Re: Celebrating Manuel Op de Coul's 233.985-cent tuning

Hello, there, Manuel Op de Coul and everyone.

In response to recent discussions of what I call the "wonder scale"
with a generator of precisely 3:2^1/3 (~233.985 cents), I would like
to do my part in clarifying some history and chronology.

Given my initial discussion of your scale and my "rediscovery" in a
long article, </tuning/topicId_22907.html#22907>, a
briefer summary of some main points concerning the chronology may be
helpful.

Especially I would like to emphasize that this is your scale, with a
role for the renowned Adriaan Fokker as you describe below, long
antedating both the recent "Miracle Scale" discussions on this forum
and my rediscovery of your 233.985-cent tuning growing out of those
discussions.

Your tuning, with a generator twice the size of the later "Miracle
Tuning" at around 3:2^1/6, is documented in the Scala archive file
temp31g3.scl: "Cycle of 31 sevenths tempered by 1/3 gamelan residue".

In view of an interest expressed here in developing a catalogue of
"Miracle" subsets, your 233.985-cent tuning -- however you choose to
name it -- should be prominently featured in such catalogues.

As you wrote of the term "gamelan residue" for the 1029:1024 ratio:

> This name comes from Fokker. I haven't seen an explanation for it,
> but it possibly stems from the fact that a chain of harmonic
> sevenths produces a slendro-type scale, perhaps not a typical one,
> but I don't see another connection.

As Graham Breed helpfully explained:

> To clarify this, a pentatonic with the following ratios

> 8:7 8:7 8:7 8:7 7:6

> has roughly equal steps, and roughly makes an octave. So you could
> call it a gamelan scale, and the 1029:1024 is what stops it being a
> perfect octave, hence "gamelan residue". I don't think it is call
> it a gamelan scale, and the 1029:1024 is what stops it being a
> perfect octave, hence "gamelan residue". I don't think it is
> connected with any real gamelan tuning, other than being roughly
> 5-equal.

This is fascinating, and in Scala I confirmed that

calculate 2/1 - (8/7^4 + 7/6)

indeed yields an interval of ~8.44 cents identified as a "gamelan
residue."

> Tempering out the 1029:1024 gives Margo's tuning, and tempering out
> a 225:224 kleisma as well gives you 7-limit Miracle temperament. I
> had thought about this pentatonic a few years back, it looks like
> Fokker must have too.

Graham, may I gently amend the beginning of your paragraph to:
"Tempering out the 1029:1024 gives Manuel's tuning, later
rediscovered by Margo with the very full benefit of sustained
discussions on a generator of almost exactly half the size"?

Curiously, the threefold division of the 1029:1024 was the last of the
three "tripartite divisions" I recognized in this tuning, maybe not so
surprising since I was coming from a Pythagorean and neo-Gothic kind
of perspective where the pure fifths were the outstanding feature.

Having followed the not inconsequential or obscure "Miracle" threads
on this forum, I was in search of some unconventional generator with a
"neo-Gothic" flavor. The idea of generator at some precise fraction of
a pure fifth, of course, would have a special appeal, given my desire
for something at once "quasi-Pythagorean" and also "offbeat."

When I tried 3:2^1/3 in Scala, and saw the neat approximations of
neo-Gothic 7-flavor ratios it produced (9:7, 7:6, 7:4, and octave
complements), I was really excited.

As I quickly realized, this was very close to 36-tET, a scale I have
enthusiastically praised here for its 7-flavor and also 17-flavor
intervals (e.g. a nice approximation of 14:17:21) -- just as the
"Miracle Scale" can be realized in 72-tET, and many in fact prefer
this realization.

The threefold division of the pure fifth was thus the defining one for
me.

Then, looking at the difference of about 30.075 cents by which five of
the 233.965-cent generators fall short of a 2:1 octave, I realized
that three of these "commas" were precisely equal to a Pythagorean
limma or diatonic semitone at 256:243 (~90.224 cents). Another
tripartite division, and one most delightful from my Pythagorean
perspective! In 7-flavor cadences, by the way, a small semitone (which
one might take as an approximation of 28:27, ~62.96 cents) is favored
equal to precisely 2/3 of a Pythagorean limma, or ~60.15 cents.

It was only when I reflected more on the 36-tET affinity, and on the
way that in 36-tET (or 36-ED2, as some would say) the same 33-1/3-cent
step takes the place both of the septimal comma at 64:63 (~27.26
cents) and the 49:48 comma (~35.70 cents), that I considered the ratio
defining the difference of these commas, 1029:1024.

Already I had noted that fifths are pure in 233.985-cent tuning, while
all 7-flavor intervals are impure by ~2.81 cents. I now realized that
this variation from pure is equal to precisely 1/3 of the 1029:1024
schisma, or whatever we choose to call the "gamelan residue" in a
neo-Gothic setting.

For me, this was poetic: three tripartite divisions on different
levels -- the fifth, the limma, and the 1029:1024 schisma.

Anyway, when I found Manuel's file documenting an identical scale in
31 notes -- a strictly proper MOS, by the way, as Dave Keenan notes --
I looked up "gamelan residue" in intnam.par and wasn't surprised to
find that it was equal to 1029:1024.

(My version, as usual, was for 24 notes mapped to two 12-note
keyboards, however less than ideal for this type of tuning.)

Contrasting Manuel's original discovery of this scale with my
"rediscovery," I remarked in my first report that 3:2^1/3 was hardly
so surprising an idea on my part after reading about 3:2^1/6 as the
main topic on this forum for some days, to say the least. Paul Erlich
made the same point even more succinctly in response to that post:

> it's just every other note of "miracle tuning", since
> the generator is exactly twice as big.

Leaving it to Manuel to choose a fitting name, I would mainly like to
help ensure that his 233.985-cent tuning is celebrated with credit
given where it is properly due, adorned also by an auspicious
association with the name of the illustrious Adriaan Fokker.

Most respectfully,

Margo Schulter
mschulter@value.net