source file: mills3.txt Date: Tue, 13 Jan 1998 22:30:02 +0100 Subject: RE: 22TET From: "Paul H. Erlich" Graham Breed wrote, }I would be very surprised if any temperament significantly }different to 22 equal supported decatonic scales. Vanishing }septimal commas is a difficult trick to pull off. You're right. If the decatonic approximation to the 3/2 is much less than 707 cents, the decatonic approximation to the 7/4 becomes too out-of-tune. If the decatonic 3/2 is larger than 711 cents, the decatonic 5/3 becomes too out-of-tune. Compare this 4-cent range with a range of about 8 cents (692-700) for the diatonic approximation to the 3/2. }The octave is the best interval to define a harmonic scale around. }Once you've done that, the octave should be very accurate. }Although you should think about the octaves, it would be annoying }if "the same" chords in different registers were out of tune with }each other. This "sameness" and "in-tune-ness" are surprisingly flexible, especially when it comes to multiple octaves. Playing the same melody on opposite ends of a piano simultaneously, you can get away with a semitone transposition without much disturbing the "sameness" or "in-tune-ness" (Yasser gives a melody to use for this, and I didn't believe it until I tried it!) > >}I tend to think that low primes should be tuned best. The more >}complex harmony you use, though, the more accuracy is required >}for _all_ intervals. > >I agree with this in a certain sense. However, perhaps returning a bit to >some of our earlier discussions, there is a sense in which harmony can become >more complex without imposing any additional requirements on tuning. For >example, consider a major 7th chord, 8:10:12:15. One might be tempted to >choose a tuning for this chord, or evalute the quality of approximation of >this chord in various tunings, by considering all its intervals, including >15:8. That would not be correct. 15:8 is a _resultant_ interval in this >chord, arising from the "stacking" or "linking" of consonant intervals. That >these consonant intervals are well-represented is enough to ensure that the >chord will have a pleasing quality. The 15:8 itself need not be tuned close >to its just value; this interval is dissonant and simply lends a roughness to >the chord that is independent of its exact tuning; this roughness is >tolerated because there are so many (5) consonant intervals in the chord. > >For example, in 12TET the major seventh is much, much closer to a 17:9 than >to a 15:8. 17:9 is not much more complex a ratio than 15:8, so that it seems >certain that the "17:9-ness" of the 12-equal major seventh is much greater >than its "15:8-ness". And yet the major seventh chord "works" just fine in >12-equal. That is because the 5 consonant intervals in the chord are >well-represented in 12-equal. For another example, take 26TET. Here the major seventh is nowhere near 15:8 -- in fact it nails 24:13, halfway between 13:7 and 11:6. Both these intervals would be much more important in describing how the 26TET major 7th is heard than 15:8. And yet, since the consonances are still OK, the major seventh chord sounds smooth, perhaps smoother than in 12TET. This is mainly because the roughness between the 2nd harmonic of the root and the fundamental of the major 7th is lessened as the major 7th gets smaller. This is one of my arguments against using anything beyond level-1 odd-limit considerations in evaluating temperaments. Also this hints at why I don't think it can be relevant whether a tuning nails 25:18. SMTPOriginator: tuning@eartha.mills.edu From: Ortgies.Ibo@t-online.de (Ortgies Ibo) Subject: My homepage (Weckman/Stellwagen/tunings and temperaments PostedDate: 13-01-98 23:10:41 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: 13-01-98 23:10:08-13-01-98 23:10:09,13-01-98 23:09:56-13-01-98 23:09:56 DeliveredDate: 13-01-98 23:09:56 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 C125658B.0079C44D; Tue, 13 Jan 1998 23:10:30 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA08536; Tue, 13 Jan 1998 23:10:41 +0100 Date: Tue, 13 Jan 1998 23:10:41 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA09740 Received: (qmail 22536 invoked from network); 13 Jan 1998 14:10:38 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 13 Jan 1998 14:10:38 -0800 Message-Id: <34BBEEEA.5250@T-Online.de> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu