source file: mills2.txt Date: Sat, 22 Mar 1997 10:31:37 -0800 Subject: Limits (Paul E) From: Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul) From: PAULE For a reference on Euler-Fokker genera, see Fokker, Adriaan D. 1975. _New Music with 31 Notes._ Verlag fur systematische Musikwissenschaft GmbH, Bonn. The prime limit definition should NOT be the default, despite its utility in describing a tuning system. The odd limit definition is not intended to describe a tuning system; it is intended to describe musical practice. Despite what Jonathan Walker said, one does not consider music in Pythagorean tuning to be a 531441-limit system under the odd definition of limit. Even under Partch's own (odd) definition, Partch's music was 11-limit, not 33-limit nor some product of powers of numbers through 11. Partch listened carefully and came up with the most useful definition for the ear and for musical practice, not for studying the mathematics of tuning systems. The definition is: A piece of music is N-limit if all intervals whose (heard) rational approximations use odd numbers up to N, and all chords containing only such intervals, are considered consonant, while intervals which cannot be so approximated, or any chord containg at least one such interval, are considered dissonant. In other words, it's how high up the harmonic series (or how far down the subharmonic series) the ear is asked to go in a particular piece. So with a few caveats (such as second-inversion triads) common-practice music is 5-limit. The difference between the prime and odd definitions really shows up at the first composite odd number, 9. As Dan Wolf and I have pointed out before, there can be a very real difference between 7-limit music, where 9 is only possible as 3*3, and 9-limit music, where 9 is independently available as a harmonic building block. I'm sure I could have said some things better here, I look forward to discussing more because this is an important issue! -Paul E. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 22 Mar 1997 19:42 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA08839; Sat, 22 Mar 1997 19:42:52 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA08828 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id KAA21646; Sat, 22 Mar 1997 10:41:09 -0800 Date: Sat, 22 Mar 1997 10:41:09 -0800 Message-Id: <199703221338_MC2-1325-9112@compuserve.com> Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu