another kind of Golden scale for Dan

I don't think I've seen a scale quite like this mentioned here before. I
found it alluded to in passing, in a dissertation on "Pairwise Well-Formed
Scales" by David Clampitt. It's the limiting case of a series of other
scales which he looks at briefly. The scale uses an *additive* generator
of phi+1, taken mod 3phi+2. Clampitt never shows the scale explicitly but
I'll give it here, first in the "generated" order:

0, phi+1, 2phi+2, 1, phi+2, 2phi+3, 2

and here reordered within the 3phi+2 octave:

0, 1, 2, phi+1, phi+2, 2phi+2, 2phi+3, (3phi+2)

the seconds are {phi-1, 1, phi},
the thirds are {phi, 2, phi+1},
the fourths are {phi+1, 2phi, phi+2}
the fifths are {phi+2, 2phi, 2phi+1}
the sixths are {2phi+1, 3phi, 2phi+2}
the sevenths are {2phi+2, 3phi+1, 2phi+3}
the 8ve is 3phi+2

In cents we get:

0 175 350 458 633 917 1092 1200
m m s m L m s

As a whole, it doesn't sound too much like anything I'm familiar with (it
does have the quarter-tone neutral third). But there's so much
self-similarity here it makes my head hurt to think about it... if we call
the 3 shades of each generic interval s m and L, then we have:

m2:s2 = L2:m2 = L3:L2 = L5:L3 = 8ve:L5 = s6:s4 = s4:s3 = phi !!

also,

8ve:L3 = m5:m2 = 2phi

and

s6:s3 = L3:L2 = phi+1

and you can find many more such relations...

And here's something neat: it looks like this could be a good example to
bolster Margo and Dave K's thesis that Golden-derived intervals tend to
maximise dissonance. Here are all the intervals smaller than an octave
available in the tuning:

108, 175, 284, 350, 459, 567, 633, 742, 850, 917, 1025, 1092 cents

These are remarkable for their lack of approximation to any small number
ratios - the closest intervals to a 3:2, 5:4 and 6:5 are off by
respectively 40, 36 and 34 cents. The 7-limit approximations are off by
52c (7:4), 17c (7:6) and 16c (7:5).

And check this out: quite a few of the intervals available in the scale,
especially 284, 350, 459 cents and their inversions, are right in the very
narrow zones defined as mediants of complexity in Margo and Dave's paper
(at http://www.egroups.com/message/tuning/12915 ), and also appear as the
local maximums of dyadic entropy in Paul E's graph's (see e.g.
http://www.egroups.com/files/tuning/perlich/o006_223.jpg - is there a
version somewhere where maxima are labelled in cents, Paul?)

Cheers --Jon

Jon Wild wrote,

>And here's something neat: it looks like this could be a good example to
>bolster Margo and Dave K's thesis that Golden-derived intervals tend to
>maximise dissonance.

I'm afraid you've confused golden fractions of an octave with golden
frequency ratios -- see my post to Joseph of a few minutes ago.

Jon Wild wrote,

>local maximums of dyadic entropy in Paul E's graph's (see e.g.
>http://www.egroups.com/files/tuning/perlich/o006_223.jpg - is there a
>version somewhere where maxima are labelled in cents, Paul?)

No, but they are:

32
144
192
213
248
291
339
362
410
472
526
600
674
728
790
838
861
909
952
987
1008
1056
1168

jon wild wrote,

> I don't think I've seen a scale quite like this mentioned here
before. I found it alluded to in passing, in a dissertation on
"Pairwise Well-Formed Scales" by David Clampitt.

Neat, thanks for taking the time to point this out.

> As a whole, it doesn't sound too much like anything I'm familiar
with

I've used a bunch of different three-stepsize scales, like these in
13-tET for example:

0 277 369 554 646 831 1015 1200
L s m s m m m

0 185 277 646 738 923 1015 1200
m s L s m s m

0 277 369 646 738 923 1015 1200
L s L s m s m

And the first of these, which I use in extensively in "With Eyes so
Blue and Dreaming", is definitely in the same general ballpark.

<http://stations.mp3s.com/stations/55/117_west_great_western.html>

These 7-tone scales that exploit the augmented second are extremely
beautiful, and 13-tET with its three-stepsize cardinality really seems
to heighten the sense of melancholic ache, or longing, that is akin to
some of the Eastern European scales that also utilize the augmented
second scale step.

thanks again,

--Dan Stearns

Hi Dan,

> I've used a bunch of different three-stepsize scales, like these in
> 13-tET for example:
>
> 0 277 369 554 646 831 1015 1200
> L s m s m m m
>
> 0 185 277 646 738 923 1015 1200
> m s L s m s m
>
> 0 277 369 646 738 923 1015 1200
> L s L s m s m
>
> And the first of these, which I use in extensively in "With Eyes so
> Blue and Dreaming", is definitely in the same general ballpark.

I like that piece by the way! Clampitt's dissertation, where I found
that golden scale, also has a 13-tET version, whose step-sizes in
13ths of an octave are <2212321>, or

m m s m L m s

The next version would be 21-tET <3323532>, then 34-tET <5535853>...
you get the picture. They get successively closer to the limiting
case:

0 175 350 458 633 917 1092 1200.

> These 7-tone scales that exploit the augmented second are extremely
> beautiful, and 13-tET with its three-stepsize cardinality really
> seems to heighten the sense of melancholic ache, or longing, that
> is akin to some of the Eastern European scales that also utilize
> the augmented second scale step.

Yes! Your scales above, I think, allow this yearning more powerfully
than Clampitt's golden scale, by having the smallest step-size
surrounding the largest--like in the 12-tET harmonic minor--or at
least directly above it. It's the combination of the wide gap, which
conveys "reaching", surrounded by the cramped intervals, which seem
to me to be attempting to repress the "reaching". You get the same
beautiful effect in some of the Greek genera with pykna, when the
tetrachords are placed conjunctly. Clampitt's golden scales can't
have this, because he is going for "pairwise well-formedness" and
can't have smallest step next to largest, in this case.

Best -Jon

Jon Wild wrote,

> I like that piece by the way! Clampitt's dissertation, where I found
that golden scale, also has a 13-tET version, whose step-sizes in
13ths of an octave are <2212321>,

Thanks Jon, I've always liked 13 equal... it's not nearly the werewolf
many make it out to be either (Carlos for example). The next thing I
hope to send off to J. Starrett's Tuning Punks site is another
microtonal set, and the first piece (a reworking of Ives' "Evidence")
is in 13 and the second 11 equal. And it's big simultaneous sonorities
and "tonality" of a strange and lovely variety, not a howling pack of
ravenous werewolves. <Not that a howling pack of ravenous werewolves
is a bad thing mind you...>

Of course as these are my opinions of my pieces they should be
immediately disregarded as such <!>, but I mention all this in the
hope of offering my bit of protest against some of the bad press and
"only good for melody" myths that seem to have grown up around these
types of tunings -- Don't believe the hype!

> Yes! Your scales above, I think, allow this yearning more powerfully
than Clampitt's golden scale, by having the smallest step-size
surrounding the largest--like in the 12-tET harmonic minor--or at
least directly above it. It's the combination of the wide gap, which
conveys "reaching",

I agree.

--Dan Stearns