planets, gravity, and Darren's one dimensional tuning space

There has recently been number of posts to this list about the dynamic
interactions between heavenly bodies and their orbits. I have also just
recently started digging around in Joe Monzo's tuning dictionary and found the
word _pain_ defined with reference to distance [of deviation from ideal]
squared. http://www.ixpres.com/interval/dict/pain.htm

My own theory or view of tuning has a bit in common with these two concepts.
It is mainly concerned with the fact that an arbitrary pitch seems to form a
recognisable interval with some other pitch, even when there is a substantial
deviation from a perfect/just interval. Compare this with the fact that any
object in nearby space is within the gravitational fields of Earth, our sun and
the moon; it is therefore under the influence of at least three heavenly
bodies. If this object is not involved in a stable orbit, one of either the
sun, moon or Earth will drag it towards itself depending on whose field
strength is strongest at the location of the object in question. So even
though the object was located at an arbitrary point, and not standing firmly
upon one of the heavenly bodies, it still formed a clear association with one
of them.

The gravitational attraction to a heavenly body (or anything for that matter)
is proportional to the mass of the body, and indirectly proportional to the
square of the distance away from the body.

I consider the realm of tuning as a one dimensional space where distance is
measured in cents, and the heavenly bodies are the rational intervals situated
at points along on a straight line. I regard the mass of each body as being
the number of common harmonics shared between the pitches forming the rational
interval, which is inversely proportional to the numerator; a perfect fourth
(4/3) has half the number of common harmonics as the interval of an octave
(2/1). (Others on this list have stated a preference for the geometric mean of
the numerator and denominator as a measure of an intervals consonance.)
Instead of measuring a force of gravitational attraction (in Newtons), I have a
unit that I have been calling _uncertainty_ which is equal to the numerator
when the interval is just, and gets higher according to the amount of pain.
Any arbitrary point along the line representing the space will have rational
intervals nearby which all engage in a gravity-like tug-of-war over that point.
The one having the lowest uncertainty wins, and that value becomes the measure
of uncertainty of that point. I have been scaling pain so that the point at
49.9 cents is claimed by unity but the point at 50.1 cents is pulled away by
another. I chose a switchover point of 50 cents because I _assume_ my ears'
pitch selectivity/fussiness/personal-taste/whatever-it's-called has been formed
by 12tET. Someone who is more fussy/selective would choose to have pain affect
uncertainty more strongly. I realise a carefully executed experiment would
provide me with a more honest pain scaling factor, but for now I am not willing
to sacrifice music/art/play time for psychology/science/research time. Maybe I
will later.

I am currently using a linear array of uncertainty values to compute the
comparative strength or deficiency of equal temperaments. I saw a plot of
step-per-octave vs deficiency by Wendy Carlos in Keyboard magazine [Nov 1986
page 65], but it only regarded maj and min chords with and without flattened
sevenths. I am interested in a more general indication of anything sweet that
a given n-tET can offer. What I have seen so far seems to support the method
since all the obvious ones stand out: 12, 19, 22, 31, 34, 53tET, and also
88CET. (I am particularly glad that last one stood out as I quite like the
sound of Gary Morrison's contribution to the list CD.)

If you plot uncertainty on a graph where my one dimensional space is the
horizontal axis, you will see a series of convex parabolas (positive
curvature). Some are low and wide, many are high and narrow. You have to get
pretty close to the narrow ones to be caught in their fields, but the wide
ones have huge fields that are difficult to get away from. Roughly translated
this means that you can be a fair way off 702 cents but despite the pain still
be convinced that it is a fifth, but get a relatively small distance away from
316 cents and you end up leaving minor third space and being captured by some
other interval.

I am now contemplating using the series of parabolas as a means of selecting
rational intervals to form scales. If you picture them as buckets catching
objects dropped from above, the objects will settle at rational interval
points. Imagine 13 equally spaced objects (representing 12 steps) falling into
an octave wide series of parabolic buckets, you would be provided with 12
pitches to tune a synth to. If the pain scaling factor used to produce the
graph is altered so that the "buckets" are wider, there will be less of them as
the more obscure ratios are overcome by the stronger ones. You would still get
the same number of pitches (assuming the buckets are not ridiculously large)
but they will be more unevenly spaced and give more normal sounding chords. Of
course you don't have to drop 13 objects but any number you please. Just space
them evenly.

I welcome any comments and/or criticisms.

DARREN McDOUGALL
Australia

Darren, your scheme sounds very similar to Paul E's concept of Harmonic
Entropy. Both have parabolic-shaped "buckets", and jiggling the input
parameters can make some buckets swallow others.

Because of complaints on this list about length of posts, a separate
H.E. list was formed a few months ago. You can find archives at:

http://www.egroups.com/archive/harmonic_entropy

I may be guilty of introducing the word "pain" to the discussion. The
words energy and entropy are equivalent, depending upon the particular
model in use. I may also be the source of Monz's reference to pain
being proportional to square of deviation from ideal. More about this
can be found at:

http://www.egroups.com/message/tuning/7890
http://www.egroups.com/message/tuning/12668

A question about your model: do you intend it to be used to create a
static scale, or to be used in dynamic (adaptive) tuning as a piece
modulates on the fly?

JdL

--- In tuning@egroups.com, "John A. deLaubenfels" <jdl@a...> wrote:
>
> I may be guilty of introducing the word "pain" to the discussion.
The
> words energy and entropy are equivalent, depending upon the
particular
> model in use. I may also be the source of Monz's reference to pain
> being proportional to square of deviation from ideal. More about
this
> can be found at:
>

And now for more of our "Probing The Masters" series:

John and Monz,

To enhance my understanding of "Pain", I would like to state that
what I have always took you to mean with this, is that it is the
level of dissonance discomfort "being proportional to square of
deviation from ideal", when harmonic timbres are under consideration.

I'm wondering if you may have ever explored the possibilities of
transposing this theory onto inharmonic timbres? It has been my
direct experience that on many inharmonic timbres, what is "ideal"
for the tuning of harmonic timbres, can become meaningless. For
instance inharmonic fifths, found in the timbre itself, which are
stretched or compressed, can sound completely resonant, where a
harmonic (JI) or ET will sound completely discordant - even wrong for
these instruments. I submit there is another rule that must apply to
this with regard to "Pain". One composition I'm working on now is
based on the spectrum of a Gong (with a Pelog-like tuning), and has a
quarter tone sharp fifth, yet sounds infinitely more consonant with
this wide fifth, than if I tune it to JI or ET (which I always like
to try, just to hear the difference).

Since harmonic instruments represent a minority on the World Stage of
music, and true harmonic timbres are very rare in nature (but can
abound in electronics), do you think that Adaptive concepts may be
transposed to the inharmonic majority, and achieve blisses of
metallic consonance never before conceived? Looks like timbral
analysis would have to become an integral part of the process.

Thanks,

Jacky Ligon

P.S. Sorry for being a Pain. : )

[Jacky Ligon wrote:]
>And now for more of our "Probing The Masters" series:
>
>John and Monz,
>
>To enhance my understanding of "Pain", I would like to state that
>what I have always took you to mean with this, is that it is the
>level of dissonance discomfort "being proportional to square of
>deviation from ideal", when harmonic timbres are under consideration.
>
>I'm wondering if you may have ever explored the possibilities of
>transposing this theory onto inharmonic timbres? It has been my
>direct experience that on many inharmonic timbres, what is "ideal"
>for the tuning of harmonic timbres, can become meaningless. For
>instance inharmonic fifths, found in the timbre itself, which are
>stretched or compressed, can sound completely resonant, where a
>harmonic (JI) or ET will sound completely discordant - even wrong for
>these instruments. I submit there is another rule that must apply to
>this with regard to "Pain". One composition I'm working on now is
>based on the spectrum of a Gong (with a Pelog-like tuning), and has a
>quarter tone sharp fifth, yet sounds infinitely more consonant with
>this wide fifth, than if I tune it to JI or ET (which I always like
>to try, just to hear the difference).
>
>Since harmonic instruments represent a minority on the World Stage of
>music, and true harmonic timbres are very rare in nature (but can
>abound in electronics), do you think that Adaptive concepts may be
>transposed to the inharmonic majority, and achieve blisses of
>metallic consonance never before conceived? Looks like timbral
>analysis would have to become an integral part of the process.

Jacky, you're quite right. I generally assume harmonic timbres, and all
the tuning I have done to date presupposes them. Bill Sethares is the
expert for the more general case, and, last I heard, he's on this list.
See, for example, my praise for his book, "Tuning, Timbre, Spectrum,
Scale" in:

http://www.egroups.com/message/tuning/5132

where, in the accompanying CD, he presents an inharmonic tuning, for
which the rules for harmony are completely different! (the sensation
alone is worth the price of the book, howbeit we're talking 50+ bucks).

In the general case, what's important is to line up the partials, and
when they are inharmonic, non-integer ratios best achieve that
alignment.

>P.S. Sorry for being a Pain. : )

Call me a "master", and you can never be a "pain"! ;->

JdL

[I wrote:]
>>See, for example, my praise for his book, "Tuning, Timbre, Spectrum,
>>Scale" in:
>>
>> http://www.egroups.com/message/tuning/5132
>>
>>where, in the accompanying CD, he presents an inharmonic tuning, for
>>which the rules for harmony are completely different! (the sensation
>>alone is worth the price of the book, howbeit we're talking 50+
>>bucks).

[Paul E:]
>Several of his pieces (including the piece in question, I believe) have
>been available on the Tuning List Tape Swap and the Tuning Punks .mp3
>page, and frankly I don't hear any new rules of harmony at work. In my
>personal opinion, tonal harmony is largely a result of the virtual
>pitch phenomenon acting on the fundamentals of the tones; the harmonics
>of the tones will either support the fundamentals, if they're nearly
>integer ratios thereof; or if not, they will sound like distinct tones
>of their own and "stick out" with their own melodic and harmonic
>implications rather than acting merely as elements of timbre.

Paul, either we're talking about completely different tracks or our
perception is completely different. I refer to tracks 2 through 5 of
Bill's CD accompanying his book, "Tuning, Timbre, Spectrum, Scale",
which compare the same small piece in four ways:

. Harmonic timbres in 12-tET
. Harmonic timbres in stretched scale
. Stretched timbres in stretched scale
. Stretched timbres in 12-tET

The first and third sound fine; the second and fourth sound awful! Yet,
to my ear, the stretched timbres don't of their own accord sound all
that strange. This stuff is a real ear-opener, IMHO!!

JdL

> I am interested in a more general indication of anything sweet that
> a given n-tET can offer. What I have seen so far seems to support the method
> since all the obvious ones stand out: 12, 19, 22, 31, 34, 53tET, and also
> 88CET. (I am particularly glad that last one stood out as I quite like the
> sound of Gary Morrison's contribution to the list CD.)

Thanks! I'm glad you liked it.

I haven't read your description here all the way through, but what you described sounds a
little bit like, speaking broad generalities, how I discovered 88CET originally. I described
that method in Xenharmonikon 13.

[I wrote:]
>>Paul, either we're talking about completely different tracks or our
>>perception is completely different. I refer to tracks 2 through 5 of
>>Bill's CD accompanying his book, "Tuning, Timbre, Spectrum, Scale",
>>which compare the same small piece in four ways:

>> . Harmonic timbres in 12-tET
>> . Harmonic timbres in stretched scale
>> . Stretched timbres in stretched scale
>> . Stretched timbres in 12-tET

>>The first and third sound fine; the second and fourth sound awful!
>>Yet, to my ear, the stretched timbres don't of their own accord sound
>>all that strange. This stuff is a real ear-opener, IMHO!!

[Paul E:]
>I would agree that if the amount of stretching is not too great, one
>certainly should match the tuning to the timbre. That doesn't
>contradict what I posted before . . .

Well, then I don't understand your earlier point at all. In any case,
we agree, do we not, that integer ratios are inapplicable for reducing
discordance in inharmonic timbres? These tracks of Bill's are VERY
unsubtle in illustrating this point.

JdL

--- In tuning@y..., ligonj@n... wrote:

/tuning/topicId_17841.html#17869

> John and Monz,
>
> To enhance my understanding of "Pain", I would like to state
> that what I have always took you to mean with this, is that
> it is the level of dissonance discomfort "being proportional
> to square of deviation from ideal", when harmonic timbres are
> under consideration.
>
> I'm wondering if you may have ever explored the possibilities of
> transposing this theory onto inharmonic timbres?

Hi Jacky,

While I'm flattered that you refer to me as a "master" (along
with John), I must defer questions about "pain" in tuning to
his expertise - the whole pain and adaptive tuning procedure
is his baby. I understand enough of it to have written the
definition that's in my Tuning Dictionary, but any more detailed
questions about it have to be answered by him, and perhaps
Paul Erlich, who was very much involved in the working-out
of the concept with John here on the Tuning List.

I will say however that the application of adaptive tuning
to inharmonic timbres is certainly a very fruitful area for
further research and experimentation, and agree with John
that Bill Sethares is probably the person who can speak most
authoritatively on that matter.

(I'm still trying to find the time to fully digest your post
on primes, so that I can give you an informed reply to that...
life has been *extremely* chaotic for me since returning home,
but it's all good and in time things will settle down and I'll
be able to concentrate more on tuning work.)

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

[I wrote:]
>>Well, then I don't understand your earlier point at all. In any case,
>>we agree, do we not, that integer ratios are inapplicable for reducing
>>discordance in inharmonic timbres?

[Paul E wrote:]
>Don't tell Harry Partch!

>Seriously, I think that they're inapplicable in the context of utonal
>chords and inharmonic timbres, but otonal chords will still have a very
>strong effect with inharmonic timbres, as we are wired to seek out
>harmonic series in everything we hear -- and in this situation we'd
>hear several. Sethares focuses on the critical band roughness component
>of discordance, while the virtual pitch ambiguity (harmonic entropy)
>and combination tone components are important as well.

OK, Paul, please tell me if the following math is correct. Bill's
examples use "2.1 stretched" octaves and/or timbres. I take this to
mean that, if a note's fundamental is at 100 Hz, its next higher
(in)harmonic will be at 210 Hz. That's 1284.47 cents instead of
1200.00, so the logarithmic stretch factor is 1.0704. Using this, the
first five (in)harmonics of a stretched scale (or timbre) starting at
100 Hz become:

100
210
324.12
441
559.98

And the differences between these frequencies, arranged in order, are:

110
114.12
116.88
118.98
224.12
231
235.86
341
349.98
459.98

What to make of these numbers? They would imply a fuzziness to any
"virtual fundamental", and, as far as I can tell, don't provide much
guidance in forming the best possible intervals between notes. On the
other hand, alignment of partials is easy and consistent; for example,
a major third would be stretched from 386.31 cents to 413.51 cents.
Starting from 100 Hz, that would be a note at 126.98 Hz; two stretched
"octaves" above that is 4.41 x 126.98 = 559.98, exactly in alignment
with the stretched partial above 100 Hz.

When I listen to the example that has both stretched timbres and
stretched scale, the alignment seems just right. I sure as HECK
wouldn't have guessed that there's something weird going on.

If using virtual fundamentals would help in making the alignment, I'm
not seeing it.

JdL