Yahoo Tuning Groups Ultimate Backup tuning The Plomp-Levelt dissonance curve

Hello:

I have just finished reading the book by Dr. William Sethares: "Tuning,
Timbre, Spectrum, Scale". Not only did I read it, but I attempted to
re-create some of his simpler results using my own synthesizer and spectrum
analyzer.

If I understand correctly, the bulk of Sethares' work is based on a computed
value for sensory dissonance of sets of complex tones. The general method
for computing the sensory dissonance for a set of complex tones is to add up
the dissonance of each pair of harmonics present in the tones. The
dissonance of the harmonic pairs is computed based on experimentation
performed by Plomp and Levelt back in 1965.

Sethares' book presents a discussion of these experiments, along with some
audio demonstrations on an accompanying CD. It is these demonstrations,
along with the interpretation, that don't match what I perceive. I was
hoping that someone would straighten me out so I can understand the rest of
Sethares' theories better. As a software engineer, I am keenly interested
in implementing the adaptive tuning algorithms that Sethares describes
towards the end of his book.

#########################################

So here we go:

Sethares presents an audio track that demonstrates "a sine wave of frequency
220 Hz together with a wave of variable frequency beginning at 220 Hz and
slowly increasing [over 60 seconds] to 470 Hz." [Page 43, TTSS]

In his discussion of this audio sample, he states that people generally
perceive "pleasant beating (at small frequency ratios), roughness (at middle
ratios), and separation into two tones (at first with roughness, and later
without) for larger ratios." [Page 44] Furthermore "the dissonance is
minimum when both sine waves are of the same frequency, increases rapidly to
its maximum somewhere near one quarter of the critical bandwidth, and then
decreases steadily back towards zero." [Page 44]

From the graphics that are connected to this text (the "Plomp-Levelt curves"
on pages 44 and 45), it is clear that the above description is meant to
apply to the entire interval from the base tone (220 Hz in this example) to
its next higher octave (440 Hz) and beyond.

An amplitude graph of the two added signals is shown at the top of page 44.
No scale is provided for the graph's x-axis, the time axis. The graph shows
an envelope with a very wide "beat" at the beginning, followed by series of
decreasing duration size beats (correponding to the increase in the
frequency difference between the two signals over time), followed eventually
by beats that are too frequent to be visible at the resolution of the graph.

Now here is my disagreement: Through different degrees of holistic vs.
analytical listening to the subject sound sample, I simply hear something
different than what Sethares (and Plomp and Levelt) describe.

When I listen to this sound sample, I do indeed hear the "fusion ->
beating -> roughness -> separation" pattern described by Sethares. (Sounds
like a recipe for a bad marriage.) However, this pattern ends when the
second (sweep) tone reaches about 300 Hz. After that, I hear the reverse
pattern: "separation -> roughness -> beating -> fusion". This reverse
pattern continues until 330 Hz (i.e. 3/2 of the base frequency), and then
the entire pattern repeats again
"fusion->beating->roughness->separation->roughness->beating->fusion", ending
at 440 Hz (i.e. 2 times the base frequency). A little bit of beating and
roughness then proceeds until 470 Hz, when the sound sample ends. (Wow, this
really sounds violent.)

At the same time, I perceive a "difference" tone. This difference tone is
not audible at first, but just about the time that beating first turns to
roughness I start to hear it (probably when the frequency difference of the
two waves exceeds 50Hz, the lowest frequency my computer's speakers will
reproduce.) The difference tone slowly increases in frequency until the
sweep tone reaches 330 Hz, at which point the difference tone is 110 Hz.
Then, as the sweep tone progresses from 330 Hz to 440 Hz, the perceived
difference tone's pitch decreases to the point that it is inaudible (just
when the "beating" phase leading up to 440 Hz begins).

At the same time, during each of the "separation" phases, I perceive a
periodic increase/decrease in another quality of the sound. I don't know
what to call this quality, but it sounds similar to pulse-width modulation.
This modulation occurs on the order of once every five seconds during the
60-second sample.

Sethares explains that Plomp and Levelt were careful to use only musical
novices in their listening experiments. The stated reason for this was that
"musically trained listeners often recognize intervals and report their
learned musical responses rather than their actual perceptions".

With the possible exception of my third observation about periodically
increasing/decreasing "pulse width modulation", I do not think that my
minimal musical training could be affecting my perceptions. However, to
make sure that this was the case, I used a spectrum analysis package to view
both the spectrum and waveform of the combined signal.

The repeating cycle of fusion -> separation -> fusion that I heard is quite
easily visible in the resulting waveform graph. I was able to produce a
graph just like the one shown at the top of page 44, except that the graph
on the top of page 44 only covers the portion of the sound sample
corresponding to the sweep tone ranging from 220 to about 250 Hz. This is
misleading, since this only shows one small section of the
fusion-separation-fusion cycle, which actually repeats two full times in the
sound sample. The 220-250 Hz region of the waveform amplitude graph does
indeed match what is shown in the Plomp-Levelt curve. The remaining
sections of the graph (not shown in Sethares' book) seem to refute the
Plomp-Levelt curve.

Let's take a specific example: Take the pair of pure sine waves at 220 Hz
and 420 Hz. This is clearly in the region of the Plomp-Levelt curve in
which the dissonance should be approaching very near zero. However, when I
listen to this pair of tones, I clearly perceive a very strong dissonance.
In other words, it is not just the intervals around the 1:1 ratio that are
very dissonant, but also the intervals around 2:1. In fact, the intervals
around 3:2 also produce dissonance, albeit not as bad. On the other hand, I
can't say I was able to hear much dissonance around the higher-numbered
intervals (4:3, 5:4, 5:3, etc.)

I am trying to figure out where this discrepancy comes from. It seems very
clear to me that dissonance does _not_ decrease continuously as the interval
increases, even for pure tones. If this is true, first of all this would
mean that Plomp and Levelt were wrong, which I doubt. Furthermore, it
throws all of Sethares' results out the window, too, since he uses the
Plomp-Levolt curves for his own calculations of sensory dissonance. Since
the other musical samples that Sethares provides with his book are
persuasive, I conclude that I must be getting something wrong.

Is the dissonance I am hearing an artifact of my sound system? I don't
think so. As I said, if I use spectrum analysis software to examine the
resulting waveforms, I can clearly see lots of amplitute beats going on
around the 1:1, 2:1, and 3:2 intervals, corresponding to the dissonance I
hear.

I have lots more specific details about the spectral analysis, but I am sure
by now I have belabored my point. Can anyone explain to me if I am
misinterpreting the Plomp-Levelt results, or what is going wrong with my
analysis? Does anyone have time to synthesize the 220 Hz plus 220 Hz-to-470
Hz sweep sine wave, listen to the result, and tell me if they hear the same
thing that I do?

Thank you very much for your time

Tim Crews
timcrews@home.com

🔗Rick McGowan <rmcgowan@apple.com>

🔗bram <bram@gawth.com>

🔗Tim Crews <timcrews@home.com>

Hi Rick:

Thanks for your response. A few re-responses are
included below:

> From: Rick McGowan <rmcgowan@apple.com>
>
> I think there's a
> nice curve in Helmholtz that shows this.

If you are thinking of the curve from "On the Sensation of
Tone", this curve was derived by Helmholtz by adding up
the dissonances of _partials_ that were present in the
complex sounds that he was comparing. While I agree that the
partials that are present in complex tones do in fact
contribute to dissonance, my claim is that even pure tones
can be dissonant with each other. BTW, the Helmholtz
curve that I am talking about is reproduced in Sethares
TTSS, bottom of page 82. Maybe you are referring to a
different curve.

The Helmholtz curve does appear to be more complex than
what I would get by creating intervals using two pure
sine waves, in all honesty. I am only _sure_ I hear
dissonance at the 1:1, 3:2, and 2:1 intervals. I hear
other phenomena which may or may not be labelled dissonance
at other rations. The Helmholtz curve has on the order of
16 peaks in dissonance. I can't hear that many, just
with sine waves.

> Go out on the net & get the CoolEdit demo. It's really easy to
> do this experiment and takes only a few moments. I'll try it out
> myself...
>

Oh, I've already spent hours and hours with a synthesizer
playing with this stuff. Although it's nice having the audio
sample CD that comes with TTSS, it's way better to be in
control of the sounds so that you can listen more carefully.

Tim Crews
timcrews@home.com

🔗Tim Crews <timcrews@home.com>

Hi Bram:

Thanks for your response. A few re-responses
are included below:

> From: bram <bram@gawth.com>
>
> I found that I couldn't really hear the dissonance curve for simple sine
> waves until I did the experiment of playing one steady pure sine wave with
> another one sliding around. Just hearing discrete points of dissonance
> makes it very hard to mentally connect everything into a smooth curve.
>

I agree that you hear a lot more complex stuff going on if you let
the tones slide against each other. Some of that I would be content
to explain as artifacts of the sliding itself. However, it is quite
easy to hear the dissonance of some intervals around 2:1, even when
the tones are fixed. My example of 220 Hz + 420 Hz (1.91 : 1) is clearly
dissonant. Another example, closer to 3:2, is 220 Hz + 320 Hz (2.9 : 2)
is also pretty clearly dissonant.

> The one thing not described in Sethares's book which I notice is a *very*
> subtle bit of a dip in dissonance around 2:1 but I have to listen for that
> on purpose, and it might just be an artifact of having a specific sampling
> rate (I did this experiment digitally.)

See the example above. Using Sethares' words, the combination of
220 Hz and 420 Hz is "not subtle" :^). Anyway, I have experimented with
various sample rates. I have also experimented with Fourier analysis
using different sample sizes and window sizes. The results are independent
of these variations.

>
> The main thing I find missing in Sethares's examples is a study of how to
> reduce the innate dissonance of a given sound - all the synthetic timbres
> are produced by just throwing together different pure tones, with no
> attention paid to the dissonance between them. Does anyone know of any
> work done to get synthetic timbres to sound more 'pure'?

TTSS, Section 11.2, discusses how to adjust spectra, i.e. move the spectral
lines around. I would think that the techniques discussed in this section
would allow you to move dissonance partials to more suitable (i.e. more
nearly harmonic) locations, so that the _inherent_ dissonance of a sound
is minimized, while hopefully retaining the basic character of the sound.
That's really what chapter 11 is all about.

Thank you for your time
Tim Crews

🔗Rosati <dante@pop.interport.net>

at the end of this email are an orc and sco file for CSound that will
generate the following:

1) a sine tone gliding from 220 to 470cps over 30 sec.
2) two sine tones: one steady at 220cps, one gliding from 220 to 470cps,
over two minutes (one minute is too quick to catch the subtleties. You can
change it to a different duration by finding the "120"s in the score file
and changing the # of seconds.)
3) two composite tones doing the same as #2. Each tone has six harmonic
partials lessening in strength as you go up.

If you don't have Csound you can get it from:

http://mitpress.mit.edu/e-books/csound/frontpage.html

save the orc part below as a text file called disscurve.orc and the score
part as disscurve.sco When you open Csound it will prompt you to locate an
.orc and a .sco file. It will also ask you where to make the .wav file and
what to call it. maybe call it disscurve.wav. Hit render and wait till its
through.

Theres no doubt that, with two sine tones, the dissonance action around
1/1 is also heard in an attenuated form around 2/1. The same effect is also
happening around 3/2. Theres more going on, but it needs more listens. I
also noticed that my ear can hear/conceptualize (with the pure sine tones)
stable intervals that "change quantum state" to the next important
ratio/scale step. I have to listen to it a few hundred more times... (one
of the quintessential "tone mandalas", I'd say)

------------------------
;disscurve.orc

sr = 44100
kr = 4410
ksmps = 10
nchnls = 1

instr 1
k1 linen p5,p6,p3,p7
a1 oscil k1,p4,p8
out a1
endin

instr 2
k1 expon p4,p3,p9
k2 linen p5,p6,p3,p7
a1 oscil k2,k1,p8
out a1
endin

-------------------------------------------------

;disscurve.sco

f1 0 1024 9 1 1 0
f2 0 1024 10 1 .9 .8 .7 .6 .5

i2 0 30 220 10000 .5 1 1 470
s
i1 1 121 220 10000 .5 1 1
i2 1 121 220 10000 .5 1 1 470

i1 122 120 220 10000 .5 1 2
i2 122 120 220 10000 .5 1 2 470

------------------------------------------------

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>

🔗Bill Alves <alves@orion.ac.hmc.edu>

>From: "Tim Crews" <timcrews@home.com>
>
>>From the graphics that are connected to this text (the "Plomp-Levelt curves"
>on pages 44 and 45), it is clear that the above description is meant to
>apply to the entire interval from the base tone (220 Hz in this example) to
>its next higher octave (440 Hz) and beyond.
>
>Sethares explains that Plomp and Levelt were careful to use only musical
>novices in their listening experiments. The stated reason for this was that
>"musically trained listeners often recognize intervals and report their
>learned musical responses rather than their actual perceptions".

This is my problem with Plomp-Levelt. They claim that two sine waves a
minor ninth apart are just as "consonant" as an octave, a fifth, or a major
third, or indeed the interval between any two sine waves greater than the
critical bandwidth. To musicians like myself who have tried this experiment
and find this description bordering on the absurd, they respond that we are
simply parroting learned responses -- hearing a minor ninth and responding
that it's dissonant because we have learned that minor ninths are
dissonant.

Perhaps that's true (though I'm skeptical). In any case, it doesn't matter
to me. I have to trust my ears when writing music. If something sounds
dissonant then it is. Whether my judgement is a result of nature or nurture
is just angels on the head of a pin when I'm composing. (In any case, I
don't think disentangling innate/learned responses is as simple as finding
"musically naive" subjects -- if there is such a thing -- or that the
judgements of musically naive subjects will necessarily lead to important
insights about our innate listening apparatus -- witness the "mel" scale.)

Though I agree that partials can have an effect on perceived dissonance, I
don't believe all partials are created equal. My own ears tell me that the
fundamental has certain special qualities and that a minor ninth has a
minor ninth-ness about it that will remain if the consituent tones are sine
waves or sawtooth waves.

Bill

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)607-7600 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

🔗David J. Finnamore <dfin@xxxxxxxxx.xxxx>

In TD 215.16 Tim Crews wrote:

> "the dissonance is
> minimum when both sine waves are of the same frequency, increases rapidly to
> its maximum somewhere near one quarter of the critical bandwidth, and then
> decreases steadily back towards zero." [Page 44]

> From the graphics that are connected to this text (the "Plomp-Levelt curves"
> on pages 44 and 45), it is clear that the above description is meant to
> apply to the entire interval from the base tone (220 Hz in this example) to
> its next higher octave (440 Hz) and beyond.

In acoustics, the term "critical bandwidth" applies to an
interval of approximately 1/3 octave (pitch-wise; i.e.,
roughly 400 c.). I know of no instance of it referring to a
full octave, as the above seems to read. (Am I missing
something?) The critical
bandwidth varies a little with frequency. But not that
much! And the idea that a minor 3rd is the most dissonant
interval that 2 sines can make seems ridiculous. Why his
graph appears to show the phenomenon over an octave beats
me (oops, NPI!). Perhaps he's using it, or some other term,
differently.

Specifically, the term as I know of it applies to the
bandwidth around any given frequency within which other
tones can at least partially mask it, and it them. "Mask"
is acoustician speak for "drown out." For most of the human
hearing spectrum, that bandwidth is somewhere around 1/3
octave, and is more-or-less bell shaped. That partially
explains why 1/3 octave graphic equalizers are pervasive in
sound reinforcement. The rest of the reason is that they
look really impressive with all those colored sliders.

One quarter of the critical bandwidth should be around 100
cents, a value which would fit most Westerners' ideas of
maximum dissonance, at least. Alternatively, he might have
meant that when the band is _centered_ on 220, moving the
other sine 1/4 of
the way from there to the edge of the band, about 50 cents
give or take a few, produces maximum dissonance. I don't
know which of those best fits other definitions/calculations
of dissonance betwixt sine waves.

In those cases, I can see where the dissonance curve for
sines might fall to near zero outside of the critical band,
even for near-Just intervals. That would be strictly
outside of, and virtually irrelevant to, a musical context,
of course. Once you start combining notes in a composition,
regardless of the spectral characteristics of the
instruments, the whole perspective on consonance/dissonance
changes to meet the theory behind the music.

If you're hearing sines beating at intervals larger than
about 6/5, it's probably due to:

1) non-linearities in your sound reproduction system (using
typical "multimedia" computer speakers guarantees this for
frequencies below about 500 Hz) generating harmonic partials
2) extraneous sounds in your listening environment -- any
fans running nearby, such as in your CPU? HVAC (Heating,
Ventilation, and Air Conditioning)? Refrigerator?

Hate to say it, but critical analytical listening for sine
waves really requires a pro-quality studio, and an
acoustician running the experiment on a (separate) subject.
I'm not saying "Don't try this at home," just don't base any
hypotheses too firmly on home test results.

David J. Finnamore

🔗bram <bram@xxxxx.xxxx>

On Fri, 11 Jun 1999, Tim Crews wrote:

> > From: bram <bram@gawth.com>
> >
> > I found that I couldn't really hear the dissonance curve for simple sine
> > waves until I did the experiment of playing one steady pure sine wave with
> > another one sliding around. Just hearing discrete points of dissonance
> > makes it very hard to mentally connect everything into a smooth curve.
>
> I agree that you hear a lot more complex stuff going on if you let
> the tones slide against each other. Some of that I would be content
> to explain as artifacts of the sliding itself.

Yes, my major worry is that the 'roughness' is really an effect of very
slightly modulating tones at a specific interval, rather than an exact
interval.

> However, it is quite
> easy to hear the dissonance of some intervals around 2:1, even when
> the tones are fixed. My example of 220 Hz + 420 Hz (1.91 : 1) is clearly
> dissonant.

I found that sounds even *slightly* off from pure sine have obvious
effects at simple ratios, and I don't mean things like sawtooth or sin^2,
I mean things like sin^1.1, or sine with the extremes cut off just a
little. I think a lot of the effects people have been describing here are
due to a lack of a good source of almost-pure sine waves. I generated them
directly as sound files, and played them on the fairly decent
voice-quality speakers on my computer (not the cheapie stock kind, more a
$200 set, although I have no sub-woofer.)

> Using Sethares' words, the combination of
> 220 Hz and 420 Hz is "not subtle" :^).

It actually is to my ears, although my ears are, as someone else described
it, 'naive' (well, more stubborn really, both a blessing and a curse.)

> TTSS, Section 11.2, discusses how to adjust spectra, i.e. move the spectral
> lines around. I would think that the techniques discussed in this section
> would allow you to move dissonance partials to more suitable (i.e. more
> nearly harmonic) locations, so that the _inherent_ dissonance of a sound
> is minimized, while hopefully retaining the basic character of the sound.
> That's really what chapter 11 is all about.

That approach is correct in spirit, but I have a feeling that there's a
much more direct way of controlling it by using the relative phases of
different harmonics - The relative phases of two pure sine waves don't
matter except close to 1:1, but with more they start to matter a lot, and
TTSS for the most part just lets them fall where they may.

One reason I like the theory in TTSS is that it generally indicates that
the number of timbres which sound fundamentally not of this earth
completely dwarfs the number which do, which is an intuitive notion I've
had for a *long* time.

-Bram

🔗Tim Crews <timcrews@xxxx.xxxx>

Hi David:

You are the first person to suggest that the
phenomena that I am hearing are a result of my sound
system. And, believe it or not, I tend to agree with
you. I just can't believe that all of these scientists
independently came up with flawed results, considering
how carefully designed the experiments were.

> In acoustics, the term "critical bandwidth" applies to an
> interval of approximately 1/3 octave (pitch-wise; i.e.,
> roughly 400 c.). I know of no instance of it referring to a
> full octave, as the above seems to read. (Am I missing
> something?)

What I meant by this was that the maximum dissonance occurs
at the critical bandwidth, and then approaches zero
continuously all the way to the octave interval and beyond.
There are no local increases in dissonance, according to the
graph, outside the region corresponding to the critical
bandwidth.

> If you're hearing sines beating at intervals larger than
> about 6/5, it's probably due to:
>
> 1) non-linearities in your sound reproduction system (using
> typical "multimedia" computer speakers guarantees this for
> frequencies below about 500 Hz) generating harmonic partials
> 2) extraneous sounds in your listening environment -- any
> fans running nearby, such as in your CPU? HVAC (Heating,
> Ventilation, and Air Conditioning)? Refrigerator?
>
> Hate to say it, but critical analytical listening for sine
> waves really requires a pro-quality studio, and an
> acoustician running the experiment on a (separate) subject.
> I'm not saying "Don't try this at home," just don't base any
> hypotheses too firmly on home test results.
>

I am willing to accept that the stuff I am hearing is due
to my sound system. I understand that perfect sound waves
are indeed difficult to generate or reproduce accurately
in a home sound system.

On the other hand, my perception of "funny stuff" going
on at the 3:2 and 2:1 intervals is not limited just to what
I can hear. I can also observe very obvious patterns in
the envelope waveform when it is displayed graphically.
(Documented in a previous post.) Something is definitely
happening at 3:2 and 2:1, which coincidentally corresponds
exactly to the two intervals at which I perceive dissonance.

Tim Crews
timcrews@home.com

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

Tim Crews wrote,

>> "the dissonance is
>> minimum when both sine waves are of the same frequency, increases rapidly
to
>> its maximum somewhere near one quarter of the critical bandwidth, and
then
>> decreases steadily back towards zero." [Page 44]

>> From the graphics that are connected to this text (the "Plomp-Levelt
curves"
>> on pages 44 and 45), it is clear that the above description is meant to
>> apply to the entire interval from the base tone (220 Hz in this example)
to
>> its next higher octave (440 Hz) and beyond.

David J. Finnamore wrote,

>In acoustics, the term "critical bandwidth" applies to an
>interval of approximately 1/3 octave (pitch-wise; i.e.,
>roughly 400 c.). I know of no instance of it referring to a
>full octave, as the above seems to read. (Am I missing
>something?)

You must be reading the above incorrectly. It refers unequivocally to the
Plomp critical bandwidth, which is usually a little under a minor third but
gets wider in the bass register.

>And the idea that a minor 3rd is the most dissonant
>interval that 2 sines can make seems ridiculous.

Of course. "one-quarter of the critical bandwidth" would be more like 70
cents.

>If you're hearing sines beating at intervals larger than
>about 6/5, it's probably due to:

>1) non-linearities in your sound reproduction system (using
>typical "multimedia" computer speakers guarantees this for
>frequencies below about 500 Hz) generating harmonic partials

Harmonic partials and combination tones would inevitably be created
together. Any nonlinearity, in either the sound reproduction system or in
the human ear (the latter is a factor unless you play the sample really
quietly), creates both combination tones and upper partials.

>Hate to say it, but critical analytical listening for sine
>waves really requires a pro-quality studio

Amen to that! What are the THD ratings of your components, Tim? And don't
forget to use very low volumes so that your ears don't create combination
tones. But second-order beating around the 2:1 is present even under
clinical conditions.