Hi Kelly,

--- In harmonic_entropy@yahoogroups.com, Kelly wrote:
>
> hi Paul (for when you come back), or others. I can't
> locate the old thread, but thanks for suggesting the
> above spelling for the chord d4-f#-b4-E5. I don't know
> if it's significant, but the intervals of this chord
> have a neat feature not shared by any other high-in-the
> series 4 note chords which I have been able to find yet
> (though math adept people could probably 'back engineer'
> one from the number pattern...)

Exactly! ;-)


> To wit, the intervals, being 5/4, 4/3, 4/3, 5/3, 16/9,
> and 20/9, have the interesting feature that the 'bottom'
> interval, 5/4, is multiplied by 9 to 'be in the series'
> at 45/36; and the 'top' interval b4-e5 (4/3) is
> multiplied by 20 to 'be in the series' at 60:80. And...
> (drum roll...) the outer interval has the ratio 20/9.
> (So I'm referring to the appearence of 20 and 9 in both
> locations.)

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

So any tetrad am : bm : cn : an has the same
interesting property you referered 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 ...

> 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.

Regards,
Yahya