source file: mills3.txt Date: Tue, 16 Dec 1997 18:44:10 +0100 Subject: stretched-octave eq temps From: "Paul H. Erlich" Gregg Gibson wrote >} I agree also that the fifths major thirds of 31-tone equal are slightly >} but noticeably smoother and more pleasing than those of the 19-tone >} equal. However, if the 19-tone octave is stretched by 2-3 cents, its >} fifths become quite as good as those of the 31-tone. > >My calculations indicate that 2-3 cents is indeed the optimal amount of >stretching for the 19-tone octave in order to maximize the overall consonance >of intervals which are approximations of ratios using numbers no higher than >6 (including the octave and double octave). This stretching results in what >can be called 18.96TET. The consonance, relative to JI, of 19TET is 94.53%, >while that of 18.96TET is 96.87%. However, 31TET is better at 97.30%, and >30.98TET (31-tone octave streched by just under a cent) is still better at >97.48%. > >By way of comparison, I get 85.07% for 12TET, and 86.70% for 12.02TET >(12-tone octave reduced by 2 cents). It appears that the ear prefers >stretched octaves as an equivalence interval anyway, especially for tones >with weak harmonic content, or piano tones where the harmonics are slighly >stretched, so adjusting the octave to increase consonance seems more viable >in cases such as 19TET, where the octave needs to be increased, than 12TET, >where the octave needs to be decreased. > >22TET gets a 92.21% score, and 22.01TET a 92.24% score, by this measure. >However, increasing the highest number allowed in the ratios from 6 to 8, we >get the following scores: > >12.00TET -- 54.48% >12.06TET -- 61.56% >19.00TET -- 79.28% >18.94TET -- 83.84% >22.00TET -- 87.68% >22.03TET -- 88.77% >31.00TET -- 98.037% >30.99TET -- 98.043% > >(15 and 27 could also fit in this sequence of increasing consonance). > >The above assumes that the ear assesses intervals with a resolution of 1%. >Changing this assumption does not change the rank-order of the results; only >the absolute consonance levels depend on this assumption. SMTPOriginator: tuning@eartha.mills.edu From: "Paul H. Erlich" Subject: series of improving equal temperaments PostedDate: 16-12-97 18:45:20 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: 16-12-97 18:43:24-16-12-97 18:43:24,16-12-97 18:43:03-16-12-97 18:43:03 DeliveredDate: 16-12-97 18:43:03 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.0061579D; Tue, 16 Dec 1997 18:45:10 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA17125; Tue, 16 Dec 1997 18:45:20 +0100 Date: Tue, 16 Dec 1997 18:45:20 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA17103 Received: (qmail 6174 invoked from network); 16 Dec 1997 09:44:28 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 16 Dec 1997 09:44:28 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu