Re: 2/3-tones?

> Message: 4
> Date: Sat, 11 Sep 1999 20:09:48 EDT
> From: Afmmjr@aol.com
> Subject: Re: Fw: 3d. tone and golden ratios
>
> Thirdtones are very large leading tones. Meantone leading tones are larger,
> but at 66.6 cents, thirdtones are larger still.

Hello, there, and I suspect that you may actually be discussing intervals
of 2/3-tone, as occur in the Renaissance temperament of 1/3-comma meantone
with pure minor thirds (Salinas), or in the almost exactly equivalent of
19-tone equal temperament (19-tet) as discussed in the same era by
Costeley.

Under such a system, the size of the whole-tone itself is somewhat
_smaller_ than the 200 cents of 12-tet; and in meantone, the diatonic
semitone (e.g. E-F or B-C) is equal in these tunings to a 2/3-tone, or two
19-tet steps. The chromatic semitone, e.g. F-F# or Bb-B, is equal to the
very narrow 1/3-tone of about 63.15 cents, as I recall, for 19-tet, close
to 28:27 (1/3-comma meantone would be very close to this).

Thus the diatonic semitone of 2/3-tone in 19-tet is equal to about 126.3
cents, indeed larger than in 1/4-comma meantone, where it is about 117.1
cents, or tertian just intonation, where it is 16:15 (a bit less than 112
cents).

For _narrow_ semitones smaller than 12-tet (100 cents) one might go to
Pythagorean, which offers a nice compact diatonic semitone of 256:243
(around 90.22 cents). This is probably the narrowest diatonic semitone
used in an historical Western European tuning, and it nicely fits
medieval chant and polyphony.

However, modern tunings such as 17-tet do offer regular diatonic semitones
of around 1/3-tone (in 17-tet, 1/17 octave or about 70.59 cents). Also,
while this would be less likely in Renaissance practice, modern practice
for 19-tet may sometimes use that narrow one-step leading-tone (in earlier
thaory, the chromatic semitone) at cadences.

Incidentally, thirdtones are mentioned around 1375 in one of the treatises
of the Berkeley Manuscript or Paris Anonymous. In describing vocal
intonation, the author says that a tone is divided into three parts, and
that one should normally sing two of these parts when intoning a usual
mi-fa interval (e.g. E-F), but should sing a narrow 1/3-tone for cadential
semitones. Oliver Ellsworth has interpreted this as a 14th-century
proposal for 19-tet, although such a reading of a treatise dealing only
finding semitones in singing, not with polyphony or keyboard instruments,
might be debatable.

Most respectfully,

Margo Schulter
mschulter@value.net