Major Thirds

The recent discussions about 5/4, 81/64 and 9/7 got me to wondering about
the limits of what can be heard as a major third. Indeed, the question is
not whether an interval >is< a major third, but rather under what
circumstances can it be heard as a major third? As far as I can tell, there
are three major factors -context, context and context!

If there is any quality of sound associated with "major thirdness" i
suppose it would be most apparent in 5/4. But this quality can also clearly
be evoked by 81/64 as well as 400c, especially if all you have to compare
it to is 6/5 and 4/3. But an examination of other intervals between 6/5 and
4/3 reveals that the question is more complex due to functional issues.

MTness can be heard in a melodic motion like 1/1, 9/8, 5/4, whereas minor
thirdness can be heard in 1/1, 9/8, 6/5. Now an interval like 49/40
(~351c) is exactly inbetween 6/5 (~316c) and 5/4 (~386c). Does 1/1, 9/8,
49/40 "sound" major or minor? If you listen to it by itself, it can sound
like either, or sound like another quality, neither major nor minor. If
you listen to it in the following sequence:

1/1, 9/8, 6/5, 9/8, 1/1, 9/8, 49/40, 9/8, 1/1, 9/8, 5/4, 9/8, 1/1, 9/8,
49/40, 9/8, 1/1....

the first time you hear it, it is right after hearing 6/5, and in
comparison to 6/5 it has a major quality. The next time you hear it, it is
after hearing 5/4, but it does not (to my ears) sound minor in comparison,
it has a unique sound, another quality of thirdness that is neither major
nor minor.

Furthermore, it seems that the choice of the second scale degree has as
leat as much influence on its soundin this context as the interval itself.
If you listen to 1/1, 10/9, 49/40, it sounds more major because of the
wider interval between the 2nd and 3rd scale degrees. But in the context of
1/1, 8/7, 49/40, it sounds decidedly minor due to the smaller step from 2nd
to 3rd.

To me, 49/40 is very slippery when trying to hear it within a diatonic
framework. Of course it has its own sound when divorced from a traditional
tonal context.

What about in a harmonic setting, that is, heard with 1/1 and 3/2? As far
as I can tell there is a continuum from 6/5 to 5/4 without there being any
point where it stops sounding minor and starts sounding major. This is
reflected in the fact that, after all, they are all >thirds<, functionally.

The principal low integer ratios in this region (besides 6/5 and 5/4) are
11/9 and 16/13. Since they are 12c apart, they are not hugely different,
and 49/40 could stand for either of them. As a group, they do sound more
like another third quality rather than either major or minor. With more
careful listening, though, 16/13 begins to have a discernable quality
distinct from 11/9. I think its also a matter of focus: if your ears are
used to resolving nothing finer than 6/5 vs. 5/4, than other intervals
sound like mistuned versions of these intervals. I keep getting an image of
a "forest and trees" analogy. all the intervals between 6/5 and 5/4 are
only individual "trees" when you can focus enough to "see" them as
individuals.

The situation is somewhat different on the other end, that is, looking at
the interval space bounded by 5/4 and 4/3. To my ears there is more of a
switchover effect where "thirdness" gives way to ":fourthness" I would put
this at around 420c, that is somewhere between 14/11 and 32/25. However,
sharper intervals can be >heard< as thirds, especially when heard after
4/3, like 9/7. In comparison to 5/4 (or even 81/64), 9/7 sounds like a
leading tone to 4/3, but hearing 9/7 after 4/3 it sounds like a resolution
of sus 4 to maj 3. 9/7 is not terribly far from being the midpoint of 5/4
and 4/3, hence the ambiguity, as with 49/40. Also, the melodic sequence
1/1, 8/7, 9/7, evokes a major third feeling by lessoning the impression of
the wideness of 9/7. So once again context is the most powerful determinant
to how something sounds.

Our ears are not tabulae rasae (latin sp?), for most of us diatonic
functional harmony is deeply engrained. It takes a major reeducation of our
ears to be able to hear some of these intervals as themselves, not as
markers for the more familiar ratios.

dante

Mike Battaglia wrote:

Para: tuning@yahoogroups.com
De: "Mike Battaglia" <battaglia01@...>
Remitente: tuning@yahoogroups.com
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Fecha: 06 Aug 2008, 07:15:47 PM
Asunto: Re: [tuning] Re: Why are scales with a minimal number of different intervals more

--------------------------------------------------------------------------------

Interesting. I do like 7 ET myself, although I don't know if I'd
gravitate towards that for music that doesn't modulate.

One thing I notice is if a singer jumps from C-E, she'll probably sing
that pretty close to a 5/4 major third. But, if she goes from C-D and
then from D-E, she might end up closer to 81/64 than 5/4. I think it's
just because singers usually have in mind a list of different
intervals - a major third, a whole step, etc... And the one they'll
pick out for the whole step is usually around 9/8, and the one they
pick out for the major third is usually 5/4. If they go from C-D and
then from D-E, they move up two whole steps. I don't often hear 9/8
tempered flat to accomodate 5/4 or 5/4 tempered sharp to accomodate
9/8, except the latter in certain cases, such as when two whole steps
are sung in a row (which isn't really a tempering at all).
--------------------------------The above message is related to the study I did recently on an imperfect scale with a major third of 1,25707872211 = K(2) P(2), where (2) are exponents of the Piagui semitone factors K and P.K=(9/8) with exponent (1/2) equals to 1,06066017178 and P =(8/9) [(2) with exponent (1/4)] gives 1,05707299111. This imperfect scale shows interesting features.The experimental major third of 1,25707872211 is the relative frequency of note E that works in Piagui III scale. The Piagui semitone factors K and P produced the Piagui variants I, II and III. I reach to the conclusion that it is not possible to get a scale better than any of the three Piagui scales, whatever the method employed. The + and minus 2 cents of Piagui II deviations from the equal tempered tones make a small discrepance between the Piagui II and the equal tempered scale. Since the imperfections of the equal tempered are small though detectable, the tone frequency differences have also to be small like the mentioned 2 cents.Lima, August 07, 2008MARIO PIZARRO< piagui@... >