Re: [tuning] Digest Number 1075

Hi Paul,

> > So therefore there has to be some other component involved
> > in our perception of consonance. Beating, or critical bands,
> > or whatever can't be the _only_ component, though it may
> > well be the most important for chords with small numbers of
> > notes.
>
> It seems to fail pretty miserable with just 4 notes, each of which has an ordinary
> harmonic timbre.

Do you have an example in mind that shows this particularly
clearly?

> > What if one were to choose the intervals for an "otonal chord"
> > from an inharmonic series of pitches, and used the virtual
> > harmonic series to make a recognisable inharmonic timbre,
> > such as the sound of bells, say.
>
> Then the effect would fall apart, since the brain is only looking for "the" harmonic
series
> (perhaps stretched a little). There's nothing to prevent you from constructing two
bells,
> with timbres which are the exact mirror-images of one another, such that one bell's
> "utonal" is the other bell's "otonal" and vice-versa with respect to their timbres.

I'm not sure - okay I could imagine one could possibly construct such bells, but one of
them would make a more "natural" bell sound than the other.

I.e. more familiar - here, maybe a church bell sound, in parts of Africa, maybe an agogo,
in Java, maybe a gamelan gong, and so on.

I'd want to listen to an actual example of it done, to see what happens.

Indeed, one could construct the mirror image bell sounds, mabye on computer anyway,
and try playing a tune on one or other, then ask which is most concordant of the
utonal and otonal chords, and see if the bell sound one has heard affects perception of
consonance of the chords based on its (in)harmonic series.

> > Or indeed, suppose one combined them to make
> > the overtone series of the inharmonic timbre itself?
>
> That _does_ lead to some consonance for the Sethares reason -- you will have lots of
> coinciding partials.

Yes, for "stretched harmonics", say, you would for the otonal, but also for the utonal
one.

Idea was, the otonal one might sound more consonant.

Also, maybe for some inharmonic timbres, the partials will be so
irregularly spaced, that combining the fundamentals to make an otonal
series of notes doesn't give many coinciding partials at all.
In theory, it could have no coinciding partials.
How irregular can the harmonics of an inharmonic partial be?

N.B. this leads into something I'm finding a bit mysterious.

You have exactly the same numbers
of shared harmonics for the diads, but it seems, if you take account
of combinations of 3 or more notes simultaneously, you actually
have _more_ coinciding groups of 3 partials simultaneously in the utonal chord:
E,g,

harmonic series on 2 3 and 5 are
2 4 6 8 10 12 14 16 ...
3 6 9 12 15 18 21 24 27 30
5 10 15 20 25 30 35 40
all three share
30, 60, 90,...

In the respective harmonic series, these are:
(2*) 15 30 45 60 ...
(3*) 10 20 30 40 ...
(5*) 6 12 18 24 ...

Series on 1/2, 1/3 and 1/5
1/2 1 3/2 2 5/2 3 7/2,...
1/3 2/3 1 4/3 5/3 2 7/3 8/3 3,...
1/5 2/5 3/5 4/5 1 6/5 7/5 8/5 9/5 2,...

All three share 1, 2, 3, 4, 5, ...
- or in the respective harmonic series:
(1/2*) 2 4 6 8 10,...
(1/3*)3 6 9 12,...
(1/5*)5 10 15 20,...

Would happen even more so with four or more notes.

E.g. add 7, and 1/7, and in the otonal series,
you get all four partials coinciding every 210
notes of the harmonic series, and in the utonal
series, evey 7 notes.

On first thought, one would think this would mean the utonal
chord should be _more_ consonant as far as beating partials
are concerned than the otonal one.

Am I missing something?

Same would happen with an inharmonic timbre if the partials
were stretched uniformly from the harmonic series partials.

(N.B. with FTS, looks as if I'll need to check for all groups of
three beating partials in Helmholtz type searches for triads,
- could first find which triads have consonant component diads,
in order to make the search manageable,
then order the ones found according to the beating partials in the
total chord)

> > Also would be interesting to see if the amount of consonance
> > depended on how familiar one was with the virtual
> > inharmonic timbre constructed. For instance, would
> > a regular gamelan player find a virtual overtone
> > series made up using one of the typical gamelan timbres
> > more consonant than others who are somewhat
> > less familiar with that timbre?
>
>
> Well some theorists have proposed that the virtual pitch phenomenon normally
> associated with the harmonic series could be a result of training (including prenatal
> training involving the mother's voice) rather than genetics; if they are right, then
there
> may be something to what you're saying -- though there is far less consistency in what
> constitutes a "typical gamelan timbre" than in what constitutes a "harmonic overtone
> series". In any case, the effect would require a very extensive training period with the
> timbre in question in order to be measurable.
>

I wonder how long it does take to get used to a new timbre. Perhaps it would
take a long time - if so, you'd have to check out the chords with players of the
instruments, and others who are unfamiliar with them.

However it is also possible that playing a totally new inharmonic timbre to someone for a
short while, enough so that they can recognise it again, might be enough to predispose
one to hear the inharmonic otonal chord as more harmonious.

Or maybe they'd need to play it on a midi keyboard for a short while, or
something like that.

I'd have thought that might possibly be enough to make it work if the effect does
happen, though of course one would expect it to be stronger if one
has played the instrument for a long time, or listened to much
music on it.

Looks as if the best experiment to do would be to see
if one had more consonance in the otonal chord than
the utonal one when using two timbres,
with few partials in common - one timbre for the
otonal /utonal pitches, and one for the actual notes.
and using an inharmonic timbre for the otonal /utonal
pitches, and either inharmonic or harmonic for the
actual notes.

I'd like to give that a try, just because on the (maybe remote)
possibility that it did work, it would be an interesting result.

Will be easy to do with FTS if I can find a nice table of partials with
their volumes for an inharmonic timbre.

Robert

Robert Walker wrote,

>> It seems to fail pretty miserable with just 4 notes, each of which has an
ordinary
>> harmonic timbre.

>Do you have an example in mind that shows this particularly
>clearly?

Well, a bunch of us on this list (and then on the Harmonic Entropy list)
have spent quite a lot of time listening to the 36 tetrads that appear at
the end of http://artists.mp3s.com/artists/140/tuning_lab.html. Check the
archives. We went through this in great detail, and it appears
incontrovertible that _otonal_ simplicity is _more_ important in explaining
our perceptions than the simpicity of the six individual intervals in each
tetrad. For example, everyone gave 4:5:6:7 a much more favorable rating than
1/7:1/6:1/5:1/4, even though the two chords contain exactly the same
intervals. Much more evidence along these lines was seen in the perceptions
of the other 34 tetrads.

>I'm not sure - okay I could imagine one could possibly construct such
bells, but one of
>them would make a more "natural" bell sound than the other.

Why would you say that?

>Indeed, one could construct the mirror image bell sounds, mabye on computer
anyway,
>and try playing a tune on one or other, then ask which is most concordant
of the
>utonal and otonal chords, and see if the bell sound one has heard affects
perception of
>consonance of the chords based on its (in)harmonic series.

>> > Or indeed, suppose one combined them to make
>> > the overtone series of the inharmonic timbre itself?
>>
>> That _does_ lead to some consonance for the Sethares reason -- you will
have lots of
>> coinciding partials.

>Yes, for "stretched harmonics", say, you would for the otonal, but also for
the utonal
>one.

>Idea was, the otonal one might sound more consonant.

I think it all comes down to the specifics of the bell timbre in question.
If the original bell timbre was closer to an integer harmonic series than to
an integer subharmonic series, then probably yes. If the original bell
timbre was closer to an integer subharmonic series than to an integer
harmonic series, its mirror-image would likely be more consonant in this
context.

>Also, maybe for some inharmonic timbres, the partials will be so
>irregularly spaced, that combining the fundamentals to make an otonal
>series of notes doesn't give many coinciding partials at all.
>In theory, it could have no coinciding partials.
>How irregular can the harmonics of an inharmonic partial be?

No Robert, as long as the inharmonic timbre has a particular spectrum in
cents, there will always be coinciding partials, e.g., the fifth partial of
the fourth member of the otonal series will always coincide with the fourth
partial of the fifth member of the otonal series, since the partial in
question is a distance from the "fundamental" of the otonal series that is
equal to the distance from the first to the fourth partial, plus the
distance from the first to fifth partial, in the inharmonic series in
question.

>You have exactly the same numbers
>of shared harmonics for the diads, but it seems, if you take account
>of combinations of 3 or more notes simultaneously, you actually
>have _more_ coinciding groups of 3 partials simultaneously in the utonal
chord:
>[...]
>On first thought, one would think this would mean the utonal
>chord should be _more_ consonant as far as beating partials
>are concerned than the otonal one.

>Am I missing something?

That's exactly right, and I've brought this up here on the list many times,
to point to the glaring inadequacies of the "coinciding partials"
explanation of consonance.

>Same would happen with an inharmonic timbre if the partials
>were stretched uniformly from the harmonic series partials.

Or even with an _arbitrary_ inharmonic timbre with a given spectrum in cents
-- the argument runs as above, but now you'll have _three_ distances between
partials to add, and that distance will be spanned in _three_ different
ways.