Re: [tuning] Re: The Other Noble Fifth

Graham Breed wrote,

> So, it happens that the perfect fourth is sqrt(2)-1 oct or 497.1
cents. The perfect fifth is 2-sqrt(2) oct or 702.9 cents.

This "silver fifth" is also very close to the mean of the 7-out-of-12
MOS generators, i.e., 29 and 41/70ths of an octave.

> I don't think these intervals have any musical significance. The
fifth is a bit wide for a schismic scale, by my tastes.

Though the above differs from your "silver fifth" and fourth, one
things that these "mean of the generators" do, is they plot note
spellings by converting MOS patterns into note spelling patterns, and
I have used them to plot various alternate alphabetized note naming
arrangements as well.

ds

The "silver fifth" scale is one of two self-similar meantone scales I
posted to the list some time ago. The relevant messages are appended.

--pH <manynote@library.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Well, so far, every time I break he runs out.
-\-\-- o But he's gotta slip up sometime . . . "

---------- Forwarded message ----------
Date: Fri, 3 May 1996 07:44:15 -0500 (CDT)
To: tuning@eartha.mills.edu
Subject: Self-similar scales (was Re: Korerup etc.)

*** SOME TEXT REMOVED --pH ***

Interesting. When John first mentioned the Golden tuning several months
ago, I thought that it was generated by setting the ratio of whole
tone/diatonic semitone to phi. (The two are equivalent, of course.)
The self-similarity that Brian mentions was apparent to me also, and it
occurred to me to speculate about an even more self-similar (by my
lights, anyway) tuning: one 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 or http://library.wustl.edu/~manynote)
O
/\ "How about that? The guy can't run six balls,
-\-\-- o and they make him president."

---------- Forwarded message ----------
Date: Thu, 26 Aug 1999 10:56:16 -0500 (CDT)
From: Paul Hahn <manynote@library.wustl.edu>
To: "'tuning@onelist.com'" <tuning@onelist.com>
Subject: Re: [tuning] RE: Fractal Scales

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