source file: mills3.txt Date: Thu, 11 Dec 1997 18:27:19 +0100 Subject: equal temperaments and pitch classes From: gbreed@cix.compulink.co.uk (Graham Breed) The discussion on 19 equal has so far ignored the possibility of 19 pitch classes tuned according to the desired harmony. For example: > I remember when I first became interested in more than 12 tones in the > octave, I liked the _idea_ of as many tones within the octave as > possible, and tended to consider 31 as the absolute minimum. But > experience has taught me that the only viable alternative to the 12-tone > equal is the 19 system. 19 notes, maybe. But why do they have to be equally tempered? > All tuning systems which are not mere noise, reduce approximately either > to the 12-tone or the 19-tone equal, melodically considered. This > follows inescapably from the principle of the melodic limen. And what is the criterion for a tuning system _being_ mere noise? That it doesn't reduce to 12 or 19 equal? I still think melodically in terms of 12 pitch classes, with diesis or comma inflections. What does this prove? > 31-tone > equal for example reduces melodically to the 12-tone equal (with a few > exceptions where the augmented tone occurs, where it has rather a flavor > of the 19-equal) 31 equal sounds nothing like either 12 or 19 equal. Melodies in 12 or 19 pitch classes can, however, be played in 31 equal. > Between C and D, for example the 31-tone equal has Dbb, C#, Db > and Cx ascending. The first of these is melodically confounded with C, > the second two are melodically confounded with each other, and the last > is confounded with D. So, the melody C-D-Dbb-Cx-Dbb-Cx-Dbb-D is the same as C-D-C-D-C-D-C-D? You don't hear the interval Dbb-Cx as a semitone? > The 31-tone equal is in fact a monstrously > complex, inefficient way of returning us to our melodic starting-point > of 12-tone equal. It is much smoother indeed, but melodically it is a > dead-end. Come on, 31 equal steps aren't _that_ complex. If you agree that it's smoother, why not use it for its harmony? It's not that difficult to convert 19 pitch classes into 19 notes from 31 equal. Usually, a string of 19 fifths will do, especially if you're working within the 5-limit. One day, I'll get a computer to do it automatically. > The 22-tone equal suffers from two grave defects however, either one of > which would be sufficient to condemn it to the category of a curiosity. > Its tuning degree is at the outermost limit of what can be reliably > distinguished in melody. This is a defect? Think of the tricks you can perform! > second defect of the 22-tone equal involves its failure to close the > cycle of fifths. I am aware that some will say to themselves - so what? > But a musical system that does not close the cycle of fifths has at a > stroke isolated itself from 99% of the music not merely of the western > 19th century, but from virtually the whole of the western tradition, and > from many other musical traditions as well. Such musical systems, like > just intonation, are mere curiosities, and are far more impoverished in > usable, aurally distinct resources even than the 12-tone equal. Yes, 22 equal certainly takes you outside of the Western tradition. It all depends how hung up you are on Western classical music. Are you saying that all other music is a mere curiosity? One of the characteristic features of this Western tradition is that everything has to be given a name, and everything without a name is considered musically unimportant. I assume be "aurally distinct resources" you mean intervals that can be given a name, and that people can be trained to recognise. What exactly does "close the cycle of fifths" mean? 1/4 comma meantone has an open cycle of fifths. If 22 equal has a cycle of fifths, it closes after 22 steps. Do you mean, rather, that there is only one size of perfect fifth? This seems to raise the diatonic notation to a principle that I do not consider valid. A system of 22 pitch classes for 5-limit JI runs into trouble because the intervals are so unequal. So, there is an argument here for equal temperament. I haven't investigated Paul Erlich's decatonic modes enough to decide if they can be treated as 10 pitch classes. I don't like the theory because it clings to the idea of diatonic harmony -- that all notes used in harmony must belong to the same key. I'd rather work out a chord sequence, and fit a melody to it. Or, define a melodic mode, write a tune in it, and harmonise it chromatically. > Your refer to the 7 and 11 limits. The latter is a mere fantasm of the > just intonationists, and is not audible to the ear as anything other > than dissonance. The septimal limit is more interesting. But it is > dissonant, however this may trouble those who aspire to forever extend > the boundaries of consonance, until someday I presume, we shall find > everything consonant, and music need trouble itself no longer with any > rules or constraints whatever... It will also no longer need to trouble > itself with pleasing an audience, for it will have none. Actually > however, the 19-tone equal is the _only_ temperament that gives the > septimal intervals in such a form that they can be melodically > distinguished from adjacent intervals. This is a consequence of the fact > that in this system only are the septimal intervals far enough away from > adjacent intervals to preserve their own unique melodic character. If you're listening to septimal intervals in 19 equal, no wonder you think they're dissonant! I outlined a listening experiment on the list before which confirmed the consonance of the 7-limit. If you disagree, can you give specific comments on it? To state my own opinion, which is the only one I trust, a just 4:6:7 chord is more consonant than a major triad in 12 equal. Extending consonance to the 7-limit means just that -- no further. You can't disprove an argument by extending it to ridiculous extremes. An example of a 31 equal mode with septimal harmony is Ab-A-C-C#-Eb- F-F#-Ab. The chord C#-F-Ab could be retuned Db-F-Ab, or left as it is. If the former, this mode is 7 from 12 pitch classes. However, it certainly sounds like a different mode in 12 equal. In 19 equal, it is recognisable, but has quite a mournful sound. It does work melodically in LucyTuning, but the harmonies are still way out. To state that 19 equal is the only temperament to do anything is clearly absurd. Sharpen one note my a millicent, and nothing important changes. Perharps you discounting out of hand all the meantone temperaments between 19 and 31 equal. From this distance, it's very difficult to tell. Now, to some details: > I agree also that the fifths major thirds of 31-tone equal are slightly > but noticeably smoother and more pleasing than those of the 19-tone > equal. However, if the 19-tone octave is stretched by 2-3 cents, its > fifths become quite as good as those of the 31-tone. The biggest problem harmonically with 13 equal is that it's fifths aren't very good. Stretching the octave for 19 equal works because both the 3rd and 5th overtones are approximated flat. I'd still much rather use 31 equal, with pure octaves and almost pure major thirds. > 55-60 cents as the limen of intervallic perception is well-established > in the literature. Pratt for example in his Meaning of Music, as I > recall, found a value "just wider than a quarter-tone", and Seashore & > Jenner found a value between 53 and 61 cents. > Naturally no one should accept these results without making trial for > himself. I find however that my own ears are fully in agreement with > these results. So there's no problem with the 54.54 cents of 22 equal, then? I'll do the experiment myself when I understand what the terms mean. I reckon one step of 41 equal is about the smallest interval that can be used melodically, but I still tend to avoid it. SMTPOriginator: tuning@eartha.mills.edu From: A440A Subject: Re: TUNING digest 1263 PostedDate: 11-12-97 19:08:28 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: 11-12-97 19:06:31-11-12-97 19:06:33,11-12-97 19:06:17-11-12-97 19:06:17 DeliveredDate: 11-12-97 19:06:17 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 C125656A.0063777F; Thu, 11 Dec 1997 19:08:22 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA12896; Thu, 11 Dec 1997 19:08:28 +0100 Date: Thu, 11 Dec 1997 19:08:28 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA13087 Received: (qmail 26747 invoked from network); 11 Dec 1997 10:08:22 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 11 Dec 1997 10:08:22 -0800 Message-Id: <2cc0edf0.3490298f@aol.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu