source file: mills2.txt Date: Wed, 2 Oct 1996 02:44:49 -0700 Subject: TUNING digest 852: _Stability_ From: Daniel Wolf <106232.3266@compuserve.com> I will write a more detailed response when I can get away from some urgent business. (1) I am rather apprehensive about the use of _stability_ to characterize a single chord. From Pauls definition, a /4:/5:/6 minor triad, lacking a power of two in the lowest voice, is less _stable_ than a Major 4:5:6. However, when placed in a progression, it may very well have the opposite reading. For example, if the tonic key is established as minor, than the Major dominant triad will _resolve_ to the tonic minor triad. Of course the root motion of the downward fifth (or upward fourth) is a strongly defining feature, but this ordering of chord qualities is well established in the _common practice_ repertoire (or else we would find more mixolydian cadences). In this case, key identity, spacing, and voice leading from one chord to the next are equally or more determinant of _stability_ than rootedness. Further, all of these determinants seem largely to be examples more of conditioning rather than of psychoacoustic properties. (The existence of alternative practices in other repertoires (eg _modal_ progressions in pop musics) both confirms the locality of the conditioning described above, and the varieties of progressions encountered in these repertoires suggest the inapplicabilty of a _stability_ concept outside of narrowly defined diachronic contexts). (2) On the other hand, _stability_, in a musical context without repertoire-defined harmonic progression may be achieved by means of extended durations or by reiterations. In this case, the conditioning is taking place in real time (Pavlov meets Stravinskys _assertion_ of tonality), and the examples of this that I find immediately suggest that lowest voice, 2^n roots are not a prerequisite to achieving stability. (3) Back to sine waves. The argument that Paul used against my sine wave subharmonics can be turned around and applied to regular harmonic-spectra harmonic pitch complexes, where the reinforcement of a single series will create masking so that individual pitches will be difficult if not impossible to discern. (This was my initial rationale for the _sine wave orchestra_; long experience with synthesized sound and precise tuning had convinced me that the situation has to be simplified in order to get any useful results at all). (4) The intuitionist (like an algorithm maker, or a student of musical cognition) proceeds in strictly chronological steps. If from the unity 1:1, the twoity is observed, and from the sum of 1+1, 2 is constructed, yielding the first harmonic interval, only then may the first subharmonic (the inversion) be constructed. In constructing the series, the subharmonic will always lag behind. Of course, another fundamental approach will have all intervals appear simultaneously, but I am lost in trying to figure out how that is described cognitively. (5) I never said that mathematics goes funny at the infinitesimal, I made the observation that the calculation in this direction is counterintuitive, and I speculated about why it should be so. Your examples of string lengths etc. I found curious, as the (pre-wave length) music theory using this instrumentality was inversionally flexible in the extreme, and ran counter to Pauls strong harmonic series approach. (Needless to say, classical harmonics was used to describe a repertoire without chords). There is indeed a (mathematical and physical) boundary problem for the infinitesimal (with null or the vacuum) that the infinite does not have. Daniel Wolf Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 2 Oct 1996 17:57 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA04020; Wed, 2 Oct 1996 16:57:48 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA04011 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id IAA00832; Wed, 2 Oct 1996 08:57:41 -0700 Date: Wed, 2 Oct 1996 08:57:41 -0700 Message-Id: <63961002154336/0005695065PK4EM@MCIMAIL.COM> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu