source file: mills3.txt Date: Sun, 14 Dec 1997 15:44:23 +0100 Subject: Stretching the 19-tone Equal From: Gregg Gibson I have found that the 19-tone equal temperament gives noticeably smoother consonant harmony if its octave is stretched by 2 to 3 cents, and the tuning degree widened slightly from 63.16 to 63.3 cents. The reason is, I take it, as follows. In 19-tone without an octave stretch (octave of exactly 1200 cents) the fifth and fourth are both tempered by 7.2 cents. The fifth is flat from just, the fourth sharp. But it is almost universally admitted that the fourth is decidedly less sensitive to mistuning than the fifth, I mean primarily in harmony, but perhaps also in melody. Indeed, before this century the fourth was more often than not considered as not so consonant even as the major third. Therefore we should be able to improve the 19-tone equal fifth by stretching the octave, without making the fourth objectionable. Indeed, merely to make the fifth equally deteriorated with the fourth, we _must_ stretch the octave. As for the octave itself, I do not imagine that many will object to a stretch so small as 2-3 cents, which is all that is possible before the fourth begins to become objectionable. Fokker, reported in Mandelbaum, used a different method to decide on how much the 31-tone equal octave should be tempered. He divided the cent total corresponding to each of the six consonances by the number of corresponding degrees in the 31-tone equal, in order to find what 31-tone equal tuning degree would give a pure value for each of the six consonances. Then he averaged these values according to divers weighting schemes in order to arrive at an ideal value for the octave. I shall perform a similar operation here for the 19-tone equal. The ideal value for the octave itself is of course 1200/19 = 63.16 ; here are the values for the other six consonances. 3:2 701.96/11 = 63.82 4:3 498.04/8 = 62.26 5:4 386.31/6 = 64.39 5:3 884.36/14 = 63.17 6:5 315.64/5 = 63.13 8:5 813.69/13 = 62.59 If one considers the consonances to be all of equal importance, one can simply take the average of the above seven figures, which is 63.22, to arrive at a figure for the octave of 1201.2. However, it seems to me that some weighting in called for, because the smoother a consonance, the more susceptible it is to mistuning. If one weights the octave at 7, the fifth at 6, the fourth at 5, the major third at 4, the major sixth at 3, the minor third at 2, and the minor sixth at 1, one arrives at a tuning degree of 63.29 cents, giving an octave of 1202.5 cents. I use a tuning degree of 63.3 cents and an octave of 1202.7 cents, and find the resulting consonant harmony to be very agreeable indeed - noticeably better than that of the 19-tone equal without an octave stretch. The fifth is 696.3, 5.7 cents flat of just, and the fourth is 506.4, 8.4 cents sharp of just. The consonances within the octave are not the only ones to be considered, however. Beyond the octave, both the major tenth 5:2 and the perfect twelfth 3:1 are well known to be scarcely less definitely fixed by beats than the octave itself. Now it so happens that as we stretch the octave we improve these two intervals, critical to harmony, even more than the major third and perfect fifth. With a tuning degree of 63.3 cents, we have a major tenth only 3.8 cents flat of just, and a perfect twelfth only 3.0 cents flat of just. The perfect eleventh 8:3 meanwhile becomes 11.4 cents sharp of just, but the perfect eleventh is a very weak consonance at most, and is of little concern. I know it can be very annoying to retune one's synth, or even to store alternate tunings, but here is one variation on the 19-tone equal that is definitely worth hearing and using. The resulting consonant harmony is very rich and brilliant. The dissonant harmonies remain extremely grinding and aggressive. SMTPOriginator: tuning@eartha.mills.edu From: Gregg Gibson Subject: 61 Intervals of Just Intonation PostedDate: 14-12-97 17:31:22 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: 14-12-97 17:29:29-14-12-97 17:29:30,14-12-97 17:29:11-14-12-97 17:29:11 DeliveredDate: 14-12-97 17:29:11 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 C125656D.005A927C; Sun, 14 Dec 1997 17:31:13 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA15013; Sun, 14 Dec 1997 17:31:22 +0100 Date: Sun, 14 Dec 1997 17:31:22 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA15017 Received: (qmail 25526 invoked from network); 14 Dec 1997 08:31:17 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 14 Dec 1997 08:31:17 -0800 Message-Id: <34946BF1.61F@ww-interlink.net> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu