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 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

Carl Lumma wrote:
>>Dave (if you are out there), or
>>Paul (if you might know):
>>
>>Why the second derivative of the "Harmonic Entropy" function?
>>What special significance is associated with differentiating?
>>
>>Curiously,  J Gill
>
>
> Was there ever a reply to this?  The post about
> justness just seems to come out of nowhere in the
> archives.

Okay, well, a just intonation point is at the bottom of a basin of a
dissonance graph.  But so is the point where one basin meets another.
So a zero first derivative tells you the either you've got a perfectly
just tuning, or a perfectly ambiguous one.

The second derivative tells you if it's a maximum or a minimum.  So a JI
point is minimum dissonance, and so should have a positive second
derivative, right?

The sharper the point is, the faster the gradient has to change.  Hence
the second derivative will tell you where the sharpest points are.


                Graham

> Dave (if you are out there), or
> Paul (if you might know):
>
> Why the second derivative of the "Harmonic Entropy" function?
> What special significance is associated with differentiating?
>
> Curiously,  J Gill

Was there ever a reply to this?  The post about
justness just seems to come out of nowhere in the
archives.

-Carl

Paul, it seems you never replied to this?  I'd love to read
any feedback you might have on it.

-Carl

--- In harmonic_entropy@yahoogroups.com, John Chalmers
<JHCHALMERS@...> wrote:
> I've been experimenting with the Tonicity function (in an old
> paper I could lay my hands on easily last weekend) and have
> some graphics, one of which I will try to upload to the file
> area.
>
> The parameters are these: S=.01, T=1200 (tet), N*D <=10000 and
> N/D reduced to one octave, the colors are arbitrary and the
> scale factors were chosen to fit on my rather small screen.
> The equation is
> (1/(sqr(2*pi)*sqr(N*D)*S))*exp(-(((log(N/D)-NT)^2)/2*S^2 ))
> where N=Num,D=Denom and NT is log of the nearest tempered note
> to N/D.  Note the peak at 3/2; the bottom line is cents marked
> every semitone.

hello harmonic entropy list.  Reading Helmholtz, and reviewing where
the semitone clashes occur in the partials of various intervals, does
anyone know if the dissonance is affected by whether the beating
semitone is diatonic (example, e-f, as the 5th and 4th partials of a
perfect 4th) or chromatic (eg, B flat - B natural, as the 7th and 5th
partial of the perfect fifth.)  Or maybe its' a moot point,
overshadowed, instead, by which semitone clash is composed of lower
pitch numbers?  thanks, Kelly

>>> It's still not clear to me, from what you and Paul have said on
>>> this subject, what intervals one actually hears with, for example,
>>> "12:15:20".
>>
>> I guess my answer is that not all chords have a singular
>> representation.  In fact, that's what harmonic entropy
>> measures in a way -- how much better any one JI representation
>> is than all the others.  It isn't clear to me exactly what
>> the entropy of 12:15:20 is... I guess I'm still waiting
>> for Paul to endorse some triadic approaches to h.e.
>
> hi Carl.  Are there any notable contender triadic approaches to
> h.e., or a post which discusses? thanks, Kelly

There are a lot of posts about it in the archives.  Exactly how
to find them is another question.

As I recall, one problem was how to partition the 2-D plot of all
triads according to JI.  The 1-D plot of JI dyads was partitioned
with the Farey series and its mediants in the original dyadic h.e.
implementation.  Paul tried voronoi cells on the 2-D plot, and you
can see some those images in the files section here, I think.  I
don't remember if a newer idea supercedes this.  IIRC, the Farey
series has been replaced with a 'product limit' in the dyadic case,
and I think the idea was to extend this for chords, like,
a*b*c for triad a:b:c.  Apparently, Paul knows what he wants to do,
and is just waiting to get around do setting up his computer for
some number-crunching.

-Carl

hi Carl.  Are there any notable contender triadic approaches to
h.e., or a post which discusses? thanks, Kelly

Carl wrote:
> I guess my answer is that not all chords have a singular
> representation.  In fact, that's what harmonic entropy
> measures in a way -- how much better any one JI representation
> is than all the others.  It isn't clear to me exactly what
> the entropy of 12:15:20 is... I guess I'm still waiting
> for Paul to endorse some triadic approaches to h.e.
>
> -Carl



> >It's still not clear to me, from what you and Paul have said on
> >this subject, what intervals one actually hears with, for example,
> >"12:15:20".  According to your mathematical argument from the
> >ealier post, we do not hear the individual reduced intervals 5/4
> >and 4/3, because 5 and 3 can not exist on the same note. On the
> >other hand, psychoacoustical properites of the chord, such as
it's
> >weak tonalness, and lack of octave reinforcement of difference
> >tones, suggest that 12:15:20 does not represent the chord
either.
> >(Remember from Paul's article on tonalness, how the signal of
some
> >high chords are overpowered by the greater tonalness of their
> >constituent intervals?)
> >
> >It seems that the two descriptions of the chord (12:15:20, and
> >1/5:4:3) are somewhat just constructs, in one case to calculate
> >difference tones, and in the other (1/5:4:3) to describe the
> >partials.  Neither seem to really try to get at what's going in
the
> >chord, in my opinion!
>
>

hi Yahya - I've had more time to study your response...

> > ....  According to your mathematical argument from
> > the ealier post, we do not hear the individual reduced intervals
5/4 > > and 4/3, because 5 and 3 can not exist on the same note.


> Not at all!  We *do* hear the relations 5:4
> between the notes of the lower dyad and
> 4:3 between the notes of the upper dyad
> (as well as the 5:3 relation between the
> outer two notes).
> If what you were saying is that the middle
> note functions *both* as a 3 in one dyad
> and as a 5 in another, I most emphatically
> agree.  But what it actually IS is neither
> a 3 nor a 5, but (an approximation to an
> ideal) single (central) frequency, perhaps
> modulated by a tremolo.  That frequency
> will not be a 3 nor a 5, but perhaps 275 Hz.

275 Hz...are you referring to the e in c4-e4-a4?  I'd be most
curious to know.

"Tremelo"...is there any elaboration of this concept?

Sorry if I sounded exaperated in my first reply ("why didn't you say
so earlier..."), but my initial post was asking whether some sort of
mixing or blending occurs on that shared note, which was roundly
dismissed, and you now seem to be suggesting similarly, by saying
there is an "approximation".

cheers, Kelly

hi Carl-

> >It seems that the two descriptions of the chord (12:15:20, and
> >1/5:4:3) are somewhat just constructs, in one case to calculate
> >difference tones, and in the other (1/5:4:3) to describe the
> >partials.  Neither seem to really try to get at what's going in
the
> >chord, in my opinion!


> I guess my answer is that not all chords have a singular
> representation.  In fact, that's what harmonic entropy
> measures in a way -- how much better any one JI representation
> is than all the others.  It isn't clear to me exactly what
> the entropy of 12:15:20 is... I guess I'm still waiting
> for Paul to endorse some triadic approaches to h.e.


Me too!

-Kelly

hi Yayha-  (I"m out of town.  Just a brief response.)


> > ....  According to your mathematical argument from
> > the ealier post, we do not hear the individual reduced intervals
5/4 > > and 4/3, because 5 and 3 can not exist on the same note.
>
> Not at all!  We *do* hear the relations 5:4
> between the notes of the lower dyad and
> 4:3 between the notes of the upper dyad
> (as well as the 5:3 relation between the
> outer two notes).
>
> If what you were saying is that the middle
> note functions *both* as a 3 in one dyad
> and as a 5 in another, I most emphatically
> agree.  But what it actually IS is neither
> a 3 nor a 5, but (an approximation to an
> ideal) single (central) frequency, perhaps
> modulated by a tremolo.  That frequency
> will not be a 3 nor a 5, but perhaps 275 Hz.

Aha!  That's what I was driving at in my original post!  Why didn't
you mention it back then!?!?!

>
> > It seems that the two descriptions of the chord (12:15:20, and
> > 1/5:4:3) are somewhat just constructs, in one case to calculate
> > difference tones, and in the other (1/5:4:3) to describe the
> > partials.
>
> You're right, they are constructs - but they
> are constructs more of our actual perceptions
> than of technical analysis.

Can you elaborate on this distinction?


> > ... Neither seem to really try to get at what's going in the
> > chord, in my opinion!
>
> What's really going on is very complex!  It
> includes the facts that the fundamental tone
> has a certain perceived pitch; that the other
> chord tones are perceived to be in harmonious
> relationships to it; and that a whole pile of
> other things are happening musically that
> a few whole numbers don't really begin to
> explain.

Right!  There's a lot going on!   thanks for your responses.

Sincerely, KElly

>It's still not clear to me, from what you and Paul have said on
>this subject, what intervals one actually hears with, for example,
>"12:15:20".  According to your mathematical argument from the
>ealier post, we do not hear the individual reduced intervals 5/4
>and 4/3, because 5 and 3 can not exist on the same note. On the
>other hand, psychoacoustical properites of the chord, such as it's
>weak tonalness, and lack of octave reinforcement of difference
>tones, suggest that 12:15:20 does not represent the chord either.
>(Remember from Paul's article on tonalness, how the signal of some
>high chords are overpowered by the greater tonalness of their
>constituent intervals?)
>
>It seems that the two descriptions of the chord (12:15:20, and
>1/5:4:3) are somewhat just constructs, in one case to calculate
>difference tones, and in the other (1/5:4:3) to describe the
>partials.  Neither seem to really try to get at what's going in the
>chord, in my opinion!

Hi Kelly,

I guess my answer is that not all chords have a singular
representation.  In fact, that's what harmonic entropy
measures in a way -- how much better any one JI representation
is than all the others.  It isn't clear to me exactly what
the entropy of 12:15:20 is... I guess I'm still waiting
for Paul to endorse some triadic approaches to h.e.

-Carl

On Fri, 17 Mar 2006 "traktus5" wrote:
>
> hi Yahya.
>
> It's still not clear to me, from what you and Paul have said on this
> subject, what intervals one actually hears with, for
> example, "12:15:20".  According to your mathematical argument from
> the ealier post, we do not hear the individual reduced intervals 5/4
> and 4/3, because 5 and 3 can not exist on the same note. On the
> other hand, psychoacoustical properites of the chord, such as it's
> weak tonalness, and lack of octave reinforcement of difference
> tones, suggest that 12:15:20 does not represent the chord either.
> (Remember from Paul's article on tonalness, how the signal of some
> high chords are overpowered by the greater tonalness of their
> constituent intervals?)


Hi Kelly,

If you thought that's what we meant, I think
you may have misunderstood us. ;-)

> ....  According to your mathematical argument from
> the ealier post, we do not hear the individual reduced intervals 5/4
> and 4/3, because 5 and 3 can not exist on the same note.

Not at all!  We *do* hear the relations 5:4
between the notes of the lower dyad and
4:3 between the notes of the upper dyad
(as well as the 5:3 relation between the
outer two notes).

If what you were saying is that the middle
note functions *both* as a 3 in one dyad
and as a 5 in another, I most emphatically
agree.  But what it actually IS is neither
a 3 nor a 5, but (an approximation to an
ideal) single (central) frequency, perhaps
modulated by a tremolo.  That frequency
will not be a 3 nor a 5, but perhaps 275 Hz.


> It seems that the two descriptions of the chord (12:15:20, and
> 1/5:4:3) are somewhat just constructs, in one case to calculate
> difference tones, and in the other (1/5:4:3) to describe the
> partials.

You're right, they are constructs - but they
are constructs more of our actual perceptions
than of technical analysis.  In this sense they
are as real as can be.


> ... Neither seem to really try to get at what's going in the
> chord, in my opinion!

What's really going on is very complex!  It
includes the facts that the fundamental tone
has a certain perceived pitch; that the other
chord tones are perceived to be in harmonious
relationships to it; and that a whole pile of
other things are happening musically that
a few whole numbers don't really begin to
explain.


Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.385 / Virus Database: 268.2.5/284 - Release Date: 17/3/06

hi Yahya.

It's still not clear to me, from what you and Paul have said on this
subject, what intervals one actually hears with, for
example, "12:15:20".  According to your mathematical argument from
the ealier post, we do not hear the individual reduced intervals 5/4
and 4/3, because 5 and 3 can not exist on the same note. On the
other hand, psychoacoustical properites of the chord, such as it's
weak tonalness, and lack of octave reinforcement of difference
tones, suggest that 12:15:20 does not represent the chord either.
(Remember from Paul's article on tonalness, how the signal of some
high chords are overpowered by the greater tonalness of their
constituent intervals?)

It seems that the two descriptions of the chord (12:15:20, and
1/5:4:3) are somewhat just constructs, in one case to calculate
difference tones, and in the other (1/5:4:3) to describe the
partials.  Neither seem to really try to get at what's going in the
chord, in my opinion!


sincerely, Kelly


--- In harmonic_entropy@yahoogroups.com, "Yahya Abdal-Aziz"
<yahya@...> wrote:
>
>
> Hi Kelly,
>
> On Thu, 16 Mar 2006, "traktus5" wrote:
> >
> > hi Yahya
> >
> > > > Hello.  Referring back to Yahya's and Paul's comments (mssgs
840
> > and > > 843) about the mathematical impossibitity of a note in a
> > chord > > representing two numbers (eg, a note which is the
upper
> > note of one > > interval and the lower note of another
interval),
> > what about chords > > which do not have a clear indication of a
> > fundamental?  For example, a > > minor triad, with its multiple
root
> > allusions, or the case which Paul > > talks about with Monz at
> > tonalsoft, where the tonalness of an > > individual dyad in a
chord
> > is stronger than the root-allusion power of > > the entire
chord?
> > In each instance, it seems that the individual > > intervals
within
> > the chord acts 'independently' of of the entire > > chord.  It
would
> > seem, in that case, then, that, in a sense, you would > > have
two
> > numbers 'co-existing'in some manner on one note.  >
> >
> > > Let's see if I can understand what you're driving at!
> > > Let's take a minor triad 10:12:15, with internal ratios
> > > 6/5, 5/4, 3/2.  Each of the first difference tones 2,
> > > 3 and 5 functions somewhat as a fundamental for one
> > > of the three dyads in the chord, but not for the whole
> > > triad.  Are we together on this?
> > >
> > > If so, we can say that the 12 is the 6th harmonic of
> > > the 2 (since 12/2 = 6) and is also the 4th harmonic of
> > > the 3 (since 12/3 = 4).
> >
> > So 12 and 2 have a harmonic relationship...but are you actually
> > saying the the difference tones themselves, in the semi-role of
> > fundamental, have their own set of partials?
>
> No, I'm saying that among the various tones you
> hear in the mix are some which have the same
> relations _as if_ they were fundamental and
> overtone.  Therefore, those higher tones which
> are multiples of the lower tend to reinforce the
> impression that the lower ARE fundamentals in
> actual fact.
>
>
> > > Is this what you mean by "a note in a chord
> > > representing two numbers"?
> >
> > No.  Actually -- I hope you don't find this too vexing -- I'm
> > resurrecting my old idea, from those messages I cited, where a
note,
> > such as the b3 in g3-b3-e4,  is represented by both 3 from the
4/3
> > of b3-e4, and 5 from the 5/4 of g-b. ...
>
> Not vexing, but I don't know what I could usefully
> add to my earlier reply.  So maybe you won't get
> much more mileage out of asking me again.
>
>
> > ...  If, in a minor triad, the
> > intervals are somewhat detached from the chord's tendancy to
suggest
> > a root, then couln'd you actually have a 6/5 on the bottom and a
5/4
> > on the top, with an 'overlapping' effect on the shared note?  If
> > partials can fuse into pitch, and intervals into the tonalness
> > effect, why can't intervals 'blend' visa vie some mechanism
that,
> > perhaps, operates in the absense of tonalness?  (I'm aware of
your
> > original mathematical objection to my idea, but am not convinced
by
> > it.)    Am I making any sense?
>
> Well, Kelly, I can't say your idea is wrong, but
> I don't find any evidence to support it.
>
> Remember, too, that partials fusing into the
> sensation of a fundamental pitch depends
> entirely on their numerical relationships to
> each other.  For example, the 12 and 2 of the
> earlier example enjoy this simple 6:1 ratio.
> Whereas the 6:5 and 5:4 ratios with a common
> tone on the 6 of the first and the 4 of the
> second have the fractional relationship 3:2.
> If we double both terms of the first ratio and
> treble those of the second, we have 12:10 and
> 15:12, so we see we must realise the chord as
> 15:12:10.  In this structure, the common element
> 12 represents the middle note by a unique number.
> And that is what makes sense to me.
>
> Someone else may be able to see what you're
> driving at.  If you find such a person, please ask
> him or her to explain it to me.
>
> Alternatively, perhaps you could find a way of
> expressing your meaning directly in music?  That
> would be nice.
>
> Regards,
> Yahya
>
> --
> No virus found in this outgoing message.
> Checked by AVG Free Edition.
> Version: 7.1.385 / Virus Database: 268.2.4/282 - Release Date:
15/3/06
>

Hi Kelly,

On Thu, 16 Mar 2006, "traktus5" wrote:
>
> hi Yahya
>
> > > Hello.  Referring back to Yahya's and Paul's comments (mssgs 840
> and > > 843) about the mathematical impossibitity of a note in a
> chord > > representing two numbers (eg, a note which is the upper
> note of one > > interval and the lower note of another interval),
> what about chords > > which do not have a clear indication of a
> fundamental?  For example, a > > minor triad, with its multiple root
> allusions, or the case which Paul > > talks about with Monz at
> tonalsoft, where the tonalness of an > > individual dyad in a chord
> is stronger than the root-allusion power of > > the entire chord?
> In each instance, it seems that the individual > > intervals within
> the chord acts 'independently' of of the entire > > chord.  It would
> seem, in that case, then, that, in a sense, you would > > have two
> numbers 'co-existing'in some manner on one note.  >
>
> > Let's see if I can understand what you're driving at!
> > Let's take a minor triad 10:12:15, with internal ratios
> > 6/5, 5/4, 3/2.  Each of the first difference tones 2,
> > 3 and 5 functions somewhat as a fundamental for one
> > of the three dyads in the chord, but not for the whole
> > triad.  Are we together on this?
> >
> > If so, we can say that the 12 is the 6th harmonic of
> > the 2 (since 12/2 = 6) and is also the 4th harmonic of
> > the 3 (since 12/3 = 4).
>
> So 12 and 2 have a harmonic relationship...but are you actually
> saying the the difference tones themselves, in the semi-role of
> fundamental, have their own set of partials?

No, I'm saying that among the various tones you
hear in the mix are some which have the same
relations _as if_ they were fundamental and
overtone.  Therefore, those higher tones which
are multiples of the lower tend to reinforce the
impression that the lower ARE fundamentals in
actual fact.


> > Is this what you mean by "a note in a chord
> > representing two numbers"?
>
> No.  Actually -- I hope you don't find this too vexing -- I'm
> resurrecting my old idea, from those messages I cited, where a note,
> such as the b3 in g3-b3-e4,  is represented by both 3 from the 4/3
> of b3-e4, and 5 from the 5/4 of g-b. ...

Not vexing, but I don't know what I could usefully
add to my earlier reply.  So maybe you won't get
much more mileage out of asking me again.


> ...  If, in a minor triad, the
> intervals are somewhat detached from the chord's tendancy to suggest
> a root, then couln'd you actually have a 6/5 on the bottom and a 5/4
> on the top, with an 'overlapping' effect on the shared note?  If
> partials can fuse into pitch, and intervals into the tonalness
> effect, why can't intervals 'blend' visa vie some mechanism that,
> perhaps, operates in the absense of tonalness?  (I'm aware of your
> original mathematical objection to my idea, but am not convinced by
> it.)    Am I making any sense?

Well, Kelly, I can't say your idea is wrong, but
I don't find any evidence to support it.

Remember, too, that partials fusing into the
sensation of a fundamental pitch depends
entirely on their numerical relationships to
each other.  For example, the 12 and 2 of the
earlier example enjoy this simple 6:1 ratio.
Whereas the 6:5 and 5:4 ratios with a common
tone on the 6 of the first and the 4 of the
second have the fractional relationship 3:2.
If we double both terms of the first ratio and
treble those of the second, we have 12:10 and
15:12, so we see we must realise the chord as
15:12:10.  In this structure, the common element
12 represents the middle note by a unique number.
And that is what makes sense to me.

Someone else may be able to see what you're
driving at.  If you find such a person, please ask
him or her to explain it to me.

Alternatively, perhaps you could find a way of
expressing your meaning directly in music?  That
would be nice.

Regards,
Yahya

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hi Yahya

> > Hello.  Referring back to Yahya's and Paul's comments (mssgs 840
and > > 843) about the mathematical impossibitity of a note in a
chord > > representing two numbers (eg, a note which is the upper
note of one > > interval and the lower note of another interval),
what about chords > > which do not have a clear indication of a
fundamental?  For example, a > > minor triad, with its multiple root
allusions, or the case which Paul > > talks about with Monz at
tonalsoft, where the tonalness of an > > individual dyad in a chord
is stronger than the root-allusion power of > > the entire chord?
In each instance, it seems that the individual > > intervals within
the chord acts 'independently' of of the entire > > chord.  It would
seem, in that case, then, that, in a sense, you would > > have two
numbers 'co-existing'in some manner on one note.  >

> Let's see if I can understand what you're driving at!
> Let's take a minor triad 10:12:15, with internal ratios
> 6/5, 5/4, 3/2.  Each of the first difference tones 2,
> 3 and 5 functions somewhat as a fundamental for one
> of the three dyads in the chord, but not for the whole
> triad.  Are we together on this?
>
> If so, we can say that the 12 is the 6th harmonic of
> the 2 (since 12/2 = 6) and is also the 4th harmonic of
> the 3 (since 12/3 = 4).

So 12 and 2 have a harmonic relationship...but are you actually
saying the the difference tones themselves, in the semi-role of
fundamental, have their own set of partials?

> Is this what you mean by "a note in a chord
> representing two numbers"?

No.  Actually -- I hope you don't find this too vexing -- I'm
resurrecting my old idea, from those messages I cited, where a note,
such as the b3 in g3-b3-e4,  is represented by both 3 from the 4/3
of b3-e4, and 5 from the 5/4 of g-b.  If, in a minor triad, the
intervals are somewhat detached from the chord's tendancy to suggest
a root, then couln'd you actually have a 6/5 on the bottom and a 5/4
on the top, with an 'overlapping' effect on the shared note?  If
partials can fuse into pitch, and intervals into the tonalness
effect, why can't intervals 'blend' visa vie some mechanism that,
perhaps, operates in the absense of tonalness?  (I'm aware of your
original mathematical objection to my idea, but am not convinced by
it.)    Am I making any sense?

thanks, Kelly

On Mon, 13 Mar 2006, "traktus5" wrote:
>
> Hello.  Referring back to Yahya's and Paul's comments (mssgs 840 and
> 843) about the mathematical impossibitity of a note in a chord
> representing two numbers (eg, a note which is the upper note of one
> interval and the lower note of another interval), what about chords
> which do not have a clear indication of a fundamental?  For example, a
> minor triad, with its multiple root allusions, or the case which Paul
> talks about with Monz at tonalsoft, where the tonalness of an
> individual dyad in a chord is stronger than the root-allusion power of
> the entire chord?  In each instance, it seems that the individual
> intervals within the chord acts 'independently' of of the entire
> chord.  It would seem, in that case, then, that, in a sense, you would
> have two numbers 'co-existing'in some manner on one note.  (Did I go
> off on another one of my diatribes in 'recombinant interals'?!  I do
> get carried away...I'm very curious about those issues, if anyone has
> time...)

Hi Kelly,

Let's see if I can understand what you're driving at!
Let's take a minor triad 10:12:15, with internal ratios
6/5, 5/4, 3/2.  Each of the first difference tones 2,
3 and 5 functions somewhat as a fundamental for one
of the three dyads in the chord, but not for the whole
triad.  Are we together on this?

If so, we can say that the 12 is the 6th harmonic of
the 2 (since 12/2 = 6) and is also the 4th harmonic of
the 3 (since 12/3 = 4).

Is this what you mean by "a note in a chord
representing two numbers"?

If so, I don't think the 12 is special in this case, for
each of the other two chord tones, 10 and 12 similarly
represents two different harmonics of two different
fundamentals.

If not, what do you mean?

Regards,
Yahya

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Hello.  Referring back to Yahya's and Paul's comments (mssgs 840 and
843) about the mathematical impossibitity of a note in a chord
representing two numbers (eg, a note which is the upper note of one
interval and the lower note of another interval), what about chords
which do not have a clear indication of a fundamental?  For example, a
minor triad, with its multiple root allusions, or the case which Paul
talks about with Monz at tonalsoft, where the tonalness of an
individual dyad in a chord is stronger than the root-allusion power of
the entire chord?  In each instance, it seems that the individual
intervals within the chord acts 'independently' of of the entire
chord.  It would seem, in that case, then, that, in a sense, you would
have two numbers 'co-existing'in some manner on one note.  (Did I go
off on another one of my diatribes in 'recombinant interals'?!  I do
get carried away...I'm very curious about those issues, if anyone has
time...)

thanks, Kelly

Hi Kelly,

You wrote:
>
> hi Yayha - this is good information, as I'm studying roughness.
> There's stuff here I've never seen textbooks at a college library.
> One initial question regarding > > 7-10-15, or> > 10-14-21 is:
>
> Isn't there a sort of 'lowness', or simplicity, about the second
> spelling, specifically, that the product of the inner intervals
> equals the outer interval, *already reduced*?  (7/5 x 3/2 = 21/10,
> as opposed to 10/7 x 3/2 = 30/14=15/7?)   Is that ever a
> consideration?

The intervals of the second chord, *before reduction*,
are 14/10, 21/14 and 21/10.  Wouldn't that also be a
consideration?  No, honestly, I don't think it makes a
whit of difference that you need to express these
intervals using compound rather than prime numbers,
since that need only arises from the need to show them
all as integers in the wider context of a chord.

Here's another way of looking at this situation.
Suppose we take the complexity of any exact ratio as
simply the largest integer in it (*).  Then:

The *reduced* intervals of the second chord are (in
order of complexity) 3/2, 7/5 and 21/10.

The *reduced* intervals of the first chord are (in
order of complexity) 3/2, 10/7 and 15/7.

Ignoring the 3/2, the complexities of all these align as:
  ... 7/5 ... 10/7 ... 15/7 ... 21/10 ...
The first chord has the two mid-most values, and the
second chord has the two outermost values, of complexity.

Those complexity numbers would (3,) 7, 10, 15, 21.  Their
sums would be 31 for the second chord and 28 for the
first.  Again, the second seems more complex.  Or is that
"interesting"? ;-)

(*) This is a very over-simplified measure of complexity
or roughness.  I recall having some conversation with Paul
Erlich on this and other measures (under what name I
don't recall) on the tuning list in, I think, around October
or Novemeber last year.


> Another possible advantage of the second spelling is -- assuming for
> the sake of argument that the 'seven-ness', from the tritone, is an
> important part of this chord's character, regardless of how the
> chord is spelled, then the second spelling, if you factor the
> primary chord tones, is more 'loaded' with sevens (7-10-15 factors
> to 7, 2x5, and 3x5, whereas 10-14-21 factors to 2x5, 2x7, and 3x7).

That 3x7 certainly acts together with the 2x7 to reinforce
the hearing of a perceived root of 1x7 for the dyad 14-21;
which also coincides with their difference tone.  Comparably,
the first chord is more "loaded" with "fiveness".  So the main
difference between the two chords may well be that the first
"feels like fives (major thirds)" whilst the second "fells like
sevens (harmonic sevenths)".  But to be sure of this,  I'd want
to tune up both chords *exactly* (not as 12-EDO approxima-
tions!), and then play them a few times.


> Finally, looking at the measures of dissonance and roughness you
> calculated, they are actaully quite close, aren't they?  The same
> species, but with slightly different values, perhaps?  Assuming
> that, then I believe the second spelling would also be considered
> more 'interesting', because it is basically just as low as the first
> spelling in roughness and dissonance, but, at the same time, reaches
> higher in the series (21 vs 15), with all the interesting
> consequences that has (such as, according to Paul, multiple
> allusions to similiarly spelled nearby chords) -- in addition to its
> greater affinity with the number 7.

More complex, certainly.  More interesting?  Depends on what
you're interested in ... ;-)


> What do you think?  thanks, Kelly

Regards,
Yahya


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hi Yayha - this is good information, as I'm studying roughness.
There's stuff here I've never seen textbooks at a college library.
One initial question regarding > > 7-10-15, or> > 10-14-21 is:

Isn't there a sort of 'lowness', or simplicity, about the second
spelling, specifically, that the product of the inner intervals
equals the outer interval, *already reduced*?  (7/5 x 3/2 = 21/10,
as opposed to 10/7 x 3/2 = 30/14=15/7?)   Is that ever a
consideration?

Another possible advantage of the second spelling is -- assuming for
the sake of argument that the 'seven-ness', from the tritone, is an
important part of this chord's character, regardless of how the
chord is spelled, then the second spelling, if you factor the
primary chord tones, is more 'loaded' with sevens (7-10-15 factors
to 7, 2x5, and 3x5, whereas 10-14-21 factors to 2x5, 2x7, and 3x7).

Finally, looking at the measures of dissonance and roughness you
calculated, they are actaully quite close, aren't they?  The same
species, but with slightly different values, perhaps?  Assuming
that, then I believe the second spelling would also be considered
more 'interesting', because it is basically just as low as the first
spelling in roughness and dissonance, but, at the same time, reaches
higher in the series (21 vs 15), with all the interesting
consequences that has (such as, according to Paul, multiple
allusions to similiarly spelled nearby chords) -- in addition to its
greater affinity with the number 7.

What do you think?  thanks, Kelly


> > They seem to have roughly equal dissonance levels (intervals
10/7,
> > 3/2, 15/7; vs 7/5, 3/2, 21/10).
> >
> > I wonder which spelling is more appropriate.  ...
>
> Traditional JI theory says that lower numbers in
> the reduced fractions mean lower dissonance.
>
> So, a rough guide would be the maximum of all the
> numbers used in the ratios between chord tones.
> For 7-10-15, that number is 15; for 10-14-21, that
> number is 21.  This calculation favours the first
> form.
>
> Another approach would be to average all the
> numerators and denominators in the ratios between
> chord tones.  The first chord has three ratios, whose
> six integers sum to 44, average 44/6 = 22/3 = 7 1/3.
> The second chord has three ratios, whose six integers
> sum to 48, average 48/6 = 8.  Again, the first chord
> has an edge.
>
> A more psychoacoustically based approach would be
> to calculate some measure of dissonance for each
> inter-dyad interval in the chord (eg Helmholtz'
> roughness or perhaps the inverse of HE), then sum
> them.  Here's a crude roughness measure for the
> interval m/n, where m > n:
>  R = m /min {n, (m-n)}
>
> (Please don't ask me to justify this!!!  It just seems
> right ...)
>
> Chord 7-10-15, interval ratios 10/7, 3/2, 15/7, has R =
> 10/3 + 3/1 + 15/7 = (70+63+45)/21 = 178/21 = 8 10/21.
>
> Chord 10-14-21, interval ratios 7/5, 3/2, 21/10, has R =
> 7/2 + 3/1 + 21/10 = (35+30+21)/10 = 86/10 = 8.6.
> This is just a little larger than that for the first chord,
> which is just under 8.5.  Again, the first chord wins.
>
>
> > ... In the latter, you have
> > the 'tautology' 7/5 x 3/2 =21/10, ...
>
> Irrelevant!  In the former, you have the similar tautology:
> 10/7 x 3/2 = 30/14 = 15/7.  These ARE tautolgies simply
> because they MUST be true - the three ratios within a triad
> ALWAYS obey such a relation.
>
>
> > ... and if that is the correct spelling,
> > why?  thanks!  KElly
>
> What is "correct" in music depends entirely
> on your objectives, I believe.
>
> Regards,
> Yahya
>
>
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>

Hi Kelly,

On Fri, 10 Mar 2006, "traktus5" wrote:
>
> Hello.  Thanks for your comments, Magnus and Yahya.  I have a
> partial response/question:
>
> > 2.  The second- and higher-order ratios, arising
> > from the strong overtone partials of the timbres
> > of those three fundamental notes.
>
> > For example, if all three are played on an
> > instrument with only odd partials,
> > the 5 gives rise to 15, 25, 35, 45, 55, 65 ...
> > the 6 gives rise to 18, 30, 42, 54, 66 ...
> > the 9 gives rise to 27, 45, 63 ...
>
> To have "only odd partials," why not just 'build' a harmonic series,
> with odd numbers only, on each primary chord note, like with the
> chord e3-g3-d4, as follows: e3-b4-g#5...;g3-d5-b5; and d4-a5-
> F#6...)?  ...

No need.  I'm talking of an _instrument_ that
produces only the first, third, fifth, seventh, etc
multiples of the fundamental tone as overtones
(all even multiples being unavailable because of
the instrument's construction.)

However, if you were using pure sine tones, you
could synthesis _something like_ this timbre by
stacking the same multiples in a chord.  The only
real difference would be in the relative strength
of each "partial".


> ... I'm not familiar with your manner of calculation.

Just multiplied each base factor (5, 6 or 9) in
turn by the odd numbers (1,) 3, 5, 7, 9, ...


> ARe you reckoning the partials as arising from each primary chord
> note, ...

Yes.


> ... or from the implied shared fundamental (ie, in may example,
> C1).

Definitely not!


> I'm weak in mathematics, so I think we have a disconnect.

Is the arithmetic clearer now?


> I will respond to the rest in a bit!  I have a hunch there are
> other "components", as well, to the sound, but need to study this.
> thanks

Pleasure!

Regards,
Yahya

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Hi...

> > I don't think this theory is a good predictor of consonance in
general > consider when two same ratios are stacked on top of each
other. This is> > generally dissonant for any ratio with prime limit
>= 5. The ratio between> > the numerators (and the denominators too)
is simply 1:1, but the sound is> > dissonant.
> >
> > Magnus

> > On Tue, 7 Mar 2006, traktus5 wrote:> > > whether the interval
6:3 is of any relavance in the chord 5:6:9 (ie,> > > the ratio of
the numerators of the intervals 6/5 and 3/2, as discussed
> > > towards the end of message #978?)

Did I say *consonance*?  There are two other aspects of my thinking
on this:

One, that the 6:3 ratio of the numerators of 5:6:9 'speaks to', in a
sense, the harmonic series, in a 'displaced' fashion, as follows.
The G in 4:5:6 (c4-e4-g4) is represented by 6 (from 6/5) and 3 (from
3/2).  So, with 4:5:9, the 3:2 'slides up' to a new postion (9:6).
That's why I see the 6:3 of 4:5:9 as expressing, in a 'recombinant'
kind of way (like genes jumping around in a chromosone!), the
harmonic series as represented by 4:5:6, and find it useful to
compare how the the 3/2 interval 'interacts' differently with it's
sister intervals in different chords.

2) Also, is it possible that having multiple 'representations' of
three in the same chord (put that way, 'cause I still don't
understand the difference between powers, logs, and doubling
series!) could have an acoustical effect?  You have 3, 3x2, and 3x3,
in the numerators of 5:6:9's intervals (6/5, 3/2, 9/5).  I know this
could be facile numerology, but have you listened to the chord e4-g4-
f#5 (20:24:45) lately*?  (Let's suppose we're hearing the same
chord, regardless of tuning.) Aside from it's very open but rough
quality, I fancy that the 9 (of 9/4) and the 6 (of 6/5), 'resonate
in some manner', probably because of the 3/2 ratio (of 9/6).

I know I'm out on a limb, but do you really believe that the unique
qualities which certain chord types possess (such as triads, seventh
chords, and their inversions), is fully described by tonalness,
difference tones, beats, partials, and roughness?

(*By the way, the musical intro to the movie Citizen Kane, by
Bernard Hermann, has a great minor ninth chord, in the scene with
the monkeys and statues.  The exact cue is the Statue of the Cat in
the mist, if you're interested.)

(I still need to respond to Yahya's comments below.)

Thanks ...

Kelly




>
> I have to agree with Magnus.  In the chord 5:6:9,
> you have -
>
> 1.  The first-order ratios 5:6, 5:9 and 2:3 (=6:9),
> arising from the three fundamentals.
>
> 2.  The second- and higher-order ratios, arising
> from the strong overtone partials of the timbres
> of those three fundamental notes.
>
> For example, if all three are played on an
> instrument with only odd partials,
> the 5 gives rise to 15, 25, 35, 45, 55, 65 ...
> the 6 gives rise to 18, 30, 42, 54, 66 ...
> the 9 gives rise to 27, 45, 63 ...
>
> 3.  The ratios of the difference tones of the
> fundamentals (1, 3, and 4) to each other, to
> the overtones and to the fundamentals.
>
> 4.  The ratios of the difference tones of the
> overtones (2, 6, 8, ...) to each other, to the
> overtones and to the fundamentals.
>
> Which ones of these actually contribute to the
> final sound is pretty much a question of timbre,
> by which I mean the relative strengths of the
> partials, and of amplitude, since soft sounds may
> have inaudible interactions.  For synthetic
> instruments, the result may also depend on phase,
> but that's not usually so for acoustic instruments.
>
> With all these different numbers in play, all of
> which arise from purely physical considerations
> there's no need to involve a *hypothetical*
> interval 6:3.  You really need to relate ALL the
> sounds of the chord to a common number, their
> GCD (Greatest Common Divisor).  For 5:6:9, this
> is 1.  Then relating this to a notional frequency,
> eg A 110 Hz, your chord is 550, 660, 990 Hz.
> These notes have overtones at integer multiples
> of those three frequencies, and difference tones
> at integer multiples of the GCD note, 110 Hz.
> (I'm assuming harmonic timbres.)
>
> However you realise your chord, it must be
> expressed as specific frequencies.  If you stick
> to exact integer ratios, those frequencies have a
> GCD which may well be in the audible range, as in
> my example using 110 Hz.  For higher integer limit
> chords, the GCD may be subsonic.  But even then,
> all of the components of the sound you hear will
> be integer multiples of that number.  You will only
> hear those ones which exceed the sensitivity of
> your ears.  Even then, you may not be aware of
> all those frequencies which your ears detect!
>
> As I understand Paul's theory of HE, it has two
> interesting consequences: smaller integer ratios
> are more consonant, and small deviations from
> exact ratios do not destroy consonance.  At least,
> that's how I read the graph ...
>
> So what I said above about exact ratios also applies
> largely to ratios that are as exact as fallible humans
> can make them.
>
> Regards,
> Yahya
>
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>

Hello.  Thanks for your comments, Magnus and Yahya.  I have a
partial response/question:

> 2.  The second- and higher-order ratios, arising
> from the strong overtone partials of the timbres
> of those three fundamental notes.

> For example, if all three are played on an
> instrument with only odd partials,
> the 5 gives rise to 15, 25, 35, 45, 55, 65 ...
> the 6 gives rise to 18, 30, 42, 54, 66 ...
> the 9 gives rise to 27, 45, 63 ...

To have "only odd partials," why not just 'build' a harmonic series,
with odd numbers only, on each primary chord note, like with the
chord e3-g3-d4, as follows: e3-b4-g#5...;g3-d5-b5; and d4-a5-
F#6...)?  I'm not familiar with your manner of calculation.

ARe you reckoning the partials as arising from each primary chord
note, or from the implied shared fundamental (ie, in may example,
C1).

I'm weak in mathematics, so I think we have a disconnect.

I will respond to the rest in a bit!  I have a hunch there are
other "components", as well, to the sound, but need to study this.
thanks

Hi Kelly,

On Thu, 09 Mar 2006, traktus5 wrote:
>
> Considering two spellings of the chord e4-Bb4-f5 (great chord: tonic,
> dominant, and subdominant combined!), the spellings being
>
> 7-10-15, or
> 10-14-21
>
> They seem to have roughly equal dissonance levels (intervals 10/7,
> 3/2, 15/7; vs 7/5, 3/2, 21/10).
>
> I wonder which spelling is more appropriate.  ...

Traditional JI theory says that lower numbers in
the reduced fractions mean lower dissonance.

So, a rough guide would be the maximum of all the
numbers used in the ratios between chord tones.
For 7-10-15, that number is 15; for 10-14-21, that
number is 21.  This calculation favours the first
form.

Another approach would be to average all the
numerators and denominators in the ratios between
chord tones.  The first chord has three ratios, whose
six integers sum to 44, average 44/6 = 22/3 = 7 1/3.
The second chord has three ratios, whose six integers
sum to 48, average 48/6 = 8.  Again, the first chord
has an edge.

A more psychoacoustically based approach would be
to calculate some measure of dissonance for each
inter-dyad interval in the chord (eg Helmholtz'
roughness or perhaps the inverse of HE), then sum
them.  Here's a crude roughness measure for the
interval m/n, where m > n:
	 R = m /min {n, (m-n)}

(Please don't ask me to justify this!!!  It just seems
right ...)

Chord 7-10-15, interval ratios 10/7, 3/2, 15/7, has R =
10/3 + 3/1 + 15/7 = (70+63+45)/21 = 178/21 = 8 10/21.

Chord 10-14-21, interval ratios 7/5, 3/2, 21/10, has R =
7/2 + 3/1 + 21/10 = (35+30+21)/10 = 86/10 = 8.6.
This is just a little larger than that for the first chord,
which is just under 8.5.  Again, the first chord wins.


> ... In the latter, you have
> the 'tautology' 7/5 x 3/2 =21/10, ...

Irrelevant!  In the former, you have the similar tautology:
10/7 x 3/2 = 30/14 = 15/7.  These ARE tautolgies simply
because they MUST be true - the three ratios within a triad
ALWAYS obey such a relation.


> ... and if that is the correct spelling,
> why?  thanks!  KElly

What is "correct" in music depends entirely
on your objectives, I believe.

Regards,
Yahya


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Considering two spellings of the chord e4-Bb4-f5 (great chord: tonic,
dominant, and subdominant combined!), the spellings being

7-10-15, or
10-14-21

They seem to have roughly equal dissonance levels (intervals 10/7,
3/2, 15/7; vs 7/5, 3/2, 21/10).

I wonder which spelling is more appropriate.  In the latter, you have
the 'tautology' 7/5 x 3/2 =21/10, and if that is the correct spelling,
why?  thanks!  KElly

On Tue, 7 Mar 2006, Magnus Jonsson wrote:
>
> Hi Kelly,
>
> I don't think this theory is a good predictor of consonance in general -
> consider when two same ratios are stacked on top of each other. This is
> generally dissonant for any ratio with prime limit >= 5. The ratio between
> the numerators (and the denominators too) is simply 1:1, but the sound is
> dissonant.
>
> Best regards
> Magnus
>
> On Tue, 7 Mar 2006, traktus5 wrote:
>
> > whether the interval 6:3 is of any relavance in the chord 5:6:9 (ie,
> > the ratio of the numerators of the intervals 6/5 and 3/2, as discussed
> > towards the end of message #978?)
> >
> > thanks, Kelly

Hi Kelly,

I have to agree with Magnus.  In the chord 5:6:9,
you have -

1.  The first-order ratios 5:6, 5:9 and 2:3 (=6:9),
arising from the three fundamentals.

2.  The second- and higher-order ratios, arising
from the strong overtone partials of the timbres
of those three fundamental notes.

For example, if all three are played on an
instrument with only odd partials,
the 5 gives rise to 15, 25, 35, 45, 55, 65 ...
the 6 gives rise to 18, 30, 42, 54, 66 ...
the 9 gives rise to 27, 45, 63 ...

3.  The ratios of the difference tones of the
fundamentals (1, 3, and 4) to each other, to
the overtones and to the fundamentals.

4.  The ratios of the difference tones of the
overtones (2, 6, 8, ...) to each other, to the
overtones and to the fundamentals.

Which ones of these actually contribute to the
final sound is pretty much a question of timbre,
by which I mean the relative strengths of the
partials, and of amplitude, since soft sounds may
have inaudible interactions.  For synthetic
instruments, the result may also depend on phase,
but that's not usually so for acoustic instruments.

With all these different numbers in play, all of
which arise from purely physical considerations
there's no need to involve a *hypothetical*
interval 6:3.  You really need to relate ALL the
sounds of the chord to a common number, their
GCD (Greatest Common Divisor).  For 5:6:9, this
is 1.  Then relating this to a notional frequency,
eg A 110 Hz, your chord is 550, 660, 990 Hz.
These notes have overtones at integer multiples
of those three frequencies, and difference tones
at integer multiples of the GCD note, 110 Hz.
(I'm assuming harmonic timbres.)

However you realise your chord, it must be
expressed as specific frequencies.  If you stick
to exact integer ratios, those frequencies have a
GCD which may well be in the audible range, as in
my example using 110 Hz.  For higher integer limit
chords, the GCD may be subsonic.  But even then,
all of the components of the sound you hear will
be integer multiples of that number.  You will only
hear those ones which exceed the sensitivity of
your ears.  Even then, you may not be aware of
all those frequencies which your ears detect!

As I understand Paul's theory of HE, it has two
interesting consequences: smaller integer ratios
are more consonant, and small deviations from
exact ratios do not destroy consonance.  At least,
that's how I read the graph ...

So what I said above about exact ratios also applies
largely to ratios that are as exact as fallible humans
can make them.

Regards,
Yahya

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Hi Kelly,

I don't think this theory is a good predictor of consonance in general -
consider when two same ratios are stacked on top of each other. This is
generally dissonant for any ratio with prime limit >= 5. The ratio between
the numerators (and the denominators too) is simply 1:1, but the sound is
dissonant.

Best regards
Magnus

On Tue, 7 Mar 2006, traktus5 wrote:

> whether the interval 6:3 is of any relavance in the chord 5:6:9 (ie,
> the ratio of the numerators of the intervals 6/5 and 3/2, as discussed
> towards the end of message #978?)
>
> thanks, Kelly

Hi Yahya - I guess this is a bit off topic.  (Do you know a good site
for misc. acoustics, not so centered on tuning?  ...though I think
there is some relevence to the question: "...  Could this chord be one
of the most consonant chords (= no > > octave doublings) you can get
with four notes?  (Maybe c4-e4-g4-b4 is > > more consonant?)  I
imagine for power and denseness they would want a > > closed position
chord....)

Since Paul's not around, do you have any interest in discussing
whether the interval 6:3 is of any relavance in the chord 5:6:9 (ie,
the ratio of the numerators of the intervals 6/5 and 3/2, as discussed
towards the end of message #978?)

thanks, Kelly

On Sun, 05 Mar 2006, "traktus5" wrote:
>
> I'm sure you've noticed that a four-note train whistle is
> approximately a minor seventh chord (something like e3-g3-b3-d4.
> Please don't ask me how it's tuned! ... though I believe the shreeking
> of it does bend the pitches...)  I wonder how they ended up with that
> chord?  Could this chord be one of the most consonant chords (= no
> octave doublings) you can get with four notes?  (Maybe c4-e4-g4-b4 is
> more consonant?)  I imagine for power and denseness they would want a
> closed position chord.
>
> How about train whistles in other countries?  Same chord?


Hi Kelly,

Here in Australia, I've only ever heard a one- or two-note
whistle; the second note arising by Doppler Shift when the
train passes you.  Whilst living in Malaysia, I was rarely ever
on or near a train, and frankly don't remember anything
remarkable.  (It WAS almost 30 years ago!)

Where can I hear a recording of the kinds of whistle you're
talking about?

Regards,
Yahya

PS  Does this topic really belong in this HE forum?  YA

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I'm sure you've noticed that a four-note train whistle is
approximately a minor seventh chord (something like e3-g3-b3-d4.
Please don't ask me how it's tuned! ... though I believe the shreeking
of it does bend the pitches...)  I wonder how they ended up with that
chord?  Could this chord be one of the most consonant chords (= no
octave doublings) you can get with four notes?  (Maybe c4-e4-g4-b4 is
more consonant?)  I imagine for power and denseness they would want a
closed position chord.

How about train whistles in other countries?  Same chord?