basic questions, harmonic entropy

Have the following been discussed before:

Whether harmonic entropy should operate independently of harmonic content?

Whether harmonic entropy should describe melodic hearing of intervals
(without simultaneous tones) as well as harmonic (with simultaneous
tones)?

Since beating and 'roughness' are not part of the definition of
harmonic entropy, I would guess the answer to the first part is 'yes'
- in which case it can only be a very partial answer to questions of
consonance in harmony.

Since harmonic entropy seems to be built on a notion of
recognizability, given a fixed uncertainty in relative pitch
perception, the answer to the second part ought to be 'yes' as well.

My personal opinion is that there is a great big hole in the middle of
the theory, because harmonic content and beats *do* matter hugely in
harmonic consonance, and in my opinion are the main factor in
conditioning what intervals Westerners do or don't recognize.

The melodic recognizability of the semitone, a highly dissonant
interval, would seem to be a bit of a problem also.

~~~T~~~

> Have the following been discussed before:
>
> Whether harmonic entropy should operate independently of
> harmonic content?

How do you mean? You mean musical context? It's not
something harmonic entropy tries to model.

> Whether harmonic entropy should describe melodic hearing of
> intervals (without simultaneous tones) as well as
> harmonic (with simultaneous tones)?

I'm not aware of a discussion on this. To the extent harmony
bears on even isolated melodic intervals, something could
perhaps be said...

> Since beating and 'roughness' are not part of the definition of
> harmonic entropy, I would guess the answer to the first part
> is 'yes' - in which case it can only be a very partial answer
> to questions of consonance in harmony.

Well since I don't understand your first question...

> Since harmonic entropy seems to be built on a notion of
> recognizability, given a fixed uncertainty in relative pitch
> perception, the answer to the second part ought to be 'yes'
> as well.

You should try

http://www.soundofindia.com/showarticle.asp?in_article_id=1905806937

which is available from my Paul Erlich theory page:

http://lumma.org/tuning/erlich

> My personal opinion is that there is a great big hole in the
> middle of the theory, because harmonic content and beats *do*
> matter hugely in harmonic consonance, and in my opinion are
> the main factor in conditioning what intervals Westerners do
> or don't recognize.

I still don't know what you mean by content, but roughness
calculations for harmonic timbres will generally agree with
harmonic entropy, except they will not distinguish between
otonal and utonal chords like harmonic entropy does.

For inharmonic timbres, it isn't clear how to apply harmonic
entropy. Paul could answer this better than I.

> The melodic recognizability of the semitone, a highly dissonant
> interval, would seem to be a bit of a problem also.

That's a very complex issue that is waay out of the scope of
a model like h.e.

-Carl

> Have the following been discussed before:> Whether harmonic entropy
should operate independently of harmonic content?

what do you mean by 'harmonic content'? Harmonic entropy measures an
intervals' fit with a series and root. Rootedness obviously does
play a part in harmony.

> Whether harmonic entropy should describe melodic hearing of
intervals> (without simultaneous tones) as well as harmonic (with
simultaneous> tones)?

I, personally, believe that there is an element of interval and
chord 'consonance' which has to do with implied melodic resolution,
but that's vary culturally specific.... This debate (simutaneities,
like Rameau, vs. step tones, ala Schenker) was raging in the 19th
century, with folks like Hauptman, and Cappellan, as I was just
reading at

The Dual Theory in Harmony
Herbert Westerby
Proceedings of the Musical Association, 29th Sess., 1902 - 1903
(1902 - 1903), pp. 21-72

> Since beating and 'roughness' are not part of the definition of
> harmonic entropy, I would guess the answer to the first part
is 'yes'
> - in which case it can only be a very partial answer to questions of
> consonance in harmony.
>
> Since harmonic entropy seems to be built on a notion of
> recognizability, given a fixed uncertainty in relative pitch
> perception, the answer to the second part ought to be 'yes' as well.
>
> My personal opinion is that there is a great big hole in the middle
of
> the theory, because harmonic content and beats *do* matter hugely in
> harmonic consonance, and in my opinion are the main factor in
> conditioning what intervals Westerners do or don't recognize.

The standard measure of chord consonance, per researchers like
Parncutt, is that root ambiguity and roughness are two, somewhat
separate factors. For example, dom 7th chord is fairly dissonance,
but is highly rooted.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > Have the following been discussed before:
> >
> > Whether harmonic entropy should operate independently of
> > harmonic content?
>
> How do you mean? You mean musical context? It's not
> something harmonic entropy tries to model.
>

Sorry if this was unclear, I meant the relative strengths of harmonics
in each tone (assumed not to be inharmonic)... timbre, if you like.

The reason for mentioning this is that it has been shown to be very
difficult to pick out pure intervals between sine waves. E.g. the
relative consonance or dissonance of (say) 695c vs 700c should in
reality depend strongly on the timbres used.

Clearly, the ranking according to 'roughness' or beating will agree
with the ranking by harmonic entropy, at least in the neighbourhood of
a recognizable consonance. But the difference in consonance is
absolutely not independent of timbre; whereas the calculated
difference in harmonic entropy is.

My suspicion is that h.e. is simply a mathematically elegant surrogate
for what actually happens in the ear, i.e. the interaction of
partials. Western ears are conditioned to recognise intervals which
occur in Western music, and these intervals occur mainly because they
approximate certain integer ratios, which makes them tunable via
interaction of partials. The consonance or dissonance is usually
registered via this interaction of partials (plus familiarity, or lack
of it). If we remove the partials, our ears can still recognise
intervals to some extent - but nowhere near so accurately.

The fact that we can't find pure fifths, etc., using sine wave tones
shows that (relative) frequency sensation alone is nowhere near enough
to account for our experiences of consonance and dissonance. I think
it very unlikely that the brain is able to recognise integer ratios of
frequency without further cues - i.e. partials.

Frequency is, crudely, distance along some part of the inner ear.
Interval between pure tones is then related in some way to the
distance between two points. Put this way, it is not surprising that
we can't pick out integer ratios unaided.

On the last point: why should recognising a semitone be considered
such a much more difficult problem (for a theory of interval
recognition) than recognising a fifth? To sing a melody you need to do
both with fair accuracy.

~~~T~~~

> Clearly, the ranking according to 'roughness' or beating will
> agree with the ranking by harmonic entropy, at least in the
> neighbourhood of a recognizable consonance. But the difference
> in consonance is absolutely not independent of timbre; whereas
> the calculated difference in harmonic entropy is.

It isn't independent of timbre, but with the exception of
some struck percussion like the piano, orchestral chimes
and tympani, and the transients of plucked strings, all
traditional pitched instruments have perfectly harmonic
spectra.

If the timbres being studyied are inharmonic but still regular
enough to evoke a clear sense of fundamental pitch, then you
can use the h.e. of their roots to model overall consonance
and an additional Sethares-type calculation to model the fine-
grained roughness on top. You can also use h.e. to model the
ability of the timbre to evoke a single pitch by plugging its
partials into a chordal version of harmonic entropy.

> My suspicion is that h.e. is simply a mathematically elegant
> surrogate for what actually happens in the ear, i.e. the
> interaction of partials.

Yet another option would be to enter all the partials being
heard at once into a chordal h.e. model.

Critical band roughness is only *one part* of consonance.
Try comparing 4:56:7:9 with 1/(4:5:6:7;9) for example.

> Western ears are conditioned to recognise intervals which
> occur in Western music,

This effect can (I believe) be overcome with training in a
reasonable amount of time.

> If we remove the partials, our ears can still recognise
> intervals to some extent - but nowhere near so accurately.

What do you think of this graph

/harmonic_entropy

as a picture of the recognizability of a dyad played with
sine tones?

> The fact that we can't find pure fifths, etc., using sine wave
> tones shows that (relative) frequency sensation alone is nowhere
> near enough to account for our experiences of consonance and
> dissonance.

On the contrary, in the real world fifths are not often not
tuned beatless by performers. And partials do more than make
accurate tuning possible via beating -- they provide more
information to the brain's virtual pitch processor.

> I think it very unlikely that the brain is able to recognise
> integer ratios of frequency without further cues - i.e. partials.

I can dig up some studies for you if you like. In bats and
cats, combination sensitive neurons in the inferior colliculus
are tuned to JI intervals...

Have you ever tried tuning a pair of sine generators? I
didn't find it as hard as you seem to be saying.

> Frequency is, crudely, distance along some part of the inner ear.

And counts of zero-crossings of waveforms on the auditory nerve.

> On the last point: why should recognising a semitone be considered
> such a much more difficult problem (for a theory of interval
> recognition) than recognising a fifth?

To be more accurate, you're asking: Why should a "semitone" be
harder to study than a 3:2? The answer is, define what a
semitone is, and then we can study it.

-Carl