source file: mills2.txt Date: Thu, 20 Mar 1997 17:44:53 -0800 Subject: RE: Prime/Odd-limits; omega functions From: Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul) Jonathan Walker writes: > Perhaps it would be better to use the Euler-Fokker notation, and simply > list all the primes [p_1,p_2,...p_n] (according to the small-omega > criterion) that are used in a given composition. That's an interesting suggestion, but an indication would be needed for the case when an incomplete genus is used, to avoid confusion with complete Euler-Fokker genera. > OK, now I'd like someone to do me a small favour: could you give me the > source in Euler for the specification of prime factors in musical > contexts. Is there a brief English-language account of the Euler-Fokker > notation? - Euler, Leonhard. _Tentamen novae theoriae musicae_. Sint Petersburg, 1739. In _Opera Omnia_ series III volumen I, Teubner, Leipzig, 1926. - Rasch, Rudolf A. (ed.) _Adriaan Daniël Fokker - Selected Compositions, with introduction by Rudolf Rasch_. Corpus Microtonale vol. 1, Diapason Press, Utrecht, 1987, 217 pages. - Mandelbaum, Joel. _Multiple Division of the Octave and the Tonal Resources of the 19-Tone Equal Temperament_. PhD Thesis, University of Indiana, 1961, 460 pages. But there are more sources in German, French and Dutch; see the bibliography. > For instance, is the convention to omit > 2 from the list of primes, or to include it? Thanks. An Euler-Fokker genus is what results if the tones of what Euler calls a "complete chord" are reduced within one octave. So the prime 2 is always omitted. Each prime is listed as many times as its multiplicity. A complete chord or genus musicum contains all the harmonics (including the first) of its fundamental, which are also subharmonics of its guide tone (i.e. the lowest common multiple of all tones). So for example the chord 1:3:5:15 is complete and 3:5:15 is not. If a lattice diagram is drawn of a complete chord it is always rectangular if you take one dimension for each prime. In a genus, any tone may be taken as the tonic. Genus [335] could be: A - E - B C - G - D | | | or | | | etc. F - C - G Ab- Eb- Bb Manuel Op de Coul coul@ezh.nl Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 21 Mar 1997 04:57 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA07546; Fri, 21 Mar 1997 04:57:45 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA07549 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id TAA21931; Thu, 20 Mar 1997 19:56:19 -0800 Date: Thu, 20 Mar 1997 19:56:19 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu