dissonance curve of composite tones

Looking at the spectrogram of the pair of composite (six-partials) tones
(one stable, one gliding from unison to octave +), you can see clearly how
the partials of the gliding tone (gt) approach and intersect the partials of
the stationary tone (st). The first partial of the gt, after leaving the the
first partial of the st (1/1), does not intersect another st partial until
the octave, when it meets the second st partial. The second partial of the
gt intersects the third partial of the st at the 3/2 point, then intersects
the fourth partial of st at 2/1. The third partial of gt intersects the
fourth partial of st at 4/3 and then intersects the fifth st partial at 5/3,
and the sixth at 2/1. And so on.

GT Partial intersections up to octave
1 1/1 2/1
2 1/1 3/2 4/2
3 1/1 4/3 5/3 6/3
4 1/1 5/4 6/4 7/4 8/4
5 1/1 6/5 7/5 8/5 9/5 10/5

It makes sense that the first gt partial would mark (intersect) the octave,
and that the second partial would make its mark at 3/2. The third partial
defines 4/3 AND 5/3. These ratios do not have the same odd limit, nor the
same prime limit, yet somehow they are related, and perhaps should therefore
receive equal values in a consonance/dissonance measuring system.

Each partial acts as a mini gliding sine wave and therefore manifests the
roughness and beating phenomena as exhibited by pure sine tone dyads in the
regions of 1/1 3/2 and 2/1. So, the composite interval consists of all the
partials of the constituent tones interacting with each other (producing
beating) dependent on where they "intersect" other partials, how close they
are to the intersection, etc. So - at the intersection: no dissonance, near
the intersection (and, apparently, near the mean of the st partials)- high
dissonance, at other points - ?.

dante

Dante Rosati wrote:

>Looking at the spectrogram of the pair of composite (six-partials) tones
>(one stable, one gliding from unison to octave +), you can see clearly how
>the partials of the gliding tone (gt) approach and intersect the partials
of
>the stationary tone (st). The first partial of the gt, after leaving the
the
>first partial of the st (1/1), does not intersect another st partial until
>the octave, when it meets the second st partial. The second partial of the
>gt intersects the third partial of the st at the 3/2 point, then intersects
>the fourth partial of st at 2/1. The third partial of gt intersects the
>fourth partial of st at 4/3 and then intersects the fifth st partial at
5/3,
>and the sixth at 2/1. And so on.
>
>
>GT Partial intersections up to octave
>1 1/1 2/1
>2 1/1 3/2 4/2
>3 1/1 4/3 5/3 6/3
>4 1/1 5/4 6/4 7/4 8/4
>5 1/1 6/5 7/5 8/5 9/5 10/5
>
>
>It makes sense that the first gt partial would mark (intersect) the octave,
>and that the second partial would make its mark at 3/2. The third partial
>defines 4/3 AND 5/3. These ratios do not have the same odd limit, nor the
>same prime limit, yet somehow they are related, and perhaps should
therefore
>receive equal values in a consonance/dissonance measuring system.

Yes! The ratios have the same denominator, and in earlier discussions on
this list, Daniel Wolf and I agreed that the size of the denominator is a
good measure of the dissonance of simple just ratios. (clearly it doesn't
work for ratios like 3001:2001).

I'm assuming here that the ratios are always written with the numerator
larger than the denominator.

I've always justified odd-limit as follows: A composer would like to
consider all octave extensions and inversions of an interval equivalent, so
that he/she could think in an octave-equivalent fashion about the
relationships between tones (clearly Partch was such a composer). Modern
music theory often refers to a set of octave-equivalent ratios as an
interval class or i. c. (e.g., in 12-tET minor third, major sixth, minor
tenth, major thirteenth, etc. are all members of i. c. 3). If an interval
has odd limit n, obviously all its octave extensions and inversions will
have an odd limit of n. Less obviously, all of its octave inversions and
extensions have a denominator n or less. So in a sense the odd limit
measures the maximum possible dissonance of any octave-specific interval in
an interval class. Since the entire interval class has the same odd limit,
one can greatly simplify things, as Partch did with his One-Footed Bride, by
simply associating a dissonance level with each interval class. The only
exception Partch made was his finitely consonant "foot" at 2:1 vs. the
apparently infinitely consonant 1:1; otherwise the graph is perfectly
symmetrical about the half-octave. Again, the numbers can't get too high; as
Dave Keenan and I discussed some time ago, around 13 or so the denominator
rule, and thus the odd-limit measure of dissonance, starts to break down
(which Partch was clearly aware of when he drew his One-Footed Bride! -- all
odd limits above 13 are assigned to the same, lowest level of consonance),
and of course by the time you get to 3001 it's totally gone.
>
>Each partial acts as a mini gliding sine wave and therefore manifests the
>roughness and beating phenomena as exhibited by pure sine tone dyads in the
>regions of 1/1 3/2 and 2/1. So, the composite interval consists of all the
>partials of the constituent tones interacting with each other (producing
>beating) dependent on where they "intersect" other partials, how close they
>are to the intersection, etc. So - at the intersection: no dissonance,
near
>the intersection (and, apparently, near the mean of the st partials)- high
>dissonance, at other points - ?.

That question is exactly what the graphs by Helmholtz, Kameoka&Kuriagawa,
and Sethares answer -- these researchers calculated the type of dissonance
you describe for every possible interval (obviously, using some small
increment) for some timbre and graphed the results. Except that they did not
consider interactions of pure sine tone dyads at regions other than that
near 1:1. However, one could try to model that by inputting a timbre into
Sethares' formula which you've modified by artificially adding amplitude to
the partials that are multiples of 2 and the partials that are multiples of
3.

-Paul Erlich

>From: "Brett Barbaro" <barbaro@noiselabs.com>
>
>
>I've always justified odd-limit as follows: A composer would like to
>consider all octave extensions and inversions of an interval equivalent, so
>that he/she could think in an octave-equivalent fashion about the
>relationships between tones (clearly Partch was such a composer). Modern
>music theory often refers to a set of octave-equivalent ratios as an
>interval class or i. c. (e.g., in 12-tET minor third, major sixth, minor
>tenth, major thirteenth, etc. are all members of i. c. 3). If an interval
>has odd limit n, obviously all its octave extensions and inversions will
>have an odd limit of n.

Isn't generating intervals by inversion abstract in the same way that
subharmonics are (may be?) abstract. I know its not especially abstract in
pitch class theory (where, of course, everthing is abstract), but from a
harmonic theory point of view? Octave equivilance preserves identity,
whereas interval inversion produces a different note. The relation between
the two different intervals is "hidden" in that it is not obvious to the
ear, as octaves are. So I think they must be two different things.

, the numbers can't get too high; as
>Dave Keenan and I discussed some time ago, around 13 or so the denominator
>rule, and thus the odd-limit measure of dissonance, starts to break down

Why would this be? If dissonance is an acoustic phenomena I don't think
measurements should break down already at 13. Even if subjective tones are
added in they should act the same way in our brains as acoustic tones, that
is, in a consistent way that is measurable.

>That question is exactly what the graphs by Helmholtz, Kameoka&Kuriagawa,
>and Sethares answer -- these researchers calculated the type of dissonance
>you describe for every possible interval (obviously, using some small
>increment) for some timbre and graphed the results. Except that they did
not
>consider interactions of pure sine tone dyads at regions other than that
>near 1:1. However, one could try to model that by inputting a timbre into
>Sethares' formula which you've modified by artificially adding amplitude to
>the partials that are multiples of 2 and the partials that are multiples of
>3.

Helmholtz's curve shows how the individual partials behave in exactly the
way I observed on the spectrogram:

GT Partial intersections up to octave
>1 1/1 2/1
>2 1/1 3/2 4/2
>3 1/1 4/3 5/3 6/3
>4 1/1 5/4 6/4 7/4 8/4
>5 1/1 6/5 7/5 8/5 9/5 10/5

His first partial only produces dissonance around 1/1 and 2/1, in proximity
to the steady tone's partials. Around 3/2 there is no dissonance at all on
his graph (for this partial). However, I agree with Tim Crews that there
is something going on around 3/2 (with two pure sine tones), and its exactly
the same as what happens around 1/1 and 2/1, just fainter. Since it is not
shown by H, nor by Sethares (according to Tim?), I wonder what exactly it
is?

dante

Dante:

A clarification on your point below: In the context
of _pure_ tone dyads, the Plomp-Levelt curve (upon which
Sethares bases all of his theoretical work in TTSS) _only_
shows dissonance around the 1:1 interval. It does not show
dissonance at the 3:2 interval or even the 2:1 interval.
In fact, the graph shows that dissonance has nearly approached
zero by the time you reach the 2:1 interval.

Actually, my analysis of the waveform produced by
the gliding tone interacting with the steady tone indicates
a different kind of envelope around the 3/2 and 2/1 intervals
than what is present at the 1/1 interval. The 1.(small)/1 interval
shows the classic "beating" waveform, in which a high-frequency
waveform is enveloped by a two sine waves that are offset
in phase by 180 degrees. The 3.(small)/2 and 2.(small)/1 intevals,
on the other hand, consist of a high-frequency waveform enveloped
by a shape consisting of the area outlined by _three_ sine
waves, each offset by 120 degrees. It's hard to explain
in words, but the effect is not so much amplitude modulation
as offset modulation (the amplitude of the envelope waveform
appears constant, but the waveform moves up and down in
a sinusoidal pattern.) The frequency of the envelope waveform's
oscillation is the same as the difference between the steady
and gliding waveform (at the 3/2 interval), but is double that
frequency at the 2/1 interval.

So the audible effect does not precisely sound like "beating"
in the typical sense of the word. However, there is definitely
a periodic change in the quality of the sound that is hard to
describe. When the period of this change is within a certain
range, the effect is a "dissonance", similar to the effect heard
when two nearly unison tones are added, but not exactly the same.

Although I can also hear other stuff going on at other intervals,
none of it is visible in the resulting displayed waveform. The
only intervals that have visually interpretable phenomena are 1:1,
3:2, and 2:1. These are also the only intervals around which I
hear sounds that I can objectively call "dissonance". The other
stuff that I hear is interesting, but not clearly dissonant.

Tim Crews
timcrews@home.com

> Helmholtz's curve shows how the individual partials behave in exactly the
> way I observed on the spectrogram:
>
> GT Partial intersections up to octave
> >1 1/1 2/1
> >2 1/1 3/2 4/2
> >3 1/1 4/3 5/3 6/3
> >4 1/1 5/4 6/4 7/4 8/4
> >5 1/1 6/5 7/5 8/5 9/5 10/5
>
> His first partial only produces dissonance around 1/1 and 2/1, in
> proximity to the steady tone's partials. Around 3/2 there is no
> dissonance at all on his graph (for this partial). However, I agree
> with Tim Crews that there is something going on around 3/2 (with two
> pure sine tones), and its exactly the same as what happens around 1/1
> and 2/1, just fainter. Since it is not shown by H, nor by Sethares
> (according to Tim?), I wonder what exactly it is?
>
> dante
>

Dante Rosati wrote,

>Isn't generating intervals by inversion abstract in the same way that
>subharmonics are (may be?) abstract. I know its not especially abstract in
>pitch class theory (where, of course, everthing is abstract), but from a
>harmonic theory point of view? Octave equivilance preserves identity,
>whereas interval inversion produces a different note.

I don't see it that way at all. The only type of interval inversion I was
talking about arises from transposing one note by one or several octaves.

>The relation between
>the two different intervals is "hidden" in that it is not obvious to the
>ear, as octaves are.

Since all you're doing is transposing one note by one or several octaves, it
is just as obvious to the ear.

>So I think they must be two different things.

Do you still think so?

>, the numbers can't get too high; as
>>Dave Keenan and I discussed some time ago, around 13 or so the denominator
>>rule, and thus the odd-limit measure of dissonance, starts to break down

>Why would this be? If dissonance is an acoustic phenomena I don't think
>measurements should break down already at 13.

They begin to because many 13-limit ratios are so close to each other that
proximity plays a larger role than limit. Dissonance is not an acoustical
phenomenon, it's a psychoacoustical phenomenon, totally dependent on the
frequency resolution of various stages of the auditory system.

>Even if subjective tones are
>added in they should act the same way in our brains as acoustic tones, that
>is, in a consistent way that is measurable.

I'm not sure what you're getting at -- does this relate to the 13-limit
thing?

>However, I agree with Tim Crews that there
>is something going on around 3/2 (with two pure sine tones), and its
exactly
>the same as what happens around 1/1 and 2/1, just fainter. Since it is not
>shown by H, nor by Sethares (according to Tim?), I wonder what exactly it
>is?

several possibilities:

first-order beating between the first-order difference tone and the
second-order difference tone
second-order beating between the first-order difference tone and the lower
tone
subjective beating between the first-order difference tone and the virtual
pitch (I think John Chalmers told me this phenomenon has been observed)
etc. etc. (see Plomp's book)

Tim Crews wrote,

>Actually, my analysis of the waveform produced by
>the gliding tone interacting with the steady tone indicates
>a different kind of envelope around the 3/2 and 2/1 intervals
>than what is present at the 1/1 interval. The 1.(small)/1 interval
>shows the classic "beating" waveform, in which a high-frequency
>waveform is enveloped by a two sine waves that are offset
>in phase by 180 degrees. The 3.(small)/2 and 2.(small)/1 intevals,
>on the other hand, consist of a high-frequency waveform enveloped
>by a shape consisting of the area outlined by _three_ sine
>waves, each offset by 120 degrees. It's hard to explain
>in words, but the effect is not so much amplitude modulation
>as offset modulation (the amplitude of the envelope waveform
>appears constant, but the waveform moves up and down in
>a sinusoidal pattern.) The frequency of the envelope waveform's
>oscillation is the same as the difference between the steady
>and gliding waveform (at the 3/2 interval), but is double that
>frequency at the 2/1 interval.

>So the audible effect does not precisely sound like "beating"
>in the typical sense of the word.

Looking at the waveform and listening to the sound are totally different
activities, and one should automatically assume that there is any kind of
correspondence between the two. For example, I could give you two totally,
utterly different-looking waveforms that sound almost exactly the same, by
simply changing the relative phases of the partials. If you are interested
in these different types of beating patterns and the presence and absense of
auditory correlates, I strongly recommend Plomp's and Roederer's books.

On Mon, 14 Jun 1999, Paul H. Erlich wrote:

> Looking at the waveform and listening to the sound are totally different
> activities, and one should [not] automatically assume that there is any
> kind of
> correspondence between the two. For example, I could give you two totally,
> utterly different-looking waveforms that sound almost exactly the same, by
> simply changing the relative phases of the partials.

The most astounding demonstration of that fact (to my eyes and ears) is to
play two tones which are just slightly off from 3:1. It looks from the
curve like there really ought to be some sort of 'beating', with the
volumes changing as the tones go from cancelling each other out to not
cancelling each other out over time, but no such effect happens.

-Bram

[Paul Erlich, TD 219.8]
>
> [Dante Rosati]
>> If dissonance is an acoustic phenomena I don't think
>> measurements should break down already at 13.
>
> They begin to because many 13-limit ratios are so close to each
> other that proximity plays a larger role than limit. Dissonance
> is not an acoustical phenomenon, it's a psychoacoustical
> phenomenon, totally dependent on the frequency resolution of
> various stages of the auditory system.

This is a very good oberservation. I have noticed myself that
several 13-limit ratios are so close to many 11-limit ratios
that in most musical contexts it's difficult to distinguish
between them.

(This does not hold for long sustained Young-style compositions)

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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Monzo [TD220.7]

>This is a very good oberservation. I have noticed myself that
>several 13-limit ratios are so close to many 11-limit ratios
>that in most musical contexts it's difficult to distinguish
>between them.

I think that the reason for this is that 11*13 = 143 which is very close
to 144 which is a wonderful harmonic number, being very closely related
to most of the ratios in the JI scale. Or, stated another way, 11/12 is
very close to 12/13 and these can be adjusted many ways by multiplying
each by a common factor such as 4/3 or 3/2 and so on.

and in the "Lattice visualization of music in real time" thread
>This is true, but Erv has used *many* other geometrical
>configurations, and one that I particularly like (and have posted
>on before) is his Logarithmic Harmonic Spiral.

Then you might be interested in the website "Music Of The Spheres"
http://websites.pagosa.net/hydrogenix/ in which the author puts the
masses of the elements around a log spiral and finds that they show some
clustering around the scale (and he has a 7/4 note :) which agrees with
my own observations.

-- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
Cycles email list -- http://www.kcbbs.gen.nz/users/af/cyc.htm
Alexandria eGroup list -- http://www.kcbbs.gen.nz/users/af/alex.htm
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