A little utonal stuff

I've recently found the utonal (undertone) series very interesting for
a kind of scale generation i have considered for a long time, but not
been able to develop it into something usable. It's all about using
the sum and difference tones of all combinations of a collection of
pitches, which can be seen as similar to Genus and CPS structures.
Except that difference tone scales always deal with absolute pitches,
that is they are not octave-repeating and must not be octave adjusted
in any way.

The utonal series is especialy usable for this because while S/D tones
(sum and difference tones) from any members of an overtone series
simply are members of the same overtone series, undertones produce
independent notes that lie outside of the original scale, but are
related by simple just ratios. (Probably an interesting subject for
lattices, which i am not concerned with myself).

I've here calculated such a scale out of the first 8 undertones (using
only the first 6 gives a fine result as well) of 4000 hz, which is an
arbitrary choice. (I realize that there are better such for
divisibility.) The "generating frequencies", rounded of to integers are:

4000 2000 1333 1000 800 667 571 500

Thus the two dimensional matrix of difference tones for them becomes:

0 2000 2666 3000 3200 3333 3428 3500
2000 0 666 1000 1200 1333 1428 1500
2666 666 0 333 533 666 761 833
3000 1000 333 0 200 333 428 500
3200 1200 533 200 0 133 228 300
3333 1333 666 333 133 0 95 166
3428 1428 761 428 228 95 0 71
3500 1500 833 500 300 166 71 0

And for sums:

8000 6000 5333 5000 4800 4666 4571 4500
6000 4000 3333 3000 2800 2666 2571 2500
5333 3333 2666 2333 2133 2000 1904 1833
5000 3000 2333 2000 1800 1666 1571 1500
4800 2800 2133 1800 1600 1466 1371 1300
4666 2666 2000 1666 1466 1333 1238 1166
4571 2571 1904 1571 1371 1238 1142 1071
4500 2500 1833 1500 1300 1166 1071 1000

Note that after having decided the generating frequency list, there
are NO divisions or multiplications involved. It is all about adding
and subtracting frequencies from each other.

Here's the full assortment of pitches, lined up in order:

71
95
133
166
200
228
300
333
428
500
533
666
761
833
1000
1071
1142
1166
1200
1238
1300
1333
1371
1428
1466
1500
1571
1600
1666
1800
1833
1904
2000
2133
2333
2500
2571
2666
2800
3000
3200
3333
3428
3500
4000
4500
4571
4666
4800
5000
5333
6000
8000

It could be turned into a .SCL file or such by adding /71 to each
number at the end.

Also, here's a "lined up" S/D frequency list for the first SIX
undertones only:

133
200
333
533
666
1000
1200
1333
1466
1600
1666
1800
2000
2133
2333
2666
2800
3000
3200
3333
4000
4666
4800
5000
5333
6000
8000

These are only one aspect of the recent S/D experiments i've been
doing, others involve using "regular" repeating musical scales as a
S/D generator including pentatonic and diatonic subsets of 12TET.

Those are made via amplitude modulation only (interference tones)
though-producing pitch tables of such irrational values is much more
complex than the lists above, and it would be a big help to be able to
convert the frequency ratios to cents which cannot be done with any
applications that i have as it requires an inverse logarithmic function.

/Ö

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

> and it would be a big help to be able to
> convert the frequency ratios to cents which cannot be done with any
> applications that i have as it requires an inverse logarithmic
>function.

inverse logarithmic function? converting ratios to cents only
requires a plain old logarithmic function (of any base):

cents = log(ratio)/log(2)*1200

Hi Mats,

You can construct these scales with Scala too, using
the ADD/SUMMATION and SUBTRACT/DIFFERENCE commands.
To combine the results, you can use MERGE.

>Those are made via amplitude modulation only (interference tones)
>though-producing pitch tables of such irrational values is much more
>complex than the lists above, and it would be a big help to be able to
>convert the frequency ratios to cents which cannot be done with any
>applications that i have as it requires an inverse logarithmic function.

Maybe the MODULATE command is what you're looking for.

Manuel

--- In tuning@y..., manuel.op.de.coul@e... wrote:
> Hi Mats,
>
> You can construct these scales with Scala too, using
> the ADD/SUMMATION and SUBTRACT/DIFFERENCE commands.
> To combine the results, you can use MERGE.

also, mats, don't forget that in reality, cubic difference tones are
usually louder than the quadratic ones you're calculating, and also
that many other-order difference tones are louder than the loudest
summation tones.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:
> also, mats, don't forget that in reality, cubic difference tones are
> usually louder than the quadratic ones you're calculating, and also
> that many other-order difference tones are louder than the loudest
> summation tones.

Sounds like an implementation detail in the mixdown process to me!

Cheers,
Jon

> > and it would be a big help to be able to
> > convert the frequency ratios to cents which cannot be done with any
> > applications that i have as it requires an inverse logarithmic
> >function.
>
> inverse logarithmic function? converting ratios to cents only
> requires a plain old logarithmic function (of any base):
>
> cents = log(ratio)/log(2)*1200

I know that, but it's very unintuitive to apply in your head while
you're playing an instrument or composing. The complexity lies in that
each 12TET (for example) _interval_ has a certain S/D tone pair with
it, that is transposed when the whole interval is tranposed. Using
only the notes within one octave there are (theoretically) 144 of sum
and difference tones EACH, so _a table of any sort is too complex to
use practically_ . There is clearly a need for a intervallic
understanding (of what S/D tones are produced by each different
interval, relative to the notes in the interval).

(Hope somebody understands the problem here...)

/Ö

> also, mats, don't forget that in reality, cubic difference tones are
> usually louder than the quadratic ones you're calculating, and also
> that many other-order difference tones are louder than the loudest
> summation tones.

Well that is irrelevant for me, i am interested in S/D as a method of
generating scale structures, and the effect those scales have when
being played in harmony (that is, of the generating pitches revealing
as an aural illusion, and the peculiar sound effect of "common
difference tone chords"), not of analyzing the difference tones
(seemingly) produced by "regular" playing of "regular" scales. Though
i do want to use 12TET instruments and melodies as a part of this
music, which is why i'm interested in that.

/Ö

--- In tuning@y..., Mats Öljare <oljare@h...> wrote:
>
> > > and it would be a big help to be able to
> > > convert the frequency ratios to cents which cannot be done with
any
> > > applications that i have as it requires an inverse logarithmic
> > >function.
> >
> > inverse logarithmic function? converting ratios to cents only
> > requires a plain old logarithmic function (of any base):
> >
> > cents = log(ratio)/log(2)*1200
>
> I know that, but it's very unintuitive to apply in your head while
> you're playing an instrument or composing.

i thought you were talking about *computer* applications that you
have. my bad. what sort of applications do you have that you run in
your hear while playing an instrument or composing? i'm confused.

> The complexity lies in that
> each 12TET (for example) _interval_ has a certain S/D tone pair with
> it, that is transposed when the whole interval is tranposed. Using
> only the notes within one octave there are (theoretically) 144 of
sum
> and difference tones EACH, so _a table of any sort is too complex to
> use practically_ . There is clearly a need for a intervallic
> understanding (of what S/D tones are produced by each different
> interval, relative to the notes in the interval).

yup. you can easily calculate the *intervals* formed by the various
combinational tones with each 12-equal dyad. then you'll have far
fewer than 144 things to remember.

> (Hope somebody understands the problem here...)

somehow, i think i'm still missing something . . . maybe not . . .

>

Hello Mats!
the scales of Mt. Meru fall into this catagory
see http://www.anaphoria.com/MERU.PDF
basically they are like recurrent sequences like the fibonacci series
the series is self generating and self referential in terms of s/d

>
> From: Mats �ljare <oljare@hotmail.com>
>
> Well that is irrelevant for me, i am interested in S/D as a method of
> generating scale structures, and the effect those scales have when
> being played in harmony (that is, of the generating pitches revealing
> as an aural illusion, and the peculiar sound effect of "common
> difference tone chords"), not of analyzing the difference tones
> (seemingly) produced by "regular" playing of "regular" scales. Though
> i do want to use 12TET instruments and melodies as a part of this
> music, which is why i'm interested in that.
>
> /�
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM 8-9PM PST

> > I know that, but it's very unintuitive to apply in your head while
> > you're playing an instrument or composing.
>
> i thought you were talking about *computer* applications that you
> have. my bad. what sort of applications do you have that you run in
> your hear while playing an instrument or composing? i'm confused.

Well i'm not really planning to use anything but computer synthesis
with sequencers or keyboard controller, but for "mental" and
compositional purposes, it would be very useful to have a more
musically sensible chart of it.

> > use practically_ . There is clearly a need for a intervallic
> > understanding (of what S/D tones are produced by each different
> > interval, relative to the notes in the interval).
>
> yup. you can easily calculate the *intervals* formed by the various
> combinational tones with each 12-equal dyad. then you'll have far
> fewer than 144 things to remember.

So how do i calculate them?

/Ö

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

> > yup. you can easily calculate the *intervals* formed by the
various
> > combinational tones with each 12-equal dyad. then you'll have far
> > fewer than 144 things to remember.
>
> So how do i calculate them?

in general, the interval formed by the quadratic difference tone
relative to the lower note of a 12-equal dyad (say, n degrees of 12-
equal) would be, in cents,

log(2^(n/12)-1)/log(2)*1200

easy enough to generalize this for cubic difference tones, summation
tones, etc. . . .