source file: mills2.txt Date: Mon, 23 Sep 1996 07:40:50 -0700 Subject: RE: Pitch spacings From: PAULE Daniel- Firstly, let me note that your assumption that "we are listening to an orchestra of sine waves" does not bode well for the perception of subharmonic complexes as such, since, as you state, the subharmonic complex is recognized "by guiding tone (the lowest pitch common to the spectra of all pitches in the complex)." I think your observation that superparticulars are easier to tune relies on the existence of difference tones, which actually beat against the virtual pitch in the case of superparticulars (tuning is easy: just eliminate this beating). However, I think you can modify the statement to say "an orchestra of generic harmonic timbres played as a reasonably loud volume" without harming your arguments. Secondly, I find myself agreeing with much of your approach, especially since I have included an algorithm for "evaluation of relative complexity," exactly as you defined it, as an appendix of a paper that will appear in Xenharmonikon. I'm relying on Goldstein's 1973 JASA paper, along with some number theory, for this algorithm. I can provide you with details of the algorithm if you like, but let me note that I have found that for listeners with slightly above-average pitch resolution, the 12-tET minor triad will be heard mainly as 10:12:15 and as 16:19:24, the winner depending on voicing, and assuming the triad is in a register that puts some audible partials in Goldstein's optimal frequency range. The latter assumption my algorithm cannot do without, so it is only good for finding the "worst-case" or most complex representations of a particular tuning's intervals and chords. If your algorithm succesfully extends this to the case of arbitrary given register, I will congratulate you. (I suggest you take a look at Goldstein's paper if you haven't yet). I would not be surprised if you are correct that 4:5:6 is happier than 10:12:15 since it requires less brain effort. However, I don't think it makes much musical difference whether the minor triad is heard as 10:12:15, or 16:19:24, or a combination of the two. My algorithm actually gives a greater salience to the p5 alone than to the minor triad, suggesting that the minor triad is actually heard most significantly as 2:x:3, where x is a non-harmonic tone not integrated into the virtual pitch sensation evoked by the 2:3. (Unfortunately, I have no justification for directly comparing a 2-note salience to a 3-note salience, but it seems to work fine.) The salient musical properties of the minor triad relative to the major triad are: (1) the p5 defining a clear root, which is not, however, reinforced by the third, and (2) a similar level of roughness and beating as is found in the major triad. The sadness of the minor triad is then due simply to its greater harmonic indefiniteness, without any extra roughness that could contribute "anger" or "fear" into the perception. -Paul Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 25 Sep 1996 09:47 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA24127; Wed, 25 Sep 1996 07:43:44 +0200 Received: from eartha.mills.edu by ns (smtpxd); id XA26025 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id WAA04669; Tue, 24 Sep 1996 22:43:42 -0700 Date: Tue, 24 Sep 1996 22:43:42 -0700 Message-Id: <199609250543.WAA26562@si.UCSC.EDU> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu