source file: mills2.txt Date: Fri, 6 Dec 1996 00:57:00 -0800 Subject: CPS From: Daniel Wolf <106232.3266@compuserve.com> Adam: What you have described is a subset of a (1,3,5,7,9,11,13 diamond (or cross set), and more specifically the 13 and /13 heptads. (Similar pairs of X-ads from the diamond can be very interesting, Partch liked to contrast 8/7 Otonality with 7/4 Utonality (The Letter, and in Oedipus) as well as 16/11 O and 11/8 U. And of course the pair 1/1 O and 1/1 U makes an authentic cadence - which HP also used: The Hell With It, I'm Going To Walk!)). Cross sets can be made by multiplying any set of numbers by any other set of numbers, and Partchian cross sets multiply a set (e.g. (1,3,5,7,9,11) by its inversion (/1,/3,/5,/7,/9,/11). Cross sets like this are monotonal, in that one central tone dominates, because the sets meet at 1 = 3/3 = 5/5 = 7/7 etc.. A CPS is a related way of working, but without having the central tone of the diamond, instead, every tone of a CPS is the tonic of an implied diamond. Given a set of n factors, (1, .... n), all combinations of p factors yields the CPS p(n - read as P out of N CPS. For example: The set of 1 factor, taken in all combinations of 1 (1 out of 1CPS) has one member, 1 The set of 2 factors (1,3), taken in all combinations of 1 (1 out of 2 CPS) has two members, 1 and 3 The set of 2 factors (1,3), taken in all combinations of 2 (2 out of 2 CPS) has one member, 1*3 = 3 Going on: 1(1,3,5) has three members: 1,3,5 2(1,3,5) has three members: 3,5,3*5 3(1,3,5) has one member: 1*3*5 So far, rather trivial, now try four factors: 1(1,3,5,7) has four members: 1,3,5,7 2(1,3,5,7) has six members; 1*3,1*5,1*7,3*5,3*7;5*7 3(1,3,5,7) has four members: 1*3*5, 1*3*7, 1*5*7, 3*5*7 4(1,3,5,7) has one member: 1*3*5*7 Clearly, the 2(1,3,5,7) genus has the most musical potential. This is what Erv calls a HEXANY, and I recommend playing around with it closely, and trying it with other sets of factors. The next really useful CPS is the EIKOSANY which is a 3 out of 6 CPS with 20 members, and following this is the monster HEBDOMEKONTANY with 4 out of 8 factors and 70 members. I have often used the EIKOSANIES 3(1,3,5,7,9,11) and 3(1,3,7,9,11,15) - the latter gives a few more melodic possibilities without really diminishing the harmonic resources. I have also tried the all prime sets 3(1,3,5,7,11,13) and 4(1,3,5,7,11,13,17,19) both of which are very interesting. The Eikosany 3(1,3, 5,7,9,11) I have used in the following arrangement, taking 1*9*11 as the tonic (to work with these materials, it is hand to write the scale out in terms of each possible tonic): (I am notating here with 32/33 indicated as arrow down) 1*9*11 = 1/1 = C 3*5*7 = 35/33 = arrow down -7D 1*3*9 = 12/11 = arrow down D 1*5*11 = 10/9 = -D 3*7*11 = 7/6 = 7Eb 1*3*5 = 40/33 = arrow down -E 5*9*11 = 5/4 = -E 1*7*9 = 14/11 = arrow down 7F 1*3*11 = 4/3 = F 3*5*9 = 15/11 = arrow down -F# 1*5*7= 140/99 = arrow down -7G 3*9*11 = 3/2 = G 1*7*11 = 14/9 = 7Ab 5*7*9 = 35/22 = arrow down -7A 3*5*11 = 5/3 = -A 1*3*7 = 56/33 = arrow down 7Bb 7*9*11 = 7/4 = 7Bb 1*5*9 = 20/11 = arrow down -B 3*7*9 = 21/11 = arrow down 7C 5*7*11 = 35/18 = -7C What is great about these structures is that all of the possible subsets are musically interesting. There are plenty of hexanies embedded here, and I am fond of the Dekanies (for example all the pitches with factor X, or all the pitches without factor X). I hope that this helps. Have some fun, Dan Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 6 Dec 1996 10:15 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA21501; Thu, 5 Dec 1996 00:17:22 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA21551 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id PAA09228; Wed, 4 Dec 1996 15:17:18 -0800 Date: Wed, 4 Dec 1996 15:17:18 -0800 Message-Id: <199612042316.PAA09175@eartha.mills.edu> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu