interval (im)purity is the most flawed part of it; it's only ameliorated slightly by enlarging the data set as seen here (in the brute-force search through key signatures in the BWV). That is, the Barnes premise itself doesn't really do much to convince people who weren't predisposed to take it as a foregone conclusion.
2. Information about the "Texas Sharpshooter Fallacy" is available here, and elsewhere:
http://www.fallacyfiles.org/texsharp.html
http://skepdic.com/texas.html
The Texas-sharpshooting method is of course made a whole lot easier by scooping various unacknowledged postings of mine to BachRecordings and BachCantatas, and then drawing the target around the results (in the permutations discussed in this article) that I presumably would find most appealing. For example, I recall a conversation with Dale Gedcke and others here where I made some remarks about the sound of Bach's D-major tonic. Sure enough, the coincidence here in this article seems rather strong. So is the coincidence that this is released on the first day of the month when the author had heard that my research is being published. But, let us move along to more interesting questions and remarks.
3. The fundamental flaw in the Sparschuhe-Zapf methodology is the premise that the loops in that drawing have anything to do with seconds of time, whatsoever.
4. The thoroughness in trying all the permutations is admirable, celebrating the author's virtuosity with computers and other electronic equipment. Nevertheless, it is a truism that all temperaments sound pretty good (and pretty close to equal temperament) if all the beat rates in the fifths are kept slow, like the rates that are common to all the temperaments investigated here. So what? Where's the proof that any one in particular, among the scattershot possibilities presented here, would have any pride of place? I see there are some "Euclidean distance to other temperaments" proposing to offer such a correlation, but it is not adequately explained. How would a man like Bach come to decide on any one of these possibilities over any of the others; what musical reasons might motivate him? What features of this particular "glyph" might swing such a balance? These things are not explained, and it doesn't appear to matter to the author that they are missing.
5. Pendulums were in use several thousands of years ago. See, for example, the discussion in the book Uriel's Machine by Knight and Lomas, where it is demonstrated that pendulum length was a unit of measurement that united time and distance, way back among megalithic-age astronomers. The claim here that Christiaan Huygens's 1656 findings validate the Sparschuhe-Zapf premise is tenuous at best.
6. To see which keys people in the 18th century really expected to sound the most harmonious, why not consult and cite standard reference works such as Owen Jorgensen's gigantic red tome Tuning and Rita Steblin's dissertation, and the many standard articles by Mark Lindley? That would seem to me useful to improve Tables 35ff, markedly. Likewise, the definition of the new term "goodness" in footnote #2 strikes me as rather arbitrary, and in need of explication.
7. Whatever happened to the esoteric temperaments allegedly derived from BWV 924, and presented in June? http://www.bach-cantatas.com/Articles/Keyboard-Temperament[Francis].htm
That temperament (or should I say "set of four temperaments" to amuse any fans of Carl Nielsen), too, would seem to deserve some attention here: at least to explain why its own author has abandoned it in favor of this new presentation. It gives me an impression once again as an example of Texas sharpshooting.
8. Have any classically trained musicians, and preferably specialists in historically-informed performance practices, tried any of these dozens of temperaments in practice to inform the musical decisions offered in the present paper? If not, why not? Or if so, why aren't they credited?
9. Why would Bach have any interest in the material presented as Tables 4 to 25, or any stake in that manner of calculation by long fractions? By his own son's admission in the historical record, Bach disliked "dry, mathematical stuff".
10. As the present paper is so long on raw undigested data, and so parsimonious on explication, it's not clear (at least to me) what it is expected that historically-informed musicians should do with these results. Are we supposed to plug these 144 different temperaments into electronic tuning devices, or what?
11. The paper doesn't really present much proof of anything, does it? It just looks to me pretty much like a warmed-over version of alt.music.j-s-bach discussions seen here from January 2001: http://tinyurl.com/3uks4 In short, the present paper gives me the distinct impression of other people's work (most notably Sparschuhe's and Zapf's), but dressed up to look impressive and new with a bunch of colors and charts. I didn't happen upon that 2001 discussion thread myself in any archives until July or August 2004, at which time I read it with amusement at the author's remarks about children scribbling. He dismissed this particular drawing (which he's now calling a "glyph") of Bach's as non-evidence, at: http://tinyurl.com/6vxlr . I'm wondering what has compelled the author to change his mind about its relevance.
12. Has Dr Sparschuhe himself been contacted about this enthusiastic further running with his work? Or, for that matter, Michael Zapf from whom that discussion thread mentioned in my point #11 originated? It would seem that they should be very excited about this.
Well, one more practical question to make it a baker's dozen:
13. When tuning or doing other maintenance on a harpsichord that has flat-topped and unthreaded (i.e. smooth) pins, when one taps a loose tuning pin more firmlinto the instrument's pinblock with the top of the tuning hammer, does the string's pitch go up or down?
Charles Francis wrote (February 1, 2005):
[To Bradley Lehman] My belated response to: http://launch.groups.yahoo.com/group/BachCantatas/message/12311
Im glad you found my article entertaining - I enjoyed yours as well and my review will follow shortly.
1. On Barnes, I have read his paper and concluded it doesn't deserve to be referenced, given his obvious contrivance to fit the historical record. Early Music should perhaps confine itself to writing entertaining pieces for musicians, rather than dabbling in
technical matters?
2. Was this ad hominem really needed?
3. See my review of your article.
4. The method in the paper is based on a mathematical analysis of the glyph and does not rely at all on the historical record or musical decisions. The historical record (e.g. Haynes book on pitch) and musical factors (e.g. correlation with historic temperaments, Bach's key usage) are only used to verify the predictions. The mathematics yields two unique Cammerton-Cornet-ton solutions corresponding to reading the glyph in both directions. Both solutions accord with the historical record. We both agree that Bach marked a specific point on the glyph indicating a particular solution for the Well Tempered Clavier.
5. The only point here is to establish that seconds were a known quantity to Bach and for the purposes of tuning sufficiently close to today's unit. Is this point in dispute?
6. There is no question of improving any tables by considering musical factors. The results derive exclusively and uniquely from mathematical analysis without any historical or musical decisions. Get a pen and paper and work out the fractions for yourself if you don't believe this!
7. You appear hung up on the idea that Bach confined himself to one temperament. Anything in the historical record to support this?
8. There are no musical decisions informing the present article - the results derive exclusively from mathematical analysis of the glyph.
9. Bach had no need to reverse-engineer the temperament he used, as he knew the reference pitch of his day. The method in my paper avoids any dependency on the historical record, however, so this information must be derived. See my review of your article for more specific comments.
10. Providing raw data to allow results to be reproduced is the norm in most scientific papers. I believe that in any field, the reader should take the trouble to understand the facts and be allowed to draw their own conclusions. Evangelism and hype have no place in the scientific process, and nothing more than fractions are needed to reproduce the fundamental results here. Bach did mark a specific solution for the Well Tempered Clavier, and we both agree on the markings. That solution corresponds to one of the Cammerton-Cornet-ton pairs derived mathematically in the paper. The tuning method is actually very simple.
11. The method of Sparschuh yields a tuning solution starting on `A', and I have analytic solutions for alternative tuningmethods which include a solution starting from A. Zapf's method,described in his spreadsheet on the Clavichord group, is based on an assumption of seconds-per-beat, not beats-per-second. I also have analytic derivations of Zapf's results. The suggestion that these the glyph was the scribbles of Bach's children came from my wife. Female intuition is not incompatible with scientific enquiry, after all. And, no I haven't "changedmy mind" about the glyph as you incorrectly suggest. Indeed, I pointed you to these finding concerning Bach glyph in April last year: http://launch.groups.yahoo.com/group/BachCantatas/message/7956
12. Not yet, although a copy of my paper has been placed on Zapf's Clavichord group for some time (I note you are a long-standing member, there). I will soon inform both Sparschuh and Zapf of my findings and point them to your paper, of course.
13. Once again, my solutions are derived exclusively from a mathematical analysis of Bach's glyph.
From Charles Francis (February 12, 2005):
New file uploaded to BachCantatas
This email message is a notification to let you know that a file has been uploaded to the Files area of the BachCantatas group.
File : /ReviewLehman.pdf
Uploaded by : bachjohann_sebastian <Francis@datacomm.ch>
Description : Review of Bach tuning article in Early Music
You can access this file at the URL: http://groups.yahoo.com/group/BachCantatas/files/ReviewLehman.pdf
To learn more about file sharing for your group, please visit: http://help.yahoo.com/help/us/groups/files
My article about Bach's tuning
Bradley Lehman wrote (February 13, 2005):
< Uploaded by: Charles Francis
Description : Review of Bach tuning article in Early Music
You can access this file at the URL:
http://groups.yahoo.com/group/BachCantatas/files/ReviewLehman.pdf >
As of yesterday morning, the first half of my article "Bach's Extraordinary Temperament: Our Rosetta Stone" is now available in the February issue of Early Music (Oxford University Press), and for download as PDF from their web site: http://em.oupjournals.org/current.dtl
The article describes JS Bach's specific keyboard tuning, its basic sound, and the historical background (in both the 18th and 20th centuries).
The second half (May 2005 issue) gets more deeply into the musical and mathematical analysis, showing how the method that Bach wrote down interacts with his music, and showing how it solves the practical "problems" that other temperaments have created.
Additionally, there are 50 pages of supplementary mathematical material to accompany the article, scheduled for publication on Oxford's web site (and promised in the footnotes); not out there yet as of today. Those supplements present analytical comparison of more than forty documented temperaments that are either contemporary with Bach's or are modern published speculations as to his methods.
My own new supplementary web site at http://www.larips.com , separately, presents various additional introductions to this research project. The practical issues here have occupied me for more than the past 15 years as a professional harpsichordist and organist, tuning these instruments for my own and other people's gigs.
Enjoy!
=====
As for a response to the review and allegations offered by "bachjohann_sebastian" (Charles Francis), I will post something at: http://www-personal.umich.edu/~bpl/freview.html when I finish drafting an appropriate response. Obviously Mr Francis was very eager to review the first half of the article within its first two days of publication, and get the word out about its existence before my own announcement of it here.
Dr Bradley Lehman, A.Mus.D. (harpsichord), University of Michigan, 1994
John Reese wrote (February 13, 2005):
I haven't had a chance to fully analyze Charles' critique, nor even Brad's original paper, so I can only offer an observation at this point.
People are always talking about the mathematical quality of music, but in actuality a purely mathematical analysis of music leads to some counter-intuitive results. Trying to explain the rules of tonal harmony, for instance, in mathematical terms, might come up with confusing and disapointing results for someone expecting to find the sort of mathematical symmetry that many non-musicians believe is there. This is because music is a balance of science and art, and, just as the ancients had to include imperfections in the great stone columns in temples, so they would look "right" to the eye, composers and other musicimake compromises between the mathematical ideal and the expectations of the ear.
To achieve mathematical symmetry in music would mean not only adopting even-temperament, but throwing out the diatonic scale and using either 12-tone, whole-tone, or octatonic scales. Any mathematical study of traditional tonal music would have to use the more general approach of discrete mathematics -- set theory, graphs and dependencies, etc.
This is why I find the graphs showing lack of symmetry unconvincing, at first glance. Although the art of tuning is somewhat more precise than the art of composition, the same principles of theory vs. practice would seem to apply.
Without more information, that's all I have to say...
The Keyboard Temperament of J. S. Bach:
Article | Music Examples | Feedback: Part 1 | Part 2 | Part 3
Discussions of Temperament / Key Character / Tuning: Part 1 | Part 2 | Part 3 | Part 4 | Part 5 | Part 6 | Meantone
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Last update: ýFebruary 13, 2005 ý14:59:55
