harmonic/subharmonic

Hello from Budapest:

Given that any collection of tones selected from a harmonic series can also be
notated as subharmonic and any selection from a subharmonic series can also be
notated as harmonic, AND -- following Sethares -- harmonic timbres are optimal
for either, (1) when does one decide to ifdentify a given chord as harmonic or
subharmonic? (2) when does a subharmonic chord become heard as a mistuned
harmonic chord?

A reasonable first guess is that whichever notation uses lower terms will be
the prevailing one (i.e. I hear 1/6:5:4 instead of 10:12:15). There is,
however, a great deal of ambiguity, depending upon whether the tones are
arpeggiated or presented as a single _Gestalt_. The latter seems to more often
lead to a sense that the chord is mistuned harmonic, while playing the
intervals melodically leads to a disambiguation of the intended notation. The
sampling period and the absolute frequency of the tones in the chord can be
_extremely_ critical. (Just compare this mistuned sound of a 14:18:21 triad
with the lowest tone below middle c with the clarity of the same triad
transposed up above the treble staff). Further, the density of the timbre can
be a factor; unfortunately, I don't know even how to begin to characterize
this.

Paul Erlich's example of the 4:5:6:7:9 vs. 1/9:7:6:5:4 (for which I posted
some .wav samples) is good case of an ambiguous situtation. In the version I
put online, with a lowest tone of 210 Hz, I find the subharmonic chord to have
a beautiful and distinct beating pattern, while at the same time the "mistuned"
harmonic interpretation satisfied when the chord is held for less than three or
four seconds. An arpeggiation reduces this ambiguity as melodic 9:7's are very
distinct. And then, a transposition of the whole chord up two octaves removes
(for me, at least) all the ambiguity -- the subharmonic is just not a mistuned
version of the harmonic, but a distinct consonance of it's own.

If there is any more interest in this topic, I'll puts some .wavs up on my web
page aftern I return to Frankfurt on Sunday evening...

Daniel Wolf

Daniel wolf (01.06.2000) wrote:
> ... any collection of tones selected from a harmonic
> series can also be notated as subharmonic and any
> selection from a subharmonic series can also be
> notated as harmonic ...

And...
Kraig Grady (27.05.2000) wrote:
> The 4 5 6 is easily found by taking the 2 out of 3 set.
> For instance 4x5 4x6 5x6, being 20 24 30 (10 12 15)

hmm... Let's check this out.
So, are these formula's right?
---------------------------------------------------
Given: To Calculate:
f1 : f2 : f3 -> 1/F1 : 1/F2 : 1/F3

F1= f2*f3
______________________
GCD(f1*f2,f1*f3,f2*f3)

Excel Notation: = (f2*f3)/(GCD(f1*f2,f1*f3,f2*f3)

F2= f1*f3
______________________
GCD(f1*f2,f1*f3,f2*f3)

Excel Notation: = (f1*f3)/(GCD(f1*f2,f1*f3,f2*f3)

F3= f1*f2
______________________
GCD(f1*f2,f1*f3,f2*f3)

Excel Notation: = (f1*f2)/(GCD(f1*f2,f1*f3,f2*f3)

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

f1*F1 = f2*F2 = f3*F3 =
The distance between harmonic fundamental and subharmonic
fundamental (WOW: between greatest common subharmonic and
least common harmonic) for multiads
is equivalent to complexity n*d for diads?

Peter Mulkers
Nijverheidsstraat 80
B-2840 Rumst
Belgium
P.Mulkers@GMX.net

Peter Mulkers wrote,

>f1*F1 = f2*F2 = f3*F3 =
>The distance between harmonic fundamental and subharmonic
>fundamental (WOW: between greatest common subharmonic and
>least common harmonic) for multiads
>is equivalent to complexity n*d for diads?

That's right, and many (such as Marion on this list) have come up with this
before. However, it's not a great measure of dissonance for multiads; e.g.,
the 4:5:6:7:9 and 1/(4:5:6:7:9) chords that Daniel Wolf recently posted
clearly differ in their level of dissonance, but have the same distance
between harmonic fundamental and guide tone (guide tone is Fokker's term for
what you call "subharmonic fundamental").

Paul!
Such things are basic properties behind all the CPS structures. 85% of all post on this
list are things that have been said before, even before this list existed!

"Paul H. Erlich" wrote:

>
>
> That's right, and many (such as Marion on this list) have come up with this
> before.

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

Yes, I think the guide tone divided by fundamental was a measure used as
early as Euler, if not earlier.

Paul!
Yes! but it was unfortunate that he failed to include 1 in his elements which caused him
to overlook the hexany when it was already contained inside one of his structures. By omitting
the 1 as a true element of a set, i am not sure how he would have seen the formula I quoted
unless he saw the subharmonic version in the 3 5 7 where 1 would not be nessicery.

"Paul H. Erlich" wrote:

> Yes, I think the guide tone divided by fundamental was a measure used as
> early as Euler, if not earlier.
>
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-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

Kraig,

We're on completely different wavelengths here. I don't know which formula
you're referring to. I was just referring to the formula Peter Mulkers
brought up.

In my opinion, Euler not seeing the hexany was simply an issue of Euler
using the rectangular lattice vs. Wilson using the triangular lattice.

-Paul

Paul!
I believe his formula was a proof/example of the one I posted to him!

"Paul H. Erlich" wrote:

> Kraig,
>
> We're on completely different wavelengths here. I don't know which formula
> you're referring to. I was just referring to the formula Peter Mulkers
> brought up.
>
> In my opinion, Euler not seeing the hexany was simply an issue of Euler
> using the rectangular lattice vs. Wilson using the triangular lattice.
>
> -Paul
>
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>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
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-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

Kraig,

Well, if it's the same, my comment stands. I'd even go further. The formula
in itself is extremely favorable to Euler's rectangular orientation, since
the formula gives a major seventh chord the same complexity as a major triad
or minor triad. However, on the triangular lattice, on which the hexany
emerges as a very compact region, the major seventh's true dissonance is
better represented

-Paul

Paul!
It is my understanding that the Euler developed the generalized lattice. 5s running
vertically, 3s horizontally and the seven diagonally. It was this very lattice that Wilson
first spotted the hexany which was visible because of the factors. The hexany lattice came
afterwards. Wilson development of Lattices is in most cases after the fact of the structure he
is illustrating. The lattice is developed to illustrate what he wants the viewer to see or
what aspect he wants to bring out. Dallesandro shows many different kind of lattices he has
developed.

"Paul H. Erlich" wrote:

>
> In my opinion, Euler not seeing the hexany was simply an issue of Euler
> using the rectangular lattice vs. Wilson using the triangular lattice.
>
> -Paul
>
>

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

OK Kraig -- points well taken. I just note that in my view, what is truly
special about the hexany is the number of 7-limit consonant intervals it
contains (12) for the number of notes it contains (6). This feature comes
out most clearly on the 3-d triangular lattice, otherwise known as the
octahedral-tetrahedral (or oct-tet) lattice, which shows each consonance as
a connecting line.