source file: mills3.txt Date: Tue, 30 Sep 1997 18:08:30 +0200 Subject: Re: Blackwood's proof From: Ben Denckla Mark Nowitzky writes: --- It was something about Helmholtz' and Ellis' attempt to use just intonation to get better harmony than equal temperament being proved impossible by Easley Blackwood in "The Structure of Recognizable Diatonic Tunings". --- James Kukula writes: --- Before I read Blackwood I had tried taking a couple of Bach harmonized chorales and figuring out how they should be played in j.t.[...] Basically the problem is just to figure out which of a set of nearby pitches to use for each note, the pitches differing by the 80/81 comma. [...] the exercise really opened up for me the structure of the Bach pieces, or seemed to anyway. I found, though, that the pieces seemed to turn on a kind of musical pun, to take advantage of the blurring of harmonic distinctions provided by temperament. As I read Blackwood, the point is not so much that e.t. or any temperament is so much better than j.t. in any universal or absolute sense, but that mainstream European musical practise at least in the 18th & 19th Centuries was really structured around temperament, and you can't just slide j.t. into the classics and make them better. A different kind of music based on j.t. might be better in some sense, but the structure of composition is strongly connected to tuning, so you can't change one without some adjustment in the other. --- After 10 months or so of research on the subject, I have come to the exact same conclusions as Mr. Kukula, and he has phrased them very nicely above. For a different phrasing, though, I must once again shamelessly plug my thesis, available in various forms (including HTML) at http://theremin.media.mit.edu/~bdenckla/thesis/main.html I believe my thesis' tuning theory is an improvement upon Blackwood's, although Blackwood's musical examples are very instructive and are the real strength of his book. On this particular subject, my thesis frames these "puns" (a nice metaphor) as reliance on what I call A1_1 enharmony, which is analogous to d2 (diminished second) enharmony, i.e. enharmony between notes like G# and Ab. Ben SMTPOriginator: tuning@eartha.mills.edu From: Carl Lumma Subject: Re:deployment PostedDate: 30-09-97 21:02:38 SendTo: CN=coul1358/OU=AT/O=EZH ReplyTo: tuning@eartha.mills.edu $MessageStorage: 0 $UpdatedBy: CN=notesrv2/OU=Server/O=EZH,CN=coul1358/OU=AT/O=EZH,CN=Manuel op de Coul/OU=AT/O=EZH RouteServers: CN=notesrv2/OU=Server/O=EZH,CN=notesrv1/OU=Server/O=EZH RouteTimes: 30-09-97 21:02:17-30-09-97 21:02:18,30-09-97 21:01:14-30-09-97 21:01:15 DeliveredDate: 30-09-97 21:01:15 Categories: $Revisions: Received: from ns.ezh.nl ([137.174.112.59]) by notesrv2.ezh.nl (Lotus SMTP MTA SMTP v4.6 (462.2 9-3-1997)) with SMTP id C1256522.006890AB; Tue, 30 Sep 1997 21:02:07 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA02724; Tue, 30 Sep 1997 21:02:38 +0200 Date: Tue, 30 Sep 1997 21:02:38 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA02733 Received: (qmail 27315 invoked from network); 30 Sep 1997 12:02:34 -0700 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 30 Sep 1997 12:02:34 -0700 Message-Id: <19970930190506296.AAA478@NIETZSCHE> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu