source file: mills2.txt Date: Fri, 20 Sep 1996 01:15:34 -0700 Subject: Pitch spacings From: Daniel Wolf <106232.3266@compuserve.com> I am currently drafting a fairly detailed text (for Xenharmonikon, possibly, but certainly for my own composition cogitation) about the spacing (or voicing) of pitch materials. This subject has only been treated in passing in harmony and orchestration texts. I will outline some of the main points, in hope of getting some feedback: Assumptions (big ones, I know, but useful in staking out the territory): (a) heard materials are mapped by the listener onto a just structure. (b) we are listening to an orchestra of sine waves (thus, the timbres of real instruments are not considered). (c) we have unlimited sample lengths (thus, perception of interval differences within the critical band is possible and harmonic rhythm is not an immediate concern). (1) General characterization of pitch spacing: (a) harmonically (larger intervals at the bottom, smaller at top), (b) neutrally (no preponderence of intervals) or (c) subharmonically. A characterization as harmonic follows when a pitch complex is analyzable at a lower position in a harmonic series than in a subharmonic series; a subharmonic characterization follow a contrary analysis; a neutral characterization is inconclusive. (2) Specification of pitch complex: (a) harmonic sonorities by fundamental, (b) neutral by central frequency - or ambiguously, between a fundamental and a guiding tone, (c) subharmonic by guiding tone (the lowest pitch common to the spectra of all pitches in the complex). (3) Evaluation of relative complexity: lowest mapping of the complex onto a harmonic series. (eg Major triad maps onto 4,5,6, minor onto 10,12,15 (or possibly 6,7,9)) divided by frequency of the fundamental in that mapping. (A method of calculating in greater detail needs to be established; possible models are in Chalmers, Divisions of the Tetrachord). (4) Cohensiveness: "gaps" in the series seem to increase instability or ambiguity in perception. (eg the interval 3/2 is easier to perceive than the interval 3/1). (Why is it easier to tune superparticulars?) Suggestion: gaps increase the uncertainty of the harmonic/subharmonic analysis. Comment: I am encouraged in making a fundamental distinction between harmonic and subharmonic materials by recent brain research identifying the emotion of happiness with lowered electrical activity in the brain. Whether this is caused by eating chocolate or the processing of small-number ratios seems all the same. It makes sense that the harmonic series, marching happily onward up to the infinite is much easier to process than the subharmonic, going inexorably to the quantum depths of the infinitesimal. Could this be part of the mechanism though which we class the Major triad as "happy" and the minor as "sad". Daniel Wolf Ludwig-Landmann-Str. 84 B 60488 Frankfurt Germany +49 69 764307 http://ourworld.compuserve.com/homepages/DJWOLF_MATERIAL Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 20 Sep 1996 10:26 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA24605; Fri, 20 Sep 1996 10:27:40 +0200 Received: from eartha.mills.edu by ns (smtpxd); id XA23701 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id BAA24340; Fri, 20 Sep 1996 01:27:30 -0700 Date: Fri, 20 Sep 1996 01:27:30 -0700 Message-Id: <32426392.284E@cavehill.dnet.co.uk> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu