source file: mills3.txt Date: Fri, 12 Dec 1997 07:00:58 +0100 Subject: More on 22-tone equal From: Gregg Gibson I wish to clarify my statements regarding the tone of 22-tone equal. This temperament does not merge 10:9 (182 cents) and 9:8 (204 cents) in the sense that it does not merge the two at a value intermediate between the two( as _do_ the harmonically usable temperaments) but it does merge them at a value quite divergent from both, namely at 218 cents. So in one sense it does not merge (or compromise between) the two species of tone, but in another sense it does merge them, at a very unfortunate excentric value. In other terms, 22-tone equal eliminates the comma only to break the back of the cycle of fifths, harmonically considered. It also reduces the diatonic semitone from 112 cents to 55, and the chromatic semitone it swells enormously from 71 cents to 164, presuming the fifths are held consonant at 709 cents. I suppose someone will announce they adore music without any consonant fifths, and if possible, without any other consonances either, but no matter. It would be hard to find a more grotesque temperament for any purpose of melody, whether diatonic, chromatic or enharmonic. On the other hand... this system is _so_ outrageous that perhaps one should give it the benefit of the doubt! Perhaps it will find wide use as an example of what exquisitely mistuned intervals sound like. All harmonically usable temperaments: 12- 19- 31- 43- 50- 55-tone equal, etc merge the two species of tone at a value intermediate between 10:9 & 9:8. This permits the cycles of the fifth, major third and minor third to be integrated within a single coherent system. These temperaments divide the tone into 2,3,5,7,8 & 9 equal parts respectively. It is interesting that no usable temperament divides the tone into four equal parts. Both 22- and 24-tone equal do so, and both are dogs, though for very different reasons. There is also a gap in the sequence at 6. It occurs to me that I might make my points concerning 22-tone equal easier to understand by giving at least the first few fifths of the system. I start on C, purely out of reverence for tradition: etc F C G D A E B F# etc 491 0 709 218 927 427 1145 654 Note the very narrow diatonic semitone between E & F, and the very wide chromatic semitone between F & F#. It occurs to me that Pythagoreans bored with their chosen system, and who want to _really_ get hold of some mistuned intervals, might love all this. SMTPOriginator: tuning@eartha.mills.edu From: "Paul H. Erlich" Subject: Gibson's modes PostedDate: 12-12-97 14:41:37 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: 12-12-97 14:39:46-12-12-97 14:39:47,12-12-97 14:39:30-12-12-97 14:39:30 DeliveredDate: 12-12-97 14:39:30 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 C125656B.004B08C4; Fri, 12 Dec 1997 14:41:30 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA13792; Fri, 12 Dec 1997 14:41:37 +0100 Date: Fri, 12 Dec 1997 14:41:37 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA13779 Received: (qmail 9685 invoked from network); 12 Dec 1997 05:41:30 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 12 Dec 1997 05:41:30 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu