source file: mills2.txt Date: Thu, 26 Sep 1996 01:52:47 -0700 Subject: More on spacing: response to PAULE From: Daniel Wolf <106232.3266@compuserve.com> (1) I cannot follow your logic regarding sine waves and the subharmonic. Since the complexes we are discussing are within a finite section of the audible frequency range, and we are probably not discussing ratios involving harmonic partial larger than, say, 81, (although I have worked with interesting - and perceivable - musical results through 128, and La Monte Young through 512, partials), whether one thinks in terms of harmonic or subharmonic is - initially - simply a matter of renotation. Thus, how is it that you are able to decide that sine waves are "distinguished and digestable" when described in harmonic context but suddenly not so when renotated as subharmonic? (2) Or do you define as subharmonic any complex without 2^n as the lowest member? A spacing theory would compare the notations to determine which characterization is more likely (if neither is, then the neutral characterization is chosen). Your algorithm - and I do wish to examine it in detail - strikes me as a root direction theory (and more in the Hindemith than in the Rameau direction). The lack of register specificity in your algorithm would however seem to be problematic in light of all experimental evidence (and in light of traditional musics and instrumentation). Both Clarence Barlow and James Tenney have profitably included absolute frequency in their agorithms. (3) You cite several psychoacoutistical certainties in support of your algorithm, I am especially curious to learn of experimental results supporting (a) "in a purely psychoacoustical sense, a chord played will evoke certain interpretations in terms of the harmonic series"; this may be what your algorithm tells me, and is a conjecture that I would like to support, but I would like to find experimental evidence in this direction (and I have encountered evidence to the contrary: see Boomsliter and Creel on the 7th partial); (b) "the earīs central pitch processor simply doesnīt go that far"; I am completely confused by this: what do you mean by "central pitch processor", and how far, exactly does it go? The harmonic progression I presented, and its Pythagorean interpretation are simply too obiquitous to be dismissed in this way. (4) The sequential procedure would be something like the following: given the complex 500 Hz, 600 Hz 750 Hz, the first ratio interpreted would be the 3/2 fifth of 750/500, the initial interpretation would be 3/2 in a harmonic series over fundamental 250Hz; however, not finding an intermediate tone between 2 and 3, successive fifths in the series are scanned to find a similar complex, examining each n(2,3) until an intermediate tone is located that matches the complex. When, for example 10,12,15 over a fundamental of 50 Hz is selected, then it is compared with its subharmonic notation /6,/5,/4. If this characterization is more simple, then it is chosen. (Need I add the fact that the difference tones of this complex are all within the spectrum of 50 Hz? (The careful reader will note that I was cautious about the register of this particular example)). (5) All of the progressions that I described work - both acoustically, are to found in existing repertoire, AND on paper. We (industrialized westerners) have come, through historical reasons (largely inertia) to the practice of using a single Pythagorean pitch notation to indicate a diversity of wildly varying intonational environments. That certain tuning systems, temperaments among them, may map onto harmonic series is probably true, but it requires a definition of intervallic tolerance that seems to me to be musically (i.e. culturally, and thus limited to particular times and places) defined, and probably outside of the domain of psychoacoustics proper: (a) tolerance may be a function of existing instrumental and room-acoustic capacities; (b) tolerance may depend upon durational conventions within a musical repertoire; (c) tolerance is a property of warp-able analog recording and digital sampling rates; (d) La Monte Young has demonstrated the perceptibility of ratios involving larger primes through the use of extended duration sound environments). (6) Your statement _the ear doesnīt continue its harmonic series math from one chord to the next_ is unsupported, and probably unsupportable. It is certainly musically untenable, unless you mean that the mechanism for relating pitches in diachronic sequences to one another is independent from the synchronic mechanism. This is possible, but seems inefficient in light of the evidence that so much of acoustical processing is done by sharing facilities. If you intend a separate _Gestalt_-type mechanism for chords and another mechanism for successions, what do you do about broken chords? (7) I cannot resist a few more remarks on the subharmonic. First, although for musical purposes we never need get past fairly small numbers, while every harmonic complex may be renotated as subharmonic and vice versa, due to one of those nice paradoxes in the fundamentals of mathematics - and beyond my ability to explain - the number of members in the harmonic series is greater than that in the subharmonic series. Second, the _descent to the infinitesimal_ represents to me a series of increasingly complex - indeed, counterintuitive - calculations, hence my speculation about increased mental activity. The ascription of quantum effects to the infinitesimal comes from the intuitionist C. Hennix, resident mathematician at the Institute for Psychoanalysis in Paris; Third, musical examples of the subharmonic are readily available in the repertoire, cf Martin Vogelīs book on the _Tristan_ chord, or this little example from the Rondo K.494, m 164-165, where the left hand sustains for two measures the chord c4 d4 f4 ab4, which moves (does not resolve) to c4 f4 a4, i.e subharmonic seventh chord from c moving to f major 6-4. Mozart uses this chord, here and elsewhere, in a voice leading distinctive from his use of diminished and dominant seventh chords. By my spacing characterization, the fourth c-f would be too important to overlook, and the traditional (in American practice) _half diminished_ description seems not very useful. If this were in just intonation, I would choose a subharmonic characterization /8 /7 /6 /5 over the harmonic 105 120 140 175. I am curious to learn how Pauls algorithm will interpret this chord (16 18 21 26? or 24 27 32 38?). Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 26 Sep 1996 17:13 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA32492; Thu, 26 Sep 1996 17:13:40 +0200 Received: from eartha.mills.edu by ns (smtpxd); id XA20847 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id IAA12290; Thu, 26 Sep 1996 08:13:03 -0700 Date: Thu, 26 Sep 1996 08:13:03 -0700 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu