Re: [tuning-math] Digest Number 1410-stretched octaves

My recollection is that Prof. A.D. Fokker proposed a slightly shrunk
31-tet octave from similar considerations, but I'd have to look up the
exact reference.

Agusto Novaro proposed several variants of 12-tet with altered octaves
and variously tempered fifths and fourths, but exactly what the
intervals are is somewhat difficult to determine since he describes them
in terms of beat rates referenced to his sonometer, which he determined
was inaccurate. For example, he found that what he laid out as the 12th
root of 1.9965 was aurally 12-tet and 12 tet was aurally the 7th root of
3/2 (a tuning that I believe was actually patented, IIRC, by someone
else). See "Sistema Natural de la Musica" for details (in Spanish).
IIRC, most of these were irregular temperaments with shrunk or stretched
octaves as well.

Ivor Darreg and Erv Wilson proposed a number of stretched and shrunk
octave tunings-- the 19th root of 3, the 49th root of 3, the 30th root
of 3, the 72nd root of 5, the 44th root of 5 and the 87th root of 7,
etc. These are intended to 'improve' the 3/2, 5/4 and 7/4 in 12, 19 and
31-tets.

--- In tuning-math@yahoogroups.com, John Chalmers <JHCHALMERS@U...> wrote:
> My recollection is that Prof. A.D. Fokker proposed a slightly shrunk
> 31-tet octave from similar considerations, but I'd have to look up the
> exact reference.

TOP tuning for 31 is the same in the 5, 7, and 11 limits, and has the
octave stretched. Given the flat fifths, this isn't surprising. The
stretch is 1.47 cents.

--- In tuning-math@yahoogroups.com, John Chalmers <JHCHALMERS@U...>
wrote:

> My recollection is that Prof. A.D. Fokker proposed a slightly shrunk
> 31-tet octave from similar considerations, but I'd have to look up the
> exact reference.

Unfortunately Fokker used an erroneous method for this -- he
inadvertantly weighted the wider intervals more heavily than the
narrower ones. We discussed this on the tuning list when Gregg Gibson
was around.