Fractal Scales

I was wondering if there were any known fractal sets that one could use to
generate a scale. Would this even work? Would it just sound like a dissonant
mess. From what I know about fractals, I imagine there'd be all sorts of
strange scales to be found inside of various fractal sets, but I imagine it'd
be a nightmare to find one that sounds somewhat appealing.

--
Christopher Miller
vogonpoet@mailandnews.com

I was wondering if there were any known fractal sets that one could use to
generate a scale. Would this even work? Would it just sound like a dissonant
mess. From what I know about fractals, I imagine there'd be all sorts of
strange scales to be found inside of various fractal sets, but I imagine it'd
be a nightmare to find one that sounds somewhat appealing.

--
Christopher Miller
vogonpoet@mailandnews.com

I was wondering if there were any known fractal sets that one could use to
generate a scale. Would this even work? Would it just sound like a dissonant
mess. From what I know about fractals, I imagine there'd be all sorts of
strange scales to be found inside of various fractal sets, but I imagine it'd
be a nightmare to find one that sounds somewhat appealing.

--
Christopher Miller
vogonpoet@mailandnews.com

I was wondering if there were any known fractal sets that one could use to
generate a scale. Would this even work? Would it just sound like a dissonant
mess. From what I know about fractals, I imagine there'd be all sorts of
strange scales to be found inside of various fractal sets, but I imagine it'd
be a nightmare to find one that sounds somewhat appealing.

--
Christopher Miller
vogonpoet@mailandnews.com

I was wondering if there were any known fractal sets that one could use to
generate a scale. Would this even work? Would it just sound like a dissonant
mess. From what I know about fractals, I imagine there'd be all sorts of
strange scales to be found inside of various fractal sets, but I imagine it'd
be a nightmare to find one that sounds somewhat appealing.

--
Christopher Miller
vogonpoet@mailandnews.com

I was wondering if there were any known fractal sets that one could use to
generate a scale. Would this even work? Would it just sound like a dissonant
mess. From what I know about fractals, I imagine there'd be all sorts of
strange scales to be found inside of various fractal sets, but I imagine it'd
be a nightmare to find one that sounds somewhat appealing.

--
Christopher Miller
vogonpoet@mailandnews.com

I was wondering if there were any known fractal sets that one could use to
generate a scale. Would this even work? Would it just sound like a dissonant
mess. From what I know about fractals, I imagine there'd be all sorts of
strange scales to be found inside of various fractal sets, but I imagine it'd
be a nightmare to find one that sounds somewhat appealing.

--
Christopher Miller
vogonpoet@mailandnews.com

I was wondering if there were any known fractal sets that one could use to
generate a scale. Would this even work? Would it just sound like a dissonant
mess. From what I know about fractals, I imagine there'd be all sorts of
strange scales to be found inside of various fractal sets, but I imagine it'd
be a nightmare to find one that sounds somewhat appealing.

--
Christopher Miller
vogonpoet@mailandnews.com

I was wondering if there were any known fractal sets that one could use to
generate a scale. Would this even work? Would it just sound like a dissonant
mess. From what I know about fractals, I imagine there'd be all sorts of
strange scales to be found inside of various fractal sets, but I imagine it'd
be a nightmare to find one that sounds somewhat appealing.

--
Christopher Miller
vogonpoet@mailandnews.com

I was wondering if there were any known fractal sets that one could use to
generate a scale. Would this even work? Would it just sound like a dissonant
mess. From what I know about fractals, I imagine there'd be all sorts of
strange scales to be found inside of various fractal sets, but I imagine it'd
be a nightmare to find one that sounds somewhat appealing.

--
Christopher Miller
vogonpoet@mailandnews.com

I was wondering if there were any known fractal sets that one could use to
generate a scale. Would this even work? Would it just sound like a dissonant
mess. From what I know about fractals, I imagine there'd be all sorts of
strange scales to be found inside of various fractal sets, but I imagine it'd
be a nightmare to find one that sounds somewhat appealing.

--
Christopher Miller
vogonpoet@mailandnews.com

I was wondering if there were any known fractal sets that one could use to
generate a scale. Would this even work? Would it just sound like a dissonant
mess. From what I know about fractals, I imagine there'd be all sorts of
strange scales to be found inside of various fractal sets, but I imagine it'd
be a nightmare to find one that sounds somewhat appealing.

--
Christopher Miller
vogonpoet@mailandnews.com

In a message dated 8/25/99 1:03:25 AM,

>From: Chris Miller <vogonpoet@MailAndNews.com>wrote:

>I was wondering if there were any known fractal sets that one could use to
>generate a scale. Would this even work? Would it just sound like a
dissonant
>mess. From what I know about fractals, I imagine there'd be all sorts of
>strange scales to be found inside of various fractal sets, but I imagine
it'd
>be a nightmare to find one that sounds somewhat appealing.

check out this website for fractal music (some the best around):

http://www-ks.rus.uni-stuttgart.de/people/schulz/fmusic/

AND this one on Chaos Theory based electronic music:

http://www.ccsr.uouc.edu/People/gmk/Papers/ChuaSndRef.html

have FUN...

Christopher Miller wrote,

>I was wondering if there were any known fractal sets that one could use to
>generate a scale. Would this even work? Would it just sound like a
dissonant
>mess. From what I know about fractals, I imagine there'd be all sorts of
>strange scales to be found inside of various fractal sets, but I imagine
it'd
>be a nightmare to find one that sounds somewhat appealing.

One (loosely speaking) fractal scale that sounds very nice (it is virtually
the optimal tuning for diatonic triadic music) is Kornerup's golden meantone
tuning. Of course a plain diatonic scale is not a fractal, but the tuning
(based on a chain of 696.21-cent fifths) can be extended indefinitely to
form a fractal-like structure, since every large interval is divided into
two smaller intervals according to the golden ratio:

The first three notes in the chain form a bare "tetrachordal" framework. The
next two form a pentatonic scale, and each of the fourths in the framework
is divided in the golden ratio into a minor third and a major second. The
next two notes form a diatonic scale, dividing each of the minor thirds in
the golden ratio into a major second and a minor second. The next five notes
form a chromatic scale, dividing each of the major seconds in the golden
ratio into a minor second and an augmented unison. The next seven notes form
a 19-tone scale, dividing each of the minor seconds in the golden ratio into
an augmented unison and a diminished second. And so on (through 31-, 50-,
81-, . . . tone scales) ad infinitum.

Thank you, this gives me something to do tonight, when I get home from work.
This sounds like a very useful scale, thanks. I'm still going to experiment
with fractal scales, see what I can find.

Thanks

>===== Original Message From tuning@onelist.com =====
>From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
>One (loosely speaking) fractal scale that sounds very nice (it is virtually
>the optimal tuning for diatonic triadic music) is Kornerup's golden meantone
>tuning. Of course a plain diatonic scale is not a fractal, but the tuning
>(based on a chain of 696.21-cent fifths) can be extended indefinitely to
>form a fractal-like structure, since every large interval is divided into
>two smaller intervals according to the golden ratio:
>
>The first three notes in the chain form a bare "tetrachordal" framework. The
>next two form a pentatonic scale, and each of the fourths in the framework
>is divided in the golden ratio into a minor third and a major second. The
>next two notes form a diatonic scale, dividing each of the minor thirds in
>the golden ratio into a major second and a minor second. The next five notes
>form a chromatic scale, dividing each of the major seconds in the golden
>ratio into a minor second and an augmented unison. The next seven notes form
>a 19-tone scale, dividing each of the minor seconds in the golden ratio into
>an augmented unison and a diminished second. And so on (through 31-, 50-,
>81-, . . . tone scales) ad infinitum.
>
>--------------------------- ONElist Sponsor ----------------------------
>
>ATTN ONELIST USERS: stay current on the latest activities,
>programs, & features at ONElist by joining our member newsletter at
><a href=" http://www.onelist.com/subscribe/onelist_announce ">Click</a>
>
>------------------------------------------------------------------------
>You do not need web access to participate. You may subscribe through
>email. Send an empty email to one of these addresses:
> tuning-subscribe@onelist.com - subscribe to the tuning list.
> tuning-unsubscribe@onelist.com - unsubscribe from the tuning list.
> tuning-digest@onelist.com - switch your subscription to digest mode.
> tuning-normal@onelist.com - switch your subscription to normal mode.

--
Christopher Miller
vogonpoet@mailandnews.com
--

Thank you, this gives me something to do tonight, when I get home from work.
This sounds like a very useful scale, thanks. I'm still going to experiment
with fractal scales, see what I can find.

Thanks

>===== Original Message From tuning@onelist.com =====
>From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
>One (loosely speaking) fractal scale that sounds very nice (it is virtually
>the optimal tuning for diatonic triadic music) is Kornerup's golden meantone
>tuning. Of course a plain diatonic scale is not a fractal, but the tuning
>(based on a chain of 696.21-cent fifths) can be extended indefinitely to
>form a fractal-like structure, since every large interval is divided into
>two smaller intervals according to the golden ratio:
>
>The first three notes in the chain form a bare "tetrachordal" framework. The
>next two form a pentatonic scale, and each of the fourths in the framework
>is divided in the golden ratio into a minor third and a major second. The
>next two notes form a diatonic scale, dividing each of the minor thirds in
>the golden ratio into a major second and a minor second. The next five notes
>form a chromatic scale, dividing each of the major seconds in the golden
>ratio into a minor second and an augmented unison. The next seven notes form
>a 19-tone scale, dividing each of the minor seconds in the golden ratio into
>an augmented unison and a diminished second. And so on (through 31-, 50-,
>81-, . . . tone scales) ad infinitum.
>
>--------------------------- ONElist Sponsor ----------------------------
>
>ATTN ONELIST USERS: stay current on the latest activities,
>programs, & features at ONElist by joining our member newsletter at
><a href=" http://www.onelist.com/subscribe/onelist_announce ">Click</a>
>
>------------------------------------------------------------------------
>You do not need web access to participate. You may subscribe through
>email. Send an empty email to one of these addresses:
> tuning-subscribe@onelist.com - subscribe to the tuning list.
> tuning-unsubscribe@onelist.com - unsubscribe from the tuning list.
> tuning-digest@onelist.com - switch your subscription to digest mode.
> tuning-normal@onelist.com - switch your subscription to normal mode.

--
Christopher Miller
vogonpoet@mailandnews.com
--

"Paul H. Erlich" <PErlich@Acadian-Asset.com> wrote:

> One (loosely speaking) fractal scale that sounds very nice (it is virtually
> the optimal tuning for diatonic triadic music) is Kornerup's golden meantone
> tuning.

Interesting. Have you a pointer to a web page or a ratio list, where I
could get formula for construction of such a tuning?

Thanks,
Rick

On Wed, 25 Aug 1999, Paul H. Erlich wrote:
> One (loosely speaking) fractal scale that sounds very nice (it is virtually
> the optimal tuning for diatonic triadic music) is Kornerup's golden meantone
> tuning. Of course a plain diatonic scale is not a fractal, but the tuning
> (based on a chain of 696.21-cent fifths) can be extended indefinitely to
> form a fractal-like structure, since every large interval is divided into
> two smaller intervals according to the golden ratio: [snip]

In a similar vein, here's an excerpt from a post of mine from several
years back:

: [snippitude] [...a tuning] in which the ratio of the octave to the
: fifth is equal to the ratio of the whole tone to the diatonic semitone.
: Mathematically, that ratio works out to sqrt(5/2)-1, or (as a continued
: fraction) 0;1,1,2. It can also therefore be represented by self-similar
: rectangles reduced by squares, thus:
:
: +-----+-----+---+------------------------+
: | | |XXX| |
: | | |XXX| |
: | | |XXX| |
: +-----+-----+---+ |
: | | |
: | | |
: | | |
: | | |
: | | |
: | | |
: | | |
: | | |
: +---------------+------------------------+
:
: In case your font is not proportioned as mine is, the X-filled
: rectangle is supposed to be similar to the large one; the other four
: spaces are supposed to be square. This is a visual representation of
: subtracting a fifth from an octave, then a fourth from a fifth, then two
: wholetones from a fourth.
:
: I think of this as my "recursive" meantone because wholetones are
: subdivided exactly as the octave is. To illustrate: if the 1/1 is
: consider position 0, three steps in the positive and negative directions
: around the spiral of fifths generate a dorian scale, thus:
:
: |-----------|------|-----------|-----------|-----------|------|-----------|
: 0 2 -3 -1 1 3 -2 0
:
: Now, if you consider what the next (nearest; smallest absolute-value)
: steps that fall between, say, -1 and 1 will be, they are:
:
: |-----------|------|-----------|-----------|-----------|------|-----------|
: -1 -13 18 6 -6 -18 13 1
:
: and the sizes of the intervals will be in the same ratios; the process
: can be continued infinitely. Unfortunately, as with Brian's schemes I
: can imagine no way to make these mathematical relationships actually
: audible in a composition.
:
: sqrt(5/2)-1 comes to about 0.5811388. This times 1200 cents/octave
: makes a fifth of 697.36660 cents. This is about two-thirds of the way
: from 1/4 comma to 1/5 comma meantone, or 3/14 comma. (It is also very
: well approximated by 74-tet.) In actual practice, I at least cannot
: distinguish this tuning from 1/5 comma meantone, so for yet another
: reason the whole house of cards comes crashing down. However, 1/5 comma
: meantone is one of my favorite tunings, so I don't mind too much. 8-)>

--pH <manynote@library.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
-\-\-- o

On Thu, 26 Aug 1999, I wrote:
> : [snippitude] [...a tuning] in which the ratio of the octave to the
> : fifth is equal to the ratio of the whole tone to the diatonic semitone.
> : Mathematically, that ratio works out to sqrt(5/2)-1, or (as a continued
> : fraction) 0;1,1,2. [snip]

A followup to that message that I always intended to make but never did
is that a positive equivalent of that scale, where the ratio of the
octave to the _fourth_ is equal to that of the whole tone to the
diatonic semitone (or closer to a limma in this case), can also be
calculated. The ratio works out to be sqrt(2)-1; in cents, 497.056...
cents; as a continued fraction, 0;2. The rectangles-&-squares diagram
comes out

+-----------+-----------+----+
| | |XXXX|
| | |XXXX|
| | |XXXX|
| | |XXXX|
| | |XXXX|
| | |XXXX|
+-----------+-----------+----+

, representing the subtraction of two fourths from an octave (the X-ed
rectangle being similar to the whole), and the zooming-in-on-the-dorian-
scale diagram looks like this:

|-----------|----|-----------|-----------|-----------|----|-----------|
0 2 -3 -1 1 3 -2 0

|-----------|----|-----------|-----------|-----------|----|-----------|
-1 11 -18 -6 6 18 -11 1

This fourth varies from just by a tad under a cent, so this tuning is
fairly close to Pythagorean tuning. Margo Schulter might like it. It
is reasonably approximated by 70TET.

--pH <manynote@library.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
-\-\-- o

I wrote,

>> One (loosely speaking) fractal scale that sounds very nice (it is
virtually
>> the optimal tuning for diatonic triadic music) is Kornerup's golden
meantone
>> tuning.

Rick McGowan wrote,

>Interesting. Have you a pointer to a web page or a ratio list, where I
>could get formula for construction of such a tuning?

Constructing the tuning is easy. Consider the pentatonic scale. The minor
third and major second are, as we know, in the golden ratio, and two minor
thirds plus three major second complete the octave. So we have the
equations:

t=s*g (where g is the golden ratio, (sqrt(5)+1)/2)
2*t+3*s=1200

solving the second equation for t

t=600-1.5*s

and plugging in to the first

600-1.5*s=s*g (now use g=1.618034)
600=s*3.118034
s=192.4289

So the perfect fifth is (1200+192.4289)/2 = 696.2145 cents.

The whole tuning is constructed by chaining this fifth over and over again.
For example, the 19 tones centered around D would be

D 0
D# 73.5013
Eb 118.9276
E 192.4289
E# 265.9303
F 311.3566
F# 384.8579
Gb 430.2842
G 503.7855
G# 577.2868
Ab 622.7132
A 696.2145
A# 769.7158
Bb 815.1421
B 888.6434
Cb 934.0697
C 1007.5711
C# 1081.0724
Db 1126.4987
(D 1200)

In-Reply-To: <935749240.7415@onelist.com>
Paul Hahn (Digest 294.10) wrote:

> A followup to that message that I always intended to make but never did
> is that a positive equivalent of that scale, where the ratio of the
> octave to the _fourth_ is equal to that of the whole tone to the
> diatonic semitone (or closer to a limma in this case), can also be
> calculated. The ratio works out to be sqrt(2)-1; in cents, 497.056...
> cents; as a continued fraction, 0;2.

This looks like a schismic temerament where 7/5 is almost just.

> and the zooming-in-on-the-dorian-
> scale diagram looks like this:
>
> |-----------|----|-----------|-----------|-----------|----|-----------|
> 0 2 -3 -1 1 3 -2 0
>
> |-----------|----|-----------|-----------|-----------|----|-----------|
> -1 11 -18 -6 6 18 -11 1

So this scale, like the negative equivalent, has true self-similarity.
The tuning I give at

http://www.cix.co.uk/~gbreed/schv12.htm

follows the same idea, in that the pattern of different-sized semitones
that make up the octave is the same as the pattern of different-sized
intervals that make up the fourth. In fact, the tuning where the two
ratios-of-intervals match is Paul's one. In this case, the ratio of large
to small semitone, and the other one, is the square root of two.

I use tau to mean a whole tone, rho to mean the smaller semitone (limma),
sigma for the larger semitone (sigma=tau-rho) and pi for the comma
(pi=sigma-rho).

Then, if sigma/rho = (rho-pi)/pi, sigma/rho = sqrt(2). You can check that
if you really want to. Also, (tau+rho)/tau = sqrt(2), so the ratio of the
Pythagorean minor third to the whole tone has this same value.

To show the two tunings are the same:

sigma/rho = sqrt(2)
(tau-rho)/rho = sqrt(2)
tau/rho -1 = sqrt(2)
tau/rho = sqrt(2)+1

Which means rho/tau=sqrt(2)-1, honest.

All this is very interesting, but I agree it's of little or no musical
value.

a fractal series multiplies real quick so you're going to get outside of the
human hearing range after a few notes.

Elodie Lauten wrote,

>a fractal series multiplies real quick so you're going to get outside of
the
>human hearing range after a few notes.

A fractal is a set whose features repeat themselves on smaller and smaller
scales. Applied to pitch, you will have an infinite number of notes within
any range, including the audible one.

In a message dated 8/30/99 4:09:09 PM, Paul H. Erlich wrote:

>From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
>Elodie Lauten wrote,
>
>>a fractal series multiplies real quick so you're going to get outside of
>the
>>human hearing range after a few notes.
>
>A fractal is a set whose features repeat themselves on smaller and smaller
>scales. Applied to pitch, you will have an infinite number of notes within
>any range, including the audible one.

---->>> attn: Paul H. Erlich.... & whoever else...

Out of curiosity, I would like to see several detailed examples of fractal
scales in
both Equal & Just scale systems ... compiling a collection of fractal scales.

Han Zhang
XenoMusicology Lab/XenoCulture Int'l.

Zhang2323@aol.com wrote,

>Out of curiosity, I would like to see several detailed examples of fractal
>scales in
>both Equal & Just scale systems ... compiling a collection of fractal
scales.

Paul Hahn and I just posted detailed examples of fractal scales. They were
neither equal-tempered (which could not be fractal) nor just (hmmm....).