source file: mills3.txt Date: Tue, 16 Dec 1997 18:47:57 +0100 Subject: 19-tone ruminations and 22-tone exhortations From: "Paul H. Erlich" Gregg Gibson wrote, >}But to return to the matter of playing these in 12-tone equal. Yes, this >}is of course possible, but the resulting modes conflate a number of >}aurally distinct modes into single, neutral modes, > >Care to provide an example of a 12-tone mode which conflates two or more >19-tone modes? > >}and lack much of the >}distinct melodic character of the 19-tone versions. > >}For an example of what I mean, take the most common form of the minor >}scale (on C for convenience): > >}C D Eb F G Ab B C > >}This can be played in 12-tone equal, but loses much of its piquancy >}thereby. This is partly because the highly characteristic augmented tone >}Ab-B is confounded with the minor third. > >True, but somehow even in 12-equal it never sounds like a minor third, always >like a dissonant interval, in the context ofthis scale (except when a late >Romantic composer is using the Ab minor chord). My high school music teacher >played the harmonic minor scale, then the augmented second interval, and >everyone in the class called the interval dissonant -- even though the very >same interval, functioning as a minor third in another scale, was deemed >consonant. Why is this so? Because tonal consonance is not merely an >acoustical phenomenon -- it is also a function of musical grammar. The >Western tonal grammar constructs consonant chords and intervals from >alternate notes of the scale. Seconds, even augmented seconds, cannot >function as harmonic consonances within this grammar. > >}Historically, the 'minor scale' >}derived much of its attractiveness from the use of the mesotonic, which >}preserves the augmented tone close to the just values (there are >}several). > >I would argue that, if so, the most significant of these values is 7:6; the >others are well beyond the 11-limit (I'm talking odd limit here) and >therefore just intonantion is irrelevant. 19-equal conflates 7:6 with 8:7, >and so if the augmented tone is to be heard as a particular ratio, 19-equal >is a poor tuning for it. In 31-equal, however, the augmented tone is a very >clear 7:6. > >}I have elsewhere referred to the fact that this is one of the >}few respects in which 31-tone equal is melodically quite distinct from >}12-tone equal. But 19-tone equal is still more distinct. > >Distinct from 12, yes; better -- well, the flexibility of interpreting the >augmented tone as a minor third in late Romanic music is an advantage of >12-tone that 19-tone does not posess, and the clear ratio-interpretation >mentioned above is an advantage of 31. > >}Again, to give a better idea of exactly why one cannot adequately play >}chromatic modes in 12-tone equal, let me offer up Yasser's old (but very >}good) analogy between 12-tone equal and 7-tone equal. One can reproduce >}the seven diatonic modes in 7-tone equal, but they are merged into a >}single, neutral mode, for 7-tone equal has no semitones. This is very >}much what occurs when a chromatic mode such as C D Eb F G Ab B C is >}played in 12-tone equal - it is largely sterilized of its unique modal >}flavor, because the two species of semitones are confounded. > >All the semitones in this mode are the same size in 19-equal: 2 degrees. And >what modes are merged? > >}Paul E[]rlich asserts that 22-tone equal is the only path to escape >}diatonicism. > >I only said it _may_ be the only path, and I explained why in a later post. > >}but 22-tone >}equal is not a temperament at all, but a mere tuning artefact that >}reproduces the worst defects of just intonation. > >But these "defects" (the non-vanishing of the syntonic comma) can only be >seen as such in the context of diatonicism! Since we are speaking of escaping >diatonicism, the relevant properties of a tuning system will be quite >different. > >}Not that I wish to >}question the validity of the just ratios as standards for musical >}thought. >}[...] >}the enharmonic genus, where harmony is not an important element. This >}involves the use of 1/3 tones, which are never written in our music, but >}which fill the living rock melos of our people. No theorist should ever >}presume to discount the importance of what the musically untutored >}produce from their own melodic inspiration. They stand in need of our >}guidance, but we also of theirs. > >Can you tell us what this genus looks like? Most of my favorite rock >1/3-tones arise from the difference between a 7:6 and a whole tone; harmony >is indeed an important element wherever such intervals arise. I suspect that >22-equal, for instance, might even "work" for your enharmonic genus when its >septimal basis is admitted. It certainly comes closer to the ancient Greek, >quarter-tonal enharmonic genus, which by the way is yet another >counter-example to your minimal melodic limen spec. > >}The 19-tone equal temperament _alone_ can give access to either of these >}two genera [the "chromatic" -- really modes of 3 Western and 1 non-western >altered diatonic scales -- >}and the "enharmonic" -- yet to be fully described by Gregg] > >We shall see. I have already mentioned that your 28 "chromatic" modes are >well-represented in 22-equal. > >}just as it alone gives adequate access to the diatonic >}genus. > >While 22-equal does not give adequate access to the diatonic genus, 19-tone >is far from the only tuning that does. The vast majority of music that we >hear today serves as proof that 12-equal is more than "adequate." 31-equal we >have already been discussing, but there's one more in-between: 26-equal. Its >harmonies are about as good as those in 12-equal, and the unusual structure >of the scale (tones 4 units wide, semitones 3 units wide) is not that hard to >get used to with the proper harmony. What's more, the tritone is distinct >from the half-octave -- in C major, the B sounds like an 11th harmonic over >an F major chord -- a clear, surprising, and unique effect. Combining two >diatonic scales a half-octave apart leads to a 14-tone set where all the >consonant triads in one diatonic scale have their septimal completion in the >other diatonic scale. The point is that there are many avenues for exploring >the diatonic genus, all with their advantages and disadvantages. > >}This follows from the principle of the melodic limen, and from >}the incredibly close harmonic congruence between just intonation and the >}19-tone equal. I have not leisure here to treat this in the depth that >}it deserves, but would like to observe something that I do not believe >}has ever been clearly noticed before. If the consonances of the senario >}are each taken as new tonics, we arrive at 19 just intervals within the >}octave: > >}1:1 25:24 16:15 10:9 9:8 16:15 5:4 32:25 4:3 25:18 36:25 3:2 25:16 8:5 >} 5:3 16:9 9:5 15:8 48:25 2:1 > >}These are the intervals that singers can actually sound accurately >}singing pure consonances, although there is some debate concerning the >}ability to reliably distinguish between the commatically separated >}species of tones. Of these, the 19-tone equal merges the minor and major >}tones, and also the two species of minor seventh, and intercalates the >}augmented tone/diminished third and the augmented sixth/diminished >}seventh. > >Thus the "incredibly close harmonic congruence" is not a congruence, or >one-to-one mapping at all -- it is a mere numerical coincidence. By >contradistinction, the many highly symmetrical just schemes that intend to >represent the 22-sruti system of India do form a one-to-one mapping with >22-equal; that is, all consonant intervals in both are represented by the >same number of steps. Similarly, the 22-tone just septimal tuning used by Ben >Johnston in his 4th string quartet is similarly congruent to 22-equal. > > >Graham Breed wrote, > >}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. > >You can't deny that most Western music derives much of its beauty and >contrapuntal versatility from having both vertical and horizontal sonorities >taken from the same pitch set. These pitch sets have characteristic dissonant >intervals which allow a "tonic" to be defined and when the pitch set changes, >so does the tonic. In this way non-diatonic notes can anticipate a change of >tonality before that tonality also arrives. > >Moreover, the pitch sets in question have a certain structure which allows >consonance and dissonance, which alone at best are only relative acoustical >qualities, to achieve absolute grammatical significance. The reality of this >is proved by the paradox of the augmented second which I mention above. The >same interval can be heard as a dissonance or a consonance depending on >whether it functions as a second or a third in the scale. > >My decatonic scales in 22-equal provide a way of extending Western >compositional techniques so that 7-limit tetrads, both otonal ("major") and >utonal ("minor") may function instead of 5-limit triads as the basic >consonant harmonies. A dodecatonic, 9-limit system is also present in >22-equal, but it is purely modal -- the conditions for a tonal style are not >fulfilled. Additionally, the consonance of 9-limit utonal pentads is >questionable under certain timbral and inversional circumstances, while that >of 11-limit otonal hexads is quite clear, so the symmetry between major and >minor seems to completely break down beyond the 7-limit. SMTPOriginator: tuning@eartha.mills.edu From: Adam Silverman Subject: Re: Partch thesis PostedDate: 16-12-97 20:42:15 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: 16-12-97 20:40:10-16-12-97 20:40:10,16-12-97 20:39:49-16-12-97 20:39:49 DeliveredDate: 16-12-97 20:39:49 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 C125656F.006C0AFE; Tue, 16 Dec 1997 20:42:03 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA17240; Tue, 16 Dec 1997 20:42:15 +0100 Date: Tue, 16 Dec 1997 20:42:15 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA17235 Received: (qmail 17121 invoked from network); 16 Dec 1997 11:42:02 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 16 Dec 1997 11:42:02 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu