RE: Prime/Odd-limits; omega functions

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 Dani�l 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