Inverted Difference Tones-any precedents??

A while ago i posted about a number of pitch sets consisting of the
sum and/or difference tones of a subharmonic series. A harmonic series
did not work, as the S/D tones are just from the same harmonic series.
But later i realized that the original subharmonic-based scales can
simply be inverted, for a scale more related to the overtone series.

This can in fact be described as the "inverted difference tones" of
the harmonic series. In fact every possible pair of two pitches, has
besides sum and difference tones, the inverted counterparts. The
functions, with X and Y being the two frequencies, are:

Inverted difference= 1/((1/X)-(1/Y)))
Inverted sum= 1/((1/x)+(1/y))

If they are to be implemented in a program, the absolute values must
be used. And watch out for zero difference(division by zero)!

I made a simple graph to show how they change with the distance
between two pitches, along with regular sum and difference for
comparison. It is in the "Oljare" folder of the files section, along
with the QBASIC script used to generate it.

Now what i want to know is if there is any precedent to the concept of
"inverted difference", or do i actually have something original here?

/Mats Öljare

On Monday 25 August 2003 21:59, Mats Öljare wrote:
> A while ago i posted about a number of pitch sets consisting of the
> sum and/or difference tones of a subharmonic series. A harmonic series

I am sorry, but I do not know what you mean by subharmonic series, am I
missing something here?

--- In tuning@yahoogroups.com, Mats Öljare <oljare@h...> wrote:

> Now what i want to know is if there is any precedent to the concept
of
> "inverted difference", or do i actually have something original
here?

since what you're doing is actually taking differences of *string
lengths* (and of periods, wavelengths, etc)., there may be a
precedent in pre-modern numerical dabblings with musical scales,
which were never based on frequencies.

--- In tuning@yahoogroups.com, Carlos <garciasuarez@y...> wrote:
> On Monday 25 August 2003 21:59, Mats Öljare wrote:
> > A while ago i posted about a number of pitch sets consisting of
the
> > sum and/or difference tones of a subharmonic series. A harmonic
series
>
>
> I am sorry, but I do not know what you mean by subharmonic series,
am I
> missing something here?

a subharmonic series is the inversion of the harmonic series; it's
also an arithmetic series in string lengths. according to the
conventional representation in terms of frequencies, it's
1:1/2:1/3:1/4:1/5:1/6:1/7 . . . . here's a subharmonic series from C4:

C4
C3
F3 -2 cents
C2
Ab1 +14 cents
F1 -2 cents
D1 + 31 cents
C1
Bb0 -4 cents
Ab0 +14 cents
Gb0 +49 cents
F0 -2 cents
etc.

> since what you're doing is actually taking differences of *string
> lengths* (and of periods, wavelengths, etc)., there may be a

That's true... i actually didn't think of it that way at all. /Ö

--- In tuning@yahoogroups.com, Mats Öljare <oljare@h...> wrote:
> Now what i want to know is if there is any precedent to the
concept of
> "inverted difference", or do i actually have something original
here?
>
> /Mats Öljare

The inversion of the first order difference tone for any interval is
the product tone of that interval. An interval can be viewed as
'hanging in space' symmetrically between these two tones.

Aaron

--- In tuning@yahoogroups.com, "pitchcolor" <Pitchcolor@a...> wrote:
> --- In tuning@yahoogroups.com, Mats Öljare <oljare@h...> wrote:
> > Now what i want to know is if there is any precedent to the
> concept of
> > "inverted difference", or do i actually have something original
> here?
> >
> > /Mats Öljare
>
> The inversion of the first order difference tone for any interval
is
> the product tone of that interval.

product tone . . . do you mean the thing known variously as the lcm
tone or lowest common harmonic or guide tone or subharmonic
fundamental? if so, how do you figure? 900 Hz and 700 Hz, the first
order difference tone is 200 Hz, mats' "inverted difference" is 1/
((1/700)+(1/900)) = 393.7500 Hz. meanwhile, the difference tone
inverted around the original interval would be 900*700/200=3150,
while the lcm tone is twice that, at 6300 Hz (shades of the MTS
discussion . . . :)

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> > The inversion of the first order difference tone for any interval
> is the product tone of that interval
> product tone . . .
> do you mean the thing known variously as the lcm
> tone or lowest common harmonic or guide tone or
subharmonic
> fundamental?

Yes, if you prefer to call it by another name. 'Product tone' is the
simplest term in my opinion. As a 'combination tone' it is
consistent with the language of 'summation tone' and 'difference
tone'.

>if so, how do you figure?

Consider the interval 8:9

The difference is 9-8 = 1
The product is 9*8 = 72

In notation, if 8 is C4 and 9 is D4 (with C4 as middle C) and we
notate the difference and product in the correct register
according to their appearance in a harmonic series, then the
difference tone is C1 and the product tone is D7. We see that
these tones are _equidistant_ from the tones of the interval,
each being 3 octaves away in opposite directions. C4 - 3 = C1
and D4 + 3 = D7.

This is a symmetrical structure; we usually say that such
structures have an inversional property; hence, the product tone
is the inversion of the difference tone. It allows us to view the
interval as derived from either a harmonic or a subharmonic
series. Here we see that the difference tone is the root of a
harmonic series from an 'overtone' point of view, and the product
is the root of a subharmonic series from the 'undertone' point of
view.

(If the above does not make sense, try reversing the numbers in
the interval so that D4 is 8 and C4 is 9 and call the product 1 and
the difference tone 72 to see how it works from a subharmonic
point of view.)

This combination tone symmetry has been known for a very long
time. Though it is obviously _not_ the same as mat's 'inverted
difference tone', I suggested it as the 'precedent' mat may have
been curious to know about.

Aaron

--- In tuning@yahoogroups.com, "pitchcolor" <Pitchcolor@a...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > The inversion of the first order difference tone for any
interval
> > is the product tone of that interval
> > product tone . . .
> > do you mean the thing known variously as the lcm
> > tone or lowest common harmonic or guide tone or
> subharmonic
> > fundamental?
>
>
> Yes, if you prefer to call it by another name. 'Product tone' is
the
> simplest term in my opinion.

why? how is it the "product"?

> As a 'combination tone' it is
> consistent with the language of 'summation tone' and 'difference
> tone'.

but you can't multiply the frequencies, say 700*900, to get it.
that's why lcm is preferable.

> >if so, how do you figure?
>
>
> Consider the interval 8:9
>
> The difference is 9-8 = 1

it doesn't work, though, unless the difference is the gcf, as it is
in your example. try 700 and 900, or 7:9. the inversion of the
difference tone, 2, around this interval, would be 31.5, not the
product 63.

> Here we see that the difference tone is the root of a
> harmonic series from an 'overtone' point of view,

not true, unless the difference is 1.

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> > Here we see that the difference tone is the root of a
> > harmonic series from an 'overtone' point of view,
>
> not true, unless the difference is 1.

Of course. To see the symmetry we use _superparticular _ratios.
I simply suggest that this is in fact an inversional difference tone
'precedent'. Perhaps this was too elementary to bother
mentioning. Mats' work extends this idea of symmetry and I find it
quite interesting.

Aaron

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> > Here we see that the difference tone is the root of a
> > harmonic series from an 'overtone' point of view,
>
> not true, unless the difference is 1.

Of course. To see the symmetry we use _superparticular _ratios.
I simply suggest that this is in fact an inversional difference tone
'precedent'. Perhaps this was too elementary to bother
mentioning. Mats' work extends this idea of symmetry and I find it
quite interesting.

Aaron

--- In tuning@yahoogroups.com, "pitchcolor" <Pitchcolor@a...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > Here we see that the difference tone is the root of a
> > > harmonic series from an 'overtone' point of view,
> >
> > not true, unless the difference is 1.
>
> Of course. To see the symmetry we use _superparticular _ratios.

or, to see the symmetry for *any* ratios, instead of the first-order
difference tone use the gcf note, or the '1' note as you'd put it.

> Perhaps this was too elementary to bother
> mentioning.

on the contrary, this can be a very confusing subject.