"chord energy"

Hi. I'm more from a music theory than acoustics background, so was
wondering...It may be partly (or wholly) a matter of beats on my
piano, but certain chord types, to my ear, seem to vibrate more
loudly than others on the piano --for example, the major seventh
chord. I'm wondering if this chord (for eg, in the simple 3-note
version c-e-b)has a certain acoustical energy because of the strong
octave componant (harmonic numbers 2,4, and 8) in its intervals
(major third x perfect 5th...5/4 x 3/2 = 15/8. In orchestration and
arranging in general, octaves brighten the textures and 'chord
color', but I was wondering if there an actual way to quantify the
effect. (One approach, I realize, is to see how the respective
overtone series of the 3 notes combine/overlap?). Does anyone
agree/disagree/comment on the idea that the power of 2 component in
the intervals somehow adds 'acoustical energy to the chord? thanks
for any thoughts!
Monty

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

> Hi. I'm more from a music theory than acoustics background, so was
> wondering...It may be partly (or wholly) a matter of beats on my
> piano, but certain chord types, to my ear, seem to vibrate more
> loudly than others on the piano

What tuning is your piano in? That can make a lot of difference. JI?

> --for example, the major seventh
> chord. I'm wondering if this chord (for eg, in the simple 3-note
> version c-e-b)has a certain acoustical energy because of the strong
> octave componant (harmonic numbers 2,4, and 8) in its intervals
> (major third x perfect 5th...5/4 x 3/2 = 15/8. In orchestration
and
> arranging in general, octaves brighten the textures and 'chord
> color', but I was wondering if there an actual way to quantify the
> effect. (One approach, I realize, is to see how the respective
> overtone series of the 3 notes combine/overlap?). Does anyone
> agree/disagree/comment on the idea that the power of 2 component in
> the intervals somehow adds 'acoustical energy to the chord? thanks
> for any thoughts!
> Monty

Hi Monty.

I would disagree with the statement -- try the chord 1:3:5:15, which
has no 2s, but seems to me to retain some of the 'energy'. I think
this special 'acoustical energy' comes from three sources:

1. Presence of a clear root/fundamental. Our nervous systems allow us
to pick out missing fundamentals to hear melodies, and this carries
over to chords.

2. The six intervals in the chord are 5:4, 6:5, 5:4, 3:2, 3:2, and
15:8. There's acoustical affinity (i.e., simple ratio concordance,
and all that implies, including coinciding partials) in five of the
six intervals in the chord.

3. The first-order difference tones formed by these six intevals
would be 1/4, 1/4, 3/8, 1/2, 5/8, and 7/8, and five of these are
octave-equivalent to notes in the chord.

You might be interested to look into the subjective comparisons a
bunch of us made for 36 tetrads, and some quantitative models stacked
up, on the harmonic_entropy list (and here before its formation). It
would be fun to revive that old topic -- Joseph Pehrson has the
tetrads on a website somewhere, if you'd like to listen . . .

Joe?

Hello Monty!
While it appears to this person that Consonance is determine by
coincidence between partials and difference tones, I have often wondered if
one could just measure the effect by measuring the volume of sonorities . i
don't know if this has been tried. Whether these numbers, which you mention
below, would cause a certain type of 'brilliance' (since one would have
difference tone separated in octives)
one would have to examine spacing that do and do not produce difference
tones in this form. Interestingly when one uses subharmonicly based chords,
which on a whole would have less octave reinforcements, ( or they would be
more evenly spaced among a series of tones) the effect is less brilliant in
sound or darker psychological. This might be just that they are more
complex in the mentioned array as in the former they would reinforce a
singular identity

traktus5 wrote:

> Hi. I'm more from a music theory than acoustics background, so was
> wondering...It may be partly (or wholly) a matter of beats on my
> piano, but certain chord types, to my ear, seem to vibrate more
> loudly than others on the piano --for example, the major seventh
> chord. I'm wondering if this chord (for eg, in the simple 3-note
> version c-e-b)has a certain acoustical energy because of the strong
> octave componant (harmonic numbers 2,4, and 8) in its intervals
> (major third x perfect 5th...5/4 x 3/2 = 15/8. In orchestration and
> arranging in general, octaves brighten the textures and 'chord
> color', but I was wondering if there an actual way to quantify the
> effect. (One approach, I realize, is to see how the respective
> overtone series of the 3 notes combine/overlap?). Does anyone
> agree/disagree/comment on the idea that the power of 2 component in
> the intervals somehow adds 'acoustical energy to the chord? thanks
> for any thoughts!
> Monty
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

Thanks fellas for the interesting and very informative comments
concerning perception of chord sonority. I would be interested to
hear more about the '36 tetrads' and 'harmonic entropy.' Good point
about spelling the major seventh chord without powers of 2 harmonic
numbers.

I'm stricken with the number mysticism bug relating to musical
intervals and chords and harmony, so I guess am sort of fixated on
the doubling and other series, for example: reading that the
logarithmic pitch space is a direct consequence of the construction
of the basilar membrane in the ear, I was wondering if that results
in some sort of perceptual difference of chords whose harmonic
numbers contain doubling series numbers, in addition to the other
factors you mentioned. (So the simple 3-note, root/closed c-e-b
position is packed with three successive doubling numbers
(3/2x5/4=15/8). If this idea is just an expression of my obsession
with Number, then I'm wondering if it's possible that, since number
is considered a basic human Archetype, that maybe it does effect
perception in music in some way for some people? -Cheers, Monty.

> Hi Monty.
>
> I would disagree with the statement -- try the chord 1:3:5:15,
which
> has no 2s, but seems to me to retain some of the 'energy'. I think
> this special 'acoustical energy' comes from three sources:
>
> 1. Presence of a clear root/fundamental. Our nervous systems allow
us
> to pick out missing fundamentals to hear melodies, and this carries
> over to chords.
>
> 2. The six intervals in the chord are 5:4, 6:5, 5:4, 3:2, 3:2, and
> 15:8. There's acoustical affinity (i.e., simple ratio concordance,
> and all that implies, including coinciding partials) in five of the
> six intervals in the chord.
>
> 3. The first-order difference tones formed by these six intevals
> would be 1/4, 1/4, 3/8, 1/2, 5/8, and 7/8, and five of these are
> octave-equivalent to notes in the chord.
>
> You might be interested to look into the subjective comparisons a
> bunch of us made for 36 tetrads, and some quantitative models
stacked
> up, on the harmonic_entropy list (and here before its formation).
It
> would be fun to revive that old topic -- Joseph Pehrson has the
> tetrads on a website somewhere, if you'd like to listen . . .
>
> Joe?

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:

> Hello Monty!
> While it appears to this person that Consonance is determine by
> coincidence between partials and difference tones, I have often
wondered if
> one could just measure the effect by measuring the volume of
sonorities . i
> don't know if this has been tried.

Hi Kraig,

The actual physical amplitude can be calculated given a set of
partials, but I'm not sure if that's what you're after. Unless
there's phase-locking, the result won't depend on which chord you
choose. You've always referred to beating sonorities as "louder", so
maybe maximum of some short-term amplitude is what you're after? Or
maybe it's a more psychological phenomenon, like the powerful effect
that causes us to hear a harmonic series as a single timbre.

Cheers,
Paul

Hi Monty,

> Thanks fellas for the interesting and very informative comments
> concerning perception of chord sonority. I would be interested to
> hear more about the '36 tetrads' and 'harmonic entropy.'

The harmonic_entropy list is located at

/harmonic_entropy/

Once you've signed on there (otherwise you won't see the graphs),
take a look at

http://tonalsoft.com/enc/harmentr.htm

You'll see the discussion about the tetrads in the group, right in
the first post which was September 2000 -- before that all the
discussion was on this group. Joseph has moved the sound examples
elsewhere . . . aha!

http://www.soundclick.com/bands/5/tuninglabmusic.htm

The 36 tetrads are all labeled "Tetrad . . .", ending with the
pitches in cents.

> I'm stricken with the number mysticism bug relating to musical
> intervals and chords and harmony, so I guess am sort of fixated on
> the doubling and other series, for example: reading that the
> logarithmic pitch space is a direct consequence of the construction
> of the basilar membrane in the ear,

Or vice versa (evolutionarily speaking) . . .

> I was wondering if that results
> in some sort of perceptual difference of chords whose harmonic
> numbers contain doubling series numbers, in addition to the other
> factors you mentioned. (So the simple 3-note, root/closed c-e-b
> position is packed with three successive doubling numbers
> (3/2x5/4=15/8).

I'm not making the connection from the logarithmic representation of
pitch to these "doubling numbers". Can you elaborate the thinking
here?

> If this idea is just an expression of my obsession
> with Number, then I'm wondering if it's possible that, since number
> is considered a basic human Archetype, that maybe it does effect
> perception in music in some way for some people? -Cheers, Monty.

If it's mysticism you're into, then you have my unfettered support
and enthusiasm. I'm more of a scientific kind of guy myself, though,
and to me numbers are just an extremely convenient way of reasoning
about all sorts of natural and artificial phenomena.

Best,
Paul

Take a look at:
http://homepages.ihug.co.nz/~ray.tomes/
wrt relative harmonics, musical scales and lots of other related things...
----- Original Message -----
From: traktus5
To: tuning@yahoogroups.com
Sent: Friday, May 21, 2004 7:29 PM
Subject: [tuning] "chord energy" and number perception

Thanks fellas for the interesting and very informative comments
concerning perception of chord sonority. I would be interested to
hear more about the '36 tetrads' and 'harmonic entropy.' Good point
about spelling the major seventh chord without powers of 2 harmonic
numbers.

I'm stricken with the number mysticism bug relating to musical
intervals and chords and harmony, so I guess am sort of fixated on
the doubling and other series, for example: reading that the
logarithmic pitch space is a direct consequence of the construction
of the basilar membrane in the ear, I was wondering if that results
in some sort of perceptual difference of chords whose harmonic
numbers contain doubling series numbers, in addition to the other
factors you mentioned. (So the simple 3-note, root/closed c-e-b
position is packed with three successive doubling numbers
(3/2x5/4=15/8). If this idea is just an expression of my obsession
with Number, then I'm wondering if it's possible that, since number
is considered a basic human Archetype, that maybe it does effect
perception in music in some way for some people? -Cheers, Monty.

> Hi Monty.
>
> I would disagree with the statement -- try the chord 1:3:5:15,
which
> has no 2s, but seems to me to retain some of the 'energy'. I think
> this special 'acoustical energy' comes from three sources:
>
> 1. Presence of a clear root/fundamental. Our nervous systems allow
us
> to pick out missing fundamentals to hear melodies, and this carries
> over to chords.
>
> 2. The six intervals in the chord are 5:4, 6:5, 5:4, 3:2, 3:2, and
> 15:8. There's acoustical affinity (i.e., simple ratio concordance,
> and all that implies, including coinciding partials) in five of the
> six intervals in the chord.
>
> 3. The first-order difference tones formed by these six intevals
> would be 1/4, 1/4, 3/8, 1/2, 5/8, and 7/8, and five of these are
> octave-equivalent to notes in the chord.
>
> You might be interested to look into the subjective comparisons a
> bunch of us made for 36 tetrads, and some quantitative models
stacked
> up, on the harmonic_entropy list (and here before its formation).
It
> would be fun to revive that old topic -- Joseph Pehrson has the
> tetrads on a website somewhere, if you'd like to listen . . .
>
> Joe?

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wallyesterpaulrus wrote:

> --- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
>
> > Hello Monty!
> > While it appears to this person that Consonance is determine by
> > coincidence between partials and difference tones, I have often
> wondered if
> > one could just measure the effect by measuring the volume of
> sonorities . i
> > don't know if this has been tried.
>
> Hi Kraig,
>
> The actual physical amplitude can be calculated given a set of
> partials, but I'm not sure if that's what you're after.

possibly . how is this calculated if

> Unless
> there's phase-locking, the result won't depend on which chord you
> choose.

let us say they are locked. but why will not this depend on the answer.
Although i can kinda see that the one way the peaks will coincide and the
volume would be louder. Etc.
But i have never noticed this with real sound .If i play the same chord
over and over at the same attack the volume does not vary even though the
phases is bound too. Was it here that there was the discussion of a sine
being really a 2 d version of a three dimensional spiral?

> You've always referred to beating sonorities as "louder", so
> maybe maximum of some short-term amplitude is what you're after?

possibly, or maybe the maximum possible amplitude

> Or
> maybe it's a more psychological phenomenon, like the powerful effect
> that causes us to hear a harmonic series as a single timbre.

Psychology plays a big part in such things. I remember talking to Doug
Leedy once about ye ol C6 (as in added 6) sounding softer and hence maybe
less dissonant than a plain c chord.

>
>
> Cheers,
> Paul
>
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

--- In tuning@yahoogroups.com, "Dan Scorpio" <Dan.Scorpio@b...> wrote:

> Take a look at:
> http://homepages.ihug.co.nz/~ray.tomes/
> wrt relative harmonics, musical scales and lots of other related
things...

Hi Dan,

Ray Tomes is a great fellow, he used to post on this list. I miss
talking with him a great deal.

I am sorry to have to say this, particularly with him not here to
defend himself. But, as far as I can tell, the information on his
website is largely incorrect, particularly in the areas of musical
history and the science of music.

The best introduction to Western tuning history was Paul Guy's, but
that seems to have disappeared from the internet.

Kyle Gann's is almost as accurate:
http://home.earthlink.net/%7Ekgann/histune.html

Musical acoustics is covered excellently here:
http://www.phys.unsw.edu.au/music/basics.html
Here's a sample article:
http://www.phys.unsw.edu.au/%7Ejw/harmonics.html

These are just a start . . .

Hi Kraig,

I found this website which seems accurate and could help inform our
discussion:

http://www.ling.upenn.edu/phonetics/docs/Amplitude.pdf

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
>
>
> wallyesterpaulrus wrote:
>
> > --- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...>
wrote:
> >
> > > Hello Monty!
> > > While it appears to this person that Consonance is
determine by
> > > coincidence between partials and difference tones, I have often
> > wondered if
> > > one could just measure the effect by measuring the volume of
> > sonorities . i
> > > don't know if this has been tried.
> >
> > Hi Kraig,
> >
> > The actual physical amplitude can be calculated given a set of
> > partials, but I'm not sure if that's what you're after.
>
> possibly . how is this calculated if

It's the kind of mathematical calculation that my freshman physics
lab books were covered with. I'd be happy to show you if you have a
particular example in mind. Amplitude of a sound wave is normally
defined as RMS (root-mean-square) of the pressure variations. So
typically you do an integral . . .

> > Unless
> > there's phase-locking, the result won't depend on which chord you
> > choose.
>
> let us say they are locked.

Then it depends on the phase. You can get constructive or destructive
interference, depending on how you line up the different sources. If
you have spacially separate sound sources which are phase-locked with
one another, then your position in space relative to the sound
sources will determine whether you have constructive or destructive
interference. Moving around the room, you will find spots where any
given common overtone get louder, and spots where it gets softer,
while the amplitude of any partial that only comes from a single
source remains the same.

> but why will not this depend on the answer.
> Although i can kinda see that the one way the peaks will coincide
and the
> volume would be louder. Etc.
> But i have never noticed this with real sound .If i play the same
chord
> over and over at the same attack the volume does not vary even
though the
> phases is bound too.

Really? I notice this variation all the time. Play octaves (about the
only pure JI interval you're likely to find on a synth) using a sound
with harmonic partials. Every time you strike the octave, you hear a
different set of partials reinforced. I've noticed this with a wide
variety of makes of synthesizer.

Of course, the perceptual importance of partials to music is often
exaggerated.

> Was it here that there was the discussion of a sine
> being really a 2 d version of a three dimensional spiral?

This is true for light but not for sound. Sound consists of
longitudinal pressure variations, so in this sense it's a one-
dimensional wave and not two-dimensional at all. You may be thinking
of Charles Lucy's stuff, which I'm sorry to say is not scientific.

> > You've always referred to beating sonorities as "louder", so
> > maybe maximum of some short-term amplitude is what you're after?
>
> possibly, or maybe the maximum possible amplitude

Well, the maximum possible amplitude would be greatest if everything
is in phase. When it's beating the amplitude is easy to calculation --
RMS just gives you the sum of the individual RMS amplitudes -- while
the amplitude for waves in phase would be higher.

On the psychology of hearing certain chords as we hear single
timbres . . . this research has recently been taken into the
neurological realm. For example, see

http://homepage.mac.com/cariani/CarianiWebsite/TramoCarianiNYAS2001.pd
f

hi Paul - uhm...I was thinking that since the 'pitch space' is
specifically log2, the doubling series, with each doubling of
frequency representing a fixed interval (the octave), that maybee the
ear/mind, just as it's 'in tune' with harmonic numbers, that it is
also somehow 'in tune' with the doubling number harmonics -- since
they overlap anyway (harmonic #1,2,4,8,16, etc)...?

"
> > I was wondering if that results
> > in some sort of perceptual difference of chords whose harmonic
> > numbers contain doubling series numbers, in addition to the other
> > factors you mentioned. (So the simple 3-note, root/closed c-e-b
> > position is packed with three successive doubling numbers
> > (3/2x5/4=15/8).
>
> I'm not making the connection from the logarithmic representation
of
> pitch to these "doubling numbers". Can you elaborate the thinking
> here?"

hi Kraig, Paul - re "I remember talking to Doug Leedy once about ye
ol C6 (as in added 6) sounding softer and hence maybe less dissonant
than a plain c chord."

I like that chord, too, and feel it sounds 'better' somehow that the
triad. Do you think it's possible that it's related to the fact that
it (ie, minor 6 chord c-e-a...using a simple 3-note version for
discussion), has intervals of lower harmonic numbers (5/4 x 4/3)
compared to the root position triad (5/4 x 6/5). And the lowest of
all, and even more 'serene' sounding (think of all its new-agey
connotations)is the major seventh chord (5/4 x 3/2).

In a way, can we consider the major seventh chord the 'numero uno'
chord, since it has, in addition to the major 3rd (the 'numero uno'
chord-building interval from the harmonic series), an added 3/2,
rather than 4/3 or 6/5?

It's also neat that it literally 'adds up' to an already simplified
fraction: 5/4 x 3/2 = 15/8! Isn't that lovely? (Compared to the
klunky 5/4 x 6/5 = 30/20=3/2 ...which does ultimately reduce to a
lower numbered fraction, however...)

Cheers, Monty

it To me the 'numbI thinkeven though the latter is--- In
tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
>
>
> wallyesterpaulrus wrote:
>
> > --- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...>
wrote:
> >
> > > Hello Monty!
> > > While it appears to this person that Consonance is
determine by
> > > coincidence between partials and difference tones, I have often
> > wondered if
> > > one could just measure the effect by measuring the volume of
> > sonorities . i
> > > don't know if this has been tried.
> >
> > Hi Kraig,
> >
> > The actual physical amplitude can be calculated given a set of
> > partials, but I'm not sure if that's what you're after.
>
> possibly . how is this calculated if
>
> > Unless
> > there's phase-locking, the result won't depend on which chord you
> > choose.
>
> let us say they are locked. but why will not this depend on the
answer.
> Although i can kinda see that the one way the peaks will coincide
and the
> volume would be louder. Etc.
> But i have never noticed this with real sound .If i play the same
chord
> over and over at the same attack the volume does not vary even
though the
> phases is bound too. Was it here that there was the discussion of a
sine
> being really a 2 d version of a three dimensional spiral?
>
> > You've always referred to beating sonorities as "louder", so
> > maybe maximum of some short-term amplitude is what you're after?
>
> possibly, or maybe the maximum possible amplitude
>
> > Or
> > maybe it's a more psychological phenomenon, like the powerful
effect
> > that causes us to hear a harmonic series as a single timbre.
>
> Psychology plays a big part in such things. I remember talking to
Doug
> Leedy once about ye ol C6 (as in added 6) sounding softer and hence
maybe
> less dissonant than a plain c chord.
>
> >
> >
> > Cheers,
> > Paul
> >
> >
> >
>
> -- -Kraig Grady
> North American Embassy of Anaphoria Island
> http://www.anaphoria.com
> The Wandering Medicine Show
> KXLU 88.9 FM WED 8-9PM PST

--- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:
> hi Paul - uhm...I was thinking that since the 'pitch space' is
> specifically log2, the doubling series, with each doubling of
> frequency representing a fixed interval (the octave),

Well that's true for any base, not just 2. For any base B, pitch
space could just as well be said to be log base B of the frequency;
and for any number C, multiplying the frequency by C is represented
by a fixed interval in "pitch space". In fact, logs base 10 were once
widely used for the former purpose -- before the advent of
electronics, when tables of logs were the norm.

> that maybee the
> ear/mind, just as it's 'in tune' with harmonic numbers, that it is
> also somehow 'in tune' with the doubling number harmonics -- since
> they overlap anyway (harmonic #1,2,4,8,16, etc)...?

Sure, that may be.

But that still seems a long way from explaining your idea below, or
at least makes for a long, circuitous, and if you don't mind my
saying so, "wishy-washy" way of explaining it, in my opinion.

Did you read my remarks about this chord? The fact that the chord has
a very clear fundamental, which is octave-equivalent to the bottom
note in the chord, incorporates the "octaves are special" idea with
one of the most basic features of our hearing -- the ability to
integrate a harmonic series as a single perceptual gestalt.

*Any* such chord, when expressed in ratios as 1/1 and some larger
ratios the way you did, will have only powers of two in the
denominators. And any chord expressible with only powers of two in
the denominators, and a 1/1 as the bottom note, will exhibit these
features. This, I feel, is a better explanation.

I think this goes a bit closer to the heart of the issue than simply
looking at the numbers in the ratios as you expressed them. The mind
doesn't "see" these numbers. But there *are* psychoacoustical
phenomena which do make simple-integer ratios "special" in various
ways. When reasoning about these matters, I believe it's more useful
to think in terms of the effect the set of frequencies will have on
the human perceptual apparatus. One reason is that there are many
equally valid ways of expressing a given JI chord in numbers. But if
it were the *numbers* which determined the chord's affect, then
expressing the chord differently would change its musical effect.
Clearly, this is absurd -- it's the *sound* that affects us, not the
numbers we may happen to use to express it.

Am I making sense? Am I being a complete @$$#*%&? Let me know.

-Paul

>
> "
> > > I was wondering if that results
> > > in some sort of perceptual difference of chords whose harmonic
> > > numbers contain doubling series numbers, in addition to the
other
> > > factors you mentioned. (So the simple 3-note, root/closed c-e-
b
> > > position is packed with three successive doubling numbers
> > > (3/2x5/4=15/8).
> >
> > I'm not making the connection from the logarithmic representation
> of
> > pitch to these "doubling numbers". Can you elaborate the thinking
> > here?"

Paul- thanks for reminding me about your comments about the
fundamental....I need to digest the rest, and learn some math... ---
In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
>
> --- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:
> > hi Paul - uhm...I was thinking that since the 'pitch space' is
> > specifically log2, the doubling series, with each doubling of
> > frequency representing a fixed interval (the octave),
>
> Well that's true for any base, not just 2. For any base B, pitch
> space could just as well be said to be log base B of the frequency;
> and for any number C, multiplying the frequency by C is represented
> by a fixed interval in "pitch space". In fact, logs base 10 were
once
> widely used for the former purpose -- before the advent of
> electronics, when tables of logs were the norm.
>
> > that maybee the
> > ear/mind, just as it's 'in tune' with harmonic numbers, that it
is
> > also somehow 'in tune' with the doubling number harmonics --
since
> > they overlap anyway (harmonic #1,2,4,8,16, etc)...?
>
> Sure, that may be.
>
> But that still seems a long way from explaining your idea below, or
> at least makes for a long, circuitous, and if you don't mind my
> saying so, "wishy-washy" way of explaining it, in my opinion.
>
> Did you read my remarks about this chord? The fact that the chord
has
> a very clear fundamental, which is octave-equivalent to the bottom
> note in the chord, incorporates the "octaves are special" idea with
> one of the most basic features of our hearing -- the ability to
> integrate a harmonic series as a single perceptual gestalt.
>
> *Any* such chord, when expressed in ratios as 1/1 and some larger
> ratios the way you did, will have only powers of two in the
> denominators. And any chord expressible with only powers of two in
> the denominators, and a 1/1 as the bottom note, will exhibit these
> features. This, I feel, is a better explanation.
>
> I think this goes a bit closer to the heart of the issue than
simply
> looking at the numbers in the ratios as you expressed them. The
mind
> doesn't "see" these numbers. But there *are* psychoacoustical
> phenomena which do make simple-integer ratios "special" in various
> ways. When reasoning about these matters, I believe it's more
useful
> to think in terms of the effect the set of frequencies will have on
> the human perceptual apparatus. One reason is that there are many
> equally valid ways of expressing a given JI chord in numbers. But
if
> it were the *numbers* which determined the chord's affect, then
> expressing the chord differently would change its musical effect.
> Clearly, this is absurd -- it's the *sound* that affects us, not
the
> numbers we may happen to use to express it.
>
> Am I making sense? Am I being a complete @$$#*%&? Let me know.
>
> -Paul
>
>
> >
> > "
> > > > I was wondering if that results
> > > > in some sort of perceptual difference of chords whose
harmonic
> > > > numbers contain doubling series numbers, in addition to the
> other
> > > > factors you mentioned. (So the simple 3-note, root/closed c-
e-
> b
> > > > position is packed with three successive doubling numbers
> > > > (3/2x5/4=15/8).
> > >
> > > I'm not making the connection from the logarithmic
representation
> > of
> > > pitch to these "doubling numbers". Can you elaborate the
thinking
> > > here?"

Hi Kelly,

> Do you think it's possible that it's related to the fact that
> it (ie, minor 6 chord c-e-a...using a simple 3-note version for
> discussion), has intervals of lower harmonic numbers (5/4 x 4/3)
> compared to the root position triad (5/4 x 6/5). And the lowest of
> all, and even more 'serene' sounding (think of all its new-agey
> connotations)is the major seventh chord (5/4 x 3/2).

I think you're on to something. Putting aside the actual question of
*tuning* (I'll assume JI), I think there are still two other factors
which have high importance if you want to compare the "concordance"
of the three chords.

One factor is the *outer* interval, which traditionally is considered
the most important interval to handle carefully in terms of
counterpoint, voice-leading, etc.

The outer interval is 3:2 for c-e-g, 5:3 for c-e-a, and 15:8 for c-e-
b. As long as the numbers are no higher than this, and you're talking
JI, it's pretty fair to rank interval discordance using the size of
the numbers. So the outer interval is most concordant for the first
chord and most discordant for the last chord. This pretty much
reflects how these chords are treated in traditional contrapuntal
music.

Another factor is how easily the chord can be heard as arising from a
single harmonic series. The three chords can be expressed as 4:5:6,
12:15:20, and 8:10:15, respectively. Now it is the middle chord, the
first-inversion minor triad, which appears most complex. It's also
the most ambiguous, since the higher the numbers, the closer the
approximation to other harmonic-series chords, as I showed on the
harmonic_entropy list. So the first-inversion minor chord may be the
most interesting to listen to of the three, as your mind shuffles
between hearing the different fundamentals implied by the chord (F,
possibly D) and by its dyad subsets (A, C) . . . So the other two may
appear more "solid" or "hard" by comparison with the "soft" yet
concordant c-e-a.

Hi Paul- very interesting analysis of those 3 chords. New material
for me to learn and consider. However, I'm out of town with no
printer and limited internet, so I haven't yet learned how to do your
type of analysis. But here it goes....

On your comment: >Another factor is how easily the chord can be heard
as arising from a single harmonic series....>

I noticed that the 'major 6th chord' (c-e-a) can also be represented
by harmonic numbers 8-10-13. Does that effect your comments
regarding the 'highness' of the harmonic numbers?

Relatedly, what if the chord represented in high harmonic numbers is,
so to speak, 'locked in' as an already-reduced simple fraction (such
as c-e-b, 5/4x3/2=15/8.) That is, in light of your idea that the c-e-
a chord is "the most ambiguous, since the higher the numbers, the
closer the
approximation to other harmonic-series chords", does a chord which
literally 'adds up' to an already-reduced higher fraction (such as
5/4x3/2=15/8), share this interesting ambiguity you discussed, as
opposed to, say, 5/4x6/5=30/20, or as compared to the c-e-a chord,
with its different representations (different representations of the
c-e-a chord discussed above?

I'm being kicked off a public computer, so this may not be coherant,
and apologize if there's ignorance from not studying/understanding
your comments, which I don't have resources at the moment to do, but
have been thinking about...(or maybe I"m dense and stubborn...)
THanks,

Kelly

I It's also
> the most ambiguous, since the higher the numbers, the closer the
> approximation to other harmonic-series chords, as I showed on the
> harmonic_entropy list. So the first-inversion minor chord may be
the
> most interesting to listen to of the three, as your mind shuffles
> between hearing the different fundamentals implied by the chord (F,
> possibly D) and by its dyad subsets (A, C) . . . So the other two
may
> appear more "solid" or "hard" by comparison with the "soft" yet
> concordant c-e-a.

Hi Kelly!

> Hi Paul- very interesting analysis of those 3 chords. New material
> for me to learn and consider. However, I'm out of town with no
> printer and limited internet, so I haven't yet learned how to do
your
> type of analysis. But here it goes....
>
> On your comment: >Another factor is how easily the chord can be
heard
> as arising from a single harmonic series....>
>
> I noticed that the 'major 6th chord' (c-e-a) can also be
represented
> by harmonic numbers 8-10-13.

That chord actually sounds a bit closer to a traditional augmented
chord. I assumed you were using JI, and when you said the adjacent
intervals were 5:4 and 4:3 (thus implying harmonic numbers 12-15-20),
I took you at your word. The chord 8-10-13 would certainly score as
more concordant or "hard" than 12-15-20 on the factor considered in
my comment you quoted above, but as more discordant on some other
factors, particularly those like roughness that depend primarily on
the concordance of the individual dyads. 13:8 and 13:10 are by no
means simple-integer dyads of the sort that exhibit the
stereotypical 'smoothness' of, say, 5:4 and 4:3. 8-10-13 is a highly
non-traditional chord, but certainly a harmonically convincing one in
certain microtonal contexts. 9-11-15 is another example that is about
equally close to 12-15-20. If you join the harmonic entropy group,
you can visually see the relationship between all these triads at

/tuning/files/Erlich/closeup.jp
g

.

> Does that effect your comments
> regarding the 'highness' of the harmonic numbers?

My comments are based on an analysis which takes *all* nearby,
similar chords, and the confusion they might lead to, into account.
It turns out that the simpler the 'harmonic numbers', the further
away the closest other harmonic chords are, and thus the lower the
confusion. As long as the numbers are not terribly high, this
relationship is a very close one and I don't see any room for effects
such as the "already-reducedness" you mention below. Look at this
graph:

/tuning/files/triadic.gif

For each harmonic-series triad a:b:c, this shows the area in triad
space of the set of points closer to a:b:c than to any other harmonic-
series triad x:y:z such that x*y*z is less than a million. Until one
gets to some quite high-numbered triads, this area is very close to
being inversely proportional to the cube root of a*b*c. So the lower
the numbers, the less the ambiguity of how the chord fits into the
harmonic series.

> Relatedly, what if the chord represented in high harmonic numbers
is,
> so to speak, 'locked in' as an already-reduced simple fraction
(such
> as c-e-b, 5/4x3/2=15/8.) That is, in light of your idea that the c-
e-
> a chord is "the most ambiguous, since the higher the numbers, the
> closer the
> approximation to other harmonic-series chords", does a chord which
> literally 'adds up' to an already-reduced higher fraction (such as
> 5/4x3/2=15/8), share this interesting ambiguity you discussed, as
> opposed to, say, 5/4x6/5=30/20, or as compared to the c-e-a chord,
> with its different representations (different representations of
the
> c-e-a chord discussed above?

I'm not sure I'm still following you, but I could mention here that
just as c-e-a or 12-15-20 has several harmonic-series subsets nearby,
so does c-e-g or 4-5-6 (9:11:13, 10:13:15, 11:14:16, 11:14:17, as you
can see on the first graph above) and c-e-b or 8-10-15 (7:9:13,
9:11:17, 10:13:19, 11:14:21, etc).

Looking forward to your return,
Paul

Hi Paul - a question regarding your earlier email...

> > Kelly wondered if chords with harmonic numbers 1,2,4,8, etc, had
more "acoustical energy", and wrote: "since the 'pitch space' is
> > specifically log2, the doubling series, with each doubling of
> > frequency representing a fixed interval (the octave)...,"

>Paul responded: "Well that's true for any base, not just 2. For any
base B, pitch
> space could just as well be said to be log base B of the frequency;
> and for any number C, multiplying the frequency by C is represented
> by a fixed interval in "pitch space". In fact, logs base 10 were
once > widely used for the former purpose -- before the advent of
> electronics, when tables of logs were the norm."

Perhaps it's my rudimentary math knowledge, but I don't understand.
Isn't our musical pitchspace specifically log 2 (the doubling
series), with frequency doubling every octave, at harmonic numbers
1,2,4,8,16...? Wouldn't log 10 imply doubling at harmonic numbers
1,10,100, etc?

Maybe I shouldnt have skimmed through that Banade text....Can you
refer me to a basic source (website or book which covers this?

thanks, Kelly

--- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:
> Hi Paul - a question regarding your earlier email...
>
>
> > > Kelly wondered if chords with harmonic numbers 1,2,4,8, etc,
had
> more "acoustical energy", and wrote: "since the 'pitch space' is
> > > specifically log2, the doubling series, with each doubling of
> > > frequency representing a fixed interval (the octave)...,"
>
>
> >Paul responded: "Well that's true for any base, not just 2. For
any
> base B, pitch
> > space could just as well be said to be log base B of the
frequency;
> > and for any number C, multiplying the frequency by C is
represented
> > by a fixed interval in "pitch space". In fact, logs base 10 were
> once > widely used for the former purpose -- before the advent of
> > electronics, when tables of logs were the norm."
>
> Perhaps it's my rudimentary math knowledge, but I don't
understand.
> Isn't our musical pitchspace specifically log 2 (the doubling
> series), with frequency doubling every octave, at harmonic numbers
> 1,2,4,8,16...? Wouldn't log 10 imply doubling at harmonic numbers
> 1,10,100, etc?

No. Doubling is always associated with harmonic numbers that
*double*, by definition. Harmonic numbers 1,10,100
represent "decupling" frequencies -- multiplying by 10.

None of this requires you to take any logs, to any base.

> Maybe I shouldnt have skimmed through that Banade text....Can you
> refer me to a basic source (website or book which covers this?
>
> thanks, Kelly

Hi Kelly,

I'm guessing that maybe you just need to brush up on the properties
of logs. All bases are equivalent, up to a constant. If have x and
you want to take the log base b of it, just take your favorite log of
x and then divide by the same kind of log of b.