source file: mills3.txt Date: Sat, 20 Dec 1997 16:32:09 +0100 Subject: RE: 22-tone equal & chromaticism From: gbreed@cix.compulink.co.uk (Graham Breed) Paul Erlich has doon a good job of proving how Gregg Gibson's chromatic genera are not restricted to meantone temperaments. Correspondents may recall that I "discovered" 6 genera that Gregg omitted. I have (at time of writing) yet to receive comment on this but, in addition to all 10 being distinct in 12 equal, they all work in JI. I use the word "genera" because Gregg did. I would much rather reserve the word for Euler genera. Anyway, here they are again: C D Eb F G Ab A C C Eb E F G A Bb C C C# E F G G# A C C Eb Fb F G Ab Cb C C C# E F F# G A C C Eb F Gb G Ab Cb C These are grouped in otonal-utonal pairs. The graph for one of a pair is the graph for the other rotated by 180?. Here, then, are the graphs for the left-hand genera: D---------A \ / \ \ / \ \ / \ \ / \ F---------C---------G \ / \ / \ / \ / \ / \ / \ / \ / Ab--------Eb [Notice that there is an out of tune fifth G-D which does not take part in any chords.] F#--------C# \ / \ \ / \ \ / \ \ / \ A---------E / \ / \ / \ / \ / \ / \ / \ / \ F---------C---------G C#--------G# / \ / / \ / / \ / / \ / A---------E / \ / \ / \ / \ / \ / \ / \ / \ F---------C---------G These last two diagrams are interesting in that, if you combine them, you get a single 8-note genus of which the four 7-note genera are subsets. I've also come across another pair: C Eb E F G Ab B C C Db E F G Ab A C E---------B / \ / / \ / / \ / / \ / F---------C---------G \ / \ / \ / \ / \ / \ / \ / \ / Ab--------Eb Which brings the total to 12. Interestingly, my 8 all have 5 consonant chords, two of which share a root. SMTPOriginator: tuning@eartha.mills.edu From: gbreed@cix.compulink.co.uk (Graham Breed) Subject: Re: More of Septimals PostedDate: 20-12-97 16:35:32 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: 20-12-97 16:33:25-20-12-97 16:33:25,20-12-97 16:33:00-20-12-97 16:33:00 DeliveredDate: 20-12-97 16:33:00 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 C1256573.00557282; Sat, 20 Dec 1997 16:35:15 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA22004; Sat, 20 Dec 1997 16:35:32 +0100 Date: Sat, 20 Dec 1997 16:35:32 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA22677 Received: (qmail 25532 invoked from network); 20 Dec 1997 07:35:29 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 20 Dec 1997 07:35:29 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu