Yahoo Tuning Groups Ultimate Backup harmonic_entropy Where in the series would...?

--- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
wrote:
>
> (BTW, I could't find 15/8 on the harmonic entropy main
> graph. How does it rank in harmonic entropy compared to 16/15?)

It depends on how you parametrize your harmonic entropy calculation.
Do you mean the particular case that's graphed on the home page of
this yahoogroup?

> But I was mainly talking about how the inverted intervals 4/3, 6/5,
> 8/5, and 16/15 (and octave equivalents) might act together as
> a 'set', to produce an aggregate effect in this chord, like they're
> all manifesting the same effect, so to speak. (The 'inversion
> effect'.)

I don't hear the supposed effect that these intervals are supposed to
have in common. Nor can I think of any theoretical reason why any
such thing should be audible.

> How helpful, anyway, are the individual dissonance
> rankings of intervals in trying to understand the sound quality of
a
> 4 note chord?

They're somewhat helpful. Also helpful is the simplicity with which
the entire chord can be expressed as a set of harmonic numbers over
some "implied fundamental". Between those two pieces of information,
you can get a pretty good sense of how dissonant the chord will be
perceived to be.

> That's why I tentatively ranked the intervals the
> way I did. Not based on dissonance, but on 'overnumberness'. :)
>
> The presense of those intervals together also reminded me loosely
of
> Partch's utonality,

Not to interrupt, but a utonality and the corresponding otonality
normally have exactly the same set of intervals . . .

> as he described it in his Genesis of Music,
> where the 'unity' (power of 1,3, 5, etc)

The 'unity' can be 1 or any number as long as it's the same for all
the notes in the utonality. I don't know why you bring up "power"
here . . .

> is in the numerators.

Of the *pitch* ratios, or interval ratios relative to a *constant*
reference pitch.

> But
> in this chord in question (b2-d4-g4-c5), there is a mixture
> of 'unities' in the numerators.

Of the *interval* ratios between different *pairs* of notes in the
chord, and only when you always measure them upwards. Totally
different scenario.

> Or maybee I'm full of shit, but at
> least I'm not watching television.

> (I was curious that, in TTSS, Sethares mentions Partch's 'One
Footed
> Bride', but not, I don't believe, the theory of Utonality. Do you
> think that is because, as I saw in a post somewhere, he (Sethares),
> does not 'believe in' the significance of subharmonics? thx, Kelly

No.

Actually, even without 'believing in' the significance or existence
of subharmonics, Sethares' theory predicts that utonalities will be
about equally consonant as their corresponding otonalities! This is
because the utonal and the otonal chord will have the exact same set
of intervals, and Sethares sums the dissonances of the individual
intervals to get the dissonance of the chord. The fact that higher-
limit otonalities are much more consonant than their corresponding
utonalities shows that there's a serious problem with Sethares's
theory, and he sort of mentions this in the second edition of his
book.

🔗wallyesterpaulrus <wallyesterpaulrus@...>

🔗traktus5 <kj4321@...>

--- In harmonic_entropy@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
> wrote:
> >
> > (BTW, I could't find 15/8 on the harmonic entropy main
> > graph. How does it rank in harmonic entropy compared to 16/15?)
>
> It depends on how you parametrize your harmonic entropy
calculation.
> Do you mean the particular case that's graphed on the home page of
> this yahoogroup?

Yes.

>
> > But I was mainly talking about how the inverted intervals 4/3,
6/5,
> > 8/5, and 16/15 (and octave equivalents) might act together as
> > a 'set', to produce an aggregate effect in this chord, like
they're
> > all manifesting the same effect, so to speak. (The 'inversion
> > effect'.)
>
> I don't hear the supposed effect that these intervals are supposed
to
> have in common. Nor can I think of any theoretical reason why any
> such thing should be audible
> > How helpful, anyway, are the individual dissonance
> > rankings of intervals in trying to understand the sound quality
of
> a
> > 4 note chord?
>
> They're somewhat helpful. Also helpful is the simplicity with
which
> the entire chord can be expressed as a set of harmonic numbers
over
> some "implied fundamental". Between those two pieces of
information,
> you can get a pretty good sense of how dissonant the chord will be
> perceived to be.
>
> > That's why I tentatively ranked the intervals the
> > way I did. Not based on dissonance, but
on 'overnumberness'. :)
> >
> > The presense of those intervals together also reminded me
loosely
> of
> > Partch's utonality,
>
> Not to interrupt, but a utonality and the corresponding otonality
> normally have exactly the same set of intervals . . .
>
> > as he described it in his Genesis of Music,
> > where the 'unity' (power of 1,3, 5, etc)
>
> The 'unity' can be 1 or any number as long as it's the same for
all
> the notes in the utonality. I don't know why you bring up "power"
> here . . .

I thought a doubling series, as in the numerators of Partch's
example of the minor chord 1/1, 6/5, 3/2, was a power.

>
> > is in the numerators.
>
> Of the *pitch* ratios, or interval ratios relative to a *constant*
> reference pitch.
>
> > But
> > in this chord in question (b2-d4-g4-c5), there is a mixture
> > of 'unities' in the numerators.
>
> Of the *interval* ratios between different *pairs* of notes in the
> chord, and only when you always measure them upwards.

What does it mean...Upward?

>Totally different scenario.

Maybe I'm just chasing the mystery of this chord, but I see a
connection, which, if I can make sense of it, I will try to discuss
here later. Also, from the Partch music I've heard (though I should
hear more), I don't quite fully appreciate the sound of his chords
as compared to, for example, the sound of Stravinsky's chords from
Le Sacre. So, at a gut level, this makes me a little skeptical of
his theory. I believe my chord explortion is more related to the
latter (Stravinsky) type of chords, where part of the chords quality
is due to their references (sometimes twisted, nostalgic, etc) to
functional harmony. (As I've mentioned, however, I'm exploring the
acoustics of the chords and progressions.)

>
> > Or maybee I'm full of shit, but at
> > least I'm not watching television.
>
> :)
>
> > (I was curious that, in TTSS, Sethares mentions Partch's 'One
> Footed
> > Bride', but not, I don't believe, the theory of Utonality. Do
you
> > think that is because, as I saw in a post somewhere, he
(Sethares),
> > does not 'believe in' the significance of subharmonics? thx,
Kelly
>
> No.
>
> Actually, even without 'believing in' the significance or
existence
> of subharmonics, Sethares' theory predicts that utonalities will
be
> about equally consonant as their corresponding otonalities! This
is
> because the utonal and the otonal chord will have the exact same
set
> of intervals, and Sethares sums the dissonances of the individual
> intervals to get the dissonance of the chord. The fact that higher-
> limit otonalities are much more consonant than their corresponding
> utonalities shows that there's a serious problem with Sethares's
> theory, and he sort of mentions this in the second edition of his
> book.
>

🔗traktus5 <kj4321@...>

> > But I was mainly talking about how the inverted intervals 4/3,
6/5,
> > 8/5, and 16/15 (and octave equivalents) might act together as
> > a 'set', to produce an aggregate effect in this chord, like
they're
> > all manifesting the same effect, so to speak. (The 'inversion
> > effect'.)
>
> I don't hear the supposed effect that these intervals are supposed
to
> have in common.

Thinking of this, I was suprised by your answer. At least that part
of your hearing that recognizes common practice harmony, don't you
hear 5:6:8 as having inverted intervals in it (esp 8/5)? Then, with
the added upper interval (say, the f3 above e2-g2-c3), more inverted
intervals! For me, the dissonance of 32/15 especially adds to
the 'inversion' effect. Just my impressions. Guess it's not
provable with know acoustics...

Nor can I think of any theoretical reason why any
> such thing should be audible.
>
> > How helpful, anyway, are the individual dissonance
> > rankings of intervals in trying to understand the sound quality
of
> a
> > 4 note chord?
>
> They're somewhat helpful. Also helpful is the simplicity with
which
> the entire chord can be expressed as a set of harmonic numbers
over
> some "implied fundamental". Between those two pieces of
information,
> you can get a pretty good sense of how dissonant the chord will be
> perceived to be.
>
> > That's why I tentatively ranked the intervals the
> > way I did. Not based on dissonance, but
on 'overnumberness'. :)
> >
> > The presense of those intervals together also reminded me
loosely
> of
> > Partch's utonality,
>
> Not to interrupt, but a utonality and the corresponding otonality
> normally have exactly the same set of intervals . . .
>
> > as he described it in his Genesis of Music,
> > where the 'unity' (power of 1,3, 5, etc)
>
> The 'unity' can be 1 or any number as long as it's the same for
all
> the notes in the utonality. I don't know why you bring up "power"
> here . . .
>
> > is in the numerators.
>
> Of the *pitch* ratios, or interval ratios relative to a *constant*
> reference pitch.
>
> > But
> > in this chord in question (b2-d4-g4-c5), there is a mixture
> > of 'unities' in the numerators.
>
> Of the *interval* ratios between different *pairs* of notes in the
> chord, and only when you always measure them upwards. Totally
> different scenario.
>
> > Or maybee I'm full of shit, but at
> > least I'm not watching television.
>
> :)
>
> > (I was curious that, in TTSS, Sethares mentions Partch's 'One
> Footed
> > Bride', but not, I don't believe, the theory of Utonality. Do
you
> > think that is because, as I saw in a post somewhere, he
(Sethares),
> > does not 'believe in' the significance of subharmonics? thx,
Kelly
>
> No.
>
> Actually, even without 'believing in' the significance or
existence
> of subharmonics, Sethares' theory predicts that utonalities will
be
> about equally consonant as their corresponding otonalities! This
is
> because the utonal and the otonal chord will have the exact same
set
> of intervals, and Sethares sums the dissonances of the individual
> intervals to get the dissonance of the chord. The fact that higher-
> limit otonalities are much more consonant than their corresponding
> utonalities shows that there's a serious problem with Sethares's
> theory, and he sort of mentions this in the second edition of his
> book.
>

🔗wallyesterpaulrus <wallyesterpaulrus@...>

In that case, 15/8 has slightly lower entropy than 16/15. The numbers
are 4.80 vs. 4.85.

> > The 'unity' can be 1 or any number as long as it's the same for
> all
> > the notes in the utonality. I don't know why you bring up "power"
> > here . . .
>
>
> I thought a doubling series, as in the numerators of Partch's
> example of the minor chord 1/1, 6/5, 3/2, was a power.

I don't get it. Can you elaborate? I'm certainly new to calling
anything like this "a power." Where do you get this terminology from?

> > > is in the numerators.
> >
> > Of the *pitch* ratios, or interval ratios relative to a
*constant*
> > reference pitch.
> >
> > > But
> > > in this chord in question (b2-d4-g4-c5), there is a mixture
> > > of 'unities' in the numerators.
> >
> > Of the *interval* ratios between different *pairs* of notes in
the
> > chord, and only when you always measure them upwards.
>
> What does it mean...Upward?

This means you're always calculating the interval from the lower note
to the upper note, and never the other way around. By contrast, when
the ratios are relative to a fixed reference pitch, they can (and
must) sometimes be upward, sometimes downward.

> >Totally different scenario.
>
> Maybe I'm just chasing the mystery of this chord, but I see a
> connection, which, if I can make sense of it, I will try to discuss
> here later. Also, from the Partch music I've heard (though I
should
> hear more), I don't quite fully appreciate the sound of his chords
> as compared to, for example, the sound of Stravinsky's chords from
> Le Sacre.

Whoa -- this is quite a jump. I think it takes a long, intense period
of listening to Partch, *particularly* if you're not already familiar
with much microtonal music, to even begin to appreciate his
sonorities in their own right. But I would agree that the sound
Partch's *utonal* chords is difficult if not impossible to appreciate
(as consonant harmony), which seems to speak loudly against dualism.

> So, at a gut level, this makes me a little skeptical of
> his theory.

Well, I'm skeptical of his dualism, but in many timbres, the
striking, special, stable quality of his *otonal* chords seems
unquestionable. Besides, there's a lot more to 'his theory' than just
a couple of kinds of chords, so you should try to narrow down a bit
what you're referring to (I know, it's only a gut feeling, but don't
gut feelings have a way of being *so* unfair sometimes)?

> > > Or maybee I'm full of shit, but at
> > > least I'm not watching television.
> >
> > :)
> >
> > > (I was curious that, in TTSS, Sethares mentions Partch's 'One
> > Footed
> > > Bride', but not, I don't believe, the theory of Utonality. Do
> you
> > > think that is because, as I saw in a post somewhere, he
> (Sethares),
> > > does not 'believe in' the significance of subharmonics? thx,
> Kelly
> >
> > No.
> >
> > Actually, even without 'believing in' the significance or
> existence
> > of subharmonics, Sethares' theory predicts that utonalities will
> be
> > about equally consonant as their corresponding otonalities! This
> is
> > because the utonal and the otonal chord will have the exact same
> set
> > of intervals, and Sethares sums the dissonances of the individual
> > intervals to get the dissonance of the chord. The fact that
higher-
> > limit otonalities are much more consonant than their
corresponding
> > utonalities shows that there's a serious problem with Sethares's
> > theory, and he sort of mentions this in the second edition of his
> > book.

Did you have no response to this? Was it unclear? Can I clarify
somehow?

🔗wallyesterpaulrus <wallyesterpaulrus@...>

🔗traktus5 <kj4321@...>

--- In harmonic_entropy@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
> wrote:
> >
> > --- In harmonic_entropy@yahoogroups.com, "wallyesterpaulrus"
> > <wallyesterpaulrus@y...> wrote:
> > >
> > > --- In harmonic_entropy@yahoogroups.com, "traktus5"
<kj4321@h...>
> > > wrote:
> > > >
> > > > (BTW, I could't find 15/8 on the harmonic entropy main
> > > > graph. How does it rank in harmonic entropy compared to
16/15?)
> > >
> > > It depends on how you parametrize your harmonic entropy
> > calculation.
> > > Do you mean the particular case that's graphed on the home
page
> of
> > > this yahoogroup?
> >
> > Yes.
>
> In that case, 15/8 has slightly lower entropy than 16/15. The
numbers
> are 4.80 vs. 4.85.
>
>
> > > The 'unity' can be 1 or any number as long as it's the same
for
> > all
> > > the notes in the utonality. I don't know why you bring
up "power"
> > > here . . .
> >
> >
> > I thought a doubling series, as in the numerators of Partch's
> > example of the minor chord 1/1, 6/5, 3/2, was a power.
>
> I don't get it. Can you elaborate? I'm certainly new to calling
> anything like this "a power." Where do you get this terminology
from?
>

I just don't know my math termininology. (I should look it up) I
thought 'power' meant logarhytms, and that doubling series were a
form of.

> > > > is in the numerators.
> > >
> > > Of the *pitch* ratios, or interval ratios relative to a
> *constant*
> > > reference pitch.
> > >
> > > > But
> > > > in this chord in question (b2-d4-g4-c5), there is a mixture
> > > > of 'unities' in the numerators.
> > >
> > > Of the *interval* ratios between different *pairs* of notes in
> the
> > > chord, and only when you always measure them upwards.
> >
> > What does it mean...Upward?
>
> This means you're always calculating the interval from the lower
note
> to the upper note, and never the other way around. By contrast,
when
> the ratios are relative to a fixed reference pitch, they can (and
> must) sometimes be upward, sometimes downward.
>
> > >Totally different scenario.

Is this process of measuring intervals part of Partch utonality, or
harmonic entropy?

I'm a little bit stuck on understanding this, maybe because I first
have to deal with a perhaps related question which I'm fixated on
(related to my musings on an 'inversion effect'): considering just
some simple intervals like 3/2, 5/3, and 5/4, is a measure of the
higher harmonic entropy (or lesser tonalness) of their corresponding
inversions (4/3, 6/5, and 8/5) a direct result (or caused by, in
some manner) the inversion --(of having a lower limit number over a
higher limit number)? (I realize there are cases where the opposite
is probably true: 9/8 is --am I correct?--less consonant than 16/9).

> >
> > Maybe I'm just chasing the mystery of this chord, but I see a
> > connection, which, if I can make sense of it, I will try to
discuss
> > here later. Also, from the Partch music I've heard (though I
> should
> > hear more), I don't quite fully appreciate the sound of his
chords
> > as compared to, for example, the sound of Stravinsky's chords
from
> > Le Sacre.
>
> Whoa -- this is quite a jump. I think it takes a long, intense
period
> of listening to Partch, *particularly* if you're not already
familiar
> with much microtonal music, to even begin to appreciate his
> sonorities in their own right. But I would agree that the sound
> Partch's *utonal* chords is difficult if not impossible to
appreciate
> (as consonant harmony), which seems to speak loudly against
dualism.
>
> > So, at a gut level, this makes me a little skeptical of
> > his theory.
>
> Well, I'm skeptical of his dualism, but in many timbres, the
> striking, special, stable quality of his *otonal* chords seems
> unquestionable. Besides, there's a lot more to 'his theory' than
just
> a couple of kinds of chords, so you should try to narrow down a
bit
> what you're referring to (I know, it's only a gut feeling, but
don't
> gut feelings have a way of being *so* unfair sometimes)?
>
> > > > Or maybee I'm full of shit, but at
> > > > least I'm not watching television.
> > >
> > > :)
> > >
> > > > (I was curious that, in TTSS, Sethares mentions
Partch's 'One
> > > Footed
> > > > Bride', but not, I don't believe, the theory of Utonality.
Do
> > you
> > > > think that is because, as I saw in a post somewhere, he
> > (Sethares),
> > > > does not 'believe in' the significance of subharmonics?
thx,
> > Kelly
> > >
> > > No.
> > >
> > > Actually, even without 'believing in' the significance or
> > existence
> > > of subharmonics, Sethares' theory predicts that utonalities
will
> > be
> > > about equally consonant as their corresponding otonalities!
This
> > is
> > > because the utonal and the otonal chord will have the exact
same
> > set
> > > of intervals, and Sethares sums the dissonances of the
individual
> > > intervals to get the dissonance of the chord. The fact that
> higher-
> > > limit otonalities are much more consonant than their
> corresponding
> > > utonalities shows that there's a serious problem with
Sethares's
> > > theory, and he sort of mentions this in the second edition of
his
> > > book.
>
> Did you have no response to this? Was it unclear? Can I clarify
> somehow?
>

I recall a statement (of his?) which corrected that notion, but I
can't remember if it was in his 2nd edition, which I read, but don't
own. It may have been in the section where he discusses your work.
I'm suprised he missed that one!

Speaking of roughness and tonalness, a nice chord (M7 inversion)
which gives me the sensation of those two phenomenon (or makes me
think about them ,at least) is 15-16-20 (and 12-15-16). In a very
impressionistic way, I hear a very rough interval (16/15) combining
with the other intervals in an interesting way. Being a number
nut, I like to imagine, in my weird science kind of way, that the
effect has something to do with the fact that, not only does 16/15 x
5/4 = 4/3, but, by cross dividing , 16/4 and 15/5 equals 3/4. It's
a sort of 'unity' (3/4 x 4/3) for me, which I like to think is
another way of "getting back to one", in addition to the fundamental
of psychoacoustic tonalness!

🔗traktus5 <kj4321@...>

--- In harmonic_entropy@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
> wrote:
> >
> > --- In harmonic_entropy@yahoogroups.com, "wallyesterpaulrus"
> > <wallyesterpaulrus@y...> wrote:
> > >
> > > --- In harmonic_entropy@yahoogroups.com, "traktus5"
<kj4321@h...>
> > > wrote:
> > > >
> > > > (BTW, I could't find 15/8 on the harmonic entropy main
> > > > graph. How does it rank in harmonic entropy compared to
16/15?)
> > >
> > > It depends on how you parametrize your harmonic entropy
> > calculation.
> > > Do you mean the particular case that's graphed on the home
page
> of
> > > this yahoogroup?
> >
> > Yes.
>
> In that case, 15/8 has slightly lower entropy than 16/15. The
numbers
> are 4.80 vs. 4.85.
>
>
> > > The 'unity' can be 1 or any number as long as it's the same
for
> > all
> > > the notes in the utonality. I don't know why you bring
up "power"
> > > here . . .
> >
> >
> > I thought a doubling series, as in the numerators of Partch's
> > example of the minor chord 1/1, 6/5, 3/2, was a power.
>
> I don't get it. Can you elaborate? I'm certainly new to calling
> anything like this "a power." Where do you get this terminology
from?
>

I just don't know my math termininology. (I should look it up) I
thought 'power' meant logarhytms, and that doubling series were a
form of.

Is this process of measuring intervals part of Partch utonality, or
harmonic entropy?

🔗traktus5 <kj4321@...>

--- In harmonic_entropy@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
> wrote:
>
> > So. are you saying that choosing which numbers to spell ones
chords
> > with is a form of tuning?!
>
> No, I don't think I'm saying that . . . (?)
>
> > >It seems you're proceeding from 12-equal as a
> > > starting point . . . am I right?
> >
> > My starting point was the above mentioned 16/15, 9/8 (and 10/9),
> > 6/5, 5/4, 4/3 .... the limited set of intervals implied by most
> > standard college music theory instruction.
>
> !!! Which college did you go to? I never met anyone who learned
> interval ratios in anything like a standard college class.

I lied. I was on a university computer, though, when I first came
across the article. (D. Canwright's "Tour of the Harmonic Series.)

> > But since I've always
> > worked on a piano, I guess it's also 12 tet.
>
> "Also?" If you've worked on a piano, you really have no idea what
> these various ratios sound like, or how consonant they are.

I agree, and remember that from previous discussions. But I'm
working with a type of functional harmony. (Chords are mostly
tertial, though some are very dissonant; and tendency tones and
progressions exist, but without obvious dominant harmony.) How
exactly a major sixth or perfect 4th is tuned does not, in this
context, I believe, really effect it's voice leading behavior, or
other effects. Or the motive power of the tritone (speaking of
functional harmony in general): wouldn't numerous variants in the
neighborhood of 600 cents suffice?

Supposing that chords and harmonies which are in a more functional
context (voice leading, progressions and other allsusions to common
practice harmony) have the ability to operate more crudely, or
flexibly, with regard to tuning, suggest to me that, by default, if
you can put it that way, you are dealing with intervals of a lower -
limit, low number 'archetype', almost. (Archetype, because
of "Occam's razor', and since it is not acutally heard as a pure
interval.) Heck, what's more pure than the thought of a musical
ratio! (I also see them in road signs around town. You might have
to go to Montana, however, where they have higher limits... )

I agree that the tuning arts you practice open up the potential for
chord sensation in interesting ways not possible on my 12 tet piano,
fine tuning partials and beats and difference tones.

> > BTW, how would you 'tune' (ie, which harmonic series numbers
would
> > you use for) 15-21-28, and 10-14-21? As written?
>
> If you had started with chords in 12-equal, this question would
make
> some sense, but it seems you're starting with harmonic series
> numbers, so I don't know how else one could conceivably reply
> than 'as written' . . . Can you elaborate what sense you meant
this
> question in?

I'm meant the latter, harmonic series numbers.

I don't practice alternate tuning, but am curious: what
considertions are there for choosing between a lower-numbered but
higher limit version (15-21-28) and a higher numbered but lower
limit version (32-45-60)?

> > > These are options but there are others of course, like 27 for
the
> > > major sixth . . .
> >
> > Do you know if there is one for the minor third, lower than 75,
> that
> > meets my criteria?
>
> If your criterion is a prime limit of 7 and a denominator which is
a
> power of 2, then I don't think so.
>

🔗wallyesterpaulrus <wallyesterpaulrus@...>

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

> > > I thought a doubling series, as in the numerators of Partch's
> > > example of the minor chord 1/1, 6/5, 3/2, was a power.
> >
> > I don't get it. Can you elaborate? I'm certainly new to calling
> > anything like this "a power." Where do you get this terminology
> from?
> >
>
> I just don't know my math termininology. (I should look it up) I
> thought 'power' meant logarhytms, and that doubling series were a
> form of.

I don't see a doubling sequence in your example, but in any case that
would be an exponential sequence, while a power sequence is something
like all the squares (1, 4, 9, 16, 25, . . .) or any Nth power of the
set of integers.

> > > > > is in the numerators.
> > > >
> > > > Of the *pitch* ratios, or interval ratios relative to a
> > *constant*
> > > > reference pitch.
> > > >
> > > > > But
> > > > > in this chord in question (b2-d4-g4-c5), there is a mixture
> > > > > of 'unities' in the numerators.
> > > >
> > > > Of the *interval* ratios between different *pairs* of notes
in
> > the
> > > > chord, and only when you always measure them upwards.
> > >
> > > What does it mean...Upward?
> >
> > This means you're always calculating the interval from the lower
> note
> > to the upper note, and never the other way around. By contrast,
> when
> > the ratios are relative to a fixed reference pitch, they can (and
> > must) sometimes be upward, sometimes downward.
> >
> > > >Totally different scenario.
>
> Is this process of measuring intervals part of Partch utonality,

There, you measure the intervals of each pitch from the "guide tone",
always in an upward sense, and the numerators will all be powers of 2.

> or
> harmonic entropy?

Well, you just have to input the data in some form the program will
understand . . . did you have a specific scenario in mind?

> I'm a little bit stuck on understanding this, maybe because I first
> have to deal with a perhaps related question which I'm fixated on
> (related to my musings on an 'inversion effect'): considering just
> some simple intervals like 3/2, 5/3, and 5/4, is a measure of the
> higher harmonic entropy (or lesser tonalness) of their
corresponding
> inversions (4/3, 6/5, and 8/5) a direct result (or caused by, in
> some manner) the inversion --(of having a lower limit number over a
> higher limit number)?

I'd say it's simply a result of the numbers in the ratio being
higher! No need to invoke 'limits' -- and certainly not "limits" on
the individual numbers in the ratio!

> (I realize there are cases where the opposite
> is probably true: 9/8 is --am I correct?--less consonant than
>16/9).

If so, not by much. The chart on the home page says 9/8 has entropy
4.774, while 16/9 is very slightly more consonant with entropy 4.765.
A truly tiny difference. What happened is that the numbers in 9/8 and
16/9 are already a bit too high for the "lower numbers = lower
entropy" equation to hold true.

> > > > > (I was curious that, in TTSS, Sethares mentions
> Partch's 'One
> > > > Footed
> > > > > Bride', but not, I don't believe, the theory of Utonality.
> Do
> > > you
> > > > > think that is because, as I saw in a post somewhere, he
> > > (Sethares),
> > > > > does not 'believe in' the significance of subharmonics?
> thx,
> > > Kelly
> > > >
> > > > No.
> > > >
> > > > Actually, even without 'believing in' the significance or
> > > existence
> > > > of subharmonics, Sethares' theory predicts that utonalities
> will
> > > be
> > > > about equally consonant as their corresponding otonalities!
> This
> > > is
> > > > because the utonal and the otonal chord will have the exact
> same
> > > set
> > > > of intervals, and Sethares sums the dissonances of the
> individual
> > > > intervals to get the dissonance of the chord. The fact that
> > higher-
> > > > limit otonalities are much more consonant than their
> > corresponding
> > > > utonalities shows that there's a serious problem with
> Sethares's
> > > > theory, and he sort of mentions this in the second edition of
> his
> > > > book.
> >
> > Did you have no response to this? Was it unclear? Can I clarify
> > somehow?
> >
>
> I recall a statement (of his?) which corrected that notion, but I
> can't remember if it was in his 2nd edition, which I read, but
don't
> own. It may have been in the section where he discusses your
work.
> I'm suprised he missed that one!

He missed which one?

> Speaking of roughness and tonalness, a nice chord (M7 inversion)
> which gives me the sensation of those two phenomenon (or makes me
> think about them ,at least) is 15-16-20 (and 12-15-16). In a very
> impressionistic way, I hear a very rough interval (16/15) combining
> with the other intervals in an interesting way.

If you listen to this in JI, it will be interesting in a
whole 'nother way than what you would hear in 12-equal.

> Being a number
> nut, I like to imagine, in my weird science kind of way, that the
> effect has something to do with the fact that, not only does 16/15
x
> 5/4 = 4/3,

Well, this is a tautology given the chord in question. The chord
couldn't even exist otherwise!

> but, by cross dividing , 16/4 and 15/5 equals 3/4.

What are you cross dividing? Can you explain?

> It's
> a sort of 'unity' (3/4 x 4/3) for me, which I like to think is
> another way of "getting back to one", in addition to the
fundamental
> of psychoacoustic tonalness!

Does any chord *not* exhibit this "getting back to one" property
you're referring to?

🔗wallyesterpaulrus <wallyesterpaulrus@...>

--- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
wrote:
>
> --- In harmonic_entropy@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
> > wrote:
> >
> > > So. are you saying that choosing which numbers to spell ones
> chords
> > > with is a form of tuning?!
> >
> > No, I don't think I'm saying that . . . (?)
> >
> > > >It seems you're proceeding from 12-equal as a
> > > > starting point . . . am I right?
> > >
> > > My starting point was the above mentioned 16/15, 9/8 (and
10/9),
> > > 6/5, 5/4, 4/3 .... the limited set of intervals implied by
most
> > > standard college music theory instruction.
> >
> > !!! Which college did you go to? I never met anyone who learned
> > interval ratios in anything like a standard college class.
>
> I lied. I was on a university computer, though, when I first came
> across the article. (D. Canwright's "Tour of the Harmonic Series.)

Still.

> > > But since I've always
> > > worked on a piano, I guess it's also 12 tet.
> >
> > "Also?" If you've worked on a piano, you really have no idea what
> > these various ratios sound like, or how consonant they are.
>
> I agree, and remember that from previous discussions. But I'm
> working with a type of functional harmony. (Chords are mostly
> tertial, though some are very dissonant; and tendency tones and
> progressions exist, but without obvious dominant harmony.) How
> exactly a major sixth or perfect 4th is tuned does not, in this
> context, I believe, really effect it's voice leading behavior, or
> other effects.

In your posts, you haven't discussed voice leading behavior, but you
have discussed "other effects", and that where these differences can
become *dramatic* with repeated listening. But voice leading behavior
*can* be very different in different tunings of the diatonic scale;
the wider the fifths, the stronger the tendency for leading tones and
tritones to resolve, until ultimately you get to a point where the
major third wants to resolve to a perfect fourth instead of the other
way around (says Ivor Darreg, and I know what he means) . . .

> Or the motive power of the tritone (speaking of
> functional harmony in general): wouldn't numerous variants in the
> neighborhood of 600 cents suffice?

Perhaps, but then it seems you should forget about ratios and just
deal with the 12-equal intervals (maybe including their harmonic
entropies) that you're actually using!

> Supposing that chords and harmonies which are in a more functional
> context (voice leading, progressions and other allsusions to common
> practice harmony) have the ability to operate more crudely, or
> flexibly, with regard to tuning, suggest to me that, by default, if
> you can put it that way, you are dealing with intervals of a lower -
> limit, low number 'archetype', almost.

To me, that supposition would suggest quite the opposite. The lower
the numbers in the ratio, the more sensitive it is to a given
mistuning.

> (Archetype, because
> of "Occam's razor', and since it is not acutally heard as a pure
> interval.) Heck, what's more pure than the thought of a musical
> ratio! (I also see them in road signs around town. You might
have
> to go to Montana, however, where they have higher limits... )
>
> I agree that the tuning arts you practice open up the potential for
> chord sensation in interesting ways not possible on my 12 tet
piano,
> fine tuning partials and beats and difference tones.

Not just for chord sensations, but for scales and voice leading, etc.

> > > BTW, how would you 'tune' (ie, which harmonic series numbers
> would
> > > you use for) 15-21-28, and 10-14-21? As written?
> >
> > If you had started with chords in 12-equal, this question would
> make
> > some sense, but it seems you're starting with harmonic series
> > numbers, so I don't know how else one could conceivably reply
> > than 'as written' . . . Can you elaborate what sense you meant
> this
> > question in?
>
>
> I'm meant the latter, harmonic series numbers.

Then I have no idea what you could be asking me.

> I don't practice alternate tuning, but am curious: what
> considertions are there for choosing between a lower-numbered but
> higher limit version (15-21-28) and a higher numbered but lower
> limit version (32-45-60)?

The fact that you consider these "versions" indicates that 12-equal
is still behind your thinking. In JI these are simply different
chords. As for the considerations, the prime limit won't tell you
much (I can concoct illustrative examples if you wish). You should
look at the size of the otonal numbers, which you are, but also at
the sizes of the numbers in the ratio for each interval. In 15-21-28,
you have 15:28, 5:7, and 3:4. In 32-45-60, you have 32:45, 15:8, and
3:4. So it looks like, either way you look at it, the second chord is
more complex and probably more discordant.

🔗traktus5 <kj4321@...>

> > I recall a statement (of his?) which corrected that notion, but
I > > can't remember if it was in his 2nd edition, which I read, but
> don't > > own. It may have been in the section where he discusses
your > work. > > I'm suprised he missed that one!

> He missed which one?

How different chords can sound if you reverse the top and bottom and
intervals.

> > Being a number nut, I like to imagine, in my weird science
kind of way, that the > > effect has something to do with the fact
that, not only does 16/15 x 5/4 = 4/3,

> Well, this is a tautology given the chord in question. The chord
> couldn't even exist otherwise!

And its a good thing I'm pointing it out!

> > but, by cross dividing , 16/4 and 15/5 equals 3/4.
>
> What are you cross dividing? Can you explain?

The numerator of the first interval (16/15) divided by the
denominator of the second interval (5/4), and visa versa (15/5),
gives 3/4. I'm trying to wade through the overlapping harmonic
series to see if there's a correlation, though it gets messy. (Do
you use a program for that, to check for overtones clashes?)

>
> > It's
> > a sort of 'unity' (3/4 x 4/3) for me, which I like to think is
> > another way of "getting back to one", in addition to the
> fundamental
> > of psychoacoustic tonalness!
>
> Does any chord *not* exhibit this "getting back to one" property
> you're referring to?

In the numerological sense I'm referring to, some more than others,
I believe (--just as some chords are more tonal, or some overtones
more easily fused.) Enjoying the number patterns, I feel that,
maybe, chords with number sequences in 'em (6:10:15, 3:5:8:13,
15:21:28) 'get back to one' in a more dramatic (maybe more musical)
way, because of their having a 'more direct' connection to higher
numbers than the intergers (and boring chords like 4:5:6).

My favorite is the 'near unity' aspect of 2/1 in the chord 5:6:9.
There's something so cool sounding about the chord (even on my 12
tet piano), that I believe it has something to do with the 2:1 ratio
of 3/2 and 6/5 (top notes). (I know you guys tried to talk me out
of the notion that I can hear two intervals in a chord --that,
because of 'fusion', the ear, instead, hears notes in harmonic
series ... but I'll address that in another post...)

(This chord 5-6-9 also, uniquely, I'm pretty sure, has exactly-equal
cubic and simple difference tones!)

My pet theory is that, like the Schenkerian melodic descent from
scale step 5 (or 3) to 1 (dealing with whole numbers) found in some
music, that there is a drive 'from below', in the fractions (in the
chords, which are mostly composed of fractions), from high numbers
to low, facilitated by appealing number patterns!

🔗traktus5 <kj4321@...>

>
> > > > But since I've always
> > > > worked on a piano, I guess it's also 12 tet.
> > >
> > > "Also?" If you've worked on a piano, you really have no idea
what
> > > these various ratios sound like, or how consonant they are.
> >
> > I agree, and remember that from previous discussions. But I'm
> > working with a type of functional harmony. (Chords are mostly
> > tertial, though some are very dissonant; and tendency tones and
> > progressions exist, but without obvious dominant harmony.) How
> > exactly a major sixth or perfect 4th is tuned does not, in this
> > context, I believe, really effect it's voice leading behavior,
or
> > other effects.
>
> In your posts, you haven't discussed voice leading behavior

A new angle on dissonance for me: the chords I work with are 'frozen
counterpoint', or 'pent-up' melodic energy, which bursts through in
texture rather than counterpoint... (What is the force in music if
you take away V7?) There once was a model of consonance based on
resolution of dissonance in voice leading...

, but you
> have discussed "other effects", and that where these differences
can
> become *dramatic* with repeated listening. But voice leading
behavior
> *can* be very different in different tunings of the diatonic
scale;
> the wider the fifths, the stronger the tendency for leading tones
and
> tritones to resolve, until ultimately you get to a point where the
> major third wants to resolve to a perfect fourth instead of the
other
> way around (says Ivor Darreg, and I know what he means) . . .
>
> > Or the motive power of the tritone (speaking of
> > functional harmony in general): wouldn't numerous variants in
the
> > neighborhood of 600 cents suffice?
>
> Perhaps, but then it seems you should forget about ratios and just
> deal with the 12-equal intervals (maybe including their harmonic
> entropies) that you're actually using!>
> > Supposing that chords and harmonies which are in a more
functional
> > context (voice leading, progressions and other allsusions to
common
> > practice harmony) have the ability to operate more crudely, or
> > flexibly, with regard to tuning, suggest to me that, by default,
if
> > you can put it that way, you are dealing with intervals of a
lower -
> > limit, low number 'archetype', almost.
>
> To me, that supposition would suggest quite the opposite. The
lower
> the numbers in the ratio, the more sensitive it is to a given
> mistuning.
>
> > (Archetype, because
> > of "Occam's razor', and since it is not acutally heard as a pure
> > interval.) Heck, what's more pure than the thought of a
musical
> > ratio! (I also see them in road signs around town. You might
> have
> > to go to Montana, however, where they have higher limits... )
> >
> > I agree that the tuning arts you practice open up the potential
for
> > chord sensation in interesting ways not possible on my 12 tet
> piano,
> > fine tuning partials and beats and difference tones.
>
> Not just for chord sensations, but for scales and voice leading,
etc.

That's a nice defence of microtonal music and it's theory. I wasn't
trying to denigrate, but just point out that there must be features
of chord sensation, and certainly of harmony and melody and other
basic musical impulses, which operate quite fine regardless of the
exact tuning. In the end, for the majority of musicians --
lamentably, I know --the tuning is a tool, like the instrument
builders, for the fullfillment of musical impulses other than
tuning. I know the ratios well from an interest in the numbers.
Whether I really hear them or not, you can not say, because I
believe in them! (Paul Erlich is a supercillious j... :) )

> > I don't practice alternate tuning, but am curious: what
> > considertions are there for choosing between a lower-numbered
but
> > higher limit version (15-21-28) and a higher numbered but lower
> > limit version (32-45-60)?
>
> The fact that you consider these "versions" indicates that 12-
equal
> is still behind your thinking.

I'm just thinking of the numbers, because I like them! They're part
of my compositional process. And I'm interested in theories of
consonance, and all the number possibilities with alternate tuning.

In JI these are simply different
> chords. As for the considerations, the prime limit won't tell you
> much (I can concoct illustrative examples if you wish). You should
> look at the size of the otonal numbers, which you are, but also at
> the sizes of the numbers in the ratio for each interval. In 15-21-
28,
> you have 15:28, 5:7, and 3:4. In 32-45-60, you have 32:45, 15:8,
and
> 3:4. So it looks like, either way you look at it, the second chord
is
> more complex and probably more discordant.

That's helpful. THanks.

🔗wallyesterpaulrus <wallyesterpaulrus@...>

--- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
wrote:
>
> > > I recall a statement (of his?) which corrected that notion, but
> I > > can't remember if it was in his 2nd edition, which I read,
but
> > don't > > own. It may have been in the section where he
discusses
> your > work. > > I'm suprised he missed that one!
>
> > He missed which one?
>
> How different chords can sound if you reverse the top and bottom
and
> intervals.

Well, he made a lot of assumptions, and though it's easy to think
they're all backed up scientifically, it only takes a little actual
listening to show otherwise . . .

> > > Being a number nut, I like to imagine, in my weird science
> kind of way, that the > > effect has something to do with the fact
> that, not only does 16/15 x 5/4 = 4/3,
>
>
> > Well, this is a tautology given the chord in question. The chord
> > couldn't even exist otherwise!
>
> And its a good thing I'm pointing it out!
>
> > > but, by cross dividing , 16/4 and 15/5 equals 3/4.
> >
> > What are you cross dividing? Can you explain?
>
> The numerator of the first interval (16/15) divided by the
> denominator of the second interval (5/4), and visa versa (15/5),
> gives 3/4.

Still don't get it. Can you elaborate? And what are these divisions
supposed to get you? They don't seem to be meaningful quantities,
though I could be missing something . . .

> I'm trying to wade through the overlapping harmonic
> series to see if there's a correlation, though it gets messy. (Do
> you use a program for that, to check for overtones clashes?)

Why not start by just calculating the set of overtones that results
when the chord is played? Your calculation above doesn't seem to have
anything to do with that . . .

> > > It's
> > > a sort of 'unity' (3/4 x 4/3) for me, which I like to think is
> > > another way of "getting back to one", in addition to the
> > fundamental
> > > of psychoacoustic tonalness!
> >
> > Does any chord *not* exhibit this "getting back to one" property
> > you're referring to?
>
> In the numerological sense I'm referring to, some more than others,
> I believe (--just as some chords are more tonal, or some overtones
> more easily fused.) Enjoying the number patterns, I feel that,
> maybe, chords with number sequences in 'em (6:10:15, 3:5:8:13,
> 15:21:28) 'get back to one' in a more dramatic (maybe more musical)
> way, because of their having a 'more direct' connection to higher
> numbers than the intergers (and boring chords like 4:5:6).

Have you looked at the combinational tones in these chords? It may be
more than just abstract number patterns . . .

> My favorite is the 'near unity' aspect of 2/1 in the chord 5:6:9.

I don't see any 2:1s in 5:6:9.

> There's something so cool sounding about the chord (even on my 12
> tet piano), that I believe it has something to do with the 2:1
ratio
> of 3/2 and 6/5 (top notes).

See, this makes no sense to me. The ratio of the top two notes of
5:6:9 is 2:3. The ratio of 3/2 and 6/5 is 5:4. So either way, I don't
know what you mean.

> (I know you guys tried to talk me out
> of the notion that I can hear two intervals in a chord

Huh? Who tried to talk you out of that?

> --that,
> because of 'fusion', the ear, instead, hears notes in harmonic
> series ... but I'll address that in another post...)

I think you may be misunderstanding whatever it is that "us guys"
were trying to say. You can certainly hear the intervals in a
chord . . .

> (This chord 5-6-9 also, uniquely, I'm pretty sure, has exactly-
equal
> cubic and simple difference tones!)

How so? I find 1, 2, 3, 4, 7, 8, and 10 among the cubic difference
tones, but only 1, 3, and 4 among the simple ones . . .

> My pet theory is that, like the Schenkerian melodic descent from
> scale step 5 (or 3) to 1 (dealing with whole numbers) found in some
> music,

That's a descent from above.

> that there is a drive 'from below', in the fractions (in the
> chords, which are mostly composed of fractions), from high numbers
> to low, facilitated by appealing number patterns!

No idea what that means.

🔗wallyesterpaulrus <wallyesterpaulrus@...>

🔗traktus5 <kj4321@...>

> > that, not only does 16/15 x 5/4 = 4/3,
> >
> >
> > > Well, this is a tautology given the chord in question. The
chord
> > > couldn't even exist otherwise!

Before the theory of special relativity, the Lorentzian equations
were considered a tautology! I'm trying to think out of the box...

> > The numerator of the first interval (16/15) divided by the
> > denominator of the second interval (5/4), and visa versa (15/5),
> > gives 3/4.
>
> Still don't get it. Can you elaborate? And what are these
divisions
> supposed to get you?

They just appeal to me in some way. Sorry I can't explain it better.

They don't seem to be meaningful quantities,
> though I could be missing something . . .
>
> > I'm trying to wade through the overlapping harmonic
> > series to see if there's a correlation, though it gets messy.
(Do
> > you use a program for that, to check for overtones clashes?)
>
> Why not start by just calculating the set of overtones that
results
> when the chord is played? Your calculation above doesn't seem to
have
> anything to do with that . . .

In general, how high up in overtones should one check for dissonance
that may be significant for the chord's character? Eg, you mentioned
that, though the minor triads have low common overtones compared to
major triads, that clashing overtones higher up in the series, for
the minor chords, account for it's dissonance. WHen I compared the
major and minor triads' overtone clashes up to the 16th harmonic (of
the lowest note in the chord), both chords (major and minor triad)
seemed to have similiar number of semitone overtone clashes in that
region (hn 8-16).

>
> > > > It's
> > > > a sort of 'unity' (3/4 x 4/3) for me, which I like to think
is
> > > > another way of "getting back to one", in addition to the
> > > fundamental
> > > > of psychoacoustic tonalness!
> > >
> > > Does any chord *not* exhibit this "getting back to one"
property
> > > you're referring to?
> >
> > In the numerological sense I'm referring to, some more than
others,
> > I believe (--just as some chords are more tonal, or some
overtones
> > more easily fused.) Enjoying the number patterns, I feel that,
> > maybe, chords with number sequences in 'em (6:10:15, 3:5:8:13,
> > 15:21:28) 'get back to one' in a more dramatic (maybe more
musical)
> > way, because of their having a 'more direct' connection to
higher
> > numbers than the intergers (and boring chords like 4:5:6).

> Have you looked at the combinational tones in these chords? It may
be > more than just abstract number patterns . . .

Yes. Like the nice sounding chord 3:8:10:15, and the matching
difference tones for the 'subset chords' 3:8:15 and 3:10:15.

> > My favorite is the 'near unity' aspect of 2/1 in the chord
5:6:9.
>
> I don't see any 2:1s in 5:6:9.

I'm just concentrating on the numerators of 6/5 and 3/2. If, despite
the ear's attempt at fusion and tonalness, we do also hear the
separate intervals, then couldn't we be hearing a 6 and a 3? (2:1
ratio.)

>
> > There's something so cool sounding about the chord (even on my
12
> > tet piano), that I believe it has something to do with the 2:1
> ratio
> > of 3/2 and 6/5 (top notes).
>
> See, this makes no sense to me. The ratio of the top two notes of
> 5:6:9 is 2:3. The ratio of 3/2 and 6/5 is 5:4.

(Excuse me...how do you get 5:4?)

So either way, I don't
> know what you mean.
>
> > (I know you guys tried to talk me out
> > of the notion that I can hear two intervals in a chord
>
> Huh? Who tried to talk you out of that?

You and Yahya, in discussion of the chord 1/5-4-3, on whether a note
in a chord can be represented by two numbers at once (eg, the lower
note of the upper inverval and the upper note of the lower interval).

> > --that,
> > because of 'fusion', the ear, instead, hears notes in harmonic
> > series ... but I'll address that in another post...)
>
> I think you may be misunderstanding whatever it is that "us guys"
> were trying to say. You can certainly hear the intervals in a
> chord . . .

I guess I took the analogy between fusion and tonalness too far,
thinking that if the ear 'successfully' places 3 notes in a series,
that the low number (or some other) quality of the intervals is
somehow lost, similar to the effect of fusion on otherwise audible
overtones !

>
> > (This chord 5-6-9 also, uniquely, I'm pretty sure, has exactly-
> equal
> > cubic and simple difference tones!)
>

> How so? I find 1, 2, 3, 4, 7, 8, and 10 among the cubic difference
> tones, but only 1, 3, and 4 among the simple ones . . .
>

Based on what I saw in a textbook, I thought cubic difference tones
(for 5:6:9) were like so: 10-6, 10-9, and 12-9. (Sorry, I should
look it up.)

> > My pet theory is that, like the Schenkerian melodic descent from
> > scale step 5 (or 3) to 1 (dealing with whole numbers) found in
some
> > music,
>
> That's a descent from above.
>
> > that there is a drive 'from below', in the fractions (in the
> > chords, which are mostly composed of fractions), from high
numbers
> > to low, facilitated by appealing number patterns!
>
> No idea what that means.

The dissonant chords found in common practice voice leading and
harmony (from scale degree 6 and 7, for example, creating passing
six and seventh chords) represent higher numbers . Their resolution
to triads and unisons, etc, represent lower numbers. And a number
sequence or series or pattern is a nice way to deviate from the
harmonic series norm!

🔗traktus5 <kj4321@...>

🔗wallyesterpaulrus <wallyesterpaulrus@...>

--- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@...>
wrote:
>
> > > that, not only does 16/15 x 5/4 = 4/3,
> > >
> > >
> > > > Well, this is a tautology given the chord in question. The
> chord
> > > > couldn't even exist otherwise!
>
> Before the theory of special relativity, the Lorentzian equations
> were considered a tautology!

I'm not sure that's a good way of putting it, but we should discuss
relativistic physics privately, or at least not on this list . . .

> I'm trying to think out of the box...
>
>
> > > The numerator of the first interval (16/15) divided by the
> > > denominator of the second interval (5/4), and visa versa
(15/5),
> > > gives 3/4.
> >
> > Still don't get it. Can you elaborate? And what are these
> divisions
> > supposed to get you?
>
> They just appeal to me in some way. Sorry I can't explain it
better.
>
> They don't seem to be meaningful quantities,
> > though I could be missing something . . .
> >
> > > I'm trying to wade through the overlapping harmonic
> > > series to see if there's a correlation, though it gets messy.
> (Do
> > > you use a program for that, to check for overtones clashes?)
> >
> > Why not start by just calculating the set of overtones that
> results
> > when the chord is played? Your calculation above doesn't seem to
> have
> > anything to do with that . . .
>
> In general, how high up in overtones should one check for
dissonance
> that may be significant for the chord's character?

Do you mean the overtones of the notes themselves, or do you mean the
overtones of an implied fundamental for which the notes of the chord
may act as overtones?

> Eg, you mentioned
> that, though the minor triads have low common overtones compared to
> major triads, that clashing overtones higher up in the series, for
> the minor chords, account for it's dissonance.

I didn't say anything like that, or at least didn't mean it. What
statement of mine are you referring to?

> WHen I compared the
> major and minor triads' overtone clashes up to the 16th harmonic
(of
> the lowest note in the chord), both chords (major and minor triad)
> seemed to have similiar number of semitone overtone clashes in that
> region (hn 8-16).

Good -- that's what I would expect.

>
> >
> > > > > It's
> > > > > a sort of 'unity' (3/4 x 4/3) for me, which I like to think
> is
> > > > > another way of "getting back to one", in addition to the
> > > > fundamental
> > > > > of psychoacoustic tonalness!
> > > >
> > > > Does any chord *not* exhibit this "getting back to one"
> property
> > > > you're referring to?
> > >
> > > In the numerological sense I'm referring to, some more than
> others,
> > > I believe (--just as some chords are more tonal, or some
> overtones
> > > more easily fused.) Enjoying the number patterns, I feel
that,
> > > maybe, chords with number sequences in 'em (6:10:15, 3:5:8:13,
> > > 15:21:28) 'get back to one' in a more dramatic (maybe more
> musical)
> > > way, because of their having a 'more direct' connection to
> higher
> > > numbers than the intergers (and boring chords like 4:5:6).
>
> > Have you looked at the combinational tones in these chords? It
may
> be > more than just abstract number patterns . . .
>
> Yes. Like the nice sounding chord 3:8:10:15, and the matching
> difference tones for the 'subset chords' 3:8:15 and 3:10:15.

So maybe you don't need recourse to numerological explanations?

> > > My favorite is the 'near unity' aspect of 2/1 in the chord
> 5:6:9.
> >
> > I don't see any 2:1s in 5:6:9.
>
> I'm just concentrating on the numerators of 6/5 and 3/2. If,
despite
> the ear's attempt at fusion and tonalness, we do also hear the
> separate intervals, then couldn't we be hearing a 6 and a 3? (2:1
> ratio.)

That makes absolutely no sense, any way I think about it.

> > > There's something so cool sounding about the chord (even on my
> 12
> > > tet piano), that I believe it has something to do with the 2:1
> > ratio
> > > of 3/2 and 6/5 (top notes).
> >
> > See, this makes no sense to me. The ratio of the top two notes of
> > 5:6:9 is 2:3. The ratio of 3/2 and 6/5 is 5:4.
>
> (Excuse me...how do you get 5:4?)

Wow -- it's just the interval between 3/2 and 6/5. (3/2)/(6/5) = (3/2)
*(5/6) = (1/2)*(5/2) = 5/4. Surely you already knew how to calculate
the interval between two pitch ratios?

> > So either way, I don't
> > know what you mean.
> >
> > > (I know you guys tried to talk me out
> > > of the notion that I can hear two intervals in a chord
> >
> > Huh? Who tried to talk you out of that?
>
> You and Yahya, in discussion of the chord 1/5-4-3, on whether a
note
> in a chord can be represented by two numbers at once (eg, the lower
> note of the upper inverval and the upper note of the lower
>interval).

We were simply pointing out that your math didn't make sense. Neither
of us made any objection to the idea that you can hear two intervals
in a chord!

> > > --that,
> > > because of 'fusion', the ear, instead, hears notes in harmonic
> > > series ... but I'll address that in another post...)
> >
> > I think you may be misunderstanding whatever it is that "us guys"
> > were trying to say. You can certainly hear the intervals in a
> > chord . . .
>
> I guess I took the analogy between fusion and tonalness too far,
> thinking that if the ear 'successfully' places 3 notes in a series,
> that the low number (or some other) quality of the intervals is
> somehow lost, similar to the effect of fusion on otherwise audible
> overtones !

The low number quality isn't lost at all, since that's what makes
this effect possible in the first place. But a certain independence
between the voices is lost, so too much "fusion" can hamper
counterpoint.

> > > (This chord 5-6-9 also, uniquely, I'm pretty sure, has exactly-
> > equal
> > > cubic and simple difference tones!)
> >
>
> > How so? I find 1, 2, 3, 4, 7, 8, and 10 among the cubic
difference
> > tones, but only 1, 3, and 4 among the simple ones . . .
> >
>
> Based on what I saw in a textbook, I thought cubic difference tones
> (for 5:6:9) were like so: 10-6, 10-9, and 12-9. (Sorry, I should
> look it up.)

Yes, those are important ones, but there's also 15-5, 11-9, and 14-6,
for a start.

🔗traktus5 <kj4321@...>

In general, how high up in overtones should one check for
> dissonance > > that may be significant for the chord's character?

> Do you mean the overtones of the notes themselves, or do you mean
the > overtones of an implied fundamental for which the notes of the
chord > may act as overtones?

The former.

(Hmm, for the latter case, where you have high numbers in chords
such as 20-24-45, and 36-45-60-80, do the numbers still represent
perceptible overtones? I'm sure there is some very interesting
tuning theory here...When I get the equimpent and software
someday...)

> > Eg, you mentioned > > that, though the minor triads have low
common overtones compared to > > major triads, that clashing
overtones higher up in the series, for > > the minor chords, account
for it's dissonance.

> I didn't say anything like that, or at least didn't mean it. What
> statement of mine are you referring to?

I can't readily find the message. I recall I mentioned that it
seemed paradoxical that major chords are considered more consonant
than minor chords, when it is the minor chords which have the low
common overtone, to which you responded (words to the effect) 'but
have you looked at [the overtone clashes and concurrences in] the
higher overtones of the minor chords', implying, I assumed, that
they were much more clashing than with major chords. Perhaps I
misunderstood.

> > > Have you looked at the combinational tones in these chords? It
> may > > be > more than just abstract number patterns . . .

> > Yes. Like the nice sounding chord 3:8:10:15, and the matching
> > difference tones for the 'subset chords' 3:8:15 and 3:10:15.

> So maybe you don't need recourse to numerological explanations?

Learning the science of psychoacoustics has really changed and
enriched my enjoyment of the chords! (The way I arrange them on the
piano, for eg, can be accounted for by roughness and beating.) But
I still believe part of the beauty of an interval is because it
approximates a ratio of whole numbers (like the energy states of an
electron; there are no gradations between them), so the numbers are
hard to ignore. But I realize I go out on a limb looking for
connections between the numbers and psychoacoustic phenemon.

(For example, there's a really nice chord, 3:7:10, dimished triad,
on strings, in the tenor/bass register, in the menuet, mozart A maj
symphony 29. It really stood out when I heard it, even though I
know there are plenty of other 'addative' chords like that.)

> > I'm just concentrating on the numerators of 6/5 and 3/2. If,
> despite > > the ear's attempt at fusion and tonalness, we do also
hear the > > separate intervals, then couldn't we be hearing a 6 and
a 3? (2:1 > > ratio.)

> That makes absolutely no sense, any way I think about it.

The ear attempts to place notes in a harmonic series
(1,2,3,4,5...). Is it not even *possible* that the ear could
isolate, in some way, the ratio 6:3 (numerators of the bottom and
top interval)?

> > > > There's something so cool sounding about the chord (even on
my > > 12 > > > > tet piano), that I believe it has something to do
with the 2:1 > > > ratio > > > > of 3/2 and 6/5 (top notes).

> > > See, this makes no sense to me. The ratio of the top two notes
of > > > 5:6:9 is 2:3. The ratio of 3/2 and 6/5 is 5:4.

> > (Excuse me...how do you get 5:4?)

> Wow -- it's just the interval between 3/2 and 6/5. (3/2)/(6/5) =
(3/2)> *(5/6) = (1/2)*(5/2) = 5/4. Surely you already knew how to
calculate > the interval between two pitch ratios?

Of course... that's my tautology (6/5 x 3/2 = 9/5)! And conversely
(inversely?) I'm accustomed to dividing the outer interval by one of
the inner intervals, to get the other inner interval. Why would one
divide the two inner intervals by each other? Aside from the
implied interval g-b natural, in the chord e4-g4-d5, what perceptual
signigicance does 5/4 have in this chord? I believe that 6:3 is
more significant in this chord than 5:4!

(Anyway, do we really know what math rules music follows?! Consider
how mysterious it's expressive effect can be! Why can't its math be
too!)

> > > > (I know you guys tried to talk me out > > > > of the notion
that I can hear two intervals in a chord

> > > > > > Huh? Who tried to talk you out of that?

> > You and Yahya, in discussion of the chord 1/5-4-3, on whether a
> note > > in a chord can be represented by two numbers at once (eg,
the lower > > note of the upper inverval and the upper note of the
lower > >interval).

> We were simply pointing out that your math didn't make sense.
Neither > of us made any objection to the idea that you can hear two
intervals > in a chord!

Ah. But it seems that we still don't really understand how we hear
two intervals in a chord. For example, in message 915, commenting
on chords which are not straitforwardly tonal, you wrote: "I think
our theories account for this just fine, in that "interesting" is a
perfect way to describe something that's different from the bland,
obvious, harmonic-series norm and yet is quite similar in terms of
critical band roughness." Beyond describing this
as 'interesting', couldn't there be un-discovered psychoacoustic
mechanisms at work here, are some sort of explanation? (And ones,
naturally, following all my numerological laws... :) )

🔗traktus5 <kj4321@...>

🔗Yahya Abdal-Aziz <yahya@...>

on Sat, 18 Feb 2006 "traktus5" wrote:
>
> > > .... Like the nice sounding chord 3:8:10:15, and the matching
> > > difference tones for the 'subset chords' 3:8:15 and 3:10:15.
>
> > So maybe you don't need recourse to numerological explanations?
>
> You're right; the difference tones sound very significant for some
> chords. In the factors Sathare's lists for "Explanations of
> Consonance and Dissonance" (small waves, fusion, virtual pitch,
> difference tones, roughness, harmonic entropy), I recall he said
> (words to the effect; I don't own the book) that difference tones are
> not a major factor, compared to the other factors.
>
> It's my experience that they are. In some chord progressions I'm
> working on , some very prominant 4-note chords, in addition to
> 3:8:10:15, are 5:8:12:15, 3:4:5:8, and 3:5:7:9, which all (though less
> for the 3rd chord) have the property mentioned above (where the subset
> 3 note chords have the same primary difference tones.)
>
> Could this be proof that difference tones are more important than he
> indicates as an explanation for consonance and dissonance?

Take 1 = C1. Then 2 = C2, 3 = G2, 4 = C3, 5 = E'3, 6 = G3,
7 = Bb`4, 8 = C4, 9 = D4, 10 = E'4, 12 = G4, 15 = B'5.

3:8:10:15 = G2:C4:E'4:B'5

5:8:12:15 = E'3:C4:G4:B'5

3:4:5:8 = G2:C3:E'3:C4

3:5:7:9 = G2:E'3:Bb`4:D4

Are these the chords you're playing?

Are you still exploring these chords on your 12-EDO piano?
Because if you are, you're not hearing anything like those
ratios you list. Which means that the conclusions you draw
would be based on evidence that is simply irrelevant to
those ratios. A small change in tuning can change the
harmonicity of a difference tone by a fair amount. Let's
look at your 3:4:5:8 chord more closely. Suppose you are
playing that as G2:C3:E3:C4 on your 12-EDO piano. Let's
assume for simplicity of calculation that your G2 is 99 Hz
- that's equivalent to tuning A4=445.5Hz, but more
importantly, to tuning C1 to 33 Hz, so all our frequencies
are multiples of 33 Hz. Then your G2:C3:E3:C4 chord
should have the fundamental frequencies 99, 132, 165, 264,
plus their overtones, with the difference tones 33, 33, 66,
99, 132, 165. The last three of these coincide with chord
tones, while the first two give a perceived fundamental of
C1, reinforced by its octave at C2.

What you actually play is a chord on the piano with the
fundamental frequencies =
33 * {2^(19/12), 2^(24/12), 2^(28/12), 2^(36/12)}, plus
overtones, with difference tones 33 * {their six
differences}:

G2C3: 33*(4 - 2*2^(7/12)) = 33*(4 - 2.99734) = 33.08778
C3E3: 33*(4*2^(4/12) - 4) = 132*(2^(4/12) -1) = 34.3332
G2E3: 33*(4*2^(4/12) - 2*2^(7/12)) = 33*(2.04306) = 67.42098
E3C4: 33*(8 - 4*2^(4/12)) = 132*(2 - 2^(4/12)) = 97.6668
C3C4: 33*(8 - 4) = 33*4 = 132
G2C4: 33*(8 - 2*2^(7/12) = 33*5.00266 = 165.08778

I've used the approximation:
2^(1/12) = 1.0595

whose powers are:
2^(2/12) = 1.12254
2^(3/12) = 1.18933
2^(4/12) = 1.26010
2^(5/12) = 1.33507
2^(6/12) = 1.41451
2^(7/12) = 1.49867
2^(8/12) = 1.58784
2^(9/12) = 1.68232
2^(10/12) = 1.78242
2^(11/12) = 1.88847

Compare the results with the desired:
G2C3: 33.08778 ....... 33 ... 0.08778
C3E3: 34.3332 .......... 33 ... 0.3332
G2E3: 67.42098 ....... 66 ... 1.42098
E3C4: 97.6668 .......... 99 ... 1.3332
C3C4: 132 ................... 132 ... 0
G2C4: 165.08778 ... 165 ... 0.91222

The third column of figures above gives the mistuning in Hertz.
There should be noticeable beating at around 1.3 Hz. This
would not occur with justly-tuned ratios 3:4:5:8.

Regards,
Yahya
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🔗traktus5 <kj4321@...>

Hi Yahya. (Thanks for your excellent comments on the rainbow and
hearing. It's veritable encyclopedia entry!)

>...like the nice sounding chord 3:8:10:15, and the matching
difference tones for the 'subset chords' 3:8:15 and 3:10:15. In
some chord progressions I'm > > working on , some very prominant 4-
note chords, in addition to > > 3:8:10:15, are 5:8:12:15, 3:4:5:8,
and 3:5:7:9, which all (though less for the 3rd chord) have the
property mentioned above (where the subset > > 3 note chords have
the same primary difference tones.)>

>> Take 1 = C1. Then 2 = C2, 3 = G2, 4 = C3, 5 = E'3, 6 = G3,
> 7 = Bb`4, 8 = C4, 9 = D4, 10 = E'4, 12 = G4, 15 = B'5.

> 3:8:10:15 = G2:C4:E'4:B'5
>
> 5:8:12:15 = E'3:C4:G4:B'5
>
> 3:4:5:8 = G2:C3:E'3:C4
>
> 3:5:7:9 = G2:E'3:Bb`4:D4
>
> Are these the chords you're playing? <<"Yes".

> Are you still exploring these chords on your 12-EDO piano? >>"Yes"

> Because if you are, you're not hearing anything like those
> ratios you list.

(I disagree! See below).

>... Which means that the conclusions you draw
> would be based on evidence that is simply irrelevant to
> those ratios.

My questions and conclusions aren't drawn totally from what my piano
is sounding, but from properties I suppose arise from the just
intervals (where number-interest and psychoacoustics coincide!).
But I"m also interested in what my piano is actually sounding, so I
find your demonstration below very interesting, and have several
questions (here, and interspersed throughout):

Ok. So, refering here to 'difference tones' as those between
overtones (5-4, 8-5, etc) and not the amplitude 'wavering' beating
between nearly identical pitches , do difference tones themselves
beat against a nearby partial or note? Is that why you make the
point about the difference tone not coinciding with the note?

(Does the fact that both beating and difference tones are a product
of the hearing system, and not actually physically present in the
sounding object, have anything to do with this?)

What if the difference tones are not audible. Do they still effect
the chord sound?

And on audibility of difference tones, why do textbooks I've seen
usually say they are only audible under special circumstances, like
high notes on the recorder? (Wow, if you can't hear it, but it's
having an effect, I wonder what's going on?)

>A small change in tuning can change the > harmonicity of a
difference tone by a fair amount.
Let's> look at your 3:4:5:8 chord more closely. Suppose you are >
playing that as G2:C3:E3:C4 on your 12-EDO piano. Let's
> assume for simplicity of calculation that your G2 is 99 Hz
> - that's equivalent to tuning A4=445.5Hz, but more
> importantly, to tuning C1 to 33 Hz, so all our frequencies
> are multiples of 33 Hz. Then your G2:C3:E3:C4 chord
> should have the fundamental frequencies 99, 132, 165, 264,
> plus their overtones, with the difference tones 33, 33, 66,
> 99, 132, 165.

AGain, is a "reinforcing effect" the absence of beating? What if
the difference tone is not audible -- which I don't believe they are
to me -- is there still a reinforcing effect? If so, how, given
that it is not audible?!?!

>The last three of these coincide with chord
> tones, while the first two give a perceived fundamental of
> C1, reinforced by its octave at C2.
>
> What you actually play is a chord on the piano with the
> fundamental frequencies =
> 33 * {2^(19/12), 2^(24/12), 2^(28/12), 2^(36/12)}, plus
> overtones, with difference tones 33 * {their six
> differences}:
>
> G2C3: 33*(4 - 2*2^(7/12)) = 33*(4 - 2.99734) = 33.08778
> C3E3: 33*(4*2^(4/12) - 4) = 132*(2^(4/12) -1) = 34.3332
> G2E3: 33*(4*2^(4/12) - 2*2^(7/12)) = 33*(2.04306) = 67.42098
> E3C4: 33*(8 - 4*2^(4/12)) = 132*(2 - 2^(4/12)) = 97.6668
> C3C4: 33*(8 - 4) = 33*4 = 132
> G2C4: 33*(8 - 2*2^(7/12) = 33*5.00266 = 165.08778
>
> I've used the approximation:
> 2^(1/12) = 1.0595
>
> whose powers are:
> 2^(2/12) = 1.12254
> 2^(3/12) = 1.18933
> 2^(4/12) = 1.26010
> 2^(5/12) = 1.33507
> 2^(6/12) = 1.41451
> 2^(7/12) = 1.49867
> 2^(8/12) = 1.58784
> 2^(9/12) = 1.68232
> 2^(10/12) = 1.78242
> 2^(11/12) = 1.88847
>
> Compare the results with the desired:
> G2C3: 33.08778 ....... 33 ... 0.08778
> C3E3: 34.3332 .......... 33 ... 0.3332
> G2E3: 67.42098 ....... 66 ... 1.42098
> E3C4: 97.6668 .......... 99 ... 1.3332
> C3C4: 132 ................... 132 ... 0
> G2C4: 165.08778 ... 165 ... 0.91222
>
> The third column of figures above gives the mistuning in Hertz.
> There should be noticeable beating at around 1.3 Hz. This
> would not occur with justly-tuned ratios 3:4:5:8.

How do you know that the margin of variation between the 12 edo
produced difference tones and the 'justly produced' difference tones
is significant? The fact that I notice, on my piano, something
special with the chords listed at top, and presuming that the effect
has something to do with these "doubly nested" difference I
describe, makes me doubt your argument that what I'm hearing is not
relavant to the just version.

The fact that I hear the presumed difference tone reinforcement
effect with the 'out of tune' values you listed makes me wonder if
there is a sort of "snap to grid" effect going on, maybe because of
the strong influence of an interval archetype, on top of any
acoustical or auditory mechanism. (Not to say that your tuning
efforts are wasted!)

🔗Yahya Abdal-Aziz <yahya@...>

Hi trak,

On Mon, 20 Feb 2006, "traktus5" wrote:
>
> Hi Yahya. (Thanks for your excellent comments on the rainbow and
> hearing. It's veritable encyclopedia entry!)

Thanks. Actually, just today I reread something I wrote
in December 2004 but never finished - a draft article on
colour spaces - and am moved to get back on the horse! :-)

> > >...like the nice sounding chord 3:8:10:15, and the matching
> > > difference tones for the 'subset chords' 3:8:15 and 3:10:15. In
> > > some chord progressions I'm working on , some very prominant 4-
> > > note chords, in addition to 3:8:10:15, are 5:8:12:15, 3:4:5:8,
> > > and 3:5:7:9, which all (though less for the 3rd chord) have the
> > > property mentioned above (where the subset 3 note chords have
> > > the same primary difference tones.)>
>
> > Take 1 = C1. Then 2 = C2, 3 = G2, 4 = C3, 5 = E'3, 6 = G3,
> > 7 = Bb`4, 8 = C4, 9 = D4, 10 = E'4, 12 = G4, 15 = B'5.
> >
> > 3:8:10:15 = G2:C4:E'4:B'5
> >
> > 5:8:12:15 = E'3:C4:G4:B'5
> >
> > 3:4:5:8 = G2:C3:E'3:C4
> >
> > 3:5:7:9 = G2:E'3:Bb`4:D4
> >
> > Are these the chords you're playing? <<"Yes".

On an instrument tuned to the just ratios, you mean??!

> > Are you still exploring these chords on your 12-EDO piano? >>"Yes"

Only, or as well as on a justly tuned instument?

> > Because if you are, you're not hearing anything like those
> > ratios you list.
>
> (I disagree! See below).

Welcome! :-)

> >... Which means that the conclusions you draw
> > would be based on evidence that is simply irrelevant to
> > those ratios.
>
> My questions and conclusions aren't drawn totally from what my piano
> is sounding, but from properties I suppose arise from the just
> intervals (where number-interest and psychoacoustics coincide!).
> But I"m also interested in what my piano is actually sounding, so I
> find your demonstration below very interesting, and have several
> questions (here, and interspersed throughout):
>
> Ok. So, refering here to 'difference tones' as those between
> overtones (5-4, 8-5, etc) and not the amplitude 'wavering' beating
> between nearly identical pitches, do difference tones themselves
> beat against a nearby partial or note? Is that why you make the
> point about the difference tone not coinciding with the note?

Alexander Ellis reports, Art. 3. (d) Beat-Notes and
Differential Tones of his Appendix XX, on p. 530 of his
translation of Hermann Helmholtz' "On the Sensations
of Tone", several experiments using tuning forks, which
confirm those reported by him made by R. Koenig, in
which Ellis detects an *inaudible* difference tone (7)
between the notes of a mistuned minor tenth (5:12), by
its beating at 4.2 [Hertz] with an auxiliary tuning fork.

The specific numbers Ellis cites are for two cases:

Mistuned minor tenth:
------------------------
Lower generating fork . . . : 223.77
Upper generating fork . . . : 539.18
Difference tone . . . . . . . . . : 315.41
Auxiliary fork . . . . . . . . . . . : 319.59
Theoretical difference . . .: 4.18
Observed beat frequency : 4.2

Mistuned octave:
-------------------
Lower generating fork . . . : 223.77
Upper generating fork . . . : 451.14
Difference tone . . . . . . . . . : 227.37
Auxiliary fork . . . . . . . . . . . : 223.77 (*)
Theoretical difference . . .: 3.6
Observed beat frequency : 3.6
(*) No auxiliary fork was necessary, the lower fork
serving.

Ellis also states that he has repeated this experiment
"several times" with similar results.

So, yes, difference tones *do* themselves beat against
a nearby partial or note.

> (Does the fact that both beating and difference tones are a product
> of the hearing system, and not actually physically present in the
> sounding object, have anything to do with this?)

Not so fast! Again according to Ellis, Art. 4. (a) Objective
beats, p. 531 op. cit.:
"Beats of a disturbed unison exist objectively as
disturbances in the air beforeit reaches the ear. They
are reinforced by resonators, they disturb sand, &c.
In the case of the beats of harmonium reeds in Appunn's
tonometer, they strongly shook the box containing the
reeds. Other beats, beat-notes and combinational tones
appear not to exist externally to the ear."

> What if the difference tones are not audible. Do they still effect
> the chord sound?

See my first comment above about Ellis' experiments.

> And on audibility of difference tones, why do textbooks I've seen
> usually say they are only audible under special circumstances, like
> high notes on the recorder? (Wow, if you can't hear it, but it's
> having an effect, I wonder what's going on?)

Ellis also uses a similar example, of very high notes on
a flageolet, but only as an extreme case that he feels
best demonstrates the existence of the difference
tones. Note * on page 153, op. cit., on combinational
tones, reads:
"I have found that combinational tones can be made
quite audible to a hundred people at once, by means
of two flageolet fifes or whistles, blown as strongly
as possible. I choose very close dissonant intervals
because the great depth of the low tone is much more
striking, being very far below anything that can be
touched by the instrument itself."

I conclude that Ellis finds that the difference notes
are real enough for lower amplitude waves and that
they arise between notes lower in pitch, exactly as
Helmholtz describes. For example, playing a 4:5 dyad
as C4:E4 produces the difference tone C2; a 4:6 dyad
as C4:G4 produces the difference tone C3; a 4:5:6
triad as C4:E4:G4 produces both of these.

> > A small change in tuning can change the harmonicity
> > of a difference tone by a fair amount. Let's look
> > at your 3:4:5:8 chord more closely. Suppose you are
> > playing that as G2:C3:E3:C4 on your 12-EDO piano. Let's
> > assume for simplicity of calculation that your G2 is 99 Hz
> > - that's equivalent to tuning A4=445.5Hz, but more
> > importantly, to tuning C1 to 33 Hz, so all our frequencies
> > are multiples of 33 Hz. Then your G2:C3:E3:C4 chord
> > should have the fundamental frequencies 99, 132, 165, 264,
> > plus their overtones, with the difference tones 33, 33, 66,
> > 99, 132, 165.
>
> AGain, is a "reinforcing effect" the absence of beating?

This doesn't follow logically. You seem to be saying:
'If it's not A, it must be B.' But couldn't it be something
else? For example, it's perfectly possible to play two notes
with exactly the same timbre and at exactly the same pitch
- so they don't beat at all - yet without reinforcement. You
have only to ensure that they are played with the opposite
phase, one lagging 180ï¿½ behind the other; then they will
completely cancel each other out.

Helmholtz was convinced that the difference tones could
themselves form differences with chord tones, thus making
second-order difference tones, third-order ... . Ellis was
properly sceptical of this.

> What if
> the difference tone is not audible -- which I don't believe they are
> to me -- is there still a reinforcing effect? If so, how, given
> that it is not audible?!?!

There is a threshold below which you will not hear a note
of a given timbre and fundamental pitch, even though it
can be detected by physical apparatus, and above which
you will hear it. However, the total amplitude of the sum
of a number of inaudible notes may well be large enough
to exceed your threshold of hearing. Suppose, for example,
you can only hear notes of that timbre and pitch when the
wave amplitude exceeds level 9X. I now play that note at
level 8X - you can't hear it. But if I now also play its
octave at level 5X, exactly in phase with the first note,
the difference tone falls on the octave and has level 2X,
say, and being exactly in phase with the first note, adds
linearly to its amplitude to give a total level of 10X for
that note - which you can now hear.

> >The last three of these coincide with chord
> > tones, while the first two give a perceived fundamental of
> > C1, reinforced by its octave at C2.

The degree of reinforcement may be less than perfect,
depending on the relative phases of the two struck notes.

> > What you actually play is a chord on the piano with the
> > fundamental frequencies =
> > 33 * {2^(19/12), 2^(24/12), 2^(28/12), 2^(36/12)}, plus
> > overtones, with difference tones 33 * {their six
> > differences}:
> >
> > G2C3: 33*(4 - 2*2^(7/12)) = 33*(4 - 2.99734) = 33.08778
> > C3E3: 33*(4*2^(4/12) - 4) = 132*(2^(4/12) -1) = 34.3332
> > G2E3: 33*(4*2^(4/12) - 2*2^(7/12)) = 33*(2.04306) = 67.42098
> > E3C4: 33*(8 - 4*2^(4/12)) = 132*(2 - 2^(4/12)) = 97.6668
> > C3C4: 33*(8 - 4) = 33*4 = 132
> > G2C4: 33*(8 - 2*2^(7/12) = 33*5.00266 = 165.08778
> >
> > I've used the approximation:
> > 2^(1/12) = 1.0595
> >
> > whose powers are:
> > 2^(2/12) = 1.12254
> > 2^(3/12) = 1.18933
> > 2^(4/12) = 1.26010
> > 2^(5/12) = 1.33507
> > 2^(6/12) = 1.41451
> > 2^(7/12) = 1.49867
> > 2^(8/12) = 1.58784
> > 2^(9/12) = 1.68232
> > 2^(10/12) = 1.78242
> > 2^(11/12) = 1.88847
> >
> > Compare the results with the desired:
> > G2C3: 33.08778 ....... 33 ... 0.08778
> > C3E3: 34.3332 .......... 33 ... 0.3332
> > G2E3: 67.42098 ....... 66 ... 1.42098
> > E3C4: 97.6668 .......... 99 ... 1.3332
> > C3C4: 132 ................... 132 ... 0
> > G2C4: 165.08778 ... 165 ... 0.91222
> >
> > The third column of figures above gives the mistuning in Hertz.
> > There should be noticeable beating at around 1.3 Hz. This
> > would not occur with justly-tuned ratios 3:4:5:8.
>
> How do you know that the margin of variation between the 12 edo
> produced difference tones and the 'justly produced' difference tones
> is significant?

Because I hear it! :-) And so do others, eg Koenig
and Ellis.

> The fact that I notice, on my piano, something
> special with the chords listed at top, and presuming that the effect
> has something to do with these "doubly nested" difference I
> describe, makes me doubt your argument that what I'm hearing is not
> relavant to the just version.

The beating of the difference tone with the chord
tone is only audible when it occurs, and that is
when either ofthem is mistuned a little.

> The fact that I hear the presumed difference tone reinforcement
> effect with the 'out of tune' values you listed makes me wonder if
> there is a sort of "snap to grid" effect going on, maybe because of
> the strong influence of an interval archetype, on top of any
> acoustical or auditory mechanism. (Not to say that your tuning
> efforts are wasted!)

I don't know whether one can actually hear things
that aren't there! It often seems that way to me
(I "hear" most of my music before I get to realise
it, either by playing it or creating a MIDI file of it).
As I wrote earlier, I tend to hear gamelan and
angklung music as timbral rather than tonal, which
could well be an effect of my learned interval
categories filtering my perceptions. Have you ever
listened to any traditional Thai court music? That
is supposed to be tuned in 7-EDO, which would play
merry hell with most of our JI interval categories ...

Also, I'd like to consider the effect of the degree
of mistuning. Remembering that this group is about
harmonic entropy (rather than colour theory or
numerology ;-)), consider the shape of the HE curve.
Isn't it a bit like a normal distribution "bell-curve"
in the region of each JI ratio? IOW, doesn't the
HE have a flattened peak spread around the exact
ratios? If so, here's a question that has long
puzzled me: what bearing does this have on how we
actually hear a mistuned interval?

Regards,
Yahya

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🔗traktus5 <kj4321@...>

> Hi trak,

Hi Yahya. Please call me Kelly...

> On Mon, 20 Feb 2006, "traktus5" wrote:

> > > >...like the nice sounding chord 3:8:10:15, and the matching
> > > > difference tones for the 'subset chords' 3:8:15 and
3:10:15. In> > > > some chord progressions I'm working on , some
very prominant 4- > > > note chords, in addition to 3:8:10:15, are
5:8:12:15, 3:4:5:8, > > > > and 3:5:7:9, which all (though less for
the 3rd chord) have the> > > > property mentioned above (where the
subset 3 note chords have> > > > the same primary difference tones.)>

> >> > > Take 1 = C1. Then 2 = C2, 3 = G2, 4 = C3, 5 = E'3, 6 = G3,
> > > 7 = Bb`4, 8 = C4, 9 = D4, 10 = E'4, 12 = G4, 15 = B'5.
> > >
> > > 3:8:10:15 = G2:C4:E'4:B'5
> > >
> > > 5:8:12:15 = E'3:C4:G4:B'5
> > >
> > > 3:4:5:8 = G2:C3:E'3:C4
> > >
> > > 3:5:7:9 = G2:E'3:Bb`4:D4
> > >
> > > Are these the chords you're playing? <<"Yes".
>
> On an instrument tuned to the just ratios, you mean??!

I'm afraid not.

> > > Are you still exploring these chords on your 12-EDO piano?
>>"Yes"

> Only, or as well as on a justly tuned instument?

Just the piano, but not just.

> > Ok. So, refering here to 'difference tones' as those between
> > overtones (5-4, 8-5, etc) and not the amplitude 'wavering'
beating> > between nearly identical pitches, do difference tones
themselves> > beat against a nearby partial or note? Is that why
you make the> > point about the difference tone not coinciding with
the note?

> Alexander Ellis reports, Art. 3. (d) Beat-Notes and
> Differential Tones of his Appendix XX, on p. 530 of his
> translation of Hermann Helmholtz' "On the Sensations
> of Tone", several experiments using tuning forks, which
> confirm those reported by him made by R. Koenig, in
> which Ellis detects an *inaudible* difference tone (7)
> between the notes of a mistuned minor tenth (5:12), by
> its beating at 4.2 [Hertz] with an auxiliary tuning fork.

> The specific numbers Ellis cites are for two cases:

> Mistuned minor tenth:
> ------------------------
> Lower generating fork . . . : 223.77
> Upper generating fork . . . : 539.18
> Difference tone . . . . . . . . . : 315.41
> Auxiliary fork . . . . . . . . . . . : 319.59
> Theoretical difference . . .: 4.18
> Observed beat frequency : 4.2

> Mistuned octave:
> -------------------
> Lower generating fork . . . : 223.77
> Upper generating fork . . . : 451.14
> Difference tone . . . . . . . . . : 227.37
> Auxiliary fork . . . . . . . . . . . : 223.77 (*)
> Theoretical difference . . .: 3.6
> Observed beat frequency : 3.6
> (*) No auxiliary fork was necessary, the lower fork
> serving.
>
> Ellis also states that he has repeated this experiment
> "several times" with similar results.
>
> So, yes, difference tones *do* themselves beat against
> a nearby partial or note.

Maybe that's what's going on my 12 edo piano with a chord like
5:8:12:15...with the somewhat unusual manner in which the difference
tones 'land' on the chord tones, doubly nested, like I
described...that would cause quite a high degree of beating between
the difference tones and the chord tones, yes? Since the piano is
so unjust to begin with, and slighty 'mistuned'...I don't
know...maybe there is a connection.

> > (Does the fact that both beating and difference tones are a
product> > of the hearing system, and not actually physically
present in the> > sounding object, have anything to do with this?)

> Not so fast! Again according to Ellis, Art. 4. (a) Objective
> beats, p. 531 op. cit.:
> "Beats of a disturbed unison exist objectively as
> disturbances in the air beforeit reaches the ear. They
> are reinforced by resonators, they disturb sand, &c.
> In the case of the beats of harmonium reeds in Appunn's
> tonometer, they strongly shook the box containing the
> reeds. Other beats, beat-notes and combinational tones
> appear not to exist externally to the ear."

Ah! I've never seen that described before.

> > And on audibility of difference tones, why do textbooks I've seen
> > usually say they are only audible under special circumstances,
like> > high notes on the recorder? (Wow, if you can't hear it, but
it's> > having an effect, I wonder what's going on?)

> Ellis also uses a similar example, of very high notes on
> a flageolet, but only as an extreme case that he feels
> best demonstrates the existence of the difference
> tones. Note * on page 153, op. cit., on combinational
> tones, reads:
> "I have found that combinational tones can be made
> quite audible to a hundred people at once, by means
> of two flageolet fifes or whistles, blown as strongly
> as possible. I choose very close dissonant intervals
> because the great depth of the low tone is much more
> striking, being very far below anything that can be
> touched by the instrument itself."

(For the first time I just heard a good demo, "Titchener Difference
Tone Training", at www.faculty.ucr.edu)

> I conclude that Ellis finds that the difference notes
> are real enough for lower amplitude waves and that
> they arise between notes lower in pitch, exactly as
> Helmholtz describes. For example, playing a 4:5 dyad
> as C4:E4 produces the difference tone C2; a 4:6 dyad
> as C4:G4 produces the difference tone C3; a 4:5:6
> triad as C4:E4:G4 produces both of these.

> > > A small change in tuning can change the harmonicity
> > > of a difference tone by a fair amount. Let's look
> > > at your 3:4:5:8 chord more closely. Suppose you are
> > > playing that as G2:C3:E3:C4 on your 12-EDO piano. Let's
> > > assume for simplicity of calculation that your G2 is 99 Hz
> > > - that's equivalent to tuning A4=445.5Hz, but more
> > > importantly, to tuning C1 to 33 Hz, so all our frequencies
> > > are multiples of 33 Hz. Then your G2:C3:E3:C4 chord
> > > should have the fundamental frequencies 99, 132, 165, 264,
> > > plus their overtones, with the difference tones 33, 33, 66,
> > > 99, 132, 165.
> >
> > AGain, is a "reinforcing effect" the absence of beating?
>
> This doesn't follow logically. You seem to be saying:
> 'If it's not A, it must be B.' But couldn't it be something
> else? For example, it's perfectly possible to play two notes
> with exactly the same timbre and at exactly the same pitch
> - so they don't beat at all - yet without reinforcement. You
> have only to ensure that they are played with the opposite
> phase, one lagging 180° behind the other; then they will
> completely cancel each other out.

> Helmholtz was convinced that the difference tones could
> themselves form differences with chord tones, thus making
> second-order difference tones, third-order ... . Ellis was
> properly sceptical of this.

> > What if> > the difference tone is not audible -- which I don't
believe they are> > to me -- is there still a reinforcing effect?
If so, how, given> > that it is not audible?!?!

> There is a threshold below which you will not hear a note
> of a given timbre and fundamental pitch, even though it
> can be detected by physical apparatus, and above which
> you will hear it. However, the total amplitude of the sum
> of a number of inaudible notes may well be large enough
> to exceed your threshold of hearing. Suppose, for example,
> you can only hear notes of that timbre and pitch when the
> wave amplitude exceeds level 9X. I now play that note at
> level 8X - you can't hear it. But if I now also play its
> octave at level 5X, exactly in phase with the first note,
> the difference tone falls on the octave and has level 2X,
> say, and being exactly in phase with the first note, adds
> linearly to its amplitude to give a total level of 10X for
> that note - which you can now hear.

> > >The last three of these coincide with chord
> > > tones, while the first two give a perceived fundamental of
> > > C1, reinforced by its octave at C2.

I see. "Coincide", "reinforce", means 'in tune'. But playing in,
or out of phase, do instrumentalists have any control over the phase
of individual notes? Or is that just in the laboratory, or midi
instrument?

> The degree of reinforcement may be less than perfect,
> depending on the relative phases of the two struck notes.

> > > What you actually play is a chord on the piano with the
> > > fundamental frequencies =
> > > 33 * {2^(19/12), 2^(24/12), 2^(28/12), 2^(36/12)}, plus
> > > overtones, with difference tones 33 * {their six
> > > differences}:
> > >
> > > G2C3: 33*(4 - 2*2^(7/12)) = 33*(4 - 2.99734) = 33.08778
> > > C3E3: 33*(4*2^(4/12) - 4) = 132*(2^(4/12) -1) = 34.3332
> > > G2E3: 33*(4*2^(4/12) - 2*2^(7/12)) = 33*(2.04306) = 67.42098
> > > E3C4: 33*(8 - 4*2^(4/12)) = 132*(2 - 2^(4/12)) = 97.6668
> > > C3C4: 33*(8 - 4) = 33*4 = 132
> > > G2C4: 33*(8 - 2*2^(7/12) = 33*5.00266 = 165.08778
> > >
> > > I've used the approximation:
> > > 2^(1/12) = 1.0595
> > >
> > > whose powers are:
> > > 2^(2/12) = 1.12254
> > > 2^(3/12) = 1.18933
> > > 2^(4/12) = 1.26010
> > > 2^(5/12) = 1.33507
> > > 2^(6/12) = 1.41451
> > > 2^(7/12) = 1.49867
> > > 2^(8/12) = 1.58784
> > > 2^(9/12) = 1.68232
> > > 2^(10/12) = 1.78242
> > > 2^(11/12) = 1.88847
> > >
> > > Compare the results with the desired:
> > > G2C3: 33.08778 ....... 33 ... 0.08778
> > > C3E3: 34.3332 .......... 33 ... 0.3332
> > > G2E3: 67.42098 ....... 66 ... 1.42098
> > > E3C4: 97.6668 .......... 99 ... 1.3332
> > > C3C4: 132 ................... 132 ... 0
> > > G2C4: 165.08778 ... 165 ... 0.91222
> > >
> > > The third column of figures above gives the mistuning in Hertz.
> > > There should be noticeable beating at around 1.3 Hz.

Why 1.3 Hz? What about the mistunings of the other intervals? Is
it an average?

So (again; sorry...) maybe the effect I hear in a chord like
5:8:12:15 (as mentioned above) is the beating which occurs around
the chord tones. (BTW, would this be considered 'fast' beating?)

> > > This would not occur with justly-tuned ratios 3:4:5:8.

> > How do you know that the margin of variation between the 12 edo
> > produced difference tones and the 'justly produced' difference
tones> > is significant?

> Because I hear it! :-) And so do others, eg Koenig> and Ellis.

And that's all I've *ever* heard on my 12 edo piano!

(snip)

> I don't know whether one can actually hear things
> that aren't there! It often seems that way to me
> (I "hear" most of my music before I get to realise
> it, either by playing it or creating a MIDI file of it).
> As I wrote earlier, I tend to hear gamelan and
> angklung music as timbral rather than tonal

tonal as in western harmony, or as in the Rameau-Erlich chord root?

, which
> could well be an effect of my learned interval
> categories filtering my perceptions. Have you ever
> listened to any traditional Thai court music?

I recall Sethares had a chapter on it.

>That> is supposed to be tuned in 7-EDO, which would play
> merry hell with most of our JI interval categories ...

> Also, I'd like to consider the effect of the degree
> of mistuning. Remembering that this group is about
> harmonic entropy (rather than colour theory or
> numerology ;-)), consider the shape of the HE curve.
> Isn't it a bit like a normal distribution "bell-curve"
> in the region of each JI ratio? IOW, doesn't the
> HE have a flattened peak spread around the exact
> ratios? If so, here's a question that has long
> puzzled me: what bearing does this have on how we
> actually hear a mistuned interval?

That sounds like a good question for Paul! My conviction --not
based on any real psychoacoustics, everyone tells me -- is that it
would 'default', somehow, to the exact just ratio.

See ya mate!

Kelly

🔗Yahya Abdal-Aziz <yahya@...>

Hi Kelly,

On Wed, 22 Feb 2006, "traktus5" wrote:
>
> > Hi trak,
>
> Hi Yahya. Please call me Kelly...
>
> > On Mon, 20 Feb 2006, "traktus5" wrote:
>
> > > > >...like the nice sounding chord 3:8:10:15, and the matching
> > > > > difference tones for the 'subset chords' 3:8:15 and
> 3:10:15. In> > > > some chord progressions I'm working on , some
> very prominant 4- > > > note chords, in addition to 3:8:10:15, are
> 5:8:12:15, 3:4:5:8, > > > > and 3:5:7:9, which all (though less for
> the 3rd chord) have the> > > > property mentioned above (where the
> subset 3 note chords have> > > > the same primary difference tones.)>
>
> > >> > > Take 1 = C1. Then 2 = C2, 3 = G2, 4 = C3, 5 = E'3, 6 = G3,
> > > > 7 = Bb`4, 8 = C4, 9 = D4, 10 = E'4, 12 = G4, 15 = B'5.
> > > >
> > > > 3:8:10:15 = G2:C4:E'4:B'5
> > > >
> > > > 5:8:12:15 = E'3:C4:G4:B'5
> > > >
> > > > 3:4:5:8 = G2:C3:E'3:C4
> > > >
> > > > 3:5:7:9 = G2:E'3:Bb`4:D4
> > > >
> > > > Are these the chords you're playing? <<"Yes".
> >
> > On an instrument tuned to the just ratios, you mean??!
>
>
> I'm afraid not.
>
>
> > > > Are you still exploring these chords on your 12-EDO piano?
> >>"Yes"
>
> > Only, or as well as on a justly tuned instument?
>
> Just the piano, but not just.
>
> > > Ok. So, refering here to 'difference tones' as those between
> > > overtones (5-4, 8-5, etc) and not the amplitude 'wavering'
> beating> > between nearly identical pitches, do difference tones
> themselves> > beat against a nearby partial or note? Is that why
> you make the> > point about the difference tone not coinciding with
> the note?
>
> > Alexander Ellis reports, Art. 3. (d) Beat-Notes and
> > Differential Tones of his Appendix XX, on p. 530 of his
> > translation of Hermann Helmholtz' "On the Sensations
> > of Tone", several experiments using tuning forks, which
> > confirm those reported by him made by R. Koenig, in
> > which Ellis detects an *inaudible* difference tone (7)
> > between the notes of a mistuned minor tenth (5:12), by
> > its beating at 4.2 [Hertz] with an auxiliary tuning fork.
>
> > > The specific numbers Ellis cites are for two cases:
>
> > Mistuned minor tenth:
> > ------------------------
> > Lower generating fork . . . : 223.77
> > Upper generating fork . . . : 539.18
> > Difference tone . . . . . . . . . : 315.41
> > Auxiliary fork . . . . . . . . . . . : 319.59
> > Theoretical difference . . .: 4.18
> > Observed beat frequency : 4.2
>
> > Mistuned octave:
> > -------------------
> > Lower generating fork . . . : 223.77
> > Upper generating fork . . . : 451.14
> > Difference tone . . . . . . . . . : 227.37
> > Auxiliary fork . . . . . . . . . . . : 223.77 (*)
> > Theoretical difference . . .: 3.6
> > Observed beat frequency : 3.6
> > (*) No auxiliary fork was necessary, the lower fork
> > serving.
> >
> > Ellis also states that he has repeated this experiment
> > "several times" with similar results.
> >
> > So, yes, difference tones *do* themselves beat against
> > a nearby partial or note.
>
> Maybe that's what's going on my 12 edo piano with a chord like
> 5:8:12:15...with the somewhat unusual manner in which the difference
> tones 'land' on the chord tones, doubly nested, like I
> described...that would cause quite a high degree of beating between
> the difference tones and the chord tones, yes? Since the piano is
> so unjust to begin with, and slighty 'mistuned'...I don't
> know...maybe there is a connection.

I'd guess that the strongest beats would be those
between the strongest partial tones in the chord.
So if any difference does coincide with, or "land
on", a partial, that should (in most cases, depending
on phase) increase the strength of that partial and
therefore of its beats.

There was an issue of Scientific American I kept
for many years featuring the inharmonicity of the
overtones (upper partials) of piano notes. From
memory, the "unjustness" of the piano manifests
mostly in the higher partials; at any rate, it was
not enough for single notes to be "very unjust",
only "interestingly coloured". Certainly, tuning the
piano in 12-EDO will result in very unjust thirds and
sixths, and no good representation of the seventh
harmonic; so that any common-practice triad or
tetrad will be significantly different than just.

> > > (Does the fact that both beating and difference tones are a
> product> > of the hearing system, and not actually physically
> present in the> > sounding object, have anything to do with this?)
>
> > Not so fast! Again according to Ellis, Art. 4. (a) Objective
> > beats, p. 531 op. cit.:
> > "Beats of a disturbed unison exist objectively as
> > disturbances in the air before it reaches the ear. They
> > are reinforced by resonators, they disturb sand, &c.
> > In the case of the beats of harmonium reeds in Appunn's
> > tonometer, they strongly shook the box containing the
> > reeds. Other beats, beat-notes and combinational tones
> > appear not to exist externally to the ear."
>
> Ah! I've never seen that described before.
>
> > > And on audibility of difference tones, why do textbooks I've seen
> > > usually say they are only audible under special circumstances,
> like> > high notes on the recorder? (Wow, if you can't hear it, but
> it's> > having an effect, I wonder what's going on?)
>
> > Ellis also uses a similar example, of very high notes on
> > a flageolet, but only as an extreme case that he feels
> > best demonstrates the existence of the difference
> > tones. Note * on page 153, op. cit., on combinational
> > tones, reads:
> > "I have found that combinational tones can be made
> > quite audible to a hundred people at once, by means
> > of two flageolet fifes or whistles, blown as strongly
> > as possible. I choose very close dissonant intervals
> > because the great depth of the low tone is much more
> > striking, being very far below anything that can be
> > touched by the instrument itself."
>
> (For the first time I just heard a good demo, "Titchener Difference
> Tone Training", at www.faculty.ucr.edu)

Tried that link, but couldn't find it that site. :-(
But I found the whole thing publicly accessible at:
http://psyche.cs.monash.edu.au/articles/schwitzgebel/
Very interesting stuff! What was your experience
of the training? Did you try the pre-test?

[snip]
> I see. "Coincide", "reinforce", means 'in tune'. ...

More than that, "reinforce" means "increase the amplitude"
and hence "increase the apparent loudness".

> ... But playing in,
> or out of phase, do instrumentalists have any control over the phase
> of individual notes? Or is that just in the laboratory, or midi
> instrument?

I think you're right.

Players of acoustical instruments have no effective
control over phase. Consider the following thought-
experiment: Under ideal conditions, a player strikes
or produces a given note repeatedly, in exactly the
same manner and with exactly the same force. His
or her instrument is capable of producing sound at
frequencies in the range 20 to 40 Hertz. Being a
very competent orchestral performer on a responsive
instrument, this player is able to deliver thirtysecond-
notes (demisemiquavers) precisely on time in a presto
tempo of 240 MM (bpm). That amounts to 4 (metrical)
beats per second, or 32 notes per second. To say that
these notes are precisely on time means that the player
can deliver them with a maximum error of much less
than half their duration, 1/64 second. (Maybe she plays
marimba, or maybe he's a drummer.) As far as I can
tell, the only way a performer can influence the phase
of the note is by timing its attack. (Conceivably a string
player might be able to do this for each of the many
tiny attacks caused by bowing, but I doubt it.) For this
paragon to control the phase of a note with frequency
32 Hz, even to the crude extent of being in a particular
quarter of the waveform's cycle, he or she must be able
to deliver the attack accurate to the nearest 1/128
second, with an error much less than half that, ie less
than 1/256 second. This is four times the accuracy we
can reasonably expect from our expert performer. Even
if the player is twice as good as I've suggested, it is still
impractical to control the phase thru timing the attack.
As for being able to control both the timing and the phase
of individual notes independently, I fear that's impossible.

Your MIDI device - given the same player accuracy - would
not do any better, I think, unless triggered by a MIDI file.
In that case we have considerably more precision in timing
the note attacks and thus their phases. It might be
instructive to write a piece with hemidemisemiquavers and
very low notes to see whether we could achieve, say, phase
cancellation.

In practice, this inability to control phase has the
interesting consequence that whenever you play a
justly tuned chord, the combination tones may vary
all the way from total reinforcement of overtones
already existing in the chord notes to their total
cancellation! You could hardly ever play the same
chord twice - as one can only ever step once into
the same river. Maybe this is one reason why those
musicians who like to have total control over the
effects of the sounds produced by their music have
gravitated towards 12-EDO - the beats of the
differences with chord partials are more predictable,
producing much the same sound each time a chord is
struck? Just an hypothesis - wouldn't be hard to
test with the right equipment.

> > The degree of reinforcement may be less than perfect,
> > depending on the relative phases of the two struck notes.
>
>
> > > > What you actually play is a chord on the piano with the
> > > > fundamental frequencies =
> > > > 33 * {2^(19/12), 2^(24/12), 2^(28/12), 2^(36/12)}, plus
> > > > overtones, with difference tones 33 * {their six
> > > > differences}:
> > > >
> > > > G2C3: 33*(4 - 2*2^(7/12)) = 33*(4 - 2.99734) = 33.08778
> > > > C3E3: 33*(4*2^(4/12) - 4) = 132*(2^(4/12) -1) = 34.3332
> > > > G2E3: 33*(4*2^(4/12) - 2*2^(7/12)) = 33*(2.04306) = 67.42098
> > > > E3C4: 33*(8 - 4*2^(4/12)) = 132*(2 - 2^(4/12)) = 97.6668
> > > > C3C4: 33*(8 - 4) = 33*4 = 132
> > > > G2C4: 33*(8 - 2*2^(7/12) = 33*5.00266 = 165.08778
> > > >
> > > > I've used the approximation:
> > > > 2^(1/12) = 1.0595
> > > >
> > > > whose powers are:
> > > > 2^(2/12) = 1.12254
> > > > 2^(3/12) = 1.18933
> > > > 2^(4/12) = 1.26010
> > > > 2^(5/12) = 1.33507
> > > > 2^(6/12) = 1.41451
> > > > 2^(7/12) = 1.49867
> > > > 2^(8/12) = 1.58784
> > > > 2^(9/12) = 1.68232
> > > > 2^(10/12) = 1.78242
> > > > 2^(11/12) = 1.88847
> > > >
> > > > Compare the results with the desired:
> > > > G2C3: 33.08778 ....... 33 ... 0.08778
> > > > C3E3: 34.3332 .......... 33 ... 0.3332
> > > > G2E3: 67.42098 ....... 66 ... 1.42098
> > > > E3C4: 97.6668 .......... 99 ... 1.3332
> > > > C3C4: 132 ................... 132 ... 0
> > > > G2C4: 165.08778 ... 165 ... 0.91222
> > > >
> > > > The third column of figures above gives the mistuning in Hertz.
> > > > There should be noticeable beating at around 1.3 Hz.
>
> Why 1.3 Hz? What about the mistunings of the other intervals? Is
> it an average?

"around 1.3 Hz" includes the 0.91, 1.33 and 1.42 Hz beats.
These may be noticeable - depending on their intensity -
if the notes are held for one to two seconds or more. The
beat at 0.33 Hz would take 3 seconds to hear one cycle;
that at 0.088 Hz would take 12 secnods to hear one cycle.
Most music doesn't have notes that long or sustained.

> So (again; sorry...) maybe the effect I hear in a chord like
> 5:8:12:15 (as mentioned above) is the beating which occurs around
> the chord tones. (BTW, would this be considered 'fast' beating?)

Why sorry? Talking to you helps me clarify my own
understanding - or lack of it! - so can be mutually
beneficial. I thank you for the opportunity.

Yes, I think the effect you hear is very likely to be
"beating around the chord tones". But you hear what
you hear, and don't let me be the one to persuade you
otherwise! ;-) You may well be hearing something that
hasn't been adequately described yet. So please do
trust your own observations, and seek explanations
of anything you don't understand until you're satisfied.
Isn't this what scientists and philosophers do?

> > > > This would not occur with justly-tuned ratios 3:4:5:8.
>
> > > How do you know that the margin of variation between the 12 edo
> > > produced difference tones and the 'justly produced' difference
> tones> > is significant?
>
> > Because I hear it! :-) And so do others, eg Koenig> and Ellis.
>
> And that's all I've *ever* heard on my 12 edo piano!

Yes, very likely.

> (snip)
>
> > I don't know whether one can actually hear things
> > that aren't there! It often seems that way to me
> > (I "hear" most of my music before I get to realise
> > it, either by playing it or creating a MIDI file of it).
> > As I wrote earlier, I tend to hear gamelan and
> > angklung music as timbral rather than tonal
>
> tonal as in western harmony, or as in the Rameau-Erlich chord root?

"tonal as in western harmony". I'm unsure what you
mean exactly by "the Rameau-Erlich chord root".

> , which
> > could well be an effect of my learned interval
> > categories filtering my perceptions. Have you ever
> > listened to any traditional Thai court music?
>
> I recall Sethares had a chapter on it.
>
> >That> is supposed to be tuned in 7-EDO, which would play
> > merry hell with most of our JI interval categories ...
>
> > Also, I'd like to consider the effect of the degree
> > of mistuning. Remembering that this group is about
> > harmonic entropy (rather than colour theory or
> > numerology ;-)), consider the shape of the HE curve.
> > Isn't it a bit like a normal distribution "bell-curve"
> > in the region of each JI ratio? IOW, doesn't the
> > HE have a flattened peak spread around the exact
> > ratios? If so, here's a question that has long
> > puzzled me: what bearing does this have on how we
> > actually hear a mistuned interval?
>
> That sounds like a good question for Paul! My conviction --not
> based on any real psychoacoustics, everyone tells me -- is that it
> would 'default', somehow, to the exact just ratio.

Well, I know that I hear an equally-tempered fifth
as pretty much an exact fifth in most contexts,
but I can't say the same for thirds, which are much
more mistuned.

> See ya mate!
> Kelly

:-)

Regards,
Yahya

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🔗traktus5 <kj4321@...>

Hi Yahya.

(excerpted)

> There was an issue of Scientific American I kept
> for many years featuring the inharmonicity of the
> overtones (upper partials) of piano notes. From
> memory, the "unjustness" of the piano manifests
> mostly in the higher partials; at any rate, it was
> not enough for single notes to be "very unjust",
> only "interestingly coloured". Certainly, tuning the
> piano in 12-EDO will result in very unjust thirds and
> sixths, and no good representation of the seventh
> harmonic; so that any common-practice triad or
> tetrad will be significantly different than just.

I remember that article, too. (I use to hang the cover illustration
on my wall!) The other very interesting aspect of piano acoustics,
to me, is how some partials (I'd have to look it up) actually
actually *increase* in amplitude over time! (ie, 3-5 seconds after
the attack.) I believe that adds to the (for me) interesting 'buzz'
which occurs as the chord sound evolves.

> > (For the first time I just heard a good demo, "Titchener
Difference > > Tone Training", at www.faculty.ucr.edu)

> Tried that link, but couldn't find it that site. :-(
> But I found the whole thing publicly accessible at:
> http://psyche.cs.monash.edu.au/articles/schwitzgebel/
> Very interesting stuff! What was your experience
> of the training? Did you try the pre-test?

I plan to.

> [snip]

> > I see. "Coincide", "reinforce", means 'in tune'. ...
>
> More than that, "reinforce" means "increase the amplitude"
> and hence "increase the apparent loudness".

(Also called 'coherence'?)

Is it noticibly louder, or just as measured in a lab? Would that be
like raising piano to mezzo piano?

> > So (again; sorry...) maybe the effect I hear in a chord like
> > 5:8:12:15 (as mentioned above) is the beating which occurs
around > > the chord tones. (BTW, would this be considered 'fast'
beating?)

> Why sorry? Talking to you helps me clarify my own
> understanding - or lack of it! - so can be mutually
> beneficial. I thank you for the opportunity.
>
> Yes, I think the effect you hear is very likely to be
> "beating around the chord tones". But you hear what
> you hear, and don't let me be the one to persuade you
> otherwise! ;-) You may well be hearing something that
> hasn't been adequately described yet. So please do
> trust your own observations, and seek explanations
> of anything you don't understand until you're satisfied.
> Isn't this what scientists and philosophers do?

Thanks for the encouragement. This list (especially Paul Erlich) has
been very helpful for understanding chords better, but I have even
more questions now than before I began this study!

> > It often seems that way to me
> > > (I "hear" most of my music before I get to realise
> > > it, either by playing it or creating a MIDI file of it).
> > > As I wrote earlier, I tend to hear gamelan and
> > > angklung music as timbral rather than tonal

> > tonal as in western harmony, or as in the Rameau-Erlich chord
root?

> "tonal as in western harmony". I'm unsure what you
> mean exactly by "the Rameau-Erlich chord root".

I meant, does Paul's concept of 'tonalness' (and isn't that a
descendent of Rameau's 'chord root'?) ever apply to chords which
occur in gamelan and angklung music? (Maybe only the tertial chords
of western harmony have that feature?)

thanks, Kelly

🔗Yahya Abdal-Aziz <yahya@...>

Hi Kelly,

Just time for the briefest reply before bed.

on Fri, 24 Feb 2006 "traktus5" wrote:

> (excerpted)
>
> > There was an issue of Scientific American I kept
> > for many years featuring the inharmonicity of the
> > overtones (upper partials) of piano notes. From
> > memory, the "unjustness" of the piano manifests
> > mostly in the higher partials; at any rate, it was
> > not enough for single notes to be "very unjust",
> > only "interestingly coloured". Certainly, tuning the
> > piano in 12-EDO will result in very unjust thirds and
> > sixths, and no good representation of the seventh
> > harmonic; so that any common-practice triad or
> > tetrad will be significantly different than just.
>
> I remember that article, too. (I use to hang the cover illustration
> on my wall!) The other very interesting aspect of piano acoustics,
> to me, is how some partials (I'd have to look it up) actually
> actually *increase* in amplitude over time! (ie, 3-5 seconds after
> the attack.) I believe that adds to the (for me) interesting 'buzz'
> which occurs as the chord sound evolves.

That late?! Wow!

> > > (For the first time I just heard a good demo, "Titchener
> Difference > > Tone Training", at www.faculty.ucr.edu)
>
> > Tried that link, but couldn't find it that site. :-(
> > But I found the whole thing publicly accessible at:
> > http://psyche.cs.monash.edu.au/articles/schwitzgebel/
> > Very interesting stuff! What was your experience
> > of the training? Did you try the pre-test?
>
> I plan to.

Let's know what happens?

> > [snip]
>
> > > I see. "Coincide", "reinforce", means 'in tune'. ...
> >
> > More than that, "reinforce" means "increase the amplitude"
> > and hence "increase the apparent loudness".
>
> (Also called 'coherence'?)

Hmmm ... I guess! Two totally coherent waves would
be exactly in phase --- just like a laser, and so would
reinforce each other.

> Is it noticibly louder, or just as measured in a lab? Would that be
> like raising piano to mezzo piano?

If the secondary, reinforcing tone is audible, it'd
have to make a noticeable difference.

> > > So (again; sorry...) maybe the effect I hear in a chord like
> > > 5:8:12:15 (as mentioned above) is the beating which occurs
> around > > the chord tones. (BTW, would this be considered 'fast'
> beating?)
>
> > Why sorry? Talking to you helps me clarify my own
> > understanding - or lack of it! - so can be mutually
> > beneficial. I thank you for the opportunity.
> >
> > Yes, I think the effect you hear is very likely to be
> > "beating around the chord tones". But you hear what
> > you hear, and don't let me be the one to persuade you
> > otherwise! ;-) You may well be hearing something that
> > hasn't been adequately described yet. So please do
> > trust your own observations, and seek explanations
> > of anything you don't understand until you're satisfied.
> > Isn't this what scientists and philosophers do?
>
> Thanks for the encouragement. This list (especially Paul Erlich) has
> been very helpful for understanding chords better, but I have even
> more questions now than before I began this study!

Not surprising! :-) Friend of mine tells the following
parable:

Think of what we know as a sphere of knowledge,
with all the ideas interconnected inside. Outside
the sphere lie all the things we don't know - a whole
universe of ignorance. The surface of the sphere is
where our understanding is challenged by unanswered
questions - and these are questions we could not even
have known how to ask when the sphere was only half
as big. The more we learn, the bigger the surface of
that sphere of knowledge grows. Which is why, the
more we know, the more we realise we don't know ...!

> > > It often seems that way to me
> > > > (I "hear" most of my music before I get to realise
> > > > it, either by playing it or creating a MIDI file of it).
> > > > As I wrote earlier, I tend to hear gamelan and
> > > > angklung music as timbral rather than tonal
>
> > > tonal as in western harmony, or as in the Rameau-Erlich chord
> root?
>
> > "tonal as in western harmony". I'm unsure what you
> > mean exactly by "the Rameau-Erlich chord root".
>
> I meant, does Paul's concept of 'tonalness' (and isn't that a
> descendent of Rameau's 'chord root'?) ever apply to chords which
> occur in gamelan and angklung music? (Maybe only the tertial chords
> of western harmony have that feature?)

I don't know.
I don't know.
I don't know.

Thereby illustrating the "sphere of knowledge" concept
quite nicely! ;-)

I'm not even sure what "Paul's concept of 'tonalness'"
includes. Help me out?

Regards,
Yahya

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🔗traktus5 <kj4321@...>

Hi Yahya,

[excerpted.]

>The other very interesting aspect of piano acoustics, > > to me, is
how some partials (I'd have to look it up) actually > > actually
*increase* in amplitude over time! (ie, 3-5 seconds after
> > the attack.) I believe that adds to the (for me)
interesting 'buzz' > > which occurs as the chord sound evolves.

> That late?!

Pardon me. The decay curve activity of the partials occurs in the
first two seconds, according to one source (in the section 'decay
curves')...

http://www.concertpitchpiano.com/PhysicsOfPiano.html

but indeed, several partials do increase in amplitude. Nonetheless,
the long decay of a loud piano chord, (eg, 5, 10, 15 seconds out,
and longer), is fascinating to listen to. I wish I had the
equipment to amplify the decay and fool around with it.

Friend of mine tells the following
> parable:
>
> Think of what we know as a sphere of knowledge,
> with all the ideas interconnected inside. Outside
> the sphere lie all the things we don't know - a whole
> universe of ignorance. The surface of the sphere is
> where our understanding is challenged by unanswered
> questions - and these are questions we could not even
> have known how to ask when the sphere was only half
> as big. The more we learn, the bigger the surface of
> that sphere of knowledge grows. Which is why, the
> more we know, the more we realise we don't know ...!

That's great. Thanks!

>
> I'm not sure what "Paul's concept of 'tonalness'"
> includes. Help me out?

From an Erlich-Monz conversation, at http://sonic-
arts.org/td/erlich/entropy.htm,

"There is a very strong propensity for the ear to try to fit what it
hears into one or a small number of harmonic series, and the
fundamentals of these series, even if not physically present, are
either heard outright, or provide a more subtle sense of overall
pitch known to musicians as the "root". As a component of
consonance, the ease with which the ear/brain system can resolve the
fundamental is known as "tonalness." I have proposed a concept
called "relative harmonic entropy" to model this component of
dissonance."

Or, from the Tonalsoft encyclopedia: "A quantity measuring the
sonance of a [musical] interval or chord, based on the degree to
which the notes approximate a harmonic series over a single
fundamental and thereby blend into a single sensation. The perceived
pitch of that single sensation corresponds to the root of the chord.
This conception of sonance is associated with the harmonic theories
in the tradition of Rameau, and finds a modern advocate in
Parncutt.)"

(Isn't Parncutt also Australian? Or was that another famous
acoustician?)

Cheers, Kelly

🔗Yahya Abdal-Aziz <yahya@...>

Hi Kelly,

On Sun, 26 Feb 2006 "traktus5" wrote:

[snip]

> >The other very interesting aspect of piano acoustics, > > to me, is
> how some partials (I'd have to look it up) actually > > actually
> *increase* in amplitude over time! (ie, 3-5 seconds after
> > > the attack.) I believe that adds to the (for me)
> interesting 'buzz' > > which occurs as the chord sound evolves.
>
> > That late?!
>
> Pardon me. The decay curve activity of the partials occurs in the
> first two seconds, according to one source (in the section 'decay
> curves')...
>
> http://www.concertpitchpiano.com/PhysicsOfPiano.html
>
> but indeed, several partials do increase in amplitude. Nonetheless,
> the long decay of a loud piano chord, (eg, 5, 10, 15 seconds out,
> and longer), is fascinating to listen to. I wish I had the
> equipment to amplify the decay and fool around with it.
>
> Friend of mine tells the following
> > parable:
> >
[snip]
> > ... the
> > more we know, the more we realise we don't know ...!
>
> That's great. Thanks!

Thought you might appreciate that one!

> > I'm not sure what "Paul's concept of 'tonalness'"
> > includes. Help me out?
>
> From an Erlich-Monz conversation, at http://sonic-
> arts.org/td/erlich/entropy.htm,
>
> "There is a very strong propensity for the ear to try to fit what it
> hears into one or a small number of harmonic series, and the
> fundamentals of these series, even if not physically present, are
> either heard outright, or provide a more subtle sense of overall
> pitch known to musicians as the "root". As a component of
> consonance, the ease with which the ear/brain system can resolve the
> fundamental is known as "tonalness." I have proposed a concept
> called "relative harmonic entropy" to model this component of
> dissonance."

Hmm. This quote definitely raises more questions
than it answers ... For example:
1. How do we know there is such a propensity?
2. How do we quantify it, or even qualify it, as
"very strong"?
3. If it exists, why does it?
4. How does it arise - acoustically in space, or
acoustically in the ear, or neurologically?

[Yul Brynner in "The King and I":]
Et cetera, et cetera, et cetera!

> Or, from the Tonalsoft encyclopedia: "A quantity measuring the
> sonance of a [musical] interval or chord, based on the degree to
> which the notes approximate a harmonic series over a single
> fundamental and thereby blend into a single sensation. The perceived
> pitch of that single sensation corresponds to the root of the chord.
> This conception of sonance is associated with the harmonic theories
> in the tradition of Rameau, and finds a modern advocate in
> Parncutt.)"

So it's a theoretical construct and measure,
rather than an acoustical or psycho-acoustical
one? If so, what makes anyone think it's real?

> (Isn't Parncutt also Australian? Or was that another famous
> acoustician?)

First two hits for "Parncutt" on Google:

Richard Parncutt: HomepageRichard Parncutt Professor of Systematic
Musicology, University of Graz, Austria ...
http://www-gewi.uni-graz.at/staff/parncutt/
www-gewi.kfunigraz.ac.at/muwi/parncutt/index.html - 19k - Cached - Similar
pages

Jessica Parncutt | Offical Website | Australian Singer / SongwriterJessica
Parncutt, Australian Singer / Songwriter - All In The Way is an instant hit,
and there are many more to come with her soulful piano based tunes.
www.jessicaparncutt.com/ - 4k - Cached - Similar pages

I think you must have meant Richard, the Austrian! :-)

Regards,
Yahya

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🔗traktus5 <kj4321@...>

hi Yahya,

On your parable about the expanding sphere of knowledge...

I like the sphere, defined by 3 axis, as a model for music. On the
upper half of the Z axis, you have melody, or whole numbers, and on
the lower part, below the middle point, you have harmony, or
fractions. On the Y axis, you have gross amplitude on one side of
the middle point, and on the other side, timbre (the volume of each
partial). And the X axis, or course, is time, with the forward
section being tempo and rhythm, and the backward part, behind the
middle point, being Form. ("Back illumination...repetition,
referring in some way to previous material, which creates form.)

(If I may digress briefly, I was thinking about that parable, and
feel it also suggests the ultimately unsatisfying nature of
rationality (for me. Mabye thought waves dissipate like sound waves
do!). As the parable suggests to me, you can also go inward to the
center, where maybe truth also resides, maybe similar to how, with
matter, concentrated 'gravitational forces' can cause a sort of
breakdown, or 'singularity,' at the center, where you transcend the
system....And taking the gravitation analogy further...given that
gravity can be felt at very long distances, maybe you can connect
with all the knowledge, regardless of how remote it is, if you're at
the center!)

> > > I'm not sure what "Paul's concept of 'tonalness'"
> > > includes. Help me out?

> > From an Erlich-Monz conversation, at http://sonic-
> > arts.org/td/erlich/entropy.htm,

> > "There is a very strong propensity for the ear to try to fit
what it> > hears into one or a small number of harmonic series, and
the> > fundamentals of these series, even if not physically present,
are> > either heard outright, or provide a more subtle sense of
overall> > pitch known to musicians as the "root". As a component of
> > consonance, the ease with which the ear/brain system can resolve
the> > fundamental is known as "tonalness." I have proposed a concept
> > called "relative harmonic entropy" to model this component of
> > dissonance."

> Hmm. This quote definitely raises more questions
> than it answers ...
For example:
> 1. How do we know there is such a propensity?

(Just to see if I've been paying attention at this list...) I
believe he is referring to laboratory findings about fusion, and
comb. tones, etc.

> 2. How do we quantify it, or even qualify it, as
> "very strong"?
> 3. If it exists, why does it?

Maybe it exists in the hearing system because harmonic overtones
exist in certain vibrating objects?

> 4. How does it arise - acoustically in space, or
> acoustically in the ear, or neurologically?

> [Yul Brynner in "The King and I":]
> Et cetera, et cetera, et cetera!

> > Or, from the Tonalsoft encyclopedia: "A quantity measuring the
> > sonance of a [musical] interval or chord, based on the degree to
> > which the notes approximate a harmonic series over a single
> > fundamental and thereby blend into a single sensation. The
perceived> > pitch of that single sensation corresponds to the root
of the chord.> > This conception of sonance is associated with the
harmonic theories> > in the tradition of Rameau, and finds a modern
advocate in> > Parncutt.)"

> So it's a theoretical construct and measure,
> rather than an acoustical or psycho-acoustical
> one? If so, what makes anyone think it's real?

Hmm, maybe Paul will comment. Myself, I've been curious what
exactly goes on in a chord whose individual intervals are more tonal
than the overall chord-signal.

Tally ho!

Kelly

(By the way, why do they call Australia "Oz"?)

🔗Yahya Abdal-Aziz <yahya@...>

Hi Kelly,

On Tue, 28 Feb 2006 "traktus5" wrote:
>
> hi Yahya,
>
> On your parable about the expanding sphere of knowledge...
>
> I like the sphere, defined by 3 axis, as a model for music. On the
> upper half of the Z axis, you have melody, or whole numbers, and on
> the lower part, below the middle point, you have harmony, or
> fractions. On the Y axis, you have gross amplitude on one side of
> the middle point, and on the other side, timbre (the volume of each
> partial). And the X axis, or course, is time, with the forward
> section being tempo and rhythm, and the backward part, behind the
> middle point, being Form.

The way I count, seems like you actually need SIX
dimensions to cover your different variables. Now
try to visualise them as being orthogonal! :-0

I particularly like your last comment here:
> Back illumination...repetition, referring in some way
> to previous material, which creates form.

It makes me think how presentation of related
material can illuminate relationships between
(musical or other) materials that might otherwise
remain hidden.

> (If I may digress briefly, I was thinking about that parable, and
> feel it also suggests the ultimately unsatisfying nature of
> rationality (for me. Mabye thought waves dissipate like sound waves
> do!). As the parable suggests to me, you can also go inward to the
> center, where maybe truth also resides, maybe similar to how, with
> matter, concentrated 'gravitational forces' can cause a sort of
> breakdown, or 'singularity,' at the center, where you transcend the
> system....And taking the gravitation analogy further...given that
> gravity can be felt at very long distances, maybe you can connect
> with all the knowledge, regardless of how remote it is, if you're at
> the center!)

Very nice!

Then again, perhaps the centre is a state of total
ignorance - or of acknowledging it. I guess that
response may suggest the ultimately unsatisfying
nature of _irrationality_ as well as of rationality.

> > > > I'm not sure what "Paul's concept of 'tonalness'"
> > > > includes. Help me out?
>
> > > From an Erlich-Monz conversation, at http://sonic-
> > > arts.org/td/erlich/entropy.htm,
>
> > > "There is a very strong propensity for the ear to try to fit
> what it> > hears into one or a small number of harmonic series, and
> the> > fundamentals of these series, even if not physically present,
> are> > either heard outright, or provide a more subtle sense of
> overall> > pitch known to musicians as the "root". As a component of
> > > consonance, the ease with which the ear/brain system can resolve
> the> > fundamental is known as "tonalness." I have proposed a concept
> > > called "relative harmonic entropy" to model this component of
> > > dissonance."
>
> > Hmm. This quote definitely raises more questions
> > than it answers ...
> > For example:
> > 1. How do we know there is such a propensity?
>
> (Just to see if I've been paying attention at this list...) I
> believe he is referring to laboratory findings about fusion, and
> comb. tones, etc.

He does say this propensity is "for the ear to try to fit
what it hears" - which suggest something that requires
no perceiver, just an ear (presumably from a cadaver
would do nicely!) Is this really just physical, or perhaps
he was indulging in a figure (organ?) of speech ...

> > 2. How do we quantify it, or even qualify it, as
> > "very strong"?
> > 3. If it exists, why does it?
>
> Maybe it exists in the hearing system because harmonic overtones
> exist in certain vibrating objects?

So, it exists externally, not in the ear?

> > 4. How does it arise - acoustically in space, or
> > acoustically in the ear, or neurologically?
>
> > [Yul Brynner in "The King and I":]
> > Et cetera, et cetera, et cetera!
>
> :)
>
> > > Or, from the Tonalsoft encyclopedia: "A quantity measuring the
> > > sonance of a [musical] interval or chord, based on the degree to
> > > which the notes approximate a harmonic series over a single
> > > fundamental and thereby blend into a single sensation. The
> perceived> > pitch of that single sensation corresponds to the root
> of the chord.> > This conception of sonance is associated with the
> harmonic theories> > in the tradition of Rameau, and finds a modern
> advocate in> > Parncutt.)"
>
> > So it's a theoretical construct and measure,
> > rather than an acoustical or psycho-acoustical
> > one? If so, what makes anyone think it's real?
>
> Hmm, maybe Paul will comment. Myself, I've been curious what
> exactly goes on in a chord whose individual intervals are more tonal
> than the overall chord-signal.
>
> Tally ho!
>
> Kelly
>
> (By the way, why do they call Australia "Oz"?)

Coz. ;-)

Akshully, in everyday speech, some laidback people
call the place "Auztralia", voicing the "s". Maybe
that's where it originated. Then again, I like to
think it's because it's a wonderful place, even if
we can't seem to find the Wizard when we want him!

Regards,
Yahya

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