overtones undertones and equaltones

By walking an undertone series into an overtone series, or vice versa,
one can create a mathematical condition that involves passing the
series through an equal zone -- i.e., an "equal series".

Where "n" is the cardinality, or distinct number of notes in a series,
and "x" is any given number, a U to O series is defined as n*x with a
sequential numerator rule of +(x-2) and a sequential denominator rule
of -1.

Letting x = 2 gives the corresponding "n"-undertone series, and
incrementally increasing the value of "x" by rationals works "n"
towards ever more accurate approximations of its corresponding
overtone series.

Letting x = sqrt(2)+2 results in a maximum underlying errors for any
given fraction of "n" that is only ~2�. This would be
indistinguishable from a corresponding n-tET in all but the most
charitable of conditions.

Relaxing x = sqrt(2)+2 so that x = 3.5 allows for the simplest
rational (or RI) interpretation of an equaltone series.

Resetting the 1:2^(1/2) as 12:17 essentially creates an RI
well-temperament relative to its corresponding n-tET.

Resetting the 1:2^(1/2) as say 5:7 or 7:10, by allowing x = 3 and x =
4, creates a condition that is more like some very colorful rational
recasting (or distortion) of the corresponding n-tET, and could
perhaps be said to represent a sort of interpretational extreme at the
two U/O borders.

Here's the equaltone series where x = 3.5 and n = 12:

1/1 87/82 9/8 31/26 24/19 99/74 17/12 3/2 27/17 37/22 57/32 117/62 2/1

Here's the 12-equaltone rotations in rounded cents:

0 102 204 305 404 504 603 702 801 900 999 1099 1200
0 101 202 302 401 501 599 698 798 897 997 1098 1200
0 101 201 300 399 498 597 696 796 895 996 1099 1200
0 100 199 298 397 496 596 695 795 895 998 1099 1200
0 99 199 298 396 496 595 695 796 898 999 1100 1200
0 99 198 297 396 496 596 696 799 900 1001 1101 1200
0 99 198 297 396 496 597 699 801 902 1001 1101 1200
0 99 198 298 397 498 601 702 803 902 1002 1101 1200
0 99 199 298 399 502 603 704 804 903 1002 1101 1200
0 99 199 300 402 504 604 704 804 903 1002 1101 1200
0 100 201 303 404 505 605 704 804 902 1001 1101 1200
0 101 203 305 405 505 604 704 803 902 1001 1100 1200

By incrementally working the value of x further away from the
sqrt(2)+2 you can increase the sense of rotational (or key) color from
that of the corresponding n-tET.

However, it is the actual blurring between and joining of logarithmic
and over/under conceptions that most interests me here, as this
conceptual framework is very much "in tune" with the way that I like
to go about things when it comes to tuning, numbers, art and music.

--Dan Stearns

Hi Dan,

This looks really interesting.

Can you explain what you mean by a sequential numerator rule of +(x-2) and a sequential
denominator rule
of -1?

> Here's the equaltone series where x = 3.5 and n = 12:

> 1/1 87/82 9/8 31/26 24/19 99/74 17/12 3/2 27/17 37/22 57/32 117/62 2/1

How, for example, do you get 87/82 (= 3*29/(2*41))
from 3.5, 12, +(x-2) and -1?

(or any of the other numbers, whichever is easiest to explain).

What order are the ratios being generated? Left to right, or by "fifths",
or some other order?

Probably seems obvious to you, but I'm not quite seeing it yet,
so hope you can explain,

Robert

Robert Walker wrote,

<<Can you explain what you mean by a sequential numerator rule of
+(x-2) and a sequential denominator rule of -1?>>

Let's say n = 7 and x = 2. The series would start with the ratio of
14/14, or n*x. So "a sequential numerator rule of +(x-2) and a
sequential denominator rule of -1" would give a second ratio of 14/13
as (n*x)+(x-2) = 14 and (n*x)-1 = 13 here... so if x = 2 and the above
process is carried all the way out to the octave, then the resulting
n-series is an n-undertone series. Incrementally increasing the value
of x by rationals works n towards ever more accurate approximations of
its corresponding n-overtone series.

The interesting point of convergence here is x = sqrt(2)+2 and its
simplest RI interpretation x = 3.5 -- the equaltone series.
(Amazingly, when x = sqrt(2)+2 the maximum difference between any
n-tET interval and its corresponding n-series interval is ~2�!)

<<How, for example, do you get 87/82 (= 3*29/(2*41))
from 3.5, 12, +(x-2) and -1?>>

n = 12 and x = 3.5, so our series without any GCD reduction is:

42/42, 43.5/41, 45/40, 46.5/39, 48/38, 49.5/37, 51/36, 52.5/35, 54/34,
55.5/33, 57/32, 58.5/31, 60/30

And 43.5/41 converts to the rational 87/82.

Hopefully this makes more sense now... ? Let me know.

thanks,

--Dan Stearns

Hi Dan,

I've found your undertone to overtone scales are great for making rather nice "fuzzy"
timbres.

To show the effect, here is a clip of a fractal tune, using bassoon, piano and marimba
fuzzy timbres and glockenspiel (normal)

http://homepage.ntlworld.com/robertwalker/fts/uo_timbre_experiment.mid
(255 Kb)

Uses the idea of your series, but with two octave range instead of 1.

Scale is one of them:
1/1 132/101 81/50 64/33 111/49 252/97 47/16 312/95 171/47 4/1
also used for the bassoon and piano partials.

The marimba partials use:
1/1 14/11 8/5 2/1 5/2 22/7 4/1

Got from x=4, notes per octave = 2, octave = 4,
and start with denom and denum =
notes*(octave-2)+notes*x
i.e. 2*(4-2) + 2*4
= 12

then use your rule of adding x-2 to denom and subtracting 1 from denum:
12/12, 14/11, 16/10, 18/9, 20/8, 22/7, 24/6.

For 2 octave range, choice of x=4 gives equal first and last steps, for any value for the
number of notes.

Made the timbres with vols. of partials decreasing linearly as you go up the
under-over-tone "harmonic
series"

The scale itself uses x=32, notes = 3, octave = 4, and so has unequal first and last
steps.

Robert

Robert Walker wrote,

>Scale is one of them:
>1/1 132/101 81/50 64/33 111/49 252/97 47/16 312/95 171/47 4/1
>also used for the bassoon and piano partials.

Robert - let me be clear on this -- you're manipulating the instruments'
partials using MIDI??? I had no idea MIDI files had this capability!! Does
it only work with certain soundcards?

Hi Paul,

>Robert - let me be clear on this -- you're manipulating the instruments'
>partials using MIDI??? I had no idea MIDI files had this capability!! Does
>it only work with certain soundcards?

Hopefully they sound like partials of a single instrument, yes. In fact,
I'm very pleased that they do.

However, what I'm actually doing is playing the partials as separate MIDI
notes.

E.g.
10 bassoons playing
1/1 132/101 81/50 64/33 111/49 252/97 47/16 312/95 171/47 4/1
simultaneously, (volume decreasing linearly as you go up the scale,
like real partials)

If you did it with real bassoons playing those notes simultaneously,
mightn't sound so much like a new timbre, but would be interesting to see
if it could be done with really fine players, who could get close to this
level of pitch resolution for inharmonic partials.

E.g. with bell sound, maybe one way to do it live would be to play the bell,
then voices come in singing all its inharmonic partials, which the singers can
pick up by ear, then mute the bell, the voices continue singing, then stop,
then all sound together again as a new note.

Would it sound like a bell?

Probably partly got the idea from a comment to TL some time
back (don't remember exactly, maybe while searching for something
else) about exactly pitched j.i. chords sounding more like timbres on
some soundcards, and some discussion of that.

So should work on any soundcard or synth with reasonably fine pitch
resolution.

Glad that it does sound like a timbre - proves the technique works!

Robert

Hi Robert.

Partials are actually sine waves -- so in general this bassoon approach
might not work if what you're trying to do is investigate inharmonic timbres
in general. Bassoons of course have upper partials of their own, and quite
prominent ones, so an inharmonic chord of bassoons would have lots of
beating and roughness, unlike an inharmonic chord of sine waves (AKA an
inharmonic timbre).

I think Joe Monzo did something where he used the "ocarina" timbres as
partials, since that was as close to a sine wave as he could find among the
usual MIDI instruments.

-Paul

Hi Paul,

>Partials are actually sine waves -- so in general this bassoon approach
>might not work if what you're trying to do is investigate inharmonic timbres
>in general. Bassoons of course have upper partials of their own, and quite
>prominent ones, so an inharmonic chord of bassoons would have lots of
>beating and roughness, unlike an inharmonic chord of sine waves (AKA an
>inharmonic timbre).

Yes, I know that. For instance, the ex. clip had bassoon, and piano playing
the partials, to quite different effect. I'm especially interested in the way the
instruments playing the partials interact with the inharmonic timbre they are playing,
to make something completely new.

For instance, you get an interesting result with pianos playing bell partials,
still somewhat piano like, but also sounding like a bell.

With Dan's scale, effect is that it sounds a little bit like the original piano / bassoon
/ marimba, but with a pleasant kind of fuzzy effect to it that transforms
it into a new timbre.

The bassoon sounds most like a new instrument perhaps. Perhaps one wouldn't
guess it was a bassoon if one didn't know, while I think one might guess the
piano and marimba.

>I think Joe Monzo did something where he used the "ocarina" timbres as
>partials, since that was as close to a sine wave as he could find among the
>usual MIDI instruments.

Sound like a good idea if one wanted to mimic the inharmonic sound as exactly
as possible.

Probably will sound most like the original instrument when the voices are
sinusoidal, except for attack, where one needs to try various Midi voices, and
see which ones combine to make an attack most like the original

Perhaps another approach would be to try several MIDI voices for the partials,
with the ones for the attack perhaps playing the fundamental and first few overtones.

Another idea to try: play some instrument on the fundamental, then add extra
inharmonic partials to it using ocarina, and see what happens.

Anyway, lots of things to try out, and getting reasonably close to finishing it,
after which anyone can have a go!

Robert

Robert,

Well I guess my point was that I wouldn't call a chord played by a bunch of
bassoons a "timbre" unless it was a perfect simple-integer otonal chord,
since that would actually sound like a timbre. Otherwise it just sounds like
a bunch of bassoons.

-Paul

Hi Paul,

What do you think of the MIDI clip then?

Maybe in theory you think it shouldn't sound like a new
timbre, but in practice it does, which may need some explanation
by theorists perhaps.

I think perhaps it is related to the way composers mimic certain
sounds, like sound of bells, using various orchestrations, or using
chords on the piano.

Even with ET notes for the partials, you can get it to some extent.
Viz, Debussy creating bell like sounds in his piano works.

Robert

Robert, let me clarify further.

You wrote,

>Scale is one of them:
>1/1 132/101 81/50 64/33 111/49 252/97 47/16 312/95 171/47 4/1
>also used for the bassoon and piano partials.

This would imply that the partials of the bassoon and piano were tuned to
these proportions, rather than to a harmonic series. But as you've
clarified, that's not what you meant.

>What do you think of the MIDI clip then?

I hear 2 bassoons playing an out-of-tune fifth at three different pitch
levels, and a lot of interesting glockenspiel and percussion in the
background . . .

>Maybe in theory you think it shouldn't sound like a new
>timbre, but in practice it does, which may need some explanation
>by theorists perhaps.

What sounds like a new timbre?

Hi Paul,

Sorry, yes, I see what has happened.

Perhaps it wasn't explained as clearly as I desired originally. That was what
I meant to say. Follows on from thread of the recorder and flute bell harmonics.

Hopefully it is clear now.

Robert

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

/tuning/topicId_18227.html#18391

> I think Joe Monzo did something where he used the "ocarina"
> timbres as partials, since that was as close to a sine wave
> as he could find among the usual MIDI instruments.

Yes, Paul... I sent this to you privately about two years ago.

Paul is talking about a MIDI-file of the "Good Times" TV theme
song, which I obtained as output after running the .wav file
of "Good Times" thru WIDI .wav-to-MIDI conversions software.

Here it is, for all to hear:

/tuning/files/monz/gdtimes2.mid

I found it truly amazing (and all of you will, too!) that
the MIDI-file, especially with the nearly "pure" ocarina timbre,
sounded so much like the .wav-file that you can almost hear the
words!

The conversion gave approximations to the vowel formants and
consonant stops that are that accurate! And this is without
using any pitch-bend!!!

I've written to the author of this software (in Russia)
encouraging him to study some tuning theory and incorporate
pitch-bend into his conversion algorithm for even more accurate
results. AFAIK, the latest version still does not.

Search the List archives for my past posts on this, or search
the web for "WIDI" to find the actual page.

The connection is apparently not so great for Americans - it took
me about 15 attempts in the middle of the night to download the
WIDI software.

-monz
http://www.monz.org
"All roads lead to n^0"