Re: Barlow's indigestibility function

Paul-

My training is in molecular biology as well as composition. As a biologist,
I had to take quite a few chemistry and physics classes, and, therefore, I
am acquainted with the scientific notion of energy. For instance, chemically
speaking you have to increase energy to go from one stable state of atoms in
a compound to another, something we call a chemical reaction. Stable states
have low energy, unstable ones have high energy.

Now, doing research on intervals I observed that if you play two tones in
alternation (keep one tone stable, move the other slowly upwards) so that
there's no acoustic overlap, you will still notice something that H. Husmann
called "Lichtpunkte" (points of light)--which correlate to simple integer
pitch ratios (SIPR; I try to avoid the term frequency for psychoacoustical
reasons). This phenomenon reveals our brains' ability to process harmony
even in absence of difference tones, beatings and other secondary harmonic
cues.

The absolute intensities of these "points of light" seem to correlates to
Barlow's harmonicity values, whereas their relative intensities seem to
increase as you approach the SIPR and decrease as you leave it. This lead me
to, first, invert Barlow's harmonicity formula to derive a harmonic energy
function (the "brighter" and stable the SIPR the lower its energy), and
second, to apply a bell curve around each SIPR. Doing this for all strong
intervals within a given interval range yields an "energy landscape" with
"dimples" and "craters."

I also took several psychoacoustical and cognitive factors, determining the
depth and width of these local minima, into consideration (and that's a
major departure from most math-based theories of
harmonicity/consonance/corcordance). I merged these factors into the term
"pitch strength."

A tuning (be it equidistant or not) consists of a set of pitches that can be
placed into local energy minima. To calculate the total energy of a tuning,
one has to compute the (combinatorial) energy of all possible interval
combinations.

I published a ranking of these tunings (relevance) with different pitch
strengths in my paper "Low Energy and Equal Spacing." What was most
important and gratifying for me, was that, despite their scientific
derivation, the results are pretty consistent with the historical evolution
of music.

A few years after writing my paper, I was exposed to the theory of
backpropagation neural networks and I learned that the term "energy" is
commonly used to describe the state of a network. It's the goal that in the
training phase (the phase during which the training values are presented to
the network), it will come as close as possible to reproducing those values
by first setting weights for all the connections randomly and then
optimizing them towards a local or, ideally, a global minimum.

It's a fact that the neurons which are responsible for harmonic perception
(lined up like organ pipes in our cortex) also respond to stimuli that are
slightly off. Ultimately, the quality of the tuning in a musical situation
depends on the response of the network of neurons, i.e. on their energy
state. That's why I never believed in purely mathematical theories of
consonance. Nonetheless, the theory of neural networks is also based on
math; everything is just a little more complicated.

Georg

>
> Thanks Georg. Now I wonder if
> you'd care to comment on your
> application of Barlow's function
> to your paper. You seem to be
> after a notion of "consonance",
> and bring up some recent
> psychoacoustic concepts in
> defining it, yet rather than
> investigating the mathematics of
> these phenomena themselves,
> you bring in Barlow's function to
> stand for consonance. Can you
> elaborate on your reasoning
> behind this move?

Hi Georg, thanks for answering. The main reason I asked is that I view with
extreme suspicion the idea that hearing a particular frequency ratio, or
approximation thereof, leads to anything like a "mental representation" of
the numbers in the ratio -- and particularly the idea that the absolute
primality of the numerator or the absolute primality of the denominator
could have perceptual relevance. I have argued these points to death on this
list, but my viewpoint is matched by the psychoacoustically derived dyadic
consonance functions of Helmholtz, Plomp, Kameoka & Kuriyagawa, Sethares,
and Parncutt (as well as, incidentally, the more abstract one of Tenney, and
the empirical "one-footed bride" of Partch) which show no special provision
for primality, and I'm rather convinced that perceptions of ratios such as
9:8, 9:5, and 15:8 are biased due to their familiarity from Western music,
and from experience with the simple _musical_ (i.e. not purely _perceptual_)
operation of stacking musical intervals on one another. That is, if you've
never heard a fourth and a fifth played in succession with one note in
common, you'd never know that a major second was reducible in terms of
simpler intervals -- in Kantian terms, you might say the reduction is
"synthetic" and not "analytic".

>I also took several psychoacoustical and cognitive factors, determining the
>depth and width of these local minima, into consideration (and that's a
>major departure from most math-based theories of
>harmonicity/consonance/corcordance). I merged these factors into the term
>"pitch strength."

If you're referring to the finite tolerance of the hearing resolution,
certainly Helmholtz, Plomp, Kameoka & Kuriyagawa, Sethares, and even Partch
produced _continuous_ consonance curves which acknowledge this finite
tolerance. You may be interested in learning about my "Harmonic Entropy"
function, a curious parallel to your "energy" function. There is an entire
newsgroup dedicated to the concept, harmonic_entropy@yahoogroups.com, I hope
you will join and look at the archives. A very outdated compilation of early
posts on the subject to this list is available at
http://www.ixpres.com/interval/td/entropy.htm -- it's sorely in need of
revision, but should give you a start. The beauty of this formulation, I
feel, is that not only does it exploit the information-theoretic concept of
"entropy" (synonymous with confusion) to model dissonance, but it does so
extremely elegantly with essentially only one free parameter -- you guessed
it -- the standard deviation of the uncertainty in the hearing resolution.

>A tuning (be it equidistant or not) consists of a set of pitches that can
be
>placed into local energy minima. To calculate the total energy of a tuning,
>one has to compute the (combinatorial) energy of all possible interval
>combinations.

Would you agree that for a 12-tone scale, you'd have 66 dyads to evaluate? A
few months ago I did a set of such optimizations for locally minimizing the
total dyadic harmonic entropy using starting points that were various N-tone
equal tempered scales. Among the interesting "relaxed" scales obtained were
pentatonic and heptatonic "diatonic" scales in approximately meantone
tuning, while for most values of N the equal tuning proved to be a local
minimum (since bettering one interval here would inevitable worsen another
interval there). I'm preparing to review the results in a little more detail
in response to Darren McDougall's query . . .

Dear Georg Hajdu,

Good to have you on the list.

You use the term simple integer pitch ratios (SIPR) and say that you try to
avoid the term frequency for psychoacoustical reasons. But you really are
talking about frequency ratios, aren't you? As I understand it the percept
of pitch is considered to have an approximately logarithmic relationship to
frequency and is affected by factors other than frequency. Also why
integers? There is no meaning attached to negative numbers. How about small
whole-number frequency ratios (SWFRs).

You wrote:

>Hence, Barlow's (scientifically unproven, but irrefutable) assumption that
an analogous mechanism also determines the
harmonicity/consonance/concordance of an interval, and the
> "metrical affinity" of simultaneous meters.

I assume you mean that, assuming Barlow's indigestibility accurately models
musicians perception of the relative consonance of dyads (vertical
intervals) with small whole-number frequency ratios (SWFRs), it is
irrefutable whether the mechanism is as he suggests. But the assertion that
Barlow's indigestibility does accurately model musicians perception of the
relative consonance of dyads is certainly refutable. And I believe it has
been adequately refuted on this list. In fact all prime-factorisation based
measures have been pretty much rejected in the case of dyads. The product
of the two numbers appears to be the best measure (or any monotonic
function of that), except that ratios where the smaller number is a power
of two appear to be at a slight advantage.

Paul Erlich and I (and possibly others) expect that prime-factor based
measures result only when musicians are simply asked to rank the consonance
of intervals from memory, rather than by a/b comparison listening to bare
dyads. The fact is that IN CHORDS, ratios whose sides have many small prime
factors are more likely to allow factors to cancel in OTHER dyads within
the chord. e.g. 8:15 is not consonant in itself but will readily take part
in chords where the other intervals are very consonant (have very simple
ratios) because it can cancel 2's, 3's and 5's.

The following URL points to a zipped Excel spreadsheet which compares (on
charts) all of the formulae we could find which purported to model the
relative consonance of SWFRs. One chart compares octave-equivalent measures
and another octave-specific.

http://www.uq.net.au/~zzdkeena/Music/HarmonicComplexity.zip

For non-SWFR intervals (continuous consonance or dissonance curves) we seem
to have 3 contenders.
Hajdu
Sethares
Erlich

More on these later.

Regards,

-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

Dave Keenan wrote,

>For non-SWFR intervals (continuous consonance or dissonance curves) we seem
>to have 3 contenders.
> Hajdu
> Sethares
> Erlich

You might want to add Helmholtz to this list . . . why not?

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
> Dave Keenan wrote,
>
> >For non-SWFR intervals (continuous consonance or dissonance curves)
we seem
> >to have 3 contenders.
> > Hajdu
> > Sethares
> > Erlich
>
> You might want to add Helmholtz to this list . . . why not?

Pardon my ignorance. Can you tell me how Helmholtz gets his continuous
function? Or is there something I can read online? Does he have to
make a somewhat arbitrary choice of which SWFRs to base it on, as
Georg does (and Sethares and Erlich don't)? Or is it like a simplified
version of Sethares'?

If either Sethares' or Hajdu's are a clear refinement of Helmholtz's,
then we can omit Helmholtz.

Regards,
-- Dave Keenan

--- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:

/tuning/topicId_18061.html#18182

>
> http://www.uq.net.au/~zzdkeena/Music/HarmonicComplexity.zip
>
>
> -- Dave Keenan
> Brisbane, Australia
> http://dkeenan.com

I'm assuming that everybody has already seen these beautiful charts
except me... I've never seen Excel charts as beautiful as these!

I'm not sure I understand them yet, but, visually, they are
stupendous...

I'm hoping Dave makes a Webpage of these charts, like he has with
some of his amazing others... so we can study them a bit more
effectively...

Don't miss them!

__________ ______ _____ _
Joseph Pehrson

>If either Sethares' or Hajdu's are a clear refinement of Helmholtz's,
>then we can omit Helmholtz.

In a sense, Sethares' is a refinement of Helmholtz's. But as I keep
mentioning, there's a problem with Sethares' . . . the "amplitudes" are
actually, Bill tells me, meant to be dB levels, but one must do some
creative fiddling to get realistic results. You should definitely purchase a
copy of the Helmholtz-Ellis book, _On the Sensations of Tone_.

To David C Keenan:

> As I understand it the percept
> of pitch is considered to have an approximately logarithmic relationship to
> frequency and is affected by factors other than frequency. Also why
> integers? There is no meaning attached to negative numbers. How about small
> whole-number frequency ratios (SWFRs).
>
"Whole numbers" is fine, although it seems more like a semantic question to
me. I think the term frequency is more questionable for reasons that Ernst
Terhardt investigated in his research.

As you know, a pitch with an irregular set of partials may still yield a
clear pitch sensation which is different from f1, the first partial. f1
might even be completely absent. The "frequency" of this (virtual) pitch can
only be determined by matching it to the frequency of a sine wave. Since the
term frequency is defined as vibration per second (and obviously, virtual
pitch doesn't satisfy this definition), it seems inappropriate to me to use
this term.

In this context it's worthwhile to mention Terhardt's early observations on
virtual pitch. He showed that three adjacent partials up to the 10th or so
partial will still yield the proper residual tone (virtual pitch). Only the
ratio 11:12:13 will confuse the brain, and the resulting residual tone will
be off.

I wonder whether the mechanism that leads to this result has already been
properly understood. Anyway, it's a symptom of how our brain deals with
certain pitch ratios, and that may give us some clues about how it might
process complex intervals with analogous properties.

>> Hence, Barlow's (scientifically unproven, but irrefutable) assumption that
>> an analogous mechanism also determines the
>> harmonicity/consonance/concordance of an interval, and the
>> "metrical affinity" of simultaneous meters.

> In fact all prime-factorisation based
> measures have been pretty much rejected in the case of dyads. The product
> of the two numbers appears to be the best measure (or any monotonic
> function of that), except that ratios where the smaller number is a power
> of two appear to be at a slight advantage.

Well, how do you explain the seemingly higher consonance of an interval such
as the small major second 10:9 (10+9=19, or 5+9=14, if you will) in
comparison to 7:5 (7+5=12)?

>
> Paul Erlich and I (and possibly others) expect that prime-factor based
> measures result only when musicians are simply asked to rank the consonance
> of intervals from memory, rather than by a/b comparison listening to bare
> dyads. The fact is that IN CHORDS, ratios whose sides have many small prime
> factors are more likely to allow factors to cancel in OTHER dyads within
> the chord. e.g. 8:15 is not consonant in itself but will readily take part
> in chords where the other intervals are very consonant (have very simple
> ratios) because it can cancel 2's, 3's and 5's.

I think that this might have something to do with fusion, a phenomenon that
also applies to complex pitches as well (only to a lesser degree).

>
> The following URL points to a zipped Excel spreadsheet which compares (on
> charts) all of the formulae we could find which purported to model the
> relative consonance of SWFRs. One chart compares octave-equivalent measures
> and another octave-specific.
>
> http://www.uq.net.au/~zzdkeena/Music/HarmonicComplexity.zip
>
>

Very interesting, I'll have look at it.

Georg

***************************************************
Dr. Georg Hajdu

Hochschule für Musik Detmold,
Abteilung Münster
Ludgeriplatz 1
D-48151 Muenster

Phone:
+49-251-48233-0 (w)
+49-2506-85758 (h)
+49-172-787-4214 (m)

Fax: +49-251-48233-30
e-mail: hajdu@uni-muenster.de
http://www.mhs-muenster.de/Hajdu.html
****************************************************

To Paul R. Ehrlich:

I want to apologize for the style of my reply. Since I am more a composer
with psychoacoustic awareness rather than a researcher with musical
awareness, the statements I am going to make will be somewhat aphoristic.

I'm not surprised that someone else has developed his own concept of
harmonic energy/entropy. The analogy to physics/chemistry is so obvious, it
was meant to happen.

The question is to what degree do perceptual and cognitive aspects of
harmonic hearing interact with and influence each other?
Please believe me, I never stop wondering whether an interval such as 32/27
is a mental reality or not. Sometimes I am inclined to think it's just a
mathematical construction with little perceptual relevance.

But on the other hand, very well-trained musicians are able to mentally
track the stacking of fifths in complex musical situations (such as
modulations) and are well aware of mistakes or deviations--so they argue
(something that would be very difficult to show with psychoacoustical
experiments). Hence, I wouldn't dismiss mental/cognitive abilities in the
processing of non SIPRs (or WNFRs), or call it synthetic as opposed to
analytical in Kantian terms.

I tend to compare this ability to a football player trying to catch a ball
while running, solely relying on his mental ability to calculate its
trajectory.

I haven't been following the discussion on the tuning site for a long time,
I'm not sure what the current agreements or disagreements are. Yet, it seems
to me that it is absolutely necessary to differentiate between simultaneous
and sequential harmonic events.

The phenomenon I'm most interested in is short-term memory or
ultra-short-term memory, which comes into play when you hear pitches in
SUCCESSION (I am NOT talking about SIMULTANEOUS dyads--no beatings and
difference tones involved). Thus, harmonic perception is purely mental,
achieved by overlap of an active synthesis process in the brain.

Time and again, I tried to verify the results shown by my continuous
consonance curve by this little experiment:
While I was playing the alternating pitch sequence I described earlier on
this list (let two tones alternate without acoustical overlap, leave one
tone stable, increase the other one gradually by one-cent increments), I
displayed the consonance curve on the computer display and redrew the curve
while the alternating note was increasing in pitch.

Now, I critically asked myself whether I could hear a correspondence between
the two. And indeed, I felt that there was a pretty good match. I
subsequently showed that Barlow and others who confirmed my observation.

This procedure may not stand scientific scrutiny. Although, it's worth
mentioning that I also perceived discrepancies, especially around 600 cents.
For me, the tempered tritone was definitely more consonant than the
"dimples" corresponding to the 7/5 and 10/7 ratios. This supports the idea
that preferences for certain intervals are culturally acquired after all.

Now, my theory is that the neural networks in our brains are constantly
"trained" with, at least, two different training materials: Firstly, a
"natural training set" which is based on the ubiquitous harmonic series and
which may also be responsible for our preference for SIPRs, and secondly, a
"culturally-determined training set" which is based on "irrational"
intervals which we get used to through repetition.

These two training sets compete with one another, and eventually,
"irrational" intervals find entry into tuning systems if they contribute to
their low energy (or entropy), as can be argued in the case of the
"irrational" 600ct tritone.

If you have a stronger "natural" interval in the vicinity of an "irrational"
one, the natural one may have the upper hand, though.

Based on these findings, one could devise the following experiment (does
anyone like to volunteer?):
Test subjects listen to the alternating pitch sequence (described above) and
use an entry device such as the arrow keys on a computer keyboard to
indicate whether they perceived a relative increase or decrease of
consonance.

I hope I'm not reinventing the wheel though.

Georg

***************************************************
Phone:
+49-251-48233-0 (w)
+49-2506-85758 (h)
+49-172-787-4214 (m)

Fax: +49-251-48233-30
e-mail: hajdu@uni-muenster.de
http://www.mhs-muenster.de/Hajdu.html
****************************************************

Georg Hajdu wrote,

>In this context it's worthwhile to mention Terhardt's early observations on
>virtual pitch. He showed that three adjacent partials up to the 10th or so
>partial will still yield the proper residual tone (virtual pitch). Only the
>ratio 11:12:13 will confuse the brain, and the resulting residual tone will
>be off.

>I wonder whether the mechanism that leads to this result has already been
>properly understood.

Harmonic entropy provides a promising way to address these issues. Although
triadic harmonic entropy has yet to be calculated, in the diadic case one
can calculate, for a given assumption about the resolution of hearing, which
ratios will evoke a clear virtual fundamental and which won't.

>Well, how do you explain the seemingly higher consonance of an interval
such
>as the small major second 10:9 (10+9=19, or 5+9=14, if you will) in
>comparison to 7:5 (7+5=12)?

I don't perceive it that way at all! But I think that the perception in
general may be due to the tritone's special function as the characteristic
dissonance in the diatonic scale (it's also the only _unique_ interval in
the diatonic scale, and it's also the only _ambiguous_ [as to size in
scale-steps] interval in the diatonic scale), signalling a need for a
resolution by contrary motion in all Western common practice music. The 7:5
is in fact beautifully consonant when divorced from this style, as those who
have played with the Bohlen-Pierce scale can attest (c'mon guys, attest!).

>I think that this might have something to do with fusion, a phenomenon that
>also applies to complex pitches as well (only to a lesser degree).

Can you speak more about your conception of "fusion"?

>To Paul R. Ehrlich:

Though I normally wouldn't care, that's a double misspelling of my name, and
it's the correct spelling of the name of a (in)famous environmentalist, not
me.

>Please believe me, I never stop wondering whether an interval such as 32/27
>is a mental reality or not. Sometimes I am inclined to think it's just a
>mathematical construction with little perceptual relevance.

>But on the other hand, very well-trained musicians are able to mentally
>track the stacking of fifths in complex musical situations (such as
>modulations) and are well aware of mistakes or deviations--so they argue
>(something that would be very difficult to show with psychoacoustical
>experiments). Hence, I wouldn't dismiss mental/cognitive abilities in the
>processing of non SIPRs (or WNFRs), or call it synthetic as opposed to
>analytical in Kantian terms.

Georg, are you saying that musicians who attempt to play some kind of
untempered, just intonation music have an acutity in detecting deviations
from 32/27 in the result of stacking three fifths that is _greater_ than the
acuity one would expect from applying the acuity with which they detect
deviations from 3/2 or 4/3 applied three times? If not, I see no need to
impose an "extra" mechanism.

>The phenomenon I'm most interested in is short-term memory or
>ultra-short-term memory, which comes into play when you hear pitches in
>SUCCESSION (I am NOT talking about SIMULTANEOUS dyads--no beatings and
>difference tones involved). Thus, harmonic perception is purely mental,
>achieved by overlap of an active synthesis process in the brain.

Well, it's good to be clear on that, since your paper didn't give that
impression. My concern has been with SIMULTANEOUS dyads -- as I believe
factors of melody, such as diatonicity, come into play whenever one heas a
SUCCESSION. Ignoring that, though, I see no need to go "beyond" what each
simple interval in the chain would imply. If 32/27 _isn't_ obtained by a
stacking of fifths, or by first-hand experience with stacking fifths, why
treat it any differently than, say, 31/26?

>This procedure may not stand scientific scrutiny. Although, it's worth
>mentioning that I also perceived discrepancies, especially around 600
cents.
>For me, the tempered tritone was definitely more consonant than the
>"dimples" corresponding to the 7/5 and 10/7 ratios. This supports the idea
>that preferences for certain intervals are culturally acquired after all.

It would also be consistent with the harmonic entropy calculations
associated with a fairly poor hearing resolution -- one sees a local minimum
_between_ 7/5 and 10/7. But your interpretation supports my assertion above
about the tritone.

>I haven't been following the discussion on the tuning site for a long time,
>I'm not sure what the current agreements or disagreements are.

It's too bad you weren't around during the heady days of this discussion.
You may want to look at the archives should you care to pick up on any of
these threads and bring up any issues with me, Dave, or anyone else.

By the way, I'm still awaiting a response to this:

>>A tuning (be it equidistant or not) consists of a set of pitches that can
be
>>placed into local energy minima. To calculate the total energy of a
tuning,
>>one has to compute the (combinatorial) energy of all possible interval
>>combinations.

>Would you agree that for a 12-tone scale, you'd have 66 dyads to evaluate?
A
>few months ago I did a set of such optimizations for locally minimizing the
>total dyadic harmonic entropy using starting points that were various
N-tone
>equal tempered scales. Among the interesting "relaxed" scales obtained were
>pentatonic and heptatonic "diatonic" scales in approximately meantone
>tuning, while for most values of N the equal tuning proved to be a local
>minimum (since bettering one interval here would inevitable worsen another
>interval there). I'm preparing to review the results in a little more
detail
>in response to Darren McDougall's query . . .

[Georg Hajdu wrote:]
>>Well, how do you explain the seemingly higher consonance of an
>>interval such as the small major second 10:9 (10+9=19, or 5+9=14, if
>>you will) in comparison to 7:5 (7+5=12)?

[Paul E:]
>I don't perceive it that way at all! But I think that the perception
>in general may be due to the tritone's special function as the
>characteristic dissonance in the diatonic scale (it's also the only
>_unique_ interval in the diatonic scale, and it's also the only
>_ambiguous_ [as to size in scale-steps] interval in the diatonic
>scale), signalling a need for a resolution by contrary motion in all
>Western common practice music. The 7:5 is in fact beautifully consonant
>when divorced from this style, as those who have played with the
>Bohlen-Pierce scale can attest (c'mon guys, attest!).

I haven't followed this thread closely, and haven't played with the
Bohlen-Pierce scale, but I most definitely _can_ attest to the
consonance and beauty of 7:5!! None of the major seconds, with the
possible exception of 8:7, can come close, to my ear.

JdL

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:

/tuning/topicId_18061.html#18352

>
> I haven't followed this thread closely, and haven't played with the
> Bohlen-Pierce scale, but I most definitely _can_ attest to the
> consonance and beauty of 7:5!! None of the major seconds, with the
> possible exception of 8:7, can come close, to my ear.
>
> JdL

Hello John...

Yes, this interval is quite beautiful in the BP scale, but, of
course, the 5:3, the BP major sixth, or BP sixth seems to "stand out"
even more from the consonance angle... at least that's how it seemed
after working with it. But I suppose that isn't really so
surprising...

____________ ______ ____ _
Joseph Pehrson

Dear Georg Hajdu,

Your description of your experiment with _alternating_ tones rather
than _simultaneous_ ones explains a lot.

I think you would have a very difficult time finding anyone who
considers 8:9 more consonant than 5:7 as vertical intervals (bare
dyads) without any other musical context, no matter whether you keep
the upper tone constant, or the lower, or some mean. But you might
well find westeners who find them so, as _melodic_ intervals.

In this case, I find your paper very misleading. You don't seem to
mention alternating tones and you call the quantity "harmonic
consonance" when it might better have been called "melodic
consonance".

We have found the product of the sides of the ratio to be the best
simple predictor of dyadic consonance for ratios whose product is not
more than about 99 and that not wider than 2 octaves.

8*9 = 72
5*7 = 35

Regarding the continuous case (for non-SWFRs), Paul Erlich's
formulation

http://www.ixpres.com/interval/td/Erlich/entropy.htm

has the delightful property that one does not need to decide in
advance which SWFRs are significant. This comes out of the
calculations when a particular pitch uncertainty (as a standard
deviation) is put in.

Regards,
-- Dave Keenan

Dave Keenan wrote,

<<We have found the product of the sides of the ratio to be the best
simple predictor of dyadic consonance for ratios whose product is not
more than about 99 and that not wider than 2 octaves.>>

While certainly handy, I think there are some pretty big problems, or
exceptions, with this... for instance, do most folks find say an 8:11
or a 7:9 dyad to be more consonant than a 9:11 dyad... ?

If not, then maybe you'll have to massage that product limit back to
"not more than about 98"!

Seriously though, I think a general rule of thumb predictor like this
has to account for cases like the neutral third a bit better than the
"product of the sides of the ratio" does...

--Dan Stearns

Dave-

Please don't jump to conclusions what my theory is concerned.

Hearing is such a complex mental process that it is necessary to isolate
certain aspects of harmony--which can be done by examining the harmonic
relationships of tones in succession.

Your distinction between harmonic consonance and melodic consonance is
artificial and defies hundreds of years of music history (if anything I
would call it horizontal and vertical harmony). Are familiar with Marc
Leman's research at all?

I personally prefer the term diagonal harmony, because this is what we
experience when we listen to most types of music.

The reason I focus on succession rather than simultaneity is obvious when
you carefully read my contributions to this list: It's the active synthesis
process, the active mental creation of harmony which interests me--an
ability "blurred" by phenomena such as difference tones and beatings, which
*reinforce* simple pitch ratios and only occur when notes are sounded
together.

> The 7:5 is in fact beautifully consonant
>> when divorced from this style, as those who have played with the
>> Bohlen-Pierce scale can attest (c'mon guys, attest!).

What my personal preference for the Bohlen-Pierce scale is concerned, please
read the following quote by Heinz Bohlen himself:

[10] Georg Hajdu: Der Sprung (1999)

The first scene of this opera, describing a dialog between two university
professors who are the victims of a shooting, is written in BP. The composer
explains: "As this scale, not having any octaves, is an intellectual
achievement in itself, it symbolizes the abstract, academic world of a
university department." This is most probably the first composition in BP
employing the human voice.
A recording of this scene is available on the Web, and a CD, containing the
whole opera, is distributed by NRW Vertrieb.

I hope this helps,

Georg

***************************************************
Dr. Georg Hajdu

Hochschule für Musik Detmold,
Abteilung Münster
Ludgeriplatz 1
D-48151 Muenster

Phone:
+49-251-48233-0 (w)
+49-2506-85758 (h)
+49-172-787-4214 (m)

Fax: +49-251-48233-30
e-mail: hajdu@uni-muenster.de
http://www.mhs-muenster.de/Hajdu.html
****************************************************

> From: tuning@yahoogroups.com
> Reply-To: tuning@yahoogroups.com
> Date: 9 Feb 2001 11:13:16 -0000
> To: tuning@yahoogroups.com
> Subject: [tuning] Digest Number 1097
>
> In this case, I find your paper very misleading. You don't seem to
> mention alternating tones and you call the quantity "harmonic
> consonance" when it might better have been called "melodic
> consonance".

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Dave Keenan wrote,
>
> <<We have found the product of the sides of the ratio to be the best
> simple predictor of dyadic consonance for ratios whose product is
not
> more than about 99 and that not wider than 2 octaves.>>
>
> While certainly handy, I think there are some pretty big problems,
or
> exceptions, with this... for instance, do most folks find say an
8:11
> or a 7:9 dyad to be more consonant than a 9:11 dyad... ?
>
> If not, then maybe you'll have to massage that product limit back to
> "not more than about 98"!
>
> Seriously though, I think a general rule of thumb predictor like
this
> has to account for cases like the neutral third a bit better than
the
> "product of the sides of the ratio" does...

All good points Dan.

I didn't say it was perfect, only "best simple". I failed to define
what I meant by "simple". Let's define it now as "requiring no more
mathematical operations (button-pushes on your calculator) than
Barlow's indigestibility". e.g. Harmonic Entropy is not simple.

Barlow's indigestibility doesn't fare any better than the product of
the sides, in regard to your ranking of 9:11 as more consonant than
7:9 or 8:11. Pulling the applicability limit back to 98 might well be
the best way of dealing with that. I think Harmonic Entropy predicts
these three will have approximately equal dissonance. Timbre would of
course affect this.

"Product of sides" (and Barlow's) has another problem that I mentioned
earlier. It doesn't seem to account for all of the consonance of
ratios whre the smaller of the two sides is a power of 2. e.g. 4:7,
8:9, 8:11, 8:13.

But I still think it's the best simple predictor we have so far.

Regards,
-- Dave Keenan

Dan wrote,

><<We have found the product of the sides of the ratio to be the best
>simple predictor of dyadic consonance for ratios whose product is not
>more than about 99 and that not wider than 2 octaves.>>

>While certainly handy, I think there are some pretty big problems, or
>exceptions, with this... for instance, do most folks find say an 8:11
>or a 7:9 dyad to be more consonant than a 9:11 dyad... ?

>If not, then maybe you'll have to massage that product limit back to
>"not more than about 98"!

>Seriously though, I think a general rule of thumb predictor like this
>has to account for cases like the neutral third a bit better than the
>"product of the sides of the ratio" does...

Well, Dan, I think the neutral third is very dissonant in a high register,
while in a lower register, one can expect a frequency resolution of roughly
1.5%, and I showed that that assumption leads to 9:11 being more consonant
than 7:9. In that realm, I think about 50 was the limit above which the
product rule fails to work. A limit of 100 comes out of the absolute
best-case scenario for register and particular listener acuity.

I wrote:

"Timbre would of course affect this."

I meant that timbre would affect the real-life perception of the
relative consonance of 7:9, 9:11, 8:11. Timbre will not affect the
harmonic entropy (HE) calculation. This is one of the weaknesses of HE
compared to Sethares' method. But then the partial "amplitudes" of
Sethares' method are somewhat mysterious and seem to greatly
exaggerate the effect of timbre.

Regards,
-- Dave Keenan

Georg wrote,

>Your distinction between harmonic consonance and melodic consonance is
>artificial and defies hundreds of years of music history

Care to elaborate? I'm sure Margo would have a lot to say on this topic too
. . .

>> The 7:5 is in fact beautifully consonant
>> when divorced from this style, as those who have played with the
>> Bohlen-Pierce scale can attest (c'mon guys, attest!).

>What my personal preference for the Bohlen-Pierce scale is concerned

Well, you can also see me "dismiss" the BP scale as a potential
generalized-diatonic system in my paper, but that doesn't take away from the
_concordance_ of 7:5 (I follow Blackwood in using "concordance" to describe
the purely sensory quality of the simultaneity, while "consonance" carries
connotations specific to Western tonal music which would be antithetical to
any "tritone".

Still awaiting your replies on other matters (assuming the list server
didn't hiccup),

Paul

Paul Erlich wrote,

<<Well, Dan, I think the neutral third is very dissonant in a high
register, while in a lower register, one can expect a frequency
resolution of roughly 1.5%, and I showed that that assumption leads to
9:11 being more consonant than 7:9. In that realm, I think about 50
was the limit above which the product rule fails to work. A limit of
100 comes out of the absolute best-case scenario for register and
particular listener acuity.>>

My feeling is that in a normalish register with normalish timbres and
under normalish conditions a 9:11 is going to be acutely more
concordant than the 8:11 and certainly less discordant than the 7:9,
and if a 'rule of thumb' doesn't admit as much, then I think it's got
some problems.

Of course I'm assuming that most would agree with my perceptions here,
and this is no doubt unwise, but a product limit shouldn't have to be
tweaked just to run problematic rascals out of town and give the
answers one hopes to see! There's got to be a better way...

--Dan Stearns

Georg,

Yes, hearing is a complex mental process. But "isolat[ing] certain
aspects of harmony" can also be done by examining simultaneous pairs
of tones. I would say that this lets us get at lower level percepts
because the presence of beating and differnece tones means that the
results are less able to be influenced by cultural conditioning. With
simultaneous notes there's "no room" between the notes for such
conditioning to come into play.

The terms horizontal and vertical harmony are fine and probably more
historically correct than mine, but I don't find the actual
distinction artificial at all. Correct me if I'm wrong, but you seem
to be saying that the findings of relative consonance of intervals for
horizontal harmony should be expected to apply equally well to
vertical harmony.

You are apparently treating beating and difference tones as if they
are not properly a part of the experience of consonance or dissonance
in vertical harmony, but merely distractions. I think that most JI
composers would strongly disagree.

I don't think I'm familiar with Marc Leman's research. Can you direct
me to a URL?

I certainly understood why you wished to focus on horizontal harmony
and agree that any theory of the evolution of scales must do this,
since the vertical has apparently only become very important
relatively recently and mainly in the west.

My only objection is to your apparent assumption (unstated in your
paper) that what works for horizontal will also work for vertical.

Regards,
-- Dave Keenan

Dan Stearns wrote:

> My feeling is that in a normalish register with normalish timbres and
> under normalish conditions a 9:11 is going to be acutely more
> concordant than the 8:11 and certainly less discordant than the 7:9,
> and if a 'rule of thumb' doesn't admit as much, then I think it's got
> some problems.

That's not what I find. 8:11 is clearly more consonant then 9:11 under
normal conditions.

> Of course I'm assuming that most would agree with my perceptions here,
> and this is no doubt unwise, but a product limit shouldn't have to be
> tweaked just to run problematic rascals out of town and give the
> answers one hopes to see! There's got to be a better way...

If something simple like a product limit gives mostly the right answers,
then go with it. We don't even have a reliable ordering of consonances
for a more complex rule to correlate with.

Graham

Graham wrote,

<<That's not what I find. 8:11 is clearly more consonant then 9:11
under normal conditions.>>

Wow... ? Ordinarily I'd never assume that my listening impressions are
anything like what "you" hear, and even here I said it was unwise, but
boy, I thought this one seemed like such an easy call! Just goes to
show ya...

<<If something simple like a product limit gives mostly the right
answers, then go with it. We don't even have a reliable ordering of
consonances for a more complex rule to correlate with.>>

I think a product limit only gives mostly the right answers if you lop
it off at the point where things start to get difficult (or
interesting). It's certainly a handy little rule of thumb, but only to
a point... and that point is right where things start to get
interesting in my opinion.

--Dan Stearns

Dave Keenan wrote,

<<You are apparently treating beating and difference tones as if they
are not properly a part of the experience of consonance or dissonance
in vertical harmony, but merely distractions.>>

That sounds about right... ! Okay, so I'm kidding, well sort of
kidding...

Generally speaking, my modus operandi is that if it works in a
scalular sense and the smallest scale steps aren't in the commatic
range -- roughly noticeably smaller than a quartertone that is -- then
it'll work vertically... "incidental harmonies" sometimes, but not
always. Okay, so I'm a primitivist... but so it is.

No, Mozart ain't gonna make too many Mozatians very happy in say 13
equal, but 13 equal ain't just "a melodic" tuning either!

--Dan Stearns

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Paul Erlich wrote,
>
> <<Well, Dan, I think the neutral third is very dissonant in a high
> register, while in a lower register, one can expect a frequency
> resolution of roughly 1.5%, and I showed that that assumption leads to
> 9:11 being more consonant than 7:9. In that realm, I think about 50
> was the limit above which the product rule fails to work. A limit of
> 100 comes out of the absolute best-case scenario for register and
> particular listener acuity.>>
>
> My feeling is that in a normalish register with normalish timbres and
> under normalish conditions a 9:11 is going to be acutely more
> concordant than the 8:11 and certainly less discordant than the 7:9,
> and if a 'rule of thumb' doesn't admit as much, then I think it's got
> some problems.

But Dan, 9*11, 8*11, and 7*9 are all greater than 50!
>
> Of course I'm assuming that most would agree with my perceptions here,
> and this is no doubt unwise, but a product limit shouldn't have to be
> tweaked just to run problematic rascals out of town and give the
> answers one hopes to see! There's got to be a better way...

Dan, I don't think anything is being arbitrarily "tweaked" here. There always has to be a product
limit since beyond a certain complexity, ratios will be confused with nearby ones. And I've
shown that a product limit follows simply from a given assumption about standard deviation of
resolution of the central pitch processor. Based on Goldstein's work, it seems that a standard
deviation of 1.5% could be considered "average" in a generalist context. And the harmonic
entropy curve associated with this standard deviation indeed shows 9:11 as relatively
concordant, since it lies in the overlap of the 4:5 valley and the 5:6 valley . . . considering that we
have not made any particular assumptions about timbre, register, or duration, I think we've done
amazingly well at explaining our perceptions -- don't you agree?

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:

> I think a product limit only gives mostly the right answers if you lop
> it off at the point where things start to get difficult (or
> interesting).

The word "limit" already carries with it the implication that you lop things off at some point. I think
what you're saying is that a product _rule_ only works up to some _limit_. That better be the
case -- I sure wouldn't want to compare 3:2 with 30001:20001, let alone an irrational ratio, using
a product rule. That's what harmonic entropy is for -- and the product rule, up to some limit, was
found using harmonic entropy. (If you recall from either this list or the harmonic entropy list, it
didn't matter whether we seeded the harmonic entropy calculation with a Farey series or a
Tenney series -- the local minima always came out obeying a product limit).

> It's certainly a handy little rule of thumb, but only to
> a point... and that point is right where things start to get
> interesting in my opinion.

And harmonic entropy comes into the picture -- so let's continue this discussion on that list, if
appropriate . . .

Paul Erlich wrote,

<<But Dan, 9*11, 8*11, and 7*9 are all greater than 50!>>

I pretty much gather that... when I say in a normalish register I'm
meaning say a C' to C'' range, and if the product rule only holds
water to about 50 there then what's the point really... ?

<<Dan, I don't think anything is being arbitrarily "tweaked" here.>>

And I quote (Dave Keenan) -- "in regard to your ranking of 9:11 as
more consonant than 7:9 or 8:11. Pulling the applicability limit back
to 98 might well be
the best way of dealing with that."

<<There always has to be a product limit since beyond a certain
complexity, ratios will be confused with nearby ones.>>

Sure, and I agree... but I think if it (the product rule truncated at
some limit) doesn't account for something like neutral thirds and
sixths (and even a bit more in my opinion) then it's just too limited.

--Dan Stearns

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Paul Erlich wrote,
>
> <<But Dan, 9*11, 8*11, and 7*9 are all greater than 50!>>
>
> I pretty much gather that... when I say in a normalish register I'm
> meaning say a C' to C'' range, and if the product rule only holds
> water to about 50 there then what's the point really... ?

What's the point of what?
>
>
> <<Dan, I don't think anything is being arbitrarily "tweaked" here.>>
>
> And I quote (Dave Keenan) -- "in regard to your ranking of 9:11 as
> more consonant than 7:9 or 8:11. Pulling the applicability limit
back
> to 98 might well be
> the best way of dealing with that."

I think Dave was joking.
>
>
> <<There always has to be a product limit since beyond a certain
> complexity, ratios will be confused with nearby ones.>>
>
> Sure, and I agree... but I think if it (the product rule truncated
at
> some limit) doesn't account for something like neutral thirds and
> sixths (and even a bit more in my opinion) then it's just too
limited.

Dan, clearly there isn't any COMPLEXITY formula, based simply on the
numbers involved in the ratios, that's going to work when the numbers
get too high -- because of the influence of nearby, simpler ratios.
This is what Dave Keenan calls TOLERANCE. Again, the three factors
seem to be COMPLEXITY, TOLERANCE, and SPAN. All that Dave and I are
saying is that the COMPLEXITY part seems better represented by a
product rule than by any other function of the numbers involved in the
ratios (except for monotonic functions of the product, e.g., Tenney's
Harmonic Distance, which is the log base 2 of th

Paul Erlich wrote,

<<What's the point of what?>>

Of a product rule that covers so (comparatively) little... remember my
responses here originated from Dave Keenan's, "we have found the
product of the sides of the ratio to be the best simple predictor of
dyadic consonance for ratios whose product is not more than about 99
and that not wider than 2 octaves" quote. Simple it is, and handy too,
but I think that there are problems that one should also mention.

<<I think Dave was joking.>>

Wanna bet... ? (Here's the bit, "Barlow's indigestibility doesn't fare
any better than the product of the sides, in regard to your ranking of
9:11 as more consonant than 7:9 or 8:11. Pulling the applicability
limit back to 98 might well be the best way of dealing with that. I
think Harmonic Entropy predicts these three will have approximately
equal dissonance. Timbre would of course affect this.")

All I'm really trying to say is that I think this breaks down way
before you reach the point where the proximity of nearby simpler
ratios comes into play, and that I've sensed a tendency to want to
massage the rules into a more "agreeable" form if they ain't lining up
quite the way one might like/expect.

--Dan Stearns

Dan is correct. I wasn't joking about cutting the limit back to 98
when using product to estimate relative consonance. It's only a rule
of thumb, since it's only the complexity component. If it doesn't work
for you, tinker with it however you like, or ditch it. But Barlow's
indigestibility is worse. I'm not sure why Paul thought I was joking,
he said himself that for some listeners in some registers it wouldn't
work for a*b > 50.

For me, 9:11 is complex enough that its consonance _is_ influenced by
its proximity to 4:5 and 5:6, but I don't find it more consonant than
8:11 or 7:9 (in artificial dyad listening experiments). I expect it
has to be tuned pretty bloody accurately to be heard _as_ a 9:11 as
opposed to any old neutral third. I personally can't tune a 7:11 or
9:11 or 9:10 by ear. But I can just barely do so with an 8:11 and an
8:15 (using sawtooth waves).

Regards,
-- Dave Keenan

Dan wrote,

>All I'm really trying to say is that I think this breaks down way
>before you reach the point where the proximity of nearby simpler
>ratios comes into play,

I disagree. I believe that for you, the 5:4 and 6:5 are coming into play,
and for most of the rest of us, they are not. Anyway, this is the hypothesis
of the harmonic entropy model, and I haven't seen any evidence against it.

>and that I've sensed a tendency to want to
>massage the rules into a more "agreeable" form if they ain't lining up
>quite the way one might like/expect.

Well, the point of harmonic entropy is to help account for what we hear, and
when a model has only one free parameter, and can conform to a given
situation by tweaking only that one parameter, while correctly predicting
(as far as I can tell) the relative "discordances" of thousand of intervals,
I'd say we have a pretty good model! (The Standard Model of particle
physics, which correctly predicted all known subatomic phenomena until last
week, has about _twenty_ free parameters, and was still considered an
outstanding model.)

>I'm not sure why Paul thought I was joking,
>he said himself that for some listeners in some registers it wouldn't
>work for a*b > 50.

I thought you were joking because going from 99 to 98 is such a small change
-- in determining the actual product limit implied by a given standard
deviation, one has even more uncertainty than this.

>For me, 9:11 is complex enough that its consonance _is_ influenced by
>its proximity to 4:5 and 5:6, but I don't find it more consonant than
>8:11 or 7:9 (in artificial dyad listening experiments).

Well that's quite the opposite of what Dan was claiming -- and seems to put
you at a standard deviation comfortably below 1.5%.

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
> >I'm not sure why Paul thought I was joking,
> >he said himself that for some listeners in some registers it
wouldn't
> >work for a*b > 50.
>
> I thought you were joking because going from 99 to 98 is such a
small change
> -- in determining the actual product limit implied by a given
standard
> deviation, one has even more uncertainty than this.

Paul,

I think the confusion here is due to you thinking of the product rule
(or its log) as an approximation of the harmonic entropy function in
the case of SWFRs, while everyone else is thinking of it as an
approximation of their experience.

Do you have a response to Dan Wolf's smallest-number rule which seems
to handily encapsulate both complexity and span in a very simple rule?

Regards,
-- Dave Keenan

Dave Keenan wrote,

>Do you have a response to Dan Wolf's smallest-number rule which seems
>to handily encapsulate both complexity and span in a very simple rule?

Although the smallest-number rule agrees quite nicely with the original
harmonic entropy formulation using a Farey, rather than Tenney, series, I'd
ask Daniel what he would find when holding the _upper_, rather than the
_lower_, note constant. I'd conjecture that he will have to invoke a rule
that is more complex than just a largest-number rule or anything like that .
. .

I wrote,
>
>> I thought you were joking because going from 99 to 98 is such a
small change
>> -- in determining the actual product limit implied by a given
standard
>> deviation, one has even more uncertainty than this.

Dave K. wrote,

>I think the confusion here is due to you thinking of the product rule
>(or its log) as an approximation of the harmonic entropy function in
>the case of SWFRs, while everyone else is thinking of it as an
>approximation of their experience.

Well, then I'd reiterate my opinion that a product rule is pretty poor on
its own, as ratios like 300001:200001 demonstrate, and irrational ratios
even more so.

But in the context of the above, I don't think you're located the source of
my confusion -- I really thought you were joking because the idea that you'd
sincerely suggest "well, there are problems when the limit is 99, but
everything is perfect and flawless when the limit is 98" seemed . . . well .
. . it seemed like there had to be a ":)" hiding somewhere.

I wrote (to Paul Erlich),

> >I think the confusion here is due to you thinking of the product
rule
> >(or its log) as an approximation of the harmonic entropy function
in
> >the case of SWFRs, while everyone else is thinking of it as an
> >approximation of their experience.

Paul replied:
> Well, then I'd reiterate my opinion that a product rule is pretty
poor on
> its own, as ratios like 300001:200001 demonstrate, and irrational
ratios
> even more so.

Try this (which is what I meant):

I think the confusion here is due to you thinking of the product
rule [with applicability limits applied to the product and to the
width of the interval] as an approximation of the harmonic entropy
function in the case of SWFRs [Small Whole-number Frequency Ratios],
while everyone else is thinking of it as an approximation of their
experience [in the case of SWFRs].

300001:200001 is of course not a SWFR.

> But in the context of the above, I don't think you're located the
source of
> my confusion -- I really thought you were joking because the idea
that you'd
> sincerely suggest "well, there are problems when the limit is 99,
but
> everything is perfect and flawless when the limit is 98" seemed . .
. well .
> . . it seemed like there had to be a ":)" hiding somewhere.

I can understand now why you thought that.

But this is exactly what one does with a _rule_of_thumb_. One adjusts
the width of ones thumb until the thing fits (not perfectly, but maybe
as well as it is ever going to).

Regards,
-- Dave Keenan