36:45:60:80

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...)

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

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)?

cheers, Kelly

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

Hi Yahya

> > 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 ...

Hmmm...I'm not sure if I would include the latter two chords in the
same category as the one I cited. And 4:6:7:14 has an octave in it,
which may make it 'too easy' to fit a pattern.

> > 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, or dom 7th
chord you site, I don't really hear the inner intervals, presumably
because they fall fairly nicely into one series. But with the c-e-g#-
b, I strongly hear the g# and b...Are there any more rigourous
psychoacoustical studies of this issue?

Thanks, Kelly

> > > 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.

But there is a correlation between the 'height' of the interval in the
series (ie, eg, 5/4 x 3 = 15/12) and the difference tone, so there
could be a connection...

Hi Kelly,

--- In harmonic_entropy@yahoogroups.com, "traktus5" wrote:
>
>
> > > > 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.
>
> But there is a correlation between the 'height' of the
> interval in the series (ie, eg, 5/4 x 3 = 15/12) and
> the difference tone, so there could be a connection...

You do *very* strange arithmetic! ;-)

You often write things like:
5/4 x 3 = 15/12

This should be, instead,
5/4 x 3/3 = 15/12

since
a) you can always multiply any number by 1 without
changing it, and
b) 2/2 = 3/3 = 4/4 = 5/5 = ... = 1 = n/n for every
natural number n.

These facts mean you cna legitimately multiply BOTH
top and bottom (numerator and denominator) of any
fraction by the same number.

Enough of the maths ... what do you mean by "height"?

Regards,
Yahya

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

--- In harmonic_entropy@yahoogroups.com, "yahya_melb" <yahya@...> wrote:
>
hi Yahya - thanks for the math correction. I'm terrible at it, though
love numbers...

By 'height', I was referring to how, for example, the major third in
the chord c-e-a is 12:15, whereas it's "height" in the chord
d4-f#4-b4-e5 (spelled one way) is 36:45. So, I still consider that
the first major third has a height, in a manner of speaking of 3
(though I realize 3/3 = 1), and the height of the second third is 9,
in a manner of speaking.

You have to admit that, according to Paul's theories, the height of a
chord in a series is related to its harmonic entropy, so I'm just
trying to 'tabulate' the 'heights of the intervals. How would you do
it?

thanks, Kelly

> Hi Kelly,
>
> --- In harmonic_entropy@yahoogroups.com, "traktus5" wrote:
> >
> >
> > > > > 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.
> >
> > But there is a correlation between the 'height' of the
> > interval in the series (ie, eg, 5/4 x 3 = 15/12) and
> > the difference tone, so there could be a connection...
>
>
> You do *very* strange arithmetic! ;-)
>
> You often write things like:
> 5/4 x 3 = 15/12
>
> This should be, instead,
> 5/4 x 3/3 = 15/12
>
> since
> a) you can always multiply any number by 1 without
> changing it, and
> b) 2/2 = 3/3 = 4/4 = 5/5 = ... = 1 = n/n for every
> natural number n.
>
> These facts mean you cna legitimately multiply BOTH
> top and bottom (numerator and denominator) of any
> fraction by the same number.
>
> Enough of the maths ... what do you mean by "height"?
>
> Regards,
> Yahya
>

--- In harmonic_entropy@yahoogroups.com, "yahya_melb" <yahya@...> wrote:
>
Hi Yayha,
> 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! ;-)

Sorry. You guys really keep me on my toes.

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 ...

Personally, I find the interval of a major seventh more consonant than
an octave, sort of a 'near octave'. Perhaps tuning disprepancies are
more of an issue with octaves.

>
> > 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?

Oh, I find your comments very rigorous! I just thought there might be
peer-reviewed research, backed up by laboratory studies, etc (like by
acousticians such as Helmholtz, Parncut, etc), which bear on this
question.

> More rigorous than what? My impressions? Your
> impressions? Your analysis?
>
> Regards,
> Yahya
>

Kelly,

--- In harmonic_entropy@yahoogroups.com, "traktus5" wrote:
[snip]
> By 'height', I was referring to how, for example, the major third
> in the chord c-e-a is 12:15, whereas it's "height" in the chord
> d4-f#4-b4-e5 (spelled one way) is 36:45. So, I still consider that
> the first major third has a height, in a manner of speaking of 3
> (though I realize 3/3 = 1), and the height of the second third is
> 9, in a manner of speaking.

What you call "height" would seem to be what
mathematicians call "Greatest Common Divisor"
or "GCD".

> You have to admit that, according to Paul's theories, the height
> of a chord in a series is related to its harmonic entropy, ...

??? I don't think I can admit what I don't understand ...!

> so I'm just trying to 'tabulate' the 'heights of the intervals.
> How would you do it?
>
> thanks, Kelly
>
> > Hi Kelly,
> >
> > --- In harmonic_entropy@yahoogroups.com, "traktus5" wrote:
> > >
> > >
> > > > > > 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.
> > >
> > > But there is a correlation between the 'height' of the
> > > interval in the series (ie, eg, 5/4 x 3 = 15/12) and
> > > the difference tone, so there could be a connection...
> >
> >
> > You do *very* strange arithmetic! ;-)
> >
> > You often write things like:
> > 5/4 x 3 = 15/12
> >
> > This should be, instead,
> > 5/4 x 3/3 = 15/12
> >
> > since
> > a) you can always multiply any number by 1 without
> > changing it, and
> > b) 2/2 = 3/3 = 4/4 = 5/5 = ... = 1 = n/n for every
> > natural number n.
> >
> > These facts mean you cna legitimately multiply BOTH
> > top and bottom (numerator and denominator) of any
> > fraction by the same number.
> >
> > Enough of the maths ... what do you mean by "height"?

Regards,
Yahya

Kelly,

--- In harmonic_entropy@yahoogroups.com, "traktus5" wrote:
[snip]
> > > ...Are there any more rigourous psychoacoustical
> > > studies of this issue?
>
> Oh, I find your comments very rigorous! I just thought
> there might be peer-reviewed research, backed up by
> laboratory studies, etc (like by acousticians such as
> Helmholtz, Parncut, etc), which bear on this question.

There might be; I don't know.

> > More rigorous than what? My impressions? Your
> > impressions? Your analysis?
> >
Regards,
Yahya