source file: mills2.txt Date: Thu, 4 Jan 1996 08:23:26 -0800 From: "John H. Chalmers" From: mclaren Subject: Paul Rapoport's article "The Notation of Equal Temperaments" --- Xenharmonikon 16, available now from Frog Peak Music or from John Chalmers, contains Paul's latest essay on non-12 notation. This is a subject littered with caltrops. To date, no one has succeeded in producing a universal notation which allows both easy performance AND analysis of xenharmonic music. Nonetheless, Paul makes many good points in this article. "Despite the considerable history of ETs, few theorists have studied their properties or created musical notations for them systematically. Because most of the ad hoc solutions in notation of one ET do not extend easily to many other ETs, such solutions obscure similarities in related temperaments." [Rapoport, P., XH 16, pg. 61] However, the worm soon munches its way free of the apple: "This article explores almost exclusively ETs of the octave and implications based on harmonics no higher than 5." The implicit presumption? That one ought to strive for a unified system of notation which relates all the especially 12-like tunings with good fifths. While this is a laudable goal, one needs must ask: why spend so much effort on the 12-like tunings? By far the most interesting equal temperaments are those which have *little or nothing* in common with traditional Pythagorean and 5-limit tunings. There exist many oddball equal temperaments which cannot be notated usefully by a 5-limit scheme, yet whose "sound" compels ever-growing interest. In this regard, 9-TET, 10-TET, 23-TET, 22-TET and 21-TET stand out in particular. 9-TET, for example, violently abjures traditional notation, yet it remains an endlessly fertile breeding ground for xenharmonic compositions. In fact Erv Wilson has called it one of his favorite tunings. Paul derives several guiding principles: [1] Additional signs must be perceptually distinct from one another. [2] Notation must reflect the determined nature of the termperament. If there's more than one way of deriving the temperament, there should be more than one way of notating it. [3] Additional signs beyond bb X, etc, should be created for kommata representing the differences in pairs of multiples of just intervals. The first 2 points are eminently sensible. Point 3 seems incomprehensible to me. If you're going to relate your equal temperament explicitly to just intonation via the notation used, you might as well abandon equal temperament and go straight to just intonation. Few equal temperaments < 48 are awfully JI-like, and all of 'em contain internal structures which inevitably frustrate the attempt to view them as this or that outgrowth of just intonation. As a result, many of the intervals which Paul defines as "kommata" do not function as such in equal temperaments below 48-TET. For instance, in 22-TET a single scale-step is numerically a fair approximation of the Pythagorean komma, but it does NOT REMOTELY function like a just interval in context.(Significantly, Paul does not discuss 22-TET.) Again, the 1/17 octave interval in 17-TET, while numerically near-indistinguishable from the difference twixt the 6/5 and the 5/4, sounds in practice like a semitone. In an actual 17-TET composition, this interval contains NO audible implications of just intonation whatever. 31-TET contains an interval quite close to the 41.059 cent "diesis," yet a single scale-step of 31 conjures up no auditory ghost of just intonation. In fast chromatic runs, the 31-TET scale-step functions like a very strange semitone, and when used as a passing-tone twixt chords its function is a somewhat small chromatic passing-tone. The idea of imposing on any but a few ETs with very good fifths (all > 48, 46, 43, 41, 39, 37, 36, 34, 31, 29, 24, 19, 17, 12) a set of just kommata purporting to represent the substructure of the tuning seems suspect to me. Many intervals numerically close to JI kommata do not function as such such in the context of the equal tempered scales below 48-TET, and the futile attempt to force them to do so produces mediocre music. Examples of such music include some of Easley Blackwood's "Twelve Microtonal Etudes For Electronic Music Media," a project only partially successful. The etudes which hew most closely to Blackwood's theoretical and notational principles prove LEAST successful as music: in particular, 17-TET, 16-TET, 19-TET and 20-TET. Blackwood jams 19-TET into a 12/oct straitjacket, and his 19-tone etude suffers greatly from his refusal to showcase 19's exotic non-Pythagorean intervals. In the 16-TET etude, Blackwood derives an awkward pseudo-tonal mode which goes against the inherently anti-tonal, non-cadential grain of the tuning, and the result sounds bad. The 17-TET etude suffers grievously from Blackwood's unwillingness to recognize & use the 5-unit neutral third as the ONLY functional triadic third, while the 20-TET etude is mangled by Blackwood's relentless insistence on generating diatonic modes from 20/oct, rather than exploiting its intercessant circles of NON-diatonic pentatonic 5ths (as he did in 15-TET). All of these problems stem from Blackwood's doomed attempt to extend 12-like structures into entirely xenharmonic tunings. He could have avoided these faux pas if he'd ignored his Pythagorean notational kommatic calculations and relied on his ear instead of v, a, k, t and p. To put it succinctly, Easley Blackwood concentrated far too much on the "harmonic" and far too little on the "xen" aspects of these xenharmonic scales. By contrast, those etudes in which Blackwood pretty much tossed out his theories and just sort of gave up & winged it seem by FAR the most successful musically: 13-TET, 23-TET, 14-TET. Moreover: In calling Blackwood's project "the most substantial non-improvised recorded project in different equal temperaments," Paul renders a decidedly peculiar definition of "notation." Most of us would argue that William Schottstaedt's and Jean-Claude Risset's and James Dashow's and John Chowning's and Richard Karpen's and Richard Boulanger's computer compositions are FULLY notated...they simply use a notation radically at odds with Paul's cherished Pythagorean common-practice-period-based paradigm. Moreover, my own MIDI compositions and those of many other xenharmonic composers use a notation which is also perfectly standardized, entirely reproducible, and which allows others to examine and analyze the compositions--we use the MIDI file format in which notes are represented as numbers from 0 to 127. Again, presumably because this notation is something that Beethoven wouldn't have used, somehow our compositions don't exist and aren't worthy of audition, analysis or (presumably) mention. Peculiar indeed. This leaves aside, of course, algorithmically composed xenharmonic music. That's a whole 'nother passel o' varmints, chilluns. Yet all the pieces of music mentioned above are carefully and painstakingly composed, note by note, with *AT LEAST* as much attention to detail as shown in Blackwood's scores. The main difference is not that "many such [electronic/computer music] works are created without a score," but rather that they use musical scores which Paul chooses *NOT* to recognize as a valid symbology. Thus, while Paul mentions "this article does not address the issue of the utility of scores," it ALSO (much more glaringly) does not address the issue of whether a piece of carefully-composed, closely- worked-out music notated in a completely untraditional way (e.g., Csound .sco file or MIDI file) ought to be treated as though it doesn't exist merely because the notation cannot be viewed as a variant of some 19th-century central European concoction. (Slippery slope time: do Gregorian chants notated with neumes qualify as musical scores? If so, why don't MIDI files printed out in piano-roll notation? And why aren't sonograms of Risset's and Chowning's and Dashow's and Schottstaedt's and Lanksy's computer music pieces *also* scores? ...We're on the slippery slope, and it's gettin' slipperier by the minute... In desperation, one MIGHT claim that 19th century musical scores allow acoustic performers to reproduce the music. This (such logic goes) distinguishes them from MIDI files or Csound .sco files. But how many live acoustic xenharmonic concerts did YOU attend last year? And didn't ALL of 'em use one-of- a-kind exotic homebuilt acoustic microtonal instruments? So what good is a 19th-century-type score if there's only one set of microtonal instruments in the world that can play 'em? ..Slippery slope time, kiddies! Let's face it: essentially no one attends or gives live acoustic concerts any more, and since 99.999999% of the music we all hear is now delivered via electronic media, this is a VERY weak and flabby and etiolated argument for ANY particular flavor of musical notation.) Paul's chart of kommata is admirably clear and his ranking exemplary; he is probably right that, for ETs with good fifths, the syntonic komma is most important for quasi-19th-century notation. Paul makes a good point in dealing with 17: "the third in question (5 u) happens to lie very close to half way between the actual just major third (386.314 cents) and just minor third (315.64 cents). It may therefore be interpreted as either or neither, depending on musical treatment of the temperament." This leads to an alternate notation which does not involve sharps or flats, and proves much superior to Easley Blackwood's notation for 17. However, those of us who've worked extensively with 17 would go even farther--many of us would contend that 17 has only ONE functional triadic third: the 5-unit third. The so-called "major" third in 17 is unbearably dissonant and useful only in melodies, or vertically as a passing tone or a cambiata. Thus many of us would urge that the so-called "major" third of 17 be notated as a species of fourth--since, like that interval, it is functionally unstable when employed vertically. Paul's 2nd notation of 53 seems as good as any other. It avoids the problem of too many flats and sharps, as usual by substituting odd new symbols. Again, this eases clutter but reduces sight-readbility. New symbols instead of the familiar sharp & flat will always prove more ambiguous for sight-readintg, since they're by definition unfamiliar. Paul's first notation for 25-TET seems less than successful, since it uses note-names E & F. 25-TET's most obvious audible characteristic is its pentatonic "mood." This, because 25 boasts not one but 5 circles of identical 5-TET fifths. The 25-TET fifth sounds unmistakably pentatonic-- it's the same 720-cent fifth found in all multiples of 5-TET up to 45-TET. Thus, a notation which implies that there are more tones than sharped- or flatted-versions of C, D, E, G and A proves less than useful. Paul's 2nd proposed notation admirably corrects this problem and exposes the five pentatonic circles of fifths, as does Paul's third suggested notation. This is a big improvement over Blackwood's notation, which retained E and F and B and C as exact enharmonic equivalents (talk about willful obfuscation!). Paul also points out that even *his* generalized notation breaks down for ETs without fifths. As an example, he gives 13...which has certainly resisted any attempt to force it into a traditional notational mold. This is inevitable. No notation can cover all equal temperaments. The main question is: where ought the notation to break down? And how? Curiously absent in this regard are tunings with good fifths but absolutely no point of contact to traditional tunings: 26-TET, 22-TET, 35-TET, etc. Paul's treatment of 13-TET as every other note of 26-TET strikes me as peculiar, inasmuch as the two tunings bear not even the most remote audible relation to one another. Any relationship is a purely augenmusik calculated-numbers-on- paper sort of thing, and does not strike me as productive. Similarly, notating 11 as every other note of 22 would be equally fruitless--the mind can calculate, but the ear cannot hear, a relationship between the two. In both cases, one must *listen* to the tuning and *throw out* the numbers if they conflict with common sense. In both cases, notation for 11-TET ought to bear NO resemblance to notation for 22, ditto notation for 13 and 26. If the two tunings sound utterly different, they should be notated utterly differently. Perhaps Paul should add this as a 4th general principle...? Paul's treatment of negative kommata (33-TET) seems eminently reasonable. By avoiding sharps and flats, many notational paradoxes are averted. Of course, dispensing with sharps and flats also eliminates much of the analytic value of a musical notation. If you can't tell at a glance whether one note is higher or lower than another, it automatically poses problems for musical analysis. In this case, one might be better off studying a printout of the MIDI note numbers, or the Csound .sco file Hz values. But for Paul's purposes it is obviously better not to raise such unsettling questions. His treatment of 14-TET proves less satisfactory. Alas, in 14 (as in 7-TET) the modes collapse back into the keys. There is no major or minor: 3 scale-steps give 257.1 cents, too small to sound or function as a minor third, while 4 scale-steps yield 342.8 cents, a neutral third antithetical to Pythagorean theory. 5 units = 423.5 cents, too large to function as any kind of recognizable major third. This situation proves so puzzling to devotees of 19th-century-style symbology that it forces the unwary notation-theorist to twist hi/rself into knots to get out of the problem. Notating 14 by taking every other note from 28 begs the problem. In fact, the issue is that 14 has two circles of 7 identical fifths, whereas 28 has 4 of them, and they ALL use neutral intervals as building-blocks. The 2-out-of-28 dodge obscures this basic fact, and tends to dupe the inexperienced xenharmonist into imagining that 14 has something like a major or a minor mode when in fact it has neither: merely two overlapping neutral 7-tone scales melding into a neutral 14-tone scale--and not a diatonic 14, either. Logic would indicate two simple overlapping sets of identical A B C D E F G symbols. Perhaps A A* B B* C C* D D* E E* F F* G G*? (Ivor Darrg's notation.) This example illustrates the problems that Paul's generalized notation creates when there are NO Pythagorean landmarks--in this case, because the multiples of 7-TET up to 42-TET are constructed from completely non-diatonic building blocks. A Pythagorean musical paradigm fails when faced with 7 anti-diatonic utterly equidistant tones: it flails like a moth caught in a searchlight. The product of a xenharmonic notation based on inappropriate diatonic kommatic assumptions is bound to falter & collapse for multiples of 7-equal. In fairness, Paul points out the problems his notation encounters with 50-TET, which is certainly no surprise: no proposed notation has dealt adroitly with such an oddball tuning. (Ditto 32, 27, 29, and particularly 35, which is probably the ultimate nightmare from notation hell!) Paul's introduction of numbers as superscripts is clearly UNsuccessful. The entire reason for using symbols to notate notes, rather than Arabic numerals, is that the human brain has evolved a marvellous pattern-recognition facility which operates at vastly greater speed than any possible number-calculation facility. Once memorized, symbols are instantly processed by a huge glob of visual cortex. Ergo, the miracle of sight-reading. Not so numbers. Numeric stuff crawls through the forebrain, where it clogs everything up and bogs everything down. Thus, it is impossible to instantly sight-read columns of numerals, whereas one can easily sight-read and musically analyze a bunch of visually striking symbols. Combining numerals with symbols forces the brain's spiffy pattern-recognition wetware to slow down to the pace of the number-recognizing forebrain (a much more recent and thus less efficient evolutionary addition), auguring ill both for the prospective sight-reader AND the would-be music theorist. Paul points out that F. R. Herf's and E. Sims' 72-TET notation is a one-off chimera, not useful for other temperaments. True, alas, and typical of all too many xenharmonic notations based on but a single tuning. The same could be said of Joseph Yasser's 19-TET notation, etc. Overall, the article is refreshing and offers excellent insights. While many of us would quarrel with the issue of what constitutes notation, Paul appears to have generally succeeded in producing a notation flexible enough to accomodate non-weird non-oddball equal temperaments below about 53-- or at least, those which boast good fifths. In my judgment the "weird" tunings like 26 and 19 and 22 and 32 demand entirely separate treatment. Ideally, thse tunings ought to have their OWN notations--preferably as distinct as possible from any others. It seems clear that sharps and flats are most useful in those tunings which *sound* as though though they have recognizable semitones. Thus, use of sharps and flats in 19 is wilfully perverse--and hellishly confusing in 9 or 10. There may be no way out of this conundrum. The issue of ETs without fifths was deftly resolved by Augusto Novaro, who simply proposed using numbers instead of noteheads on a single staff line. Incidentally, by proposing a notation for 171-TET Paul has also notated the non-octave scale Carlos Gamma, since it is audibly identical to every 5th note out of 171-TET. A much larger issue is the quesion of whether 7 basic note-names is or should be the be-all and end-all of music notation. Miller's article "The Magic Number Seven, Plus or Minus Two" (J. Psychol., 1956) makes it clear that the human brain can efficaciously process as few as 5 or as many as 9 different note-names. Yet there have, to date, been almost NO suggestions for eugmenting the basic A B C D E F G seven note names by including up to two more--say, H and I (NOT to be confused with the German H for "B flat"). There have also been NO discussions whatever to my knowledge about 6-note or 8-note modes, especially in prime-number ETs. (How about it, Mnauel? Any chance of your writing a computer program to find & list all the 5-, 6-, 7-, 8-, and 9-note modes of every relatively prime ET with fifths less than 21 cents away from 3/2, from 5/oct through 53/oct?) Why such 12-centric thinking? Why must ALL xenharmonic equal temperaments employ always and only SEVEN note names? Why must ALL xenharmonic musical modes use always and only 5 or 7 notes? True, 5 and 7 are relatively prime to 12--and so what? 6 and 8 and 9 are relatively prime to 19, or 17, or 29. Why only 5- or 7-note modes when we move outside of 12 tones/oct? Why not 6? Why not 8? Why not 9? Paul will probably take issue with some of these points, particularly where music-theory students analyzing pieces of xenharmonic music are concerned. However, I would point out that so few xenharmonic pieces of music have been composed--and so few music students have gotten together to analyze them!-- that to date the issue remains a pie-in-the-sky abstraction at best. It is entirely possible that xenharmonic composition will demand such a schismatic break with the past that previous 19th-century notational paradigms must be thrown out. However, we must be wary of such proposals. John Cage and other foolish folk made similar noises in the 50s about *their* brand of foofaraw, and--as the magazine "The Wire" put it so concisely in its November 1995 issue--"John Cage's music was intensely theoretical and centered around the cult of personality of John Cage, and as a result most of it is today unlistenable." Claims that "THIS musical revolution requires a COMPLETE break with the past!" are perennial, and have never proven true. Thus we must view with the utmost skepticism any such pronouncements made on behalf of microtonality. For the moment, until this issue is resolved, Paul's article seems an admirable advance in the state of the art of microtonal notation. --mclaren Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 5 Jan 1996 08:43 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id XAA29081; Thu, 4 Jan 1996 23:43:23 -0800 Date: Thu, 4 Jan 1996 23:43:23 -0800 Message-Id: <01HZMVBYZ5YA9D7TAS@delphi.com> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu