source file: mills3.txt Date: Sat, 20 Dec 1997 10:23:50 +0100 Subject: 22-tone Equal and the Genera From: Gregg Gibson As I have abundantly shown in previous posts, this temperament (and analogous ones such as 29- 41- & 53-tone equal,) is impossible for the diatonic, as well as for the chromatic modes (see the four chromatic modal genera or scales I have contributed; these include several of the forms of the 'minor' of the West) which obviously require thirds and sixths vis-?-vis the tonic, it being granted that the fifth and/or fourth must also be present. The case of the enharmonic is less clear-cut, and requires more detailed consideration. For after all, one might ask, surely the divers enharmonic modes have fewer thirds or sixths vis-?-vis the tonic, and so the inconsistency between the fifth cycle and those of the thirds would appear to be less troublesome; in certain enharmonic modes it might never arise at all. For example, take one form of the enharmonic: C C# Db F G G# Ab C (not the 22-tone equal notation) or in 22-tone equal: 0 55 109 491 709 764 818 1200 There is a consonant major third between Db & F, and another between Ab & C, and a consonant minor third between F & Ab. In reality, of course, the notation of the 22-tone differs from the above notation: C Db B# F G Ab Fx C which BTW is not a mere artefact of notation, but on the contrary explains very succinctly the utterly non-diatonic and non-chromatic nature of this system, but that is beside the point. The effect of this particular Greek enharmonic mode - the most famous of them as I recollect - is in 22-tone not wholly dissimilar to its effect in 19-tone equal, except that the latter is much more singable because of the wider 63.3 cent interval. Other enharmonic modes in 22-tone equal such as: C D D# F G G# Ab C (not 22-tone equal notation) 0 218 273 491 709 764 818 1200 have problems however. Here D-F is dissonant, and this would tend to disagregate the already weak tonality of the enharmonic. If D is lowered to make D-F consonant, D-G becomes dissonant. This is similar to the break in the fifths that besets just intonation itself. This is not a trivial matter; it is very noticeable to the ear. Therefore ? and I am very glad to be able to give Paul Ehrlich some reason for at least toying with the 22-tone equal ? some of its enharmonic modes seem usable, so long as modulation, harmony, or the drone are rigidly forbidden. As soon as these enter, the enharmonic itself evolves toward the chromatic or the diatonic, with 1/3 tones used not as basic modal members, but only as occasional ornaments. (And then, in the diatonic and chromatic, the 22-tone equal becomes unusable). And this indeed has always seemed to me ultimately the most valuable r?le for the enharmonic; not as a fully independent genus, but as an adjunct ? a very important adjunct ? to the chromatic and diatonic. But I here hesitate, and have long hesitated, to pronounce with any kind of certainty; if the Greeks used the enharmonic as a fully independent genus, and they appear to have done so, there may well be a way to avoid the difficulty, only I do not see it. In fine, observe that _all_ enharmonic modes, whether infiltrated by the chromatic or diatonic or kept pure, are available in the 19-tone equal, and because of the 63.3 cent interval are much more easily sung in the therein than in the 22-tone equal. The difference of 9 cents between 54.5 and 63.3 cents is very important, because it occurs near the melodic threshold or limen of perception. SMTPOriginator: tuning@eartha.mills.edu From: mr88cet@texas.net (Gary Morrison) Subject: A Higher-Level Comment on Gregg Gibson's Limen PostedDate: 20-12-97 11:00:20 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: 20-12-97 10:58:15-20-12-97 10:58:16,20-12-97 10:57:50-20-12-97 10:57:51 DeliveredDate: 20-12-97 10:57:51 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 C1256573.0036C231; Sat, 20 Dec 1997 11:00:02 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA22593; Sat, 20 Dec 1997 11:00:20 +0100 Date: Sat, 20 Dec 1997 11:00:20 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA22544 Received: (qmail 18997 invoked from network); 20 Dec 1997 02:00:13 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 20 Dec 1997 02:00:13 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu