Re : Re: High third experiment

Jerry Eskelin:

>>BTW, You still haven't responded to my "off the wall" suggestion that the
>>shift in tuning may have to do with the combination of overtones (or
>>undertones) in the tonic and dominant fundamentals. Might that have any
>>relation to this discussion. (If that is a naive notion, just say so and
>>I'll forget it.)

Paul Erlich's reply:

> I did respond to that -- it was the first time I suggested 1/24:1/19:1/16.
> The common overtone of these notes is the dominant, four octaves higher. As
> for undertones, the "undertone series" does not occur in nature as a
> simultaneous event, the way the overtone series does.

What about combinational tones? Helmholtz explains on page 216 of the Ellis
translation that the minor triad C - Eb - G (10:12:15 in JI terms) produces
a low Ab as the lowest of the combinational tones and therefore cannot be as
consonant as the major triad C - E - G (4:5:6) which produces a low C. In
the major triad which Paul suggests, the minor third is a 19:16. If we
consider this to be C- Eb than the combinational tone will be a low G, which
is more acceptable as the Ab from Helholtz's point of view. I'm a poor
singer but I can simulate a kind of locking-in effect by tuning a string on
my guitar against a drone in order to get only one single note right. Tuning
Eb over a C-G drone ended up in a pitch slightly lower as the tempered Eb,
which may perfectly correspond to the 19:16 (297,51 cents).

Xavier Charles offered me generously his preparation to a thesis
"Contribution � l'�tude des probl�mes de la gamme et de la justesse dans la
musique occidentale, vers une autre th�orie des hauteurs?" ( Contribution
the the study of the problems of the scale and 'being in tune' in Western
music, toward another pitch theory?) In this paper he abondantly describes
the minor third 19:16 and gives a profound analysis of Mozart's Ave verum
corpus, KV618, using the 19:16 minor third. On 11 November 1999 he wrote to
us:

> I think that singers do 16-19-24 for a really minor chord if the three
> notes sound together. If the tonic isn't here, for separate work for
> example, 24/19 would be too near of 5/4 and singers would have very
> difficult to do 24/19. But I have no experience of that, it's only that
> I suppose.
>
> But with my violin I have long years of practice. When I play a minor
> third, I always make a choice between 6/5, 19/16 and 7/6 with the help
> of difference tone (for diatonic music, chromatism is another problem).
> With bB/G, I hear bE for 6/5, D for 19/16, C for 7/6. For 6/5 and 7/6
> it's quite easy, for 19/16, if the third is enough high, it's possible,
> if not, I play a "little" 6/5 which "seams" minor...

Xavier used the term difference tone for Ellis' combinational tone. IMO it
might be very usefull to investigate the phenomenon 'combinational tones' to
determine which is the interval that *locks* the best.

--Wim

I wrote,

>> I did respond to that -- it was the first time I suggested
1/24:1/19:1/16.
>> The common overtone of these notes is the dominant, four octaves higher.
As
>> for undertones, the "undertone series" does not occur in nature as a
>> simultaneous event, the way the overtone series does.

Wim Hoogewerf wrote,

>What about combinational tones? Helmholtz explains on page 216 of the Ellis
>translation that the minor triad C - Eb - G (10:12:15 in JI terms) produces
>a low Ab as the lowest of the combinational tones and therefore cannot be
as
>consonant as the major triad C - E - G (4:5:6) which produces a low C. In
>the major triad which Paul suggests, the minor third is a 19:16. If we
>consider this to be C- Eb than the combinational tone will be a low G,
which
>is more acceptable as the Ab from Helholtz's point of view. I'm a poor
>singer but I can simulate a kind of locking-in effect by tuning a string
on
>my guitar against a drone in order to get only one single note right.
Tuning
>Eb over a C-G drone ended up in a pitch slightly lower as the tempered Eb,
>which may perfectly correspond to the 19:16 (297,51 cents).

Very true, but we were discussing the major triad. As far as the major triad
is concerned, combinational tones clearly favor the 4:5:6 tuning, as you
yourself indicate above.

>IMO it
>might be very usefull to investigate the phenomenon 'combinational tones'
to
>determine which is the interval that *locks* the best.

Agreed. As I've said there are three phenomena important in "locking":
combination tones, overtone beating/roughness, and virtual pitch sensation.