source file: mills2.txt Date: Tue, 13 Aug 1996 07:12:54 -0700 Subject: From Brian McLaren From: John Chalmers From: mclaren Subject: Erv Wilson's CPS tunings -- John Chalmers and Paul Erlich have perceptively pointed out some issues in regard to Erv Wilson's CPS scales. Those of you not familiar with such tunings would be well advised either to go back and look at Topics 1, 2, 3 and 4 from Tuning Digest 30, Topic 2 from Tuning Digest number 31, or order MusicWorks 60 from Gayle Young and read Paul Rapoport's article "Just Shape, Nothing Central." Paul Rapoport is to date one of the keenest minds and best writers to have explained these types of tunings in detail, along with the inestimable John Chalmers. Kraig Grady's 1989 1/1 article "Erv Wilson's Hexany" deals with a subset of the Wilson CPS scales and does so in some detail, but does not concern itself with many of the implications of Wilson's CPS tunings, nor with the many operations which can be performed on Wilson CPS tunings (stellation, cross-product sets, etc.). Paul Rapoport has explained these ideas with admirable clarity, and John Chalmers has also explained them very well. Warren Burt's Topic 2 of Digest 31 is the clearest explanation of cross-product sets written to date. -- Paul Erlich pointed out that my lists of Wilson CPS tunings sometimes added a "confusing and extraneous "1/1. It is probably true that adding the implied 1/1 was confusing--so let me take a stab at explaining why Erv Wilson doesn't, and why I sometimes do. The Wilson CPS tunings can be highly tonal but never have a 1/1 in the form Erv Wilson generates them. Erv typically starts with 1 and takes some mutually prime set of numbers and generates an abstract set of points in ratio space by combinatorial methods. (See Tuning Digest 30 if you're not familiar with this process.) This abstract set of points in ratio space is not strictly speaking a tuning, but a framework within which individual tunings can "collapse" down into acoustic space. What I'm talking about here is akin conceptually to the process by which an equal-tempered chromatic scale generates a melodic mode. Depending on which note of the chromatic scale one starts on, quite different-sounding modes can be generated. Thus, phrygian sounds entirely different lydian or dorian--yet all are theoretically contained within the 12-TET chromatic scale. Similarly, the process of getting individual tunings (as on a guitar or a synthesizer or a set of tubulongs) from abstract points in ratio space is a process of projection from a higher- dimensional ratio space into a lower-dimensional acoustic space. This process is familiar to students of analytic geometry: the family of conic curves is usually derived by explaining them as projections of 3 dimensional lines inscribed on the surface of a 3-D cone down onto the two-dimensional space of graph paper. Depending on the angle of the cone to the 2-D graph paper, one can obtain an hyperbola, a parabola, etc. A classic example of projective geometry--one of the more fascinating byways of mathematics. Erv Wilson's CPS structures in ratio space exhibit a number of paradoxical properties. If low prime numbers are used as generators, the tunings will strongly imply tonality-- because the root motions between three- pitch "facets" (i.e., triads) in ratio space are ratios like 3/2, 4/3, &c. Yet, paradoxically, the CPS structure itself in ratio space has no 1/1 and thus no tonal center--until it is collapsed down into acoustical space by dividing the entire set of ratio space points by the ratio of a single ratio space point. (I.e., by choosing some ratio as a 1/1 and dividing all the other ratios by that ratio.) This process has the effect both of centering the acoustic representation around one particular point in ratio space--in effect, making some arbitrary point in the n-space lattice the origin of the ratio space coordinates--and it also has the effect of making the individual acoustic intervals more complex than they were in the original abstract ratio-space representation because they're now all multiplied by the inverse of the ratio chosen as 1/1 (that is, divided by the ratio chosen as 1/1). Erv Wilson's CPS tunings are not really atonal when low integer generators are used. They become increasingly atonal-sounding as higher and ever higher integers are used as members of the generating set. However, because any other point of the ratio space lattice can be chosen at any time as a different 1/1, the Wilson CPS tunings allow maximum modulatory flexibility with a minimum set of pitches. Partch's diamond is less efficient in modulatory terms than Wilson's CPS structures in ratio space because moving from one section of the two-dimensional Partch ratio-space diamond to the next does not produce a maximum change in triadic notes with a minimum length of ratio-space movement. You could think of this in projective geometry terms: the length of the geodesic in n-space between two points separated by more than 2 dimensions will always be shorter than the length of the geodesic on a plane between the same two points projected on a plane. This is nothing more complicated that the very familiar problem of projecting 3-D objects onto a 2-D plane. Map-makers have for millenia known that projecting coordinates on a sphere down onto a 2-D plane creates large distortions in some of the distances between some of the coordinates. Some distances which are short in the 3-D original are unnaturally and deceptively long on the 2-D plane; and so it is with ratio space projected onto a plane--as with Partch's ratio diamond, which can be (as has been, by Erv Wilson) visualized as a much more compact higher-dimensional ratio-space structure. Another way of thinking of the Wilson CPS method is that it provides a ratio-space coordinate system within which acoustic structures can be rotated with maximum symmetry. To put it another way, as the size of the Partch diamond increases, the amount of symmetry-breaking increases as the square as the number of prime generators. By "amount of symmetry-breaking" I mean here the number of anisometric points in the grid, each of which has the potential to break symmetry upon rotation of a musical structure across the grid. Example: In the 4 5 6 7 1/4 1/5 1/6 1/7 Partch diamond with 4 generators there are 4 1/1's--1/1, 3/3, 5/5 and 7/7. This anisometric diagonal of 1/1s acts in effect like an axis. It has the effect of breaking the symmetry of 2-D ratio space structures when they're rotated through the 2-D ratio space of the Partch diamond. But in a Wilson CPS *there is no 1/1*, and thus there's *never* any example of symmetry-breaking when substructures are rotated through ratio space. (Wilson CPS structures are inherently symmetrical: thus their symmetry cannot collapse in ratio space but only when they are projected into some anisometric lower-dimensional coordinate system.) Physicists know that many physical processes can be described in terms of broken symmetry: the nucleation of snowflakes in a supersaturated air, the spontaneous transformation of isotherms into tornadic weather patterns, the formation of new crystals through screw dislocations, and the fracture of the single unified force after the intial few femtoseconds of the Big Bang into the 4 fundamental universal forces of gravitation, electromagnetism, strong and weak nuclear forces, are all classic examples of symmetry- breaking which leads to emergent order. In the same way, symmetry breaking is required to generate an emergent order out of the abstract ratio space structures of the Wilson CPS or the Partch diamond. This leads us to a concise explanation of why I sometimes add an "extraneous" 1/1 to the Wilson CPS. Seen from the perspective of symmetry breaking as a means of generating scalar order, it's not obvious why choosing one of the CPS lattice points as a 1/1 is significantly different from adding a 1/1 point at the lowest-dimensional point of the structure in ratio space. Both procedures break the symmetry of the abstract ratio-space CPS; both procedures collapse the higher-dimensional structure down into a lower-dimensional acoustic space. The advantages of adding a 1/1 at the lowest- dimensional point of the ratio space Wilson CPS are: [A] the acoustic ratios are maximally simplified and so it's especially easy to hear the underlying harmonic progressions. [B] the acoustic distance from the 1/1 to any given scale-step is minimized. The disadvantages of adding such an "extraneous" 1/1 in ratio space are: [C] modulation is no longer maximally efficient--that is, modulations no longer produce maximum change in number of chord components for minimum change in root position; [D] it is no longer simple to modulate to another substructure embedded within the ratio space CPS. These advantages and disadvantages must be weighed carefully. In fact, Erv Wilson often adds "extraneous" pitches to his CPS tunings--there is no reason whatever why I can't add "extraneous" pitches too. The point is that the process is not black-&- white, good-or-bad. Simply because Erv Wilson doesn't do something (and even Erv sometimes refers to the "nonexistent" 1/1 at the lowest-dimensional point in ratio space in the process of theorizing; he regards as the null operation in which all axes of the ratio space are raised to the zeroth power) doesn't mean it can't or oughtn't to be done. --mclaren Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 13 Aug 1996 21:16 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA02160; Tue, 13 Aug 1996 21:17:01 +0200 Received: from eartha.mills.edu by ns (smtpxd); id XA02166 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id MAA17431; Tue, 13 Aug 1996 12:16:58 -0700 Date: Tue, 13 Aug 1996 12:16:58 -0700 Message-Id: <13960813191331/0005695065PK5EM@MCIMAIL.COM> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu