source file: mills2.txt Date: Sun, 4 May 1997 17:42:20 -0700 Subject: equal temperament From: James Kukula A fundamental property of an equal tempered scale is symmetry. The intervals available at any note are the same as those from any other note in the scale. This property really defines equal temperament. This property impacts the design of multi-stringed fretted instruments. As long as each string is tuned to a note in the scale, the frets will need to be in essentially the same place for each of the strings. (Ignoring effects due to string diameter etc.) But the symmetry property has far vaster impact than instrument design. I think I picked this up from Danielou. Mainstream European music is built on modulation, key changes. The tonal center moves around. So the function of a note is always shifting, it depends on the current key context. Since every note can play every role as the context shifts, then the uniform availability of intervals at each note is required. So modulation pushes one to equal temperament. If one gives up modulation, if each note has a constant and specific function, then the intervals available at that note can reflect that specific function, hence equal temperament becomes actually inappropriate. Danielou argues that by holding the context fixed and building it up over the course of a whole piece of music, then subtle differences of meaning can be given to each note - a single note can express rich meaning - whereas the constant context shifting of modulated music implies that the meaning of a note is ambiguous. One needs several notes to express a clear meaning. Of course, that very ambiguity can be used artfully, creating puns etc. See the analysis in Blackwood's THE STRUCTURE OF REGULAR DIATONIC TUNINGS. Jim Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Mon, 5 May 1997 02:55 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA04625; Mon, 5 May 1997 02:55:38 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA04630 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id RAA13514; Sun, 4 May 1997 17:45:58 -0700 Date: Sun, 4 May 1997 17:45:58 -0700 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu