source file: mills2.txt Date: Mon, 14 Jul 1997 23:52:08 +0200 Subject: RE: Basic dimensionality, and related issues From: "Paul H. Erlich" Graham wrote, >On dissonance of composites in general: it is much easier to form >consonant chords using composite numbers than prime numbers of >about the same size. As an octave invariant example, 1:3:5:15 >includes the interval 15/1. Try finding an equally consonant 4 >note octave invariant chord including 13/1. If the consonance of >an interval reflects its propensity to form consonant chords, then >composites are more consonant than primes and, in some >circumstances, the metric should handicap high primes accordingly. In some circumstances, but not in others (such as when intervals are sounded alone). Graham, how does your metric address this? The consonance of a chord depends on all intervals within it, not just the most dissonant one. 5/6 of the intervals in 1:3:5:15 are consonant within the 5-limit, which will not be true of any 4-not chord containing 13. That should be enough of a handicap against 13. I admit that there might not a single metric that will give the consonance of intervals as well as of chords. The consonance of chords is a combination of roughness issues (see Plomp & Levelt, Kameoka & Kuriyagawa, Sethares) and tonalness issues (Parncutt, preliminary model by me (posted once), upcoming model by me (if I live long enough) based on harmonic entropy concept). 3:5:9:15 is at least as consonant as 1:3:5:15 although the rectangular 5-limit lattice favors the latter. The triangular lattice seems better in all circumstances. >An unrelated (or is it) point. Recently, Gary Morrison wrote: >> By fibonacci, what I'm referring to is J. Yasser's fibonacci-like >>sequence of tunings (5 7 12 19 31 50 81...). (By fibonacci-like I mean >>that the next number in the sequence is the sum of the previous two.) 12 >Firstly, wasn't it Kornerup who first identified this sequence? In a very different context. Kornerup found a subset of ETs consistent with meantone notation and composition, and approaching a meantone tuning (often called golden tuning) where the tone, the minor third, the perfect fourth, and the minor sixth are all divided into two parts by a golden section. I don't think he included 5 or 7. Yasser thought that at any stage in the evolution of tonality, the total number of pitches in use is one number in the sequence and the number of diatonic pitches is the previous number. He starts his sequence with 3, 2, 5, 7, . . . I tend to side with Kraehenbuehl and Schmidt that 12 out of 22 is a more self-delimiting (i.e., logically closed) scale than 12 out of 19 and a more logical outcome of 12-tone chromaticism. K&S dealt with JI, so my agreement with them may be coincidental, although we both rely on the introduction of the number 7 into consonant frequency ratios. Yasser's frequency ratios use primes up to 13 and are unbelievably out of tune with 19-tet. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 15 Jul 1997 00:03 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA23407; Tue, 15 Jul 1997 00:04:33 +0200 Date: Tue, 15 Jul 1997 00:04:33 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA23329 Received: (qmail 1038 invoked from network); 14 Jul 1997 21:53:44 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 14 Jul 1997 21:53:44 -0000 Message-Id: <199707142150.OAA25521@well.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu