source file: mills3.txt Date: Thu, 29 Jan 1998 20:04:12 +0100 Subject: RE: Tuning of decatonic modes From: "Paul H. Erlich" >}I've worked out that Paul Erlich's Pentachordal modes can be >}written as 10 JI notes plus three commas of 50/49. This >}interval is larger than a syntonic comma, so some tempering is >}advisable, but it should be small enough for the scales to >}function as 10 pitch classes. > >In my paper, I mention that in JI, two of the ten notes of the pentachordal >modes would each require two versions a 50:49 apart. An earlier version of >the paper (such as the one on John Starrett's web page) had 49:48 instead of >50:49. That was an error. However, the number two is correct. You don't need >three commas, just two, and the two are located at scale steps a 6:5 from one >another. > >The symmetrical decatonic modes, however, require four 50:49 commas, and two >64:63 commas. > > >For those confused but interested, the pentachordal scale is > >s s s L s s s L s s > >and the symmetrical scale is > >s s s s L s s s s L > >and the template (1,4,7,9) is supposed to produce complete 7-limit tetrads on >most scale degrees, just like, in the diatonic scale, the pattern (1,3,5) >produces complete 5-limit triads on most scale degrees. Unlike Partch, I >think the most stable root of these chords is the lower note of the 3:2 >interval found within them, and this root will always be the (1) in these >templates. > >}As 10 fixed pitches, the comma of 50/49 means that the tritones >}10/7 and 7/5 must be equal. With just octaves, they must both >}be an exact half-octave. As these are the worst 7-limit >}intervals in 22-equal, that means there is no better way of >}tuning decatonic modes unless you allow more than 10 notes. > >Yuppers. This is all in the paper. SMTPOriginator: tuning@eartha.mills.edu From: "Paul H. Erlich" Subject: Re: 60 vs 45 / GUITAR PostedDate: 29-01-98 20:35:33 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: 29-01-98 20:34:45-29-01-98 20:34:46,29-01-98 20:33:30-29-01-98 20:33:31 DeliveredDate: 29-01-98 20:33:31 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 C125659B.006B88EE; Thu, 29 Jan 1998 20:34:34 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA01850; Thu, 29 Jan 1998 20:35:33 +0100 Date: Thu, 29 Jan 1998 20:35:33 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA01848 Received: (qmail 13561 invoked from network); 29 Jan 1998 11:34:46 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 29 Jan 1998 11:34:46 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu