Hi again Kelly,

--- In harmonic_entropy@yahoogroups.com, "traktus5" wrote:
[snip]
> > The tetrad am : bm : cn : dn has the property
> > that dn/am = n/m if and only if a = d. (Here
> > juxtaposition means multiplication: am = a x m
> > etc.) m and n play the roles of your 9 and 20.
> > That means a and b correspond to 4 and 5, and
> > that c and d correspond to 3 and 4:
> > 36 : 45 : 60 : 80
> > = 4 x 9 : 5 x 9 : 3 x 20 : 4 x 20
> >
> > 45/36 = 5 x 9 / 4 x 9 = 5/4
> >
> > 80/60 = 4 x 20 / 3 x 20 = 4/3
> >
> > 80/36 = 4 x 20 / 4 x 9 = 20/9

Errata: in the above, I should have written:
45/36 = (5 x 9) / (4 x 9) = 5/4
80/60 = (4 x 20) / (3 x 20) = 4/3
80/36 = (4 x 20) / (4 x 9) = 20/9

according to the usual expression-writing rules.


> > So any tetrad am : bm : cn : an has the same
> > interesting property you [referred] to
> > Eg 2x2 : 2x3 : 1x7 : 2x7
> > = 4 : 6 : 7 : 14 is an example with a low limit.
> > Another example, engineered from yours, is
> > 4 x 7 : 5 x 7 : 3 x 17 : 4 x 17
> > = 28 : 35 : 51 : 68.
> > Or again,
> > 3 x 7 : 5 x 7 : 2 x 18 : 3 x 18
> > = 21 : 35 : 36 : 54.
> > This latter example has a highly dissonant 35:36,
> > so ...
>
> Hmmm...I'm not sure if I would include the latter two
> chords in the same category as the one I cited...

This was *exactly* my point in the comment marked ***
below!


> ... And 4:6:7:14 has an octave in it, which may make
> it 'too easy' to fit a pattern.

You make it a bit hard to follow your thinking when you
keep changing the rules! ;-) I just chose very small
values for c and a, namely 1 and 2. If you don't want
octaves, you can of course bar any of a:b, b:c and c:a
from having that ratio.


> > > Do you think this could have any acoustical significance?
> > > It's a very nice sounding chord!
>

***
> > ... no, I don't think it guarantees an overall
> > "nice" sound.
***

>
> > > Also, speaking of 'bottom' and 'top' intervals: in a four
> > > note chord, do intervals formed by adjacent notes (eg,
> > > 5/4 and 4/3, from our chord) have any more prominance to
> > > our hearing system than do intervals formed by non-
> > > adjacent notes (eg, the 5/3 and 16/9 in the above chord)?
> >
*****
> > It's my impression that middle voices and
> > intervals are usually the hardest to hear.
> > After the melody, most people pick up the
> > bass.
*****
>
> > Try this experiment: play a Cma7 chord:
> > C E G B
> > then alter it to Cmima7:
> > C Eb G B
> > and C #5 ma7:
> > C E G# B.
> > They all have a family resemblance, don't
> > they?
> >
> > Now play Cdom7:
> > C E G Bb
> > and Cm7:
> > C Eb G Bb.
> > It's a different family, right?
> >
> > Wait! I can already hear the objection! ;-)
> > "The two minor chords make one family,
> > and the rest make another." Well, yes.
> > The third above the root is usually very
> > salient in determining mood and mode. But
> > apart from the third (of whatever size, but
> > clearly more than a second and less than a
> > fourth) above the root, if present, the most
> > salient interval seems to me to be usually
> > the outside one.
>
> For me, with 'consonant' chords like the maj 7th chord,

I find the major 7th chord very pleasantly dissonant.
YMMV ...


> or dom 7th chord you site, I don't really hear the
> inner intervals, ...

This was the point I made at the note marked *****
above.


> ... presumably because they fall fairly nicely into
> one series.

Seems like a fair analysis.


> ... But with the c-e-g#-b, I strongly hear the g#
> and b...

... presumably because they *don't* fall fairly nicely
into one series?


> ...Are there any more rigourous psychoacoustical
> studies of this issue?

More rigorous than what? My impressions? Your
impressions? Your analysis?

Regards,
Yahya