source file: mills2.txt Date: Tue, 8 Oct 1996 07:47:39 -0700 Subject: Last answer to Paul From: Daniel Wolf <106232.3266@compuserve.com> 1) I have three degrees in music, so perhaps you can bear with a bit of the history of theory. In western tonal theory, there are two major lines of thought one, which we will call _voice leading theory_ and identified with Fux, and continuing through Shenkerian and other methods, bases analyses upon the existing chord structure, described from the sounding bass tone; the other we can call _fundamental bass_ theory, identified with Rameau, is based upon the discovery of roots that my or may not be present. In the first model, rule systems are based primarily on melodic movement within single voices and then between them in counterpoint, while the second model is based upon a harmonic relationships. (Your example of a Rameau rule about seconds in the Fundamental bass is not present in all versions of his theories, which he changed periodically). In practice the division between these models is not entirely straightforward, but comparing the work of a voice leading composer like Brahms with a fundamental bass composer like D-Indy or Faure is instructive. It is interesting that the both the Riemann chord classification scheme - an essentially just intonation model, perhaps familiar to those who know the work of Vogel - and Schoenbergs functional harmony hedge their bets by making it clear that voice leading (via species counterpoint) is prior to harmonic theory. My attempt to locate your work within this theoretical framework is an attempt to locate its utility for musical analysis. 2) In that only pairs of tones will share harmonic series in a subharmonic chord, the masking will be at a lower amplitude than for a harmonic chord where all tones share a single series. 3) A division of a string into aliquot parts (subharmonic) is finite. Successive harmonic divisions are not. Example: the division of the string into four equal parts yields four pitches of the subharmonic sequence 1/1, 4/3, 2/1, 4/1 and no more. 4) The period (for example, one second) is a function of our notation. If you have an alternative notion please describe it. 5) Huygens was the founder of wave theory and arguably the first modern physicist; his methods were truly counterintuitive for his time and remains so. (I am not saying that he was wrong, just that the physics and the cognition of music are different and depend upon different apprehensions - hence my attraction to intutionist models where temporal perception is prior to spatial, and computation is placed in temporal sequence). Daniel Wolf Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 8 Oct 1996 18:39 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA00558; Tue, 8 Oct 1996 18:40:56 +0200 Received: from eartha.mills.edu by ns (smtpxd); id XA00551 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id JAA01761; Tue, 8 Oct 1996 09:40:37 -0700 Date: Tue, 8 Oct 1996 09:40:37 -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