two questions

hello Entropy group (Paul, Carl, et al...)

1. In the chord c-e-a, with the intervals 5/4, 4/3, and 5/3: taking,
for instance, the center note e: it is represented by the number 3, as
the lower note of 4/3, and by the number 5, as the upper note of
5/4. Does this mean, in some sense, that there is a sort of mixing
of waveforms in that note (of the third and fifth harmonic)?

2. In the minor 9th chord C2-Eb3-D4, spelled with the intervals 12/5,
15/8, and 9/2, the primary difference tones, if I'm correct, all have
a numerator of 21 (21/40, 21/10, and 21/8; or they all equal seven? -
12-7, 15-8, and 9-2). This seems a little bit unusual, and I was
wondering if there is any psychoacoustical effect you guys are aware
of that results from a congruence of difference tones.

cheers, Kelly

--- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...> wrote:

> hello Entropy group (Paul, Carl, et al...)
>
> 1. In the chord c-e-a, with the intervals 5/4, 4/3, and 5/3: taking,
> for instance, the center note e: it is represented by the number 3,
as
> the lower note of 4/3, and by the number 5, as the upper note of
> 5/4. Does this mean, in some sense, that there is a sort of mixing
> of waveforms in that note (of the third and fifth harmonic)?

You can understand the whole chord as 1/5:1/4:1/3; therefore, the 5th
harmonic of the lower note, the 4th harmonic of the middle note, and
the 3rd harmonic of the upper note all coincide in pitch. Other than
that, I have no idea what you could mean.

> 2. In the minor 9th chord C2-Eb3-D4, spelled with the intervals 12/5,
> 15/8, and 9/2, the primary difference tones, if I'm correct, all have
> a numerator of 21 (21/40, 21/10, and 21/8; or they all equal seven?

Tones are tones and interval ratios are interval ratios -- don't
confuse the two. If you want to express the difference tones as ratios,
you need to specify a reference pitch from which these ratios will be
measured. The easiest way is to express the chord you mention as
10:24:45 (which puts the fundamental 1 at Ab-2, 4 octaves below Ab2),
and then calculate the difference tones: 24-10 = 14, 45-10 = 35, and 45-
24 = 21. Relative to the chord in question, these notes would be a flat
Gb2, a flat Bb3 (a wolf fifth up from the Eb), and a flat Db3.

> 12-7, 15-8, and 9-2). This seems a little bit unusual, and I was
> wondering if there is any psychoacoustical effect you guys are aware
> of that results from a congruence of difference tones.

This chord doesn't have coinciding difference tones, but does have
difference tones which are the second, third, and fifth harmonics of a
flat Gb1. Hence, if the chord is played in true JI, due to
psychoacoustic effects I might expect to hear a suggestion of a flat
Gb1 pitch emerging when the chord is ramped up in amplitude. Very
interesting! Of course, there are other difference tones that are often
louder than these 'primary' ones . . .

Hi Kelly,

You wrote:
> hello Entropy group (Paul, Carl, et al...)
>
> 1. In the chord c-e-a, with the intervals 5/4, 4/3, and 5/3: taking,
> for instance, the center note e: it is represented by the number 3, as
> the lower note of 4/3, and by the number 5, as the upper note of
> 5/4. Does this mean, in some sense, that there is a sort of mixing
> of waveforms in that note (of the third and fifth harmonic)?
>
> 2. In the minor 9th chord C2-Eb3-D4, spelled with the intervals 12/5,
> 15/8, and 9/2, the primary difference tones, if I'm correct, all have
> a numerator of 21 (21/40, 21/10, and 21/8; or they all equal seven? -
> 12-7, 15-8, and 9-2). This seems a little bit unusual, and I was
> wondering if there is any psychoacoustical effect you guys are aware
> of that results from a congruence of difference tones.

I think Paul has given a good answer to the second question. He has also
answered the first one, but I have a slightly different answer - you're
confused! :-) Let me explain why I think so.

Look at "the chord c-e-a, with the intervals 5/4, 4/3, and 5/3". What
does this really mean? Let's draw up a table of the numbers involved -

c : e : a
= 4 : 5 : x <-- interval 4:5 or 5/4
= y : 3 : 4 <-- interval 3:4 or 4/3
= 12 : 15 : 20 <-- see below
= 3 : z : 5 <-- interval 3:5 or 5/3

This is important: All the lines in this table express the same ratios.
So I've written = signs to connect them all.

The major third c:e has the ratio 4:5.

The fourth e:a has the ratio 3:4. Since we couldn't write it as a simple
ratio 5: some integer, I just put down x for the time being in the first
row.

Similarly, I put down y for the number such that y:3 is the major third
4:5. And z for the number such that 3:z is also the major third.

Now I could solve immediately for x, y and z, just by rearranging terms
in the three independent equations 5:x = 3:4, y:3 = 4:5 and 3:z = 4:5.

But it's more _fun_ to write down the LCM (least common multiple) of
the two numbers we know in the middle column - that's 15, then multiply
the first two rows by 15/those two numbers in turn, that's by 15/5=3
for the first row, and by 15/3=5 for the second row, but ignoring the
unknowns. This gives the numbers in the third row -
4*3=12, 5*3=15; also 3*5=15, 4*5=20.

If we ignore the middle note e, the remaining two numbers are 12 and
20, which have a common factor of 4, so are in the ratio 3:5. So the
fourth row expresses that, confirming that a major sixth is the ratio
3:5, and leaving z for the corresponding number for e.

From a mathematical point of view, it's interesting that you can use a
similar technique to find the interval ratio between any two notes which
are separated by a set of known interval ratios, staying entirely within
the realm of integers. This is only really important if you are a
mathematician of a particular logical persuasion - those who "don't
believe in" fractions (rational numbers) or any other numbers except
the natural counting numbers. I was told such people exist - I forget
the name for them (it might be Amish?). However, for mere mortals who
get confused at times by the arithmetic of fractions, it's a useful
way of dealing mostly with whole numbers instead.

Now, where I think you might have been confused is where you said "the
center note e ... is represented by the number 3, as the lower note of
4/3, and by the number 5, as the upper note of 5/4." Please note that it
can't be represented by both numbers _at the same time_. Numbers
don't work like that! :-)

The note e is represented by 3, _only when_ the note a is represented
by 4; that is, on the second row in the table.

And the note e is represented by 5, _only when_ the note c is
represented by 4; that is, on the first row in the table.

If both these representations were true at the same time, then both c
and a would be represented by the same number 4 - even though they
are different notes!

Each row of the table is a different representation of the chord c:e:a.
The most informative is the one with no unknowns in it and all in
integers, that is 12:15:20. In this row, the note e is represented by 15,
and we can easily reduce the ratios 12:15 and 15:20 by removing
common factors, to 4:5 and 3:4 respectively, confirming that we do
indeed have a third below and a fourth above the note e.

We can get another representation - Paul's - by dividing this one
through by their LCM, which is 60, getting 1/5:1/4:1/3. (This represents
the minor triad by the reciprocals of the numbers in the major triad
g:c:e 's representation as 3:4:5.) As Paul rightly commented, the fifth
harmonic of the lowest note c, the fourth harmonic of the middle note e,
and the third harmonic of the highest note a, will all coincide at the
same note, two octaves above the e of your chord. This follows because
the chord formed by these harmonics is 5*1/5:4*1/4:3*1/3, which
equals 1:1:1, a unison chord.

Your question "Does this mean, in some sense, that there is a sort of
mixing of waveforms in that note (of the third and fifth harmonic)?"
can be answered by saying, "Yes, the fifth harmonic _of the lowest
chord tone_ and the third harmonic _of the highest chord tone_ will
coincide with the fourth harmonic (the double octave) of the middle
chord tone. Further, if they happen to be in phase, they may reinforce
each other, making the double octave of the middle chord tone more
prominent; however, if they happen to be opposite in phase, they may
cancel each other, making the double octave of the middle chord tone
less prominent or even absent."

We really do need to qualify what the third and fifth harmonics are
harmonics _of_, as in the answer above. Suppose we had any
fundamental tone, say the note R=100 Hz. Its fiRst, Second, Third,
foUrth and fifth harmonics are the notes R, S=200 Hz, T=300 Hz,
U=400 Hz and V=500 Hz. The chord T:U:V = 300:400:500 which in turn
= 3:4:5. So T:U is a fourth and U:V is a major third, and the chord is
just the same as the C major triad in second inversion, G:C:E, only
transposed into "the key of U". The note names don't matter, but the
ratios do. The fundamental tone - R in our example in the key of U - is
implied by the 3:4:5 major triad, in the sense that it is the highest note
of which all the chord notes are harmonics.

I wonder whether perhaps, when you asked your question, you were
thinking of the third and fifth harmonics of the fundamental tone in
a 3:4:5 major triad? Because in the context of your example of the
A minor triad in second inversion, C:E:A, you have a different set of
ratios, namely 1/5:1/4:1/3, or in whole numbers, 12:15:20. The
fundamental tone implied by these three integers is the one of which
the note C is the 12th harmonic, the note E is the 15th harmonic and the
note A is the 20th harmonic. That makes it three octaves and a fifth
below the C, that is, the note F. I don't at all suppose you were thinking
of the third and fifth harmonics of that F.

---

What were the numbers x, y and z?

If you insist on representing the note e by 5, then you need to use the
representation 4:5:x - the first row of the table, and find out what
value of x makes 5:x the same ratio as 3:4. That is, what x is when
5/x=3/4. The answer is that x=5*4/3=20/3.

But if you want to represent the note e by 3, then you need to use the
representation y:3:4 - the second row of the table, and find out what
value of y makes y:3 the same ratio as 4:5. That is, what y is when
y/3=4/5. The answer is that y=3*4/5=12/5.

(Of course you can't use both these representations at the same time.)

From the last row, it turns out that z=3*5/4=15/4. There's a curious
pattern here - All three integers 3, 4 and 5 of your ratios appear once
each in x, y and z, and they all have the form (3*4*5)/(n*n), where n is
in turn each of 3, 4 and 5. I guess this pattern would generalise to
other sets of relatively prime integers, and perhaps to any 3 integers
p, q, r if we replace the product (p*q*r) by their LCM, thus -
LCM(p, q, r)/(n*n), where n is in turn each of p, q and r.

Regards,
Yahya

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Wow! what an amazing answer, Yahya! There's a lot there for me to
think about...and much that you opened up. (Thanks to you, too,
Paul.) I get so 'microscopically' into listening to the chords,
trance-like, writing progressions with them, trying to imagine what
the wave forms might look like, and what the numbers might have to
do with it, that I guess I play amateur scientist here, which math
laymen tend to do with number patterns. So I feel rather foolish
posting here. (I do it because I have learned some things here.)
But I can see you really made an attempt to make sense of where I
was going, or not going!, which I appreciate. (I can see you like
number patters too! )

The pattern that most interested me with this lovely chord c-e-a
(and then I'll drop it) is when you represent each note (in my
fallacious fashion, but please hear me out) with a fraction, giving
4/3, 5/3, and 5/4 (corresponding to c, e, and a, in the fashion
discussed, with each fraction representing the overlap of the two
intervals on that note.) I looked at many many chords in that
fashion, and I believe c-e-a is the only one that has it's 'harmonic
numbers' 3-4-5 represented in the fractions (starting with the
demoninators, and moving up to the numerators, ie, 3,3,4,4,5,5.) I
imagine there is an obvious reason for this, embedded in your
answer, which I promise to study and respond to (--summer reading
for my upcoming vaction!)...but can you think of any reason why the
chord c-e-a would be the only chord which has that feature? (ISn't
this the only traditional type chrod which can be represented by
numbers 5 and under?) (I understand if you consider this to not
even be significant feature to comment on, due to a flawed method!)

(The pattern also appealed to me because it reminded me of the
ancient but highly accurate approximation of pi: 355/113, which, as
Martin Gardner pointed out, has a simple number pattern (1,3,5) read
from left to right, denominators to numerators.)

I will work on learning your method, and ...

cheers, Kelly

@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> Hi Kelly,
>
> You wrote:
> > hello Entropy group (Paul, Carl, et al...)
> >
> > 1. In the chord c-e-a, with the intervals 5/4, 4/3, and 5/3:
taking,
> > for instance, the center note e: it is represented by the number
3, as
> > the lower note of 4/3, and by the number 5, as the upper note of
> > 5/4. Does this mean, in some sense, that there is a sort of
mixing
> > of waveforms in that note (of the third and fifth harmonic)?
> >
> > 2. In the minor 9th chord C2-Eb3-D4, spelled with the intervals
12/5,
> > 15/8, and 9/2, the primary difference tones, if I'm correct, all
have
> > a numerator of 21 (21/40, 21/10, and 21/8; or they all equal
seven? -
> > 12-7, 15-8, and 9-2). This seems a little bit unusual, and I
was
> > wondering if there is any psychoacoustical effect you guys are
aware
> > of that results from a congruence of difference tones.
>
>
> I think Paul has given a good answer to the second question. He
has also
> answered the first one, but I have a slightly different answer -
you're
> confused! :-) Let me explain why I think so.
>
> Look at "the chord c-e-a, with the intervals 5/4, 4/3, and 5/3".
What
> does this really mean? Let's draw up a table of the numbers
involved -
>
> c : e : a
> = 4 : 5 : x <-- interval 4:5 or 5/4
> = y : 3 : 4 <-- interval 3:4 or 4/3
> = 12 : 15 : 20 <-- see below
> = 3 : z : 5 <-- interval 3:5 or 5/3
>
> This is important: All the lines in this table express the same
ratios.
> So I've written = signs to connect them all.
>
> The major third c:e has the ratio 4:5.
>
> The fourth e:a has the ratio 3:4. Since we couldn't write it as a
simple
> ratio 5: some integer, I just put down x for the time being in the
first
> row.
>
> Similarly, I put down y for the number such that y:3 is the major
third
> 4:5. And z for the number such that 3:z is also the major third.
>
> Now I could solve immediately for x, y and z, just by rearranging
terms
> in the three independent equations 5:x = 3:4, y:3 = 4:5 and 3:z =
4:5.
>
> But it's more _fun_ to write down the LCM (least common multiple)
of
> the two numbers we know in the middle column - that's 15, then
multiply
> the first two rows by 15/those two numbers in turn, that's by
15/5=3
> for the first row, and by 15/3=5 for the second row, but ignoring
the
> unknowns. This gives the numbers in the third row -
> 4*3=12, 5*3=15; also 3*5=15, 4*5=20.
>
> If we ignore the middle note e, the remaining two numbers are 12
and
> 20, which have a common factor of 4, so are in the ratio 3:5. So
the
> fourth row expresses that, confirming that a major sixth is the
ratio
> 3:5, and leaving z for the corresponding number for e.
>
> From a mathematical point of view, it's interesting that you can
use a
> similar technique to find the interval ratio between any two notes
which
> are separated by a set of known interval ratios, staying entirely
within
> the realm of integers. This is only really important if you are a
> mathematician of a particular logical persuasion - those who "don't
> believe in" fractions (rational numbers) or any other numbers
except
> the natural counting numbers. I was told such people exist - I
forget
> the name for them (it might be Amish?). However, for mere mortals
who
> get confused at times by the arithmetic of fractions, it's a useful
> way of dealing mostly with whole numbers instead.
>
> Now, where I think you might have been confused is where you
said "the
> center note e ... is represented by the number 3, as the lower
note of
> 4/3, and by the number 5, as the upper note of 5/4." Please note
that it
> can't be represented by both numbers _at the same time_. Numbers
> don't work like that! :-)
>
> The note e is represented by 3, _only when_ the note a is
represented
> by 4; that is, on the second row in the table.
>
> And the note e is represented by 5, _only when_ the note c is
> represented by 4; that is, on the first row in the table.
>
> If both these representations were true at the same time, then
both c
> and a would be represented by the same number 4 - even though they
> are different notes!
>
> Each row of the table is a different representation of the chord
c:e:a.
> The most informative is the one with no unknowns in it and all in
> integers, that is 12:15:20. In this row, the note e is
represented by 15,
> and we can easily reduce the ratios 12:15 and 15:20 by removing
> common factors, to 4:5 and 3:4 respectively, confirming that we do
> indeed have a third below and a fourth above the note e.
>
> We can get another representation - Paul's - by dividing this one
> through by their LCM, which is 60, getting 1/5:1/4:1/3. (This
represents
> the minor triad by the reciprocals of the numbers in the major
triad
> g:c:e 's representation as 3:4:5.) As Paul rightly commented, the
fifth
> harmonic of the lowest note c, the fourth harmonic of the middle
note e,
> and the third harmonic of the highest note a, will all coincide at
the
> same note, two octaves above the e of your chord. This follows
because
> the chord formed by these harmonics is 5*1/5:4*1/4:3*1/3, which
> equals 1:1:1, a unison chord.
>
> Your question "Does this mean, in some sense, that there is a sort
of
> mixing of waveforms in that note (of the third and fifth
harmonic)?"
> can be answered by saying, "Yes, the fifth harmonic _of the lowest
> chord tone_ and the third harmonic _of the highest chord tone_ will
> coincide with the fourth harmonic (the double octave) of the middle
> chord tone. Further, if they happen to be in phase, they may
reinforce
> each other, making the double octave of the middle chord tone more
> prominent; however, if they happen to be opposite in phase, they
may
> cancel each other, making the double octave of the middle chord
tone
> less prominent or even absent."
>
> We really do need to qualify what the third and fifth harmonics are
> harmonics _of_, as in the answer above. Suppose we had any
> fundamental tone, say the note R=100 Hz. Its fiRst, Second, Third,
> foUrth and fifth harmonics are the notes R, S=200 Hz, T=300 Hz,
> U=400 Hz and V=500 Hz. The chord T:U:V = 300:400:500 which in turn
> = 3:4:5. So T:U is a fourth and U:V is a major third, and the
chord is
> just the same as the C major triad in second inversion, G:C:E, only
> transposed into "the key of U". The note names don't matter, but
the
> ratios do. The fundamental tone - R in our example in the key of
U - is
> implied by the 3:4:5 major triad, in the sense that it is the
highest note
> of which all the chord notes are harmonics.
>
> I wonder whether perhaps, when you asked your question, you were
> thinking of the third and fifth harmonics of the fundamental tone
in
> a 3:4:5 major triad? Because in the context of your example of the
> A minor triad in second inversion, C:E:A, you have a different set
of
> ratios, namely 1/5:1/4:1/3, or in whole numbers, 12:15:20. The
> fundamental tone implied by these three integers is the one of
which
> the note C is the 12th harmonic, the note E is the 15th harmonic
and the
> note A is the 20th harmonic. That makes it three octaves and a
fifth
> below the C, that is, the note F. I don't at all suppose you were
thinking
> of the third and fifth harmonics of that F.
>
> ---
>
> What were the numbers x, y and z?
>
> If you insist on representing the note e by 5, then you need to
use the
> representation 4:5:x - the first row of the table, and find out
what
> value of x makes 5:x the same ratio as 3:4. That is, what x is
when
> 5/x=3/4. The answer is that x=5*4/3=20/3.
>
> But if you want to represent the note e by 3, then you need to use
the
> representation y:3:4 - the second row of the table, and find out
what
> value of y makes y:3 the same ratio as 4:5. That is, what y is
when
> y/3=4/5. The answer is that y=3*4/5=12/5.
>
> (Of course you can't use both these representations at the same
time.)
>
> From the last row, it turns out that z=3*5/4=15/4. There's a
curious
> pattern here - All three integers 3, 4 and 5 of your ratios appear
once
> each in x, y and z, and they all have the form (3*4*5)/(n*n),
where n is
> in turn each of 3, 4 and 5. I guess this pattern would generalise
to
> other sets of relatively prime integers, and perhaps to any 3
integers
> p, q, r if we replace the product (p*q*r) by their LCM, thus -
> LCM(p, q, r)/(n*n), where n is in turn each of p, q and r.
>
> Regards,
> Yahya
>
>
>
> --
> No virus found in this outgoing message.
> Checked by AVG Anti-Virus.
> Version: 7.0.344 / Virus Database: 267.10.17/84 - Release Date:
29/8/05

--- In harmonic_entropy@yahoogroups.com, "Yahya Abdal-Aziz"

> Your question "Does this mean, in some sense, that there is a sort
of
> mixing of waveforms in that note (of the third and fifth harmonic)?"
> can be answered by saying, "Yes, the fifth harmonic _of the lowest
> chord tone_ and the third harmonic _of the highest chord tone_ will
> coincide with the fourth harmonic (the double octave) of the middle
> chord tone. Further, if they happen to be in phase, they may
reinforce
> each other, making the double octave of the middle chord tone more
> prominent; however, if they happen to be opposite in phase, they may
> cancel each other, making the double octave of the middle chord tone
> less prominent or even absent."

On real acoustic instruments, none of the interval ratios will be
*exact* (even the best tuners commit small errors, and often
temperament is desirable for musical reasons), so one will tend to
hear a *loud beating pattern* at the double octave of the middle
chord. It can be more complex that a simple wa-wa-wa beat, since
*three* slightly different sine waves are involved.

> I wonder whether perhaps, when you asked your question, you were
> thinking of the third and fifth harmonics of the fundamental tone in
> a 3:4:5 major triad? Because in the context of your example of the
> A minor triad in second inversion, C:E:A,

That's first inversion ;)

--- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
wrote:
> Wow! what an amazing answer, Yahya! There's a lot there for me
to
> think about...and much that you opened up. (Thanks to you, too,
> Paul.) I get so 'microscopically' into listening to the chords,
> trance-like, writing progressions with them, trying to imagine what
> the wave forms might look like, and what the numbers might have to
> do with it, that I guess I play amateur scientist here, which math
> laymen tend to do with number patterns. So I feel rather foolish
> posting here. (I do it because I have learned some things here.)
> But I can see you really made an attempt to make sense of where I
> was going, or not going!, which I appreciate. (I can see you like
> number patters too! )
>
> The pattern that most interested me with this lovely chord c-e-a
> (and then I'll drop it) is when you represent each note (in my
> fallacious fashion, but please hear me out) with a fraction, giving
> 4/3, 5/3, and 5/4 (corresponding to c, e, and a, in the fashion
> discussed, with each fraction representing the overlap of the two
> intervals on that note.) I looked at many many chords in that
> fashion, and I believe c-e-a is the only one that has
it's 'harmonic
> numbers' 3-4-5 represented in the fractions (starting with the
> demoninators, and moving up to the numerators, ie, 3,3,4,4,5,5.)

How about c-f-a, which is 3:4:5?

>(ISn't
> this the only traditional type chrod which can be represented by
> numbers 5 and under?)

You can express F,-F-c-f-a as 1:2:3:4:5.

>> I looked at many many chords in that
> > fashion, and I believe c-e-a is the only one that has
> it's 'harmonic
> > numbers' 3-4-5 represented in the fractions (starting with the
> > demoninators, and moving up to the numerators, ie, 3,3,4,4,5,5.)
>
> How about c-f-a, which is 3:4:5?

Yes, in whole numbers. My survey just looked at -- you would call
it? -- the utonal chords? with fractions, where that pattern (not that
I'm arguing it means anytning scientifically!) is apparently much
rarer ...(maybee all my searching for 'number purity' in the c-e-a
chord is a way to 'defend' it over the triad, per my own chord
preferences and theories.)

In your work, does 1/5:1/4:1/3 (c-e-a) score higher in consonance than
it's 'reflection' 3:4:5, the latter being a 6.4 chord? (I know from
diff. tones, etc, that's not the case ... but ...Hmm, I've never
inquired if voice leading expectations have anything to do with your
models of consonance. ..I guess such a model would need to include
cultural conditionaing...(and if that's the case, if there's
conditioning, then wouldn't that make it part of the psychoacoustical
apparatus?)

> > ia this the only traditional type chrod which can be represented
by
> > numbers 5 and under?)
>
> You can express F,-F-c-f-a as 1:2:3:4:5.

But with chord defined as 3 notes, no duplicates, closed position, and
based on thirds ('traditional')?

Maybe part of the specialness (to my ear) of the c-e-a chord --
speculating here -- is its 4/3 interval, which is the
first 'melodically active' interval in the series (ie, voiceleading
tendency, in this case, scale going 4-3, or 4-flat 3). Maybee --
really going out on limb here -- the interval 3:4 in the series, with
it's strong leading tone tendency (-- I know this is all cultural
conditioning!) as it occurs in the chords 3:4:5, and 1/5:1/4:1/3, is a
sort of 'gateway' between the otonal and utonal series. I just notice
how the two chords (eg, g-c-e//g-b-e) very easily flip back and forth,
hinged on the leading tone, sort of like, in chemistry,
stereoisomers. (Also, 3:4:5/1 is purer than 4:5:6, since 6 is sort
of 'second generation'...just a double of 3.)

Question: In surveys of music around the world and through time, and
if it is possible to restrict interval definition roughly to half step
or whole step as are commonl defined, is there any sort of prevalance
of one over the other (half step vs whole step)? I was listening to
some electroacoustical music the other night (Xenakis, Legend of Eer),
and noticed the prevelence (maybe by his design) of minor thirds
(produced scale wise, half step-whole step.) Just another pet notion
of mine, on the quality of 'minor-ness, that the minor interval can be
composed of a whole step and half step, sort of paralleling the big
third, little third construction of the triad. I don't know...do you
think there could be anything to that, that a whole step and half step
are more stable in some way than two whole steps (ie, minor third vs
major third?) From my limited listening to other cultures' music, I
just sort of get the impression that 'minor-ness' prevails.

Thanks as usual for the input. cheers, Kelly

--- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
wrote:
> >> I looked at many many chords in that
> > > fashion, and I believe c-e-a is the only one that has
> > it's 'harmonic
> > > numbers' 3-4-5 represented in the fractions (starting with the
> > > demoninators, and moving up to the numerators, ie, 3,3,4,4,5,5.)
> >
> > How about c-f-a, which is 3:4:5?
>
> Yes, in whole numbers. My survey just looked at -- you would call
> it? -- the utonal chords?

Yes.

> with fractions, where that pattern (not that
> I'm arguing it means anytning scientifically!) is apparently much
> rarer

Rarer? Meaning the total waveform repeats more rarely?

...(maybee all my searching for 'number purity' in the c-e-a
> chord is a way to 'defend' it over the triad, per my own chord
> preferences and theories.)

What do you mean? C-e-a is a triad. Over?

> In your work, does 1/5:1/4:1/3 (c-e-a) score higher in consonance
than
> it's 'reflection' 3:4:5, the latter being a 6.4 chord?

Not really -- my work doesn't recognize the supposed 'dissonance' of
the 6.4 chord -- it addresses 'discordance' rather than 'dissonance',
the latter being very context- and culture-specific.

> (I know from
> diff. tones, etc, that's not the case ... but ...Hmm, I've never
> inquired if voice leading expectations have anything to do with
your
> models of consonance. ..

No, but they have something to do with my scale theories . . .

>I guess such a model would need to include
> cultural conditionaing...(and if that's the case, if there's
> conditioning, then wouldn't that make it part of the
psychoacoustical
> apparatus?)

Yup!

> > > ia this the only traditional type chrod which can be
represented
> by
> > > numbers 5 and under?)
> >
> > You can express F,-F-c-f-a as 1:2:3:4:5.
>
> But with chord defined as 3 notes, no duplicates, closed position,
and
> based on thirds ('traditional')?

c-f-a is 3:4:5, and is 'based on thirds' just as much as c-e-a.

> Maybe part of the specialness (to my ear) of the c-e-a chord --

To me it's special because of the common overtone, only two octaves
above the e -- most chords have their first common overtone (if any)
much higher up.

> speculating here -- is its 4/3 interval, which is the
> first 'melodically active' interval in the series (ie, voiceleading
> tendency, in this case, scale going 4-3, or 4-flat 3). Maybee --
> really going out on limb here -- the interval 3:4 in the series,
with
> it's strong leading tone tendency (-- I know this is all cultural
> conditioning!) as it occurs in the chords 3:4:5, and 1/5:1/4:1/3,
is a
> sort of 'gateway' between the otonal and utonal series. I just
notice
> how the two chords (eg, g-c-e//g-b-e) very easily flip back and
forth,
> hinged on the leading tone, sort of like, in chemistry,
> stereoisomers.

I don't see why you can't treat any of the other intervals in the
chord in the same exact way, especially if you allow inversion.

> (Also, 3:4:5/1

That would be c-f-a or some transposition thereof.

> is purer than 4:5:6, since 6 is sort
> of 'second generation'...just a double of 3.)
>
> Question: In surveys of music around the world and through time,
and
> if it is possible to restrict interval definition roughly to half
step
> or whole step as are commonl defined,

There are far too many sesquisteps in world music to restrict
interval definition thus.

> I was listening to
> some electroacoustical music the other night (Xenakis, Legend of
Eer),
> and noticed the prevelence (maybe by his design)

You can count on it! Have you heard his microtonal music?

> of minor thirds
> (produced scale wise, half step-whole step.) Just another pet
notion
> of mine, on the quality of 'minor-ness, that the minor interval can
be
> composed of a whole step and half step, sort of paralleling the big
> third, little third construction of the triad. I don't know...do
you
> think there could be anything to that, that a whole step and half
step
> are more stable in some way than two whole steps (ie, minor third
vs
> major third?)

I think there's a huge difference between the stability of the major
third in 12-equal (standard tuning) and that of a pure 4:5 major
third.

Paul,

You wrote:

> --- In harmonic_entropy@yahoogroups.com, "Yahya Abdal-Aziz"
>
> > Your question "Does this mean, in some sense, that there is a sort
> > of
> > mixing of waveforms in that note (of the third and fifth harmonic)?"
> > can be answered by saying, "Yes, the fifth harmonic _of the lowest
> > chord tone_ and the third harmonic _of the highest chord tone_ will
> > coincide with the fourth harmonic (the double octave) of the middle
> > chord tone. Further, if they happen to be in phase, they may
> > reinforce
> > each other, making the double octave of the middle chord tone more
> > prominent; however, if they happen to be opposite in phase, they may
> > cancel each other, making the double octave of the middle chord tone
> > less prominent or even absent."
>
> On real acoustic instruments, none of the interval ratios will be
> *exact* (even the best tuners commit small errors, and often
> temperament is desirable for musical reasons), so one will tend to
> hear a *loud beating pattern* at the double octave of the middle
> chord. It can be more complex that a simple wa-wa-wa beat, since
> *three* slightly different sine waves are involved.

Reasonable. On a purely electronically-tuned instrument, I'd expect
tuning errors small enough that beats would take several seconds to
evolve, and that the double octave would by strongly affected by
phase, as I outlined.

> > I wonder whether perhaps, when you asked your question, you were
> > thinking of the third and fifth harmonics of the fundamental tone in
> > a 3:4:5 major triad? Because in the context of your example of the
> > A minor triad in second inversion, C:E:A,
>
> That's first inversion ;)

But of course! :-)

Regards,
Yahya

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Kelly,

You wrote:

> Wow! what an amazing answer, Yahya! There's a lot there for me to
> think about...and much that you opened up. (Thanks to you, too,
> Paul.) ...

Do listen to Paul! He makes more sense than I do - even to me! :)

> ... I get so 'microscopically' into listening to the chords,
> trance-like, writing progressions with them, trying to imagine what
> the wave forms might look like, and what the numbers might have to
> do with it, that I guess I play amateur scientist here, which math
> laymen tend to do with number patterns. ...

And not only laymen. Many mathematicians see pattern as being the
core subject matter of mathematics, number "merely" providing some
elegant examples.

> ... So I feel rather foolish
> posting here. (I do it because I have learned some things here.)
> But I can see you really made an attempt to make sense of where I
> was going, or not going!, which I appreciate. (I can see you like
> number patters too! )

Right on both counts.

> The pattern that most interested me with this lovely chord c-e-a
> (and then I'll drop it) is when you represent each note (in my
> fallacious fashion, but please hear me out) with a fraction, giving
> 4/3, 5/3, and 5/4 (corresponding to c, e, and a, in the fashion
> discussed, with each fraction representing the overlap of the two
> intervals on that note.) ...

I'd say, rather, with each fraction being the only interval in the
chord that does not involve that note.

Let's make a chord by stacking two intervals, a:b and c:d, where
a < b and c < d. For concreteness, we play the chord on the bass note
R=100 Hz. So the second note is at 100*(b/a) Hz, and the third note
is at 100*(b/a)*(d/c). The interval from the first note to the third
is therefore 100*(b/a)*(d/c)/100, or (b/a)*(d/c). If we use any
other bass note in place of R, the interval will still always be the same
ratio (b/a)*(d/c).

In the case of the c:e:a chord, the intervals are 4:5, 3: 4 and 3:5.
That is, the interval ratios are 5/4, 4/3 and (5/4)*(4/3)=5/3. In
this product, you see that the two middle terms cancel, since they
have the same value, namely 4. A similar thing would happen whenever
the two middle terms in the product (b/a)*(d/c) for our general triad
(comprising intervals a:b and c:d) are equal - that is, whenever a=d.
We'd have the three intervals a:b, c:a and c:b. So this pattern quite
neatly generalises your 3, 4, 5 chord - and works for any three
integers c < a < b. As you pointed out in a later message to Paul, you
can also build a chord like this using 4, 5, 6. But you could equally
well choose to use any other triple of integers at all. Try this triple:
(5, 12, 13) - inspired by your Pythagorean example of (3, 4, 5) :-)
Or this: (1, 2, 3) - this gives you a very ancient consonance involving
only an octave, a fifth and a twelfth - not a third in sight! Or again,
(2, 3, 4) - a fifth, a fourth and an octave. Good Lord, we've invented
organum ... But your first and simplest "traditional" (tertian) chord
does arise from the triple (3, 4, 5) as you say below. If you like small
numbers, this one's hard to beat.

> ... I looked at many many chords in that
> fashion, and I believe c-e-a is the only one that has it's 'harmonic
> numbers' 3-4-5 represented in the fractions (starting with the
> demoninators, and moving up to the numerators, ie, 3,3,4,4,5,5.) I
> imagine there is an obvious reason for this, embedded in your
> answer, which I promise to study and respond to (--summer reading
> for my upcoming vaction!)...but can you think of any reason why the
> chord c-e-a would be the only chord which has that feature? (ISn't
> this the only traditional type chrod which can be represented by
> numbers 5 and under?) ...

See above.

> ...
> (The pattern also appealed to me because it reminded me of the
> ancient but highly accurate approximation of pi: 355/113, which, as
> Martin Gardner pointed out, has a simple number pattern (1,3,5) read
> from left to right, denominators to numerators.)

I don't ascribe any particular significance to this digit pattern -
what would it be if we counted in eights or twelves instead of tens?
8-0

> I will work on learning your method, and ...

Gosh, it was just a _pattern_ I saw ...:-)

> cheers, Kelly

All the best
Yahya

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> Rarer? Meaning the total waveform repeats more rarely?

No. The number pattern is less commom. What exactly do you mean by
waveform?

> ...(maybee all my searching for 'number purity' in the c-e-a
> > chord is a way to 'defend' it over the triad, per my own chord
> > preferences and theories.)

> What do you mean? C-e-a is a triad. Over?

Sorry. I should use other terminology, as I distinguish inversions
from root position. (I'll explain why later.)

> c-f-a is 3:4:5, and is 'based on thirds' just as much as c-e-a.

I agree. I was just defining 'chord', esp. in regard to open vs.
closed position, for eg, 10ths collapse to thirds, but ninths do not
collapse to 2nds!

> > Maybe part of the specialness (to my ear) of the c-e-a chord --

> To me it's special because of the common overtone, only two
octaves above the e -- most chords have their first common overtone
(if any) much higher up.

That's interesting. I"m starting to look at chords that way! (But
I still haven't found anyone who looks at chords they way I do, sob
sob...)

> [xenakis] You can count on it! Have you heard his microtonal music?

Can you suggest a title?

Hello

Concerning c-e-a, Paul wrote that it is "best understood" as 5:4:3/1,
whereas Yahya wrote that 12:15:20 is "most informative." Someone here
(Carl?) wrote that it was better to go with the lower-numbered version
(eg, 6:5:4/1 vs. 10-12-15). What do you suggest?

Just an observation: the 'flipped over-and-up' (ie, third inversion)
minor seventh chord, 3:5:8:13 -- it's difference tones, because of the
fabinnocci (sp) sequence, sort of curl inward on itself: 13-5=8, and,
8-3=5. I've only just started looking at chords as integers, and
don't know if that's unique. (Isn't it a great chord? So perfectly
spaced...)

(Yahya, thanks for your reply about the 'simplest...tertian chord').

best, Kelly

--- In harmonic_entropy@yahoogroups.com, "Yahya Abdal-Aziz"
<yahya@m...> wrote:
>
> Paul,
>
> You wrote:
>
> > --- In harmonic_entropy@yahoogroups.com, "Yahya Abdal-Aziz"
> >
> > > Your question "Does this mean, in some sense, that there is a
sort
> > > of
> > > mixing of waveforms in that note (of the third and fifth
harmonic)?"
> > > can be answered by saying, "Yes, the fifth harmonic _of the
lowest
> > > chord tone_ and the third harmonic _of the highest chord tone_
will
> > > coincide with the fourth harmonic (the double octave) of the
middle
> > > chord tone. Further, if they happen to be in phase, they may
> > > reinforce
> > > each other, making the double octave of the middle chord tone
more
> > > prominent; however, if they happen to be opposite in phase,
they may
> > > cancel each other, making the double octave of the middle chord
tone
> > > less prominent or even absent."
> >
> > On real acoustic instruments, none of the interval ratios will be
> > *exact* (even the best tuners commit small errors, and often
> > temperament is desirable for musical reasons), so one will tend
to
> > hear a *loud beating pattern* at the double octave of the middle
> > chord. It can be more complex that a simple wa-wa-wa beat, since
> > *three* slightly different sine waves are involved.
>
> Reasonable. On a purely electronically-tuned instrument, I'd expect
> tuning errors small enough that beats would take several seconds to
> evolve, and that the double octave would by strongly affected by
> phase, as I outlined.

Depends on the instrument. Unfortunately, many electronically-tuned
instruments don't have the resolution that you need for the beating
to be reliably this slow.

--- In harmonic_entropy@yahoogroups.com, "Yahya Abdal-Aziz"
<yahya@m...> wrote:
>
> Kelly,
>
> You wrote:
>
> > Wow! what an amazing answer, Yahya! There's a lot there for me
to
> > think about...and much that you opened up. (Thanks to you, too,
> > Paul.) ...
>
> Do listen to Paul! He makes more sense than I do - even to me! :)

And Yahya makes more sense to me than just about anyone.

>> > ... I get so 'microscopically' into listening to the chords,
>> > trance-like, writing progressions with them, trying to imagine
what
>> > the wave forms might look like, and what the numbers might have
to
>> > do with it, that I guess I play amateur scientist here, which
math
>> > laymen tend to do with number patterns. ...
>
>> And not only laymen. Many mathematicians see pattern as being the
>> core subject matter of mathematics, number "merely" providing some
>> elegant examples.

This is a key point, and an excellent amplification of your earlier
replies to Kelly.

> > The pattern that most interested me with this lovely chord c-e-a
> > (and then I'll drop it) is when you represent each note (in my
> > fallacious fashion, but please hear me out) with a fraction,
giving
> > 4/3, 5/3, and 5/4 (corresponding to c, e, and a, in the fashion
> > discussed, with each fraction representing the overlap of the
>two
> > intervals on that note.) ...
>
> I'd say, rather, with each fraction being the only interval in the
> chord that does not involve that note.

Only this latter description makes any sense to me, so if it's
correct, thanks Yahya!

--- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
wrote:
> > Rarer? Meaning the total waveform repeats more rarely?
>
> No. The number pattern is less commom.

Less common where?

> What exactly do you mean by
> waveform?

The pattern of compression and rarefaction of the air or the pattern
of movement of the speaker cone or of your eardrum, as a function of
time. You know, sine waves, sawtooth waves, triangle waves, square
waves . . .

> > ...(maybee all my searching for 'number purity' in the c-e-a
> > > chord is a way to 'defend' it over the triad, per my own chord
> > > preferences and theories.)
>
> > What do you mean? C-e-a is a triad. Over?
>
> Sorry. I should use other terminology, as I distinguish inversions
> from root position. (I'll explain why later.)
>
>
> > c-f-a is 3:4:5, and is 'based on thirds' just as much as c-e-a.
>
> I agree. I was just defining 'chord', esp. in regard to open vs.
> closed position, for eg, 10ths collapse to thirds, but ninths do
not
> collapse to 2nds!

I still don't see the distinction, if you're trying to draw one.

> > [xenakis] You can count on it! Have you heard his microtonal >
>music?
>
> Can you suggest a title?

_Pleaides_ is interesting, though I may have misspelled it :)

--- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...> wrote:
> Hello
>
> Concerning c-e-a, Paul wrote that it is "best understood" as 5:4:3/1,

Not quite -- 1/(5:4:3).

> whereas Yahya wrote that 12:15:20 is "most informative."

Sorry, I didn't mean to favor one over the other. In terms of actual
psychoacoustics, I believe 1/(5:4:3) is most convenient and most
telling when you're focused on the interaction between partials, while
12:15:20 makes it easier to see the combinational tones and such. If
you're working with pure sine waves, there aren't any partials and the
1/(5:4:3) ceases to be of much psychoacoustic relevance.

> Just an observation: the 'flipped over-and-up' (ie, third inversion)
> minor seventh chord, 3:5:8:13

In what kind of scale or tuning system does this kind of minor seventh
chord arise for you?

> -- it's

its

> difference tones, because of the
> fabinnocci (sp)

Fibonacci

> sequence, sort of curl inward on itself: 13-5=8, and,
> 8-3=5. I've only just started looking at chords as integers, and
> don't know if that's unique.

Starting with any two notes, a and b, the chord a : b : a+b : a+2b will
have this property, won't it?

--- In harmonic_entropy@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
> wrote:
> > > Rarer? Meaning the total waveform repeats more rarely?
> >
> > No. The number pattern is less commom.
>
> Less common where?

In my 'survey' of utonal chords, now debunked, where I looked at the
patterns of integers in the three fractions.

> > What exactly do you mean by
> > waveform?
>
> The pattern of compression and rarefaction of the air or the
pattern
> of movement of the speaker cone or of your eardrum, as a function
of
> time. You know, sine waves, sawtooth waves, triangle waves, square
> waves . . .

Ok. (I thought you may have been specifically referring to, for
example, where 9/8 is described -- if I can recall Partch's wording -
- 3 vibrations thrice, and two vibrations 4 times. Incidentally,
that sort of description of vibrations is what led me to speculate
that the middle note in a 3 note chord somehow 'mixes the adjoining
frequencies of the adjacent intervals....)

> > > ...(maybee all my searching for 'number purity' in the c-e-a
> > > > chord is a way to 'defend' it over the triad, per my own
chord
> > > > preferences and theories.)
> >
> > > What do you mean? C-e-a is a triad. Over?
> >
> > Sorry. I should use other terminology, as I distinguish
inversions
> > from root position. (I'll explain why later.)
> >
> >
> > > c-f-a is 3:4:5, and is 'based on thirds' just as much as c-e-a.
> >
> > I agree. I was just defining 'chord', esp. in regard to open
vs.
> > closed position, for eg, 10ths collapse to thirds, but ninths do
> not
> > collapse to 2nds!
>
> I still don't see the distinction, if you're trying to draw one.

Root positions and their inversions are the same for you?

>
> > > [xenakis] You can count on it! Have you heard his microtonal >
> >music?
> >
> > Can you suggest a title?
>
> _Pleaides_ is interesting, though I may have misspelled it :)

--- In harmonic_entropy@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
wrote:
> > Hello
> >
> > Concerning c-e-a, Paul wrote that it is "best understood" as
5:4:3/1,
>
> Not quite -- 1/(5:4:3).

...oops...(dah)

> > whereas Yahya wrote that 12:15:20 is "most informative."
>
> Sorry, I didn't mean to favor one over the other. In terms of
actual
> psychoacoustics, I believe 1/(5:4:3) is most convenient and most
> telling when you're focused on the interaction between partials,
while
> 12:15:20 makes it easier to see the combinational tones and such.
If
> you're working with pure sine waves, there aren't any partials and
the
> 1/(5:4:3) ceases to be of much psychoacoustic relevance.
>
> > Just an observation: the 'flipped over-and-up' (ie, third
inversion)
> > minor seventh chord, 3:5:8:13
>
> In what kind of scale or tuning system does this kind of minor
seventh
> chord arise for you?

My piano tuner is my tuning system! I'm still rather 'asleep' to
tuning system issues. I'm really making an effort to learn, reading
your articles, Partch's Genesis of Music, etc, to better interact
with the list, but by practice, (ie, writing progressions and
listening to chords), I'm more and more believing in intervals (and
by extensions) chords as Types, as they operate in my type of
harmony. (I have a rather extensive theory I will share here
later.) But obviously we (tuning math, [and performer? composer?]
background, vs. a more traditional product of university theory/comp
(ucberkeley) curriculum like myself have some interests in common.

>
> > -- it's
>
> its
>
> > difference tones, because of the
> > fabinnocci (sp)
>
> Fibonacci
>
> > sequence, sort of curl inward on itself: 13-5=8, and,
> > 8-3=5. I've only just started looking at chords as integers,
and
> > don't know if that's unique.
>
> Starting with any two notes, a and b, the chord a : b : a+b : a+2b
will
> have this property, won't it?

I can't even do a simple equation like that. I think there's a sort
of 'equation' dyslexia (...giving myself the benefit that I"m not
stupid in all matters math)...so can you give an example, please?

Hi Paul.

paul wrote>In the minor 9th chord C2-Eb3-D4...you mention as
> 10:24:45 (which puts the fundamental 1 at Ab-2, 4 octaves below
Ab2),
> and then calculate the difference tones: 24-10 = 14, 45-10 = 35,
and 45- 24 = 21. Relative to the chord in question, these notes
would be a flat > Gb2, a flat Bb3 (a wolf fifth up from the Eb), and
a flat Db3.
> > > 12-7, 15-8, and 9-2). > This chord doesn't have coinciding
difference tones, but does have > difference tones which are the
second, third, and fifth harmonics of a > flat Gb1. Hence, if the
chord is played in true JI, due to > psychoacoustic effects I might
expect to hear a suggestion of a flat > Gb1 pitch emerging when the
chord is ramped up in amplitude. Very interesting! >

So, that's a unique effect? Are you referring to how deep the Gb
is? There seem to be many chords (like all the utonal chords, and
those higher up in the series), whose difference tone sets, like the
one above, are, in themselves chords (ie, tertial).

--- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
wrote:

> > > > ...(maybee all my searching for 'number purity' in the c-e-a
> > > > > chord is a way to 'defend' it over the triad, per my own
> chord
> > > > > preferences and theories.)
> > >
> > > > What do you mean? C-e-a is a triad. Over?
> > >
> > > Sorry. I should use other terminology, as I distinguish
> inversions
> > > from root position. (I'll explain why later.)
> > >
> > >
> > > > c-f-a is 3:4:5, and is 'based on thirds' just as much as c-e-
a.
> > >
> > > I agree. I was just defining 'chord', esp. in regard to open
> vs.
> > > closed position, for eg, 10ths collapse to thirds, but ninths
do
> > not
> > > collapse to 2nds!
> >
> > I still don't see the distinction, if you're trying to draw one.
>
> Root positions and their inversions are the same for you?

No, but neither of these chords are in root position, and in regard
to what you wrote above, both are in closed position (i.e., close-
voiced).

--- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
wrote:

> > > Just an observation: the 'flipped over-and-up' (ie, third
> inversion)
> > > minor seventh chord, 3:5:8:13
> >
> > In what kind of scale or tuning system does this kind of minor
> seventh
> > chord arise for you?
>
> My piano tuner is my tuning system!

Then there probably isn't anything close to 3:5:8:13 on there.

> I'm still rather 'asleep' to
> tuning system issues.

Well, get yourself a piano tuning wrench and wake up!

> I'm really making an effort to learn, reading
> your articles, Partch's Genesis of Music, etc, to better interact
> with the list, but by practice, (ie, writing progressions and
> listening to chords), I'm more and more believing in intervals (and
> by extensions) chords as Types, as they operate in my type of
> harmony.

What tunings are you listening to them in? When you're talking about
difference tones and the like it makes a *huge* difference (even
small changes in the tuning do).

> > > -- it's
> >
> > its
> >
> > > difference tones, because of the
> > > fabinnocci (sp)
> >
> > Fibonacci
> >
> > > sequence, sort of curl inward on itself: 13-5=8, and,
> > > 8-3=5. I've only just started looking at chords as integers,
> and
> > > don't know if that's unique.
> >
> > Starting with any two notes, a and b, the chord a : b : a+b :
a+2b
> will
> > have this property, won't it?
>
> I can't even do a simple equation like that. I think there's a
sort
> of 'equation' dyslexia (...giving myself the benefit that I"m not
> stupid in all matters math)...so can you give an example, please?

examples:
1:2:3:5
1:3:4:7
2:3:5:8
1:4:5:9
3:4:7:11
1:5:6:11
2:5:7:12
3:5:8:13
1:6:7:13
5:6:11:17
1:7:8:15
2:7:9:16
3:7:10:17
4:7:11:18
5:7:12:19
6:7:13:20

etc.

--- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
wrote:
>> Hi Paul.
>
>> paul wrote>In the minor 9th chord C2-Eb3-D4...you mention as
>> > 10:24:45 (which puts the fundamental 1 at Ab-2, 4 octaves below
>> Ab2),
>> > and then calculate the difference tones: 24-10 = 14, 45-10 = 35,
>> and 45- 24 = 21. Relative to the chord in question, these notes
>> would be a flat > Gb2, a flat Bb3 (a wolf fifth up from the Eb),
and
>> a flat Db3.
>> > > > 12-7, 15-8, and 9-2). > This chord doesn't have coinciding
>> difference tones, but does have > difference tones which are the
>> second, third, and fifth harmonics of a > flat Gb1. Hence, if the
>> chord is played in true JI, due to > psychoacoustic effects I
>might
>> expect to hear a suggestion of a flat > Gb1 pitch emerging when
>the
>> chord is ramped up in amplitude. Very interesting! >
>
> So, that's a unique effect?

No, but it's not that common either.

> Are you referring to how deep the Gb
> is?

Not especially.

> There seem to be many chords (like all the utonal chords, and
> those higher up in the series), whose difference tone sets, like
the
> one above, are, in themselves chords (ie, tertial).

Indeed.

Hi Yahya (and the group)- when you wrote, in your first response to
my "Two questions", that the middle note of c-e-a can not be
represented by both both the number 5 and 3 (representing the upper
note of 5/4 and the lower note of 4/3), that 'numbers don't work that
way', I don't know how to reconcile that with writings of Paul's where
he describes how, in hearing a chord, the brain "shuffles ...between
the harmonic series and roots implied by the various dyad subsets ."
(From a post here May 04). Or in his "On Harmonic Entropy" (c. 1997),
where he wrote, concerning 10:12:15, that "these numbers are already
too high for the entropy of the entire signal to be low enough to
complete with the low entropy of the [various dyad subsets]." This
seems to imply, to me, that the hearing mechanism is, in some way,
hearing multiple numbers for the e, in c-e-a. Can you comment?

thanks, kElly

--- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...> wrote:

> Hi Yahya (and the group)- when you wrote, in your first response to
> my "Two questions", that the middle note of c-e-a can not be
> represented by both both the number 5 and 3 (representing the upper
> note of 5/4 and the lower note of 4/3), that 'numbers don't work that
> way', I don't know how to reconcile that with writings of Paul's
where
> he describes how, in hearing a chord, the brain "shuffles ...between
> the harmonic series and roots implied by the various dyad subsets ."

Yes, in a sense, the middle note can be heard as both the 3rd harmonic
of A, and also as the 5th harmonic of C; thus either A or C may be
heard as "the root" of this chord. A minor? C6? But there's no real
dissonance in the sound, it's really smooth. Perhaps it's kind of
unique as a very smooth, yet unrooted, sonority.

> (From a post here May 04). Or in his "On Harmonic Entropy" (c.
1997),
> where he wrote, concerning 10:12:15, that "these numbers are already
> too high for the entropy of the entire signal to be low enough to
> complete with the low entropy of the [various dyad subsets]."

Occasionally (such as when you have distortion and just intonation),
they're not too high, and you hear the chord as the 12th, 15th, and
20th harmonics of a low F. Other times, though, the notes c and a might
be heard as the 3rd and 5th harmonics of F (with e non-harmonic).

> This
> seems to imply, to me, that the hearing mechanism is, in some way,
> hearing multiple numbers for the e, in c-e-a. Can you comment?

It may be hearing multiple numbers for every note. If C is the root, c
and e are the 4th and 5th harmonics, and a is non-harmonic. If A is the
root, e and a are the 3rd and 4th harmonics, and c is non-harmonic. In
both cases, e is harmonic. Not suprising, given the prominent common
overtone at e'' that we talked about before. But if F is the perceived
root, c and a can be the 3rd and 5th harmonics, and e is the "least
harmonic" note in the chord. So possibly each note is heard in
two "numerical" or harmonic senses.

Research seems to indicate that the 3rd, 4th, and 5th harmonics are
most important to the ear (more even than the fundamental!) in tracking
pitch. So perhaps this (1/(5:4:3)) is the best chord for eliciting an
ambiguous response in the overall pitch-tracking (allowing each note to
ring out independently from the sonority as a whole), while at the same
time avoiding any harsh conflicts in the overlapping pattern of
overtones.

Thanks for your interesting and informative comments, Paul.

> Research seems to indicate that the 3rd, 4th, and 5th harmonics are
> most important to the ear (more even than the fundamental!) in
tracking > pitch.

This is why I need to review the Plomp-Sethares model, because, from
my incomplete view of the literature, jumping from Plomp's mid-80's
writing on frequency analysis of *complex tones* ("Sensation of
Pitch") to your work on discordance in *dyads*, there seems to be a
tantalyzing similarity between the frequency analsis of harmonics in a
complex tone and the assignation of harmonic series and root in a dyad
or triad, as if the ear treats both signals (complex tone, and chord)
similiarly. (I will probably clear some of this up in my future
reading.)

That's part of the reason I had the 'off the wall' idea that, in c-e-
a, the 3rd harmonic of a and the 5th harmonic of c sort
of 'heterodyne' in the middle note of the chord, trying to draw a
parallel with how, in a complex tone, the sinusoidal components of
individual harmonics 'mix' to form the complex wave form.

--- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
wrote:
> Thanks for your interesting and informative comments, Paul.
>
>
> > Research seems to indicate that the 3rd, 4th, and 5th harmonics
are
> > most important to the ear (more even than the fundamental!) in
> tracking > pitch.
>
>
> This is why I need to review the Plomp-Sethares model, because,
from
> my incomplete view of the literature, jumping from Plomp's mid-80's
> writing on frequency analysis of *complex tones* ("Sensation of
> Pitch") to your work on discordance in *dyads*, there seems to be a
> tantalyzing similarity between the frequency analsis of harmonics
in a
> complex tone and the assignation of harmonic series and root in a
dyad
> or triad, as if the ear treats both signals (complex tone, and
chord)
> similiarly.

Right, but this has nothing to do with the Plomp-Levelt-Sethares
discordance model that was brought up. Instead, it's part of the
justification for harmonic entropy.

> (I will probably clear some of this up in my future
> reading.)
>
> That's part of the reason I had the 'off the wall' idea that, in c-
e-
> a, the 3rd harmonic of a and the 5th harmonic of c sort
> of 'heterodyne' in the middle note of the chord, trying to draw a
> parallel with how, in a complex tone, the sinusoidal components of
> individual harmonics 'mix' to form the complex wave form.

Unfortunately I don't see the parallel. Every note in the chord seems
to have the numerical requirements for the phenomenon you seem to
have pointed out, not just the middle note. Unless you're considering
octave equivalence, in which case the middle note is equivalent to
the lowest common overtone. In either case, this doesn't seem to have
anything to do with the important justification for harmonic entropy
that you brought up -- hearing a chord is 'like' hearing a complex
tone in many ways, inasmuch as hearing a root in a chord may
be 'like' hearing a fundamental pitch in a set of sine-wave frequency
components and may be mediated by the very same neurological
processes.

Hi Kelly,

On Fri, 16 Sep 2005, you wrote, and Paul replied:
> > Hi Yahya (and the group)- when you wrote, in your first response to
> > my "Two questions", that the middle note of c-e-a can not be
> > represented by both both the number 5 and 3 (representing the upper
> > note of 5/4 and the lower note of 4/3), that 'numbers don't work that
> > way', I don't know how to reconcile that with writings of Paul's
> > where
> > he describes how, in hearing a chord, the brain "shuffles ...between
> > the harmonic series and roots implied by the various dyad subsets ."
>
> Yes, in a sense, the middle note can be heard as both the 3rd harmonic
> of A, and also as the 5th harmonic of C; thus either A or C may be
> heard as "the root" of this chord. A minor? C6? But there's no real
> dissonance in the sound, it's really smooth. Perhaps it's kind of
> unique as a very smooth, yet unrooted, sonority.
>
> > (From a post here May 04). Or in his "On Harmonic Entropy" (c.
> > 1997),
> > where he wrote, concerning 10:12:15, that "these numbers are already
> > too high for the entropy of the entire signal to be low enough to
> > complete with the low entropy of the [various dyad subsets]."
>
> Occasionally (such as when you have distortion and just intonation),
> they're not too high, and you hear the chord as the 12th, 15th, and
> 20th harmonics of a low F. Other times, though, the notes c and a might
> be heard as the 3rd and 5th harmonics of F (with e non-harmonic).
>
> > This
> > seems to imply, to me, that the hearing mechanism is, in some way,
> > hearing multiple numbers for the e, in c-e-a. Can you comment?
>
> It may be hearing multiple numbers for every note. If C is the root, c
> and e are the 4th and 5th harmonics, and a is non-harmonic. If A is the
> root, e and a are the 3rd and 4th harmonics, and c is non-harmonic. In
> both cases, e is harmonic. Not suprising, given the prominent common
> overtone at e'' that we talked about before. But if F is the perceived
> root, c and a can be the 3rd and 5th harmonics, and e is the "least
> harmonic" note in the chord. So possibly each note is heard in
> two "numerical" or harmonic senses.
>
> Research seems to indicate that the 3rd, 4th, and 5th harmonics are
> most important to the ear (more even than the fundamental!) in tracking
> pitch. So perhaps this (1/(5:4:3)) is the best chord for eliciting an
> ambiguous response in the overall pitch-tracking (allowing each note to
> ring out independently from the sonority as a whole), while at the same
> time avoiding any harsh conflicts in the overlapping pattern of
> overtones.

Sorry for taking so long to reply - I've been working (for money!), so
haven't had time to attend much to mailing lists.

In any case, I don't think I could possibly improve on Paul's reply.
All the more so, as he's _the_ proponent of the "harmonic entropy"
approach, which I do not pretend to fully understand.

I do think Paul's explanation of the roots implied by various dyad
subsets makes sense, don't you? And given that any triad has three
such subsets, and any tetrad has six, gives the brain a wealth of
potential roots to use. No wonder that, very often, we simply can't
make up our minds which is the "real" root of a chord. And this very
ambiguity provides the composer with an important tool, which,
carefully used, can heighten or lessen the tensions and uncertainty
we feel as a piece progresses, helping to shape the piece into a more
satisfying experience.

As for two numbers representing the same note, my point was simply
that they can't do so _at the same time_. When we are uncertain
exactly how to analyse the chord, our perceptions of that note may
switch back-and-forth quite rapidly, even several times a second,
between two or even more interpretations. In one interpretation,
we use one of the numbers, and in another, another.

If we take a chord superficially quite similar to your c-e-a, namely
e-a-c#, we have a situtation where the fourth (3:4) from e to a,
and the major third (4:5) from a to c#, use exactly the same
number (4) to represent the middle note a. The whole chord is
3:4:5; taking any dyad from it, whether 3:4, 4:5, or 3:5, we find
that each of them implies the same root (1), two octaves below our
a. This second inversion of the major triad thus gives a very firmly
rooted impression; it's hard to think of it as anything _but_ an A
major chord. This is a very different effect from your c-e-a
chord (or any other first inversion of a minor triad, which must
have the same ratios).

Regards,
Yahya

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