source file: mills2.txt Date: Wed, 9 Jul 1997 21:29:23 +0200 Subject: Dimensions, etc. From: John Chalmers Paul E. is correct (as usual) that the best fifths of 24- and 34-tets is not cyclic through the whole gamut. However, there are other intervals that are, and in this respect, ET's are 1-D. In fact, in all ET's with more than 6 tones, there is at least one pair of intervals other than 1 and N-1 that is a cyclic generator. (For 6-tet, the pair 1-5 is the only cyclic generator). Paul is also correct that meantone rolls the 3x5 plane into an infinite cylinder. This is true for any non-cyclic temperament which defines one harmonic dimension in terms of a cycle of another generating interval. Thus a meantone-like tuning which defined the 7/4 by 10 ascending fifths would roll up (compactify?) the 3x7 plane. A temperament which tweaked the third to obtain the just 7th would roll up the 5 x 7 plane (disregarding the Fifth for the moment). I imagine 1/3 comma meantone, with its just 6/5-5/3, would roll up a plane lying obliquely to the coordinate axes. Paul? In summary, the dimensionality of a scale or tonal space depends somewhat on how we conceive or use it. Gerry Balzano described tunings of 12, 20, 30, n(n-1) tones in terms of triads whose component "thirds" spanned n and n-1 degrees and scales of 2*n-1 tones. Fokker showed how regions of tonal spaces correspond various to ET's and these regions induce different dimensionalities on ET's of the same number of tones. Thus dimensionality is in the mind of the theorist. My other major point is, as I think Graham first mentioned, that one can restrict oneself to various subspaces of these tonal spaces. Fokker in New Music with 31 Notes (translated by Leigh Gerdine from Fokker's German original, published by Orpheus), discusses Diamond-like struct „ in the 3x5, 3x7 and 5x7 planes generated by mirroring triads such as 1.3.5, 1.3.7, and 1.5.7 A skew Diamond may be generated by mirroring the 3.5.7 triad. These scales are equivalent to portions of the 7 limit Partch Diamond and are perhaps visualized most easily by constructing cross-sets (to use E.W.'s term) of the 1.3.7 triad by its inversion, the sub-1.3.7., etc. For the 3.5.7, multiply 1/1 5/3 7/6 by 1/1 6/5 12/7. Alternatively, one may construct a 7-prime-limit Diamond and delete the notes containing the omitted factors. Other scales are possible in subspace (that has a nice SF'nal sound to it). A very nice pseudo-diatonic scale in the (2)x3x7 subspace is the septimal or subminor scale 1/1 9/8 7/6 4/3 3/2 14/9 7/4 2/1. Five-less chord progressions can sound very consonant and I recommend trying them. A final and minor point: I believe Erv's 22-tone constant structure was based on the 1.3.7.9.11.15 Eikosany with 2 additional tones to complete the gamut. In CPS, the 1 (2, 4, etc.) factor axis may be an essential conceptual dimension, even if it can be omitted (compactified?) when plotting notes on paper. It is the presence of small "kommata" that inevitably occur when one uses complex intervals or moves very far from the origin and which are less than single ET degrees that makes it impossible to do 1-to-1 mappings of some scales into some ET's. --John Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 9 Jul 1997 21:58 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA22297; Wed, 9 Jul 1997 21:58:34 +0200 Date: Wed, 9 Jul 1997 21:58:34 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA22227 Received: (qmail 25303 invoked from network); 9 Jul 1997 19:56:40 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 9 Jul 1997 19:56:40 -0000 Message-Id: <199707091549_MC2-1A8F-80CA@compuserve.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu