source file: mills2.txt Date: Sat, 16 Sep 1995 09:45:27 -0700 Subject: 88CET #10: Note Doubling From: Gary Morrison <71670.2576@compuserve.com> Probably the easiest way in a traditional tuning to extend a chord is by doubling one or more notes in octaves from its basic position. One of the main goals of Sethares' 88CET-optimized partial mappings is to create a pseudo-octave with a pitch-duplicating sensation kind of like the real octave's. That 14-step pseudo-octave works great for doubling on these mapped timbres. On approximately harmonic timbres however, this and the other false perfect consonances don't work at all for doubling. The rest of this posting addresses doubling in reference to the usual unmapped timbres. Can the octave-doubling principle be extended to tunings that don't even have an octave, by doubling by an interval other than an octave? Brian McLaren, who has worked with a variety of nonoctave tunings, believes that whatever your cycle-interval may be can produce a doubling effect. He reports, for example, that doubling notes in 3:1 perfect twelfths in Pierce-Bohlen tuning works just great. What exactly the cycle interval is in 88CET, or in Carlos' alpha and beta tunings for that matter, is not immediately apparent. The step size for these tunings, unlike Pierce-Bohlen, are not based on making any particular interval exactly just. They are instead based upon making several target intervals as accurate as possible. To my way of thinking, for a tone added to a chord to be thought of as a double, that tone should not have much effect on the underlying character of the entire chord. If it changes the character of the chord, then it clearly has a significant function in the new chord, so it doesn't make sense to think of it as a functionally equivalent double of a another chord tone. The idea of nonoctave doubling then has meaning provided that: 1. The interval by which you double a note is more "bland" than the intervals in the chord itself. Consider for example the traditional 4:5:6 close- voiced, root-position, major triad. Doubling the root an octave higher does not significantly change the underlying character of the resultant chord from its undoubled form. The octave added into that chord is comparatively bland; doesn't attract our ears' attention like the thirds. On the other hand, if we were to instead to try to double its root a 7:4 ratio higher, the underlying character of the chord does significantly change. It becomes a dominant seventh chord. The subminor seventh attracts attention as much as the thirds in the chord, if not more. 2. No other resultant interval is more interesting than the intervals in the undoubled chord. Using the above example of doubling the root of 4:5:6 major triad up an octave, the resultant chord has the ratio 4:5:6:8. That clearly creates the interval of an octave between the root and its double (4:8), but it also creates a perfect fourth (6:8), and a minor sixth (5:8). The underlying character of the chord does not change also because those intervals are no more attention-getting than the major and minor thirds already in the undoubled chord. Obviously you would get a very different sensation if you were to start with a 3:4:5 second-inversion triad, and attempt to double the bass up a perfect fifth. Although the interval of doubling, the fifth, is bland in comparison with the other intervals in the original chord, this also produces a 9:8 major second, which is clearly anything but bland, so it definitely attracts attention away from the constituents of the original chord. It changes the chord's underlying character. Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 16 Sep 1995 20:23 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id LAA26065; Sat, 16 Sep 1995 11:23:55 -0700 Date: Sat, 16 Sep 1995 11:23:55 -0700 Message-Id: <950916182135_71670.2576_HHB42-7@CompuServe.COM> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu