tonalness of minor triads

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

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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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

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.

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
>

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

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>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

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

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 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. 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!
>
>

>>> 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