5:6:9

Hello. I noticed that in the chord 5:6:9, both its simple and cubic
difference tones are exactly the same (1,3,4). I haven't found this
feature in any other 5-limit closed-position chord. (A few others may
have this feature -- I don't have my notes with me-- but only through
octave equivalence.) I;ve always found this chord to be especially
smooth sounding, and wonder if this has anything to with it. (I've
always liked the fact that the numerators, 6/5, 3/2, are in a 2:1
ratio.)

thanks, Kelly

--- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...> wrote:

> Hello. I noticed that in the chord 5:6:9, both its simple and cubic
> difference tones are exactly the same (1,3,4).

Neat!

> I;ve always found this chord to be especially
> smooth sounding, and wonder if this has anything to with it.

It might, but I thought you hadn't listened to any JI chords yet. You
should calculate the quadratic and cubic difference tones of the 12-
equal chord if that's what you're listening to. You might be surprised
what pitches they land on.

hi Paul-
> > I;ve always found this chord to be especially
> > smooth sounding, and wonder if this has anything to with it.
> It might, but I thought you hadn't listened to any JI chords yet.
You

It's true, I haven't. (I will probably contact a synthesist for a
demontration of alternate tunings, rather than re-tuning my piano.)
But perhaps there's a quality of the chord, perhaps based on it's
interval types, apart from the acoustical processes we've been
discussing.

> should calculate the quadratic and cubic difference tones of the 12-
> equal chord if that's what you're listening to. You might be
surprised
> what pitches they land on.

Ok. And, to the extent that combinations may play a part in pitch
perception, are certain types (eg, simple, cubic) more important than
others?

I had a few other questions/comments.

- I'm still struggling with understanding (accepting?) utonality, but
was wondering...it seems that the utonal conception of chords (--
because it's based on subharmonics?--) is not compatible with the
concept of Tonalness. I'm thinking of how you describe how low entropy
dyads (eg, 3/2, 5/4) can prevent the perception of the signal of the
overall chord (eg 10:12:15) they are part of, so am assuming that
1/6:5:4 would be even more suseptible to being overpowered by the low
entropy of it's individual dyads.

- Given that harmonics 3, 4, and 5 are possibly even more important
than the fundamental itself in pitch perception, can it be argued that
a chord, such as c-e-a, which suggests a 'small number of
fundamentals", and in which the multiple suggested fundamentals,
themselves, comprise a low-entropy interval, could have even stronger
Tonalness (as you have defined it) than a chord such as the major
triad, which 'points strait down' to the fundamental?

thanks, Kelly

--- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
wrote:
> hi Paul-
> > > I;ve always found this chord to be especially
> > > smooth sounding, and wonder if this has anything to with it.
> > It might, but I thought you hadn't listened to any JI chords
yet.
> You
>
> It's true, I haven't. (I will probably contact a synthesist for a
> demontration of alternate tunings, rather than re-tuning my
piano.)
> But perhaps there's a quality of the chord, perhaps based on it's
> interval types, apart from the acoustical processes we've been
> discussing.

Sure . . .

> > should calculate the quadratic and cubic difference tones of the
12-
> > equal chord if that's what you're listening to. You might be
> surprised
> > what pitches they land on.
>
> Ok. And, to the extent that combinations may play a part in pitch
> perception, are certain types (eg, simple, cubic) more important
than
> others?

As far as I know, none of them play a part in pitch perception,
though perhaps I'm missing your meaning.

> I had a few other questions/comments.
>
> - I'm still struggling with understanding (accepting?) utonality,
but
> was wondering...it seems that the utonal conception of chords (--
> because it's based on subharmonics?--) is not compatible with the
> concept of Tonalness.

Kind of correct.

> I'm thinking of how you describe how low entropy
> dyads (eg, 3/2, 5/4) can prevent the perception of the signal of
the
> overall chord (eg 10:12:15) they are part of, so am assuming that
> 1/6:5:4 would be even more suseptible to being overpowered by the
low
> entropy of it's individual dyads.

I don't understand how 1/6:5:4 could even make sense as an
interpretation in this context, where supposedly you're hearing
everything as a set of harmonics. You'd have to (at least be willing
to) rewrite 1/6:5:4 as 10:12:15 first, it seems to me, for it to have
any meaning in this context.

> - Given that harmonics 3, 4, and 5 are possibly even more important
> than the fundamental itself in pitch perception, can it be argued
that
> a chord, such as c-e-a, which suggests a 'small number of
> fundamentals", and in which the multiple suggested fundamentals,
> themselves, comprise a low-entropy interval, could have even
stronger
> Tonalness (as you have defined it) than a chord such as the major
> triad, which 'points strait down' to the fundamental?

No, because a single implied fundamental is always more "tonal" than
multiple implied fundamentals.

The fundamental is where our conscious attention is drawn, while the
harmonics that imply it are generally not heard directly, even if
only the harmonics, and not the fundamental, are physically present.

When given an ambiguous signal, such as specially designed sets of
inharmonic partials, the brain can be tricked into hearing one or the
other of the multiple implied fundamentals by various means, but it
never consciously hears them all at the same time.

> > to the extent that combinations may play a part in pitch
> > perception, are certain types (eg, simple, cubic) more important
> than > > others?

> As far as I know, none of them play a part in pitch perception,
> though perhaps I'm missing your meaning.

Sorry. Not pitch perception. I meant to say, to the extent that
combination tones are audible and part of the chord sensation (as I
believe you have attensted to), with beating patterns and other
effects (heavily influenced by tuning system, Iknow, and not to be
confused with root tracking!), which combination tones should I look
at? (I'm starting to look at these for 4-note chords, and have to
do it manually, or with a spreadsheet.)

For a few 4-note chords I've looked at (chords which, as usual I'm
attracted to, and trying to understand why!) are these simple
difference tone patterns unusual?

1) the preponderence of odd numbers in the simple difference tones
4,5,7,9,11, 16, for the dissonant chord a#2-c#4-f#4-b4 ( chord 5-12-
16-21).

2) again, odd numbers, but also having the same difference tone
value twice, for the major seventh chord inversion: d2-g3-b3-f#4 (
chord 3-8-10-15), with the simple difference tones being 2,5,5,7,7,
and 12. (That seems pretty sweet to me! but I haven't looked at
many of these, and don't know if it's significant.)

thanks for your insights, Kelly

--- In harmonic_entropy@yahoogroups.com, "traktus5" <kj4321@h...>
wrote:
>
> > > to the extent that combinations may play a part in pitch
> > > perception, are certain types (eg, simple, cubic) more
important
> > than > > others?
>
> > As far as I know, none of them play a part in pitch perception,
> > though perhaps I'm missing your meaning.
>
> Sorry. Not pitch perception. I meant to say, to the extent that
> combination tones are audible and part of the chord sensation (as I
> believe you have attensted to), with beating patterns and other
> effects (heavily influenced by tuning system, Iknow, and not to be
> confused with root tracking!), which combination tones should I
look
> at? (I'm starting to look at these for 4-note chords, and have to
> do it manually, or with a spreadsheet.)

The relative and absolute loudnesses of the combinational tones (and
even the question of whether they're audible at all) depend in a
sensitive way on the loudnesses of the original tones. It's almost
impossible to make any fair generalizations about this without a
particular scenario, including the physical distance between listener
and instrument (or loudspeaker). If you want to get into the data,
I'd suggest starting with Plomp's book _Aspects of Tone Sensation_
and then searching for more recent articles.

> For a few 4-note chords I've looked at (chords which, as usual I'm
> attracted to, and trying to understand why!) are these simple
> difference tone patterns unusual?
>
> 1) the preponderence of odd numbers in the simple difference tones
> 4,5,7,9,11, 16, for the dissonant chord a#2-c#4-f#4-b4 ( chord 5-12-
> 16-21).

If you start with two even and two odd numbers, the set of six
differences will necessarily contain two even and four odd numbers.
Nothing unusual here.

> 2) again, odd numbers, but also having the same difference tone
> value twice, for the major seventh chord inversion: d2-g3-b3-f#4 (
> chord 3-8-10-15), with the simple difference tones being 2,5,5,7,7,
> and 12. (That seems pretty sweet to me! but I haven't looked at
> many of these, and don't know if it's significant.)

It could be significant in giving the chord (if tuned in JI) a
certain relative coherence at the loud volumes at which the
difference tones are relevant; thought the prevalence of 7 might
simply cloud or confuse the perception of the original chord. Really,
no words can describe well what one hears and feels when actually
playing around with musical ideas in different tunings and scales.