source file: mills3.txt Date: Sun, 11 Jan 1998 08:49:40 +0100 Subject: septimal keys From: gbreed@cix.compulink.co.uk (Graham Breed) You will recall Paul Erlich's blues scale: 9---------0---------13 \ / \ \ / \ \ / \ \ / \ 6---------19--------10 I'll rewrite that with different names. Remember that the two D's are different: D---------A---------E \ / \ \ / \ \ / \ \ / \ C---------G--------D I played around with this trying to get the difference between the D's to be two commas. Then I decided the commas could be switched to a ration of 36/35, what I call a septimal diesis. I ended up with the following scale: / \ / \ / / C\--------G\--------D\ / \ / \ / D---------A---------E \ / \ \ / \ \ / \ \ / \ C---------G This is a representation of a 3-D lattice, with D-C\ the subminor seventh 7/4. D and D\ are separated by a septimal comma which is too small. Between C\ and C, and G\ and G, there are septimal dieses, though. This looks like a septimal equivalent of a diatonic scale: 7 notes plus a comma and 6 consonant triads. Unfortunately, two of those triads share a root. I've been playing this with my schismic mapping, and it sounds fine melodically. D\ isn't used much and, for the loss of a chord, you could remove it from the key. In 22 equal, the comma vanishes anyway: A C\ C D\ D E G\ G A 5 1 3 0 4 5 1 3 =22 This is a kind of minor. There's also a major: A---------E \ / \ \ / \ \ / \ \ / \ C---------G---------D / \ / \ / D/-/---\--A/-/---\--E/ / / \ / \ / C D D/ E E/ G A A/ C 4 0 3 1 5 3 1 5 =22 The following scales, and their inversions, also have 7 notes and 6 consonant triads in 22-equal. / \ / \ / / C\--------G\--------D\ / \ / \ / D---------A---------E \ / \ / \ / \ / \ / \ / \ / \ / F---------C / \ / \ / / C\--------G\--------D\ / \ / \ / D---------A---------E / \ / / \ / / \ / / \ / Bb--------F This last one is a subset of a decatonic key: A Bb C\ D\ D E F G\ A 2 3 4 0 4 2 3 4 =22 2 3 2+2 2+2 2 3 2+2 Alternate Pentachordal Major The rest aren't because they include steps of a septimal diesis. All of these scales have more than two step sizes, so they don't fulfil Paul Erlich's melodic criteria. So far, I've only listened to the first type. A---------E \ / \ \ / \ \ / \ \ / \ C---------G / \ / \ / \ / \ ---Gb\-------Db\--- \ \ / / \ -\--Eb-/- \ \ / \ \ / \ Bbb\------Fb\ In meantone temperaments, Fb\=E and Bbb\=A. So, this has 7 notes and 10 7-limit triads, or 7 triads plus a tetrad. It includes the more dissonant 7-limit chords, but it's still pretty good. C C# Eb E F# G A C 2 6 2 5 3 5 8 =31 All these 7 note keys should work with diatonic harmony. I thought it couldn't be done, but I was wrong. SMTPOriginator: tuning@eartha.mills.edu From: Carl Lumma Subject: Clearer PostedDate: 11-01-98 16:11:15 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: 11-01-98 16:10:41-11-01-98 16:10:42,11-01-98 16:10:32-11-01-98 16:10:32 DeliveredDate: 11-01-98 16:10:32 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 C1256589.00535F51; Sun, 11 Jan 1998 16:11:07 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA23217; Sun, 11 Jan 1998 16:11:15 +0100 Date: Sun, 11 Jan 1998 16:11:15 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA23904 Received: (qmail 10605 invoked from network); 11 Jan 1998 07:11:07 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 11 Jan 1998 07:11:07 -0800 Message-Id: <3.0.32.19980111004824.007cba80@nni.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu