source file: mills3.txt Date: Tue, 23 Dec 1997 23:44:53 +0100 Subject: More on Just Intonation 2 From: Gregg Gibson A singer who gets all the notes perfect, and who is proceeding along swimmingly, but then gets to a note that he just can't seem to sing, and who then totally loses his way, and starts to use notes that sound vaguely wrong, and then hits another really bad note, and starts to use notes that sound definitely wrong and off-key, so that the audience starts whispering and fidgetting ? this is the effect of the just intervals when untempered. Theorists who have never actually tried to play much music in just intonation take one look at the 'movable second' of this system (both above and below the tonic) and say to themselves: "No matter, what is all the silly fuss about? let us remain just, pure and holy - let us simply add the requisite additional tone." In so doing they fail to understand that in music one crack in the dike is enough to flood the village fifty feet deep beneath our old enemy, the sea. For as soon as our unfortunate too-perfectly-skilled singer (or instrumentalist) reaches that movable tone (and this quite apart from any question of modulation or musical style), all hell breaks loose. Entirely new notes are needed, and then still more, which do not fit into the old tonal fabric. This is not the momentary tension of modulation, but the creaking sound of the ocean liner before it splits in two. That a handful of presumptuous people aggressively declare that they love this sound, and are able to make great masterpieces with it, is as may be ? but so far they have made no masterpieces, but only curiosities wholly irrelevant to music as an art. All these difficulties were completely and intimately known to the musicians and theorists of 400 years ago. It was also known long before to musicians, but the theorists, in the grip of their worship of one particular Greek school of theory, the Pythagorean, preferred to pretend no difficulty existed. Quite strained arguments have been made to the effect that the ancients universally defined even the thirds as dissonant - hence the hopeful attempts of a long series of Western theorists to gradually extend the number of consonances to include the septimals. But this is definitely true only for the Pythagorean tradition, not for Greek musical theory as a whole, whose conclusions are largely lost to us. Pythagorean intonation is simply a variety of just intonation in which the fifth/fourth cycle is alone used, resulting in difficulties just as excruciating as those of the true just intonation, although of a different kind. To pretend that a singer can accurately produce an interval four or five perfect fifths above the tonic is nonsense; this is a commatic disjunct dissonance, and not accurately singable. These matters are widely known ? they are also widely ignored or denied or imperfectly known. The ignorant very typically imagine that We Clever Moderns have invented a new music in which all the Stupid Old Ideas of the dead past don't matter. Every generation thinks like this ? and every generation is wrong. The poor, itinerant singers ? many of them great geniuses of a kind that are largely absent from our academies ? to whom our people deliberately go listen, know little of temperament. But their profession prevents them from falling into the excesses of the just intonationists, who imagine that some clever new mathematics will somehow find a way to eliminate the necessity of tempering, or permit us to use with success some temperament even more self-inconsistent than just intonation itself. But the best mathematical minds in Europe once busied themselves very largely ? I almost said 'primarily' ? with escaping from temperament, and never found a way. Popular singers are also preserved from the extravagances of the academics who cultivate such monstrosities as 22- 29- or 41-tone equal, which merely reproduce the worst deformities of just intonation, with very few of its benefits. The irony is that just intonation itself is considerably poorer in usable melodic resources than the 19-tone equal, of which matter I omit to treat again here. Now an important consequence of the observation that a singer who uses absolutely pure intervals involves himself in fearsome practical difficulties, is that singers do indeed temper intervals. But how? I have presented evidence to suggest that the seven consonances and twelve tonal dissonances form a tonal fabric to which 19-tone equal temperament corresponds far more closely than any other intonational system. The 19-tone equal intervals are with _no_ exceptions very close to the 19 melodic pitch classes into which just intonation tends to resolve. I am not alone in this opinion. This by the way, is _not_ a circular argument, which I anticipate someone will suspect it to be ? and should so suspect as a matter of normal skepticism. The 19 just intervals were not invented or chosen arbitrarily in order to favor the 19-tone equal; they follow ineluctably from the consonances, and from the nature of singers' ability to sing dissonances vis-?-vis the tonic. I was dumbfounded when I first noticed this correspondance; I expected that 31-tone equal would offer a better correspondance; but this is not the case. 31-tone equal corresponds better to the atonal dissonances, but much less well to the tonal ones. Another possible source of circularity in my arguments might be supposed to be the choice of melodic limen of circa 60 cents, which is quite near the 19-tone equal tuning degree. But in fact a melodic limen as narrow as circa 50 cents or as wide as circa 70 cents, still results in much the same melodic pitch classes when applied to the 19 just intervals. Finally, it is to be noted that even if onesuppose the atonal dissonances to be perceptibly part of the tonal fabric, which I doubt, they are melodically confused with the tonal dissonances, and more objectionably yet, with the consonances. I have also remarked that all microtonal systems other than 19-tone equal tend to reduce to about 12 melodic pitch classes in the octave. This may well help to explain why the 12-tone equal, which was known for centuries by our ancestors, but nearly always rejected by them with the most sovereign contempt, has achieved such a peculiar destiny. It is that intonational system toward which ignorance and error converge; it is that system which a random choice of tuning, in violation of the canons of just intonation and the human voice, and therefore in violation of the musical understanding, must finally produce. Error is manifold, truth one. This is not a statement that arises from religio-dogmatical habits of thought, but from science, which usually supplies many incorrect conclusions, but only one correct answer, or group of answers. The 19-tone equal temperament is the only closed, fully cyclic system toward which just intonation, in its practically audible aspect, tends; it is therefore the only correct system. Having now given sufficient information to, at the least, propel the open-minded to a close examination of the 19-tone equal temperament, having also perhaps discouraged a few frivolous triflers from disgracing the 19-tone equal temperament by any use of it, and finally, having communicated my discovery of the octave stretch (which noticeably improves the harmony of this temperament,) I shall abate very considerably the frequency of my postings, not wishing to interfere with the learned debates of others, and unwilling to devote too much of my time, which is to me very precious, to correcting vulgar errors or trivial disapprobations. SMTPOriginator: tuning@eartha.mills.edu From: Robert C Valentine Subject: Re: 19, 22, 29, etc... PostedDate: 24-12-97 08:29:11 SendTo: CN=coul1358/OU=AT/O=EZH ReplyTo: tuning@eartha.mills.edu $MessageStorage: 0 $UpdatedBy: CN=notesrv2/OU=Server/O=EZH,CN=coul1358/OU=AT/O=EZH,CN=Manuel op de Coul/OU=AT/O=EZH RouteServers: CN=notesrv2/OU=Server/O=EZH,CN=notesrv1/OU=Server/O=EZH RouteTimes: 24-12-97 08:27:02-24-12-97 08:27:02,24-12-97 08:26:32-24-12-97 08:26:32 DeliveredDate: 24-12-97 08:26:32 Categories: $Revisions: Received: from ns.ezh.nl ([137.174.112.59]) by notesrv2.ezh.nl (Lotus SMTP MTA SMTP v4.6 (462.2 9-3-1997)) with SMTP id C1256577.0028E9AC; Wed, 24 Dec 1997 08:28:49 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA24941; Wed, 24 Dec 1997 08:29:11 +0100 Date: Wed, 24 Dec 1997 08:29:11 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA24802 Received: (qmail 5224 invoked from network); 23 Dec 1997 23:29:09 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 23 Dec 1997 23:29:09 -0800 Message-Id: <9712240725.AA85822@ilx170.iil.intel.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu