source file: mills3.txt Date: Fri, 19 Dec 1997 18:46:46 +0100 Subject: Re: Septimal Intervals From: gbreed@cix.compulink.co.uk (Graham Breed) Gary Morrison, at least, seems to be listening to the same chords as me: > That's interesting. I've always thought of 4:6:7 as fairly > self-content, but 4:5:7 is most likely dissonant. (Then again dissonance > is context-related whereas discordance is not, but that's a different > discussion altogether.) However, I did actually specify 31 equal, where the situation is more complicated. In lieu of Gregg Gibson discovering why, I'll respond to his other points: > The perfect fifth 3:2 is comprised of the minor third 6:5 and major > third 5:4. This in turn means that the consonant chords of the 3-limit > and those of the 5-limit are congruent, that is to say, they can exist > together without producing dissonant intervals. Yeah, yeah, yeah. This is taking a feature of 5-limit and turning it into a principle. I don't buy it. > But when we reach the septimals, we find that the perfect fifth 3:2 is > comprised of 7:6 and the undoubtedly dissonant 9:7. The other possible > septimal bisection of the fifth, 8:7 and 21:16, also includes an > undoubted dissonance (21:16) although the closeness of 8:7 makes it too, > undoubtedly dissonant. Yes, bisecting the fifth is a bad idea with septimals. > Now a chord with one dissonant interval, is dissonant. Someone may try > to deny this, but I am afraid he will fail. I believe your fears are unjustified. > There do exist a few septimal triads that seem to escape this argument, > e.g. 5:6:7, 4:5:7, 4:6:7. But a few triads do not begin to compare with > the richness and variety of the harmonies of the senario. Exactly what are you trying to say, Gregg? You come up with an "undeniable" argument, and then deny it! Or, am I misunderstanding? To my ears, these chords are certainly more concordant than their constituent dyads. Three examples are quite enough to disprove that little theory. What is "the senario(sic)" supposed to mean? If 5-limit consonance, we are talking about 6 triads within the octave -- 3:4:5, 4:5:6, 5:6:8, 1/3:1/4:1/5, 1/4:1/5:1/6 and 1/5:1/6:1/8. Granted, septimal chords don't work well in inverted forms with an 8/7. However, I am prepared to admit the utonal versions. That gives us ... 6 chords. So, we've doubled the number available from the 5-limit. > If follows that the septimals, even if one admits them to be weakly > consonant - which I do _not_ - certainly give rise, for the most part, > to undoubtedly dissonant chords. They have only the most tenuous > relation with the harmony of the senario, which on the contrary, forms a > closed, well-ordered system. There is certainly doubt as to the first statement. Unless you consider all chords containing a septimal interval, when it is true but septimals are no different to quintals in this respect. There is no question that the 7-limit is a different sonority to the 5-limit. Why this is an argument for rejecting 7-limit consonances is beyond me. Well-ordered systems are really dreadfully early 20th Century. > All this was observed centuries ago. Mersenne is full of highly > ingenious arguments regarding the septimals - nihil sub sole novum. My ignorance on the subject prevents me form getting into a historical discussion. > So today, some people still take it as a personal > insult if an interval they fancy is called dissonant, and - very > absurdly - try to make consonance and dissonance purely relative ideas > with no basis in physics. Certainly, consonance and dissonance are relative concepts. By which I mean that a chord is only consonant or dissonant relative to another chord. Why is this absurd? > But I - just as about 99% of all the Western musical theorists who have > ever lived - do really hate and wish to banish a certain narrow class of > dissonances - those which are so close to consonances that they beat > badly, but yet are melodically confounded with the consonances. Well, that's Western musical theorists for you. Do these include the people who advocated 12 note well temperaments? We're clearly going to get nowhere in a consonance/dissonance argument. I will, however, make the following assertions. 7-limit chords have a character which makes them closely akin to 5-limit consonances. 7-limit chords are sensitive to tuning and so, if they are to be used, they must be considered in choosing a temperament. A well tuned 4:6:7 chord fulfils all the usual criteria of concordance. Some 5-limit chords, spread over more than an octave, sound frankly dissonant -- see Helmholtz for more on this. In the right circumstances, a 7-limit chord can function as a consonance. SMTPOriginator: tuning@eartha.mills.edu From: gbreed@cix.compulink.co.uk (Graham Breed) Subject: Pythagoreanism PostedDate: 19-12-97 18:47:23 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: 19-12-97 18:45:14-19-12-97 18:45:15,19-12-97 18:44:50-19-12-97 18:44:50 DeliveredDate: 19-12-97 18:44:50 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 C1256572.00618509; Fri, 19 Dec 1997 18:47:06 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA18869; Fri, 19 Dec 1997 18:47:23 +0100 Date: Fri, 19 Dec 1997 18:47:23 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA18103 Received: (qmail 10395 invoked from network); 19 Dec 1997 09:46:39 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 19 Dec 1997 09:46:39 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu