source file: m1371.txt Date: Tue, 31 Mar 1998 13:43:39 -0800 Subject: RE: reply to Bill Alves From: alves@orion.ac.hmc.edu (Bill Alves) Paul wrote: >>}>"Given that the diminished fifth resolves inward and the augmented >>}>fourth resolves outward, wouldn't one expect that, ideally, the >>}>augmented fourth to be a larger interval than the diminished fifth?" I answered: >>}Yes. >> >>}You really do believe that? So now tell me, what is, ideally, the dividing >>line between minor sevenths }which tend to resolve inwards and those that >>tend to resolve outwards? As I said, I think that the shortest path to resolution tends to be the most "natural." A resolution of a leading tone to the tonic is journey of a semitone, and more natural than the resolution of the sub-tonic to the tonic, a whole tone. By extension, a resolution by 76 cents, say, is slightly more natural than one by 117 cents (though I do agree that is just one factor in the musical context). I believe that string players, for example, have a natural tendency to slightly sharpen or flatten notes in the direction of the resolution. >>}Yes, though there are different sized tritones in a given meantone tuning >>}of 12 keys. >> >>I'm not sure what you mean. In a given meantone tuning, there is only one >>size of augmented fourth, and one size of diminished fifth. Enharmonic >>equivalents (in the 12-tone sense) cannot be used in meantone temperament. Handel's 16-key organ was the exception, not the rule, and the vast majority of meantone keyboards had 12 keys. Thus musicians had to decide whether to tune a given tritone to an augmented fourth or diminished fifth, and so they might encounter music where they had to play a Db when they had tuned an C#. Thus enharmonics, while theoretically distinct, had to be treated as the same for practical purposes on most keyboards. Bill ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^ Bill Alves email: alves@hmc.edu ^ ^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^ ^ 301 E. Twelfth St. (909)607-4170 (office) ^ ^ Claremont CA 91711 USA (909)607-7600 (fax) ^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^