Tuning puzzle -- What is this tuning system?

Hello, there, everyone, and what follows is a tuning "puzzle" in the
tradition, more or less, of Paul Erlich -- although, you, Paul, and
others, will have to judge how freely I have adapted that tradition to
my own tastes.

The object of this game is to identify the tuning scheme described in
the puzzle through a set of clues; you can also consider the known
propensities of the author, and some allusions in the clues to prior
threads on this list. As far as I know, this specific overall system
hasn't been discussed here previously, but I'm open to correction on
that point.

An additional object of this game is to decide, or to debate, whether
this tuning system is a "middle path" -- either before or after the
"official" solution is deduced/intuited by some participant or group
of participants (cooperative puzzle-solving is encouraged), or as a
last resort revealed by the author.

Anyone is free to ask questions and seek "hints"; I'll try to be as
helpful as I can without making the solution overly obvious at too
early a stage.

Also, part of the fun is for people so inclined to discuss the clues
and cooperate in seeking a solution. This can be an educational
process in itself, as well as a strategy for a faster solution:
pooling knowledge and intuitions is what this list is about.

-------------------------------------------------
Puzzle: Can you identify this "microtemperament"?
-------------------------------------------------

You are brought into a room with a keyboard instrument having a
certain number of notes per octave greater than 12, and invited to try
"my latest tuning -- just possibly a new one." Moving your fingers
around the keyboards with their conventional 12-note arrangements, and
measuring interval sizes flawlessly, you get some curious clues.

1. You find that an approximation of 12:14:18:21 or 14:18:21:24 is
available at 9 locations with a sum of squares variation from just of
around 8.92 cents, giving 2:3 (or its octave complement of 3:4) a
double weighting, Paul Erlich style, since it occurs twice.

Your result brings the comment, "Yes, it actually outdoes 36-tET, with
better accuracy on every interval, although it's not as accurate
overall as Graham Breed's 135-tET."

2. Pressing the keys C-E on one of the manuals, you get a major third
with a ratio of almost precisely a just 100:127 -- a possible oxymoron
under a Dave Keenan definition of "just," by the way. You confirm that
this interval is built in a usual eventone fashion, and that the
tuning is an array of eventone MOS sets.

3. Pressing the keys Eb-F# on any of the manuals, you get a minor kind
of third which brings the comment, "Notice that this interval is about
3.76 cents more accurate than its nearest counterpart in 22-tET --
although lots of people on the Alternate Tuning List might view it the
other way because they're interested in a different ratio, one
included in the decatonic tetrad of 4:5:6:7."

You are offered an additional clue to what ratio is "3.76 cents more
accurate than in 22-tET." One of the integer terms for this dyad is
equal to the largest Pythagorean MOS discussed in late medieval
European theory (e.g. Prosdocimus, Ugolino); the other is equal to the
number of notes per octave on a split-key organ sometimes cited as the
earliest known practical example, dating from around the era of Ramos
and taken by Mark Lindley as an early meantone instrument.

4. Starting with C# on one of the keyboards as the lowest note, you
find that you can play a kind of approximation of 14:18:21:24 by
pressing the same set of key levers that you'd use for the nearest
approximation in 1/4-comma meantone -- but with not all these levers
located on the same manual. Rather, you find yourself combining two of
the levers from one manual with two from another. The resulting
approximation has a variation from pure of about 2.70 cents narrow for
the lower 7:9, and 4.19 cents narrow for the outer 7:12; the upper 7:8
is off by 5.68 cents, also in the narrow direction.

This prompts the comment, "This has a certain meantone-like feel, but
I should assure you that this tuning system wasn't derived by mistake
while attempting a subset of Vicentino's cycle -- just in case there
might be any possible rumors on the list."

5. Who should enter the room but Dan Stearns, who does his own quick
test and excitedly comments: "This tuning has a very accurate
emulation of the 20-tET approximation of 28:27." He also notes that
this near-28:27 is tempered by twice as much as the best approximation
of 9:7, and that there's a closer approximation of 28:27 in five
locations tempered by about 1.20 cents in the opposite direction.

6. For a change of pace, you try appropriate keys for an approximation
of the Bohlen-Pierce triad at 3:5:7, and find the 3:5 about 3.20 cents
wide, and the 5:7 narrow by an identical amount.

However, your puzzler adds, "This near-3:5:7 is only available in
four positions in the overall tuning -- as Dave Keenan might observe
when gauging utility."

By the way, you note that the near-3:5 has the same key positions as
in an eventone or "Pythagorean" arrangement of 22-tET -- for example,
C-G# -- but with the two notes played on different keyboards.

7. You also find nine near-9:11 intervals about 2.22 cents wider than
just.

8. At 18 positions in the overall tuning, you find approximations of
the 11:13 ratio at about 0.45 cents wide.

9. Some recent talk on the list about unison vectors and the like
leads to the comment: "Each MOS set disperses the 351:352 comma, and
the overall tuning very closely represents the 77:78 comma. However,
that isn't to imply that this is a periodicity block -- let Paul
Erlich or someone decide that."

10. As a curiosity, you find three locations in the overall tuning
approximating 4:5:6:7 with a near-4:5 at about 391.01 cents, prompting
the comment, "That's the least accurate interval in this tetrad, with
5:6 off in the narrow direction by around 3.20 cents." The position
for one of these tetrads involving pressing the keys Eb-F#-Bb-C in
ascending order -- but with two notes on one manual, and two on
another.

Will someone be able to identify the system based on these clues?
People should feel free to ask more questions -- about interval sizes
and approximations, or about what intervals are produced by pressing
certain keys, etc. You can also ask about the size of the tuning, and
I have a clue ready for that. "Yes or no" questions in "Twenty
Questions" fashion are also fine.

If and when someone or some group of participants correctly identifies
the system, I'll confirm this; if not, I'll let the search for clues
continue for a reasonable time before announcing the "official"
solution.

Also, it may provide a clue that I regard the "middle path" status of
the _overall_ system as something of a debatable question.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

> 1. You find that an approximation of 12:14:18:21 or 14:18:21:24 is
> available at 9 locations with a sum of squares variation from just
of
> around 8.92 cents,

You mean 8.92 cents _squared_, not 8.92 cents, right? If the sum-of-
squares is 8.92 cents squared, and six intervals are being compared
here, then the mean-square is 1.49 cents squared, and the root-mean-
squared variation from just is, enfin, 1.22 cents.
>
> 2. Pressing the keys C-E on one of the manuals, you get a major
third
> with a ratio of almost precisely a just 100:127 -- a possible
oxymoron
> under a Dave Keenan definition of "just," by the way. You confirm
that
> this interval is built in a usual eventone fashion, and that the
> tuning is an array of eventone MOS sets.

So the fifth is about 703.45 cents. The squared deviation from a just
fifth is about 2.23 cents squared. So the fifth accounts for 4.46
cents squared of the 8.92 total. Exactly half -- and yet there are
four more intervals to take into account here: two 7:6s, one 7:4, and
one 9:7. I must conclude, therefore, that you are using just 7:6s!
>
> 3. Pressing the keys Eb-F# on any of the manuals, you get a minor
kind
> of third which brings the comment, "Notice that this interval is
about
> 3.76 cents more accurate than its nearest counterpart in 22-tET --
> although lots of people on the Alternate Tuning List might view it
the
> other way because they're interested in a different ratio, one
> included in the decatonic tetrad of 4:5:6:7."
>
> You are offered an additional clue to what ratio is "3.76 cents more
> accurate than in 22-tET." One of the integer terms for this dyad is
> equal to the largest Pythagorean MOS discussed in late medieval
> European theory (e.g. Prosdocimus, Ugolino);

17

> the other is equal to the
> number of notes per octave on a split-key organ sometimes cited as
the
> earliest known practical example, dating from around the era of
Ramos
> and taken by Mark Lindley as an early meantone instrument.

14? . . . 17:14 is 336.13 cents; 22-tET has an interval of 327.28
cents; so 331.03 cents is your interval.
>
> 4. Starting with C# on one of the keyboards as the lowest note, you
> find that you can play a kind of approximation of 14:18:21:24 by
> pressing the same set of key levers that you'd use for the nearest
> approximation in 1/4-comma meantone -- but with not all these levers
> located on the same manual. Rather, you find yourself combining two
of
> the levers from one manual with two from another. The resulting
> approximation has a variation from pure of about 2.70 cents narrow
for
> the lower 7:9,

Weird . . . 7:9, based on my calculation above, should have the same
mistuning as the fifth, namely, 1.49 cents, and in the sharp
direction. So this must be a different approximation to 14:18:21:24
than the "good" one mentioned first, with a difference of 4.19 cents.

> and 4.19 cents narrow for the outer 7:12;

OK . . . this was just before . . . so again a difference of 4.19
cents pops up . . .

> the upper 7:8
> is off by 5.68 cents, also in the narrow direction.

Hmm . . . so the lower 3:2 is 1.49 cents wide, making it a "regular"
fifth in the system . . . the "regular" 7:8 would be 1.49 cents
narrow, so once again the difference is 4.19 cents . . . and in this
chord, the central 7:6 is wide by x where

(-2.70) + x + (-5.68) = -4.19
x = 4.19

4.19 cents, once again!

> 5. Who should enter the room but Dan Stearns, who does his own quick
> test and excitedly comments: "This tuning has a very accurate
> emulation of the 20-tET approximation of 28:27." He also notes that
> this near-28:27 is tempered

Relative to 20-tET, or relative to JI?

> by twice as much as the best approximation
> of 9:7,

In 20-tET, or in this tuning?

> and that there's a closer approximation of 28:27 in five
> locations tempered by about 1.20 cents in the opposite direction.

Hmm . . . I'm pretty sure the rest of your clues should be enough
information to solve the puzzle . . . I'll leave this alone for now
so others may jump in . . . how'm I doing so far, Margo?

Hello, there, Paul, and please let me clarify some of my imprecise
language, much helped by your alert questions and queries. I hope that
this will assist others also.

>> 1. You find that an approximation of 12:14:18:21 or 14:18:21:24 is
>> available at 9 locations with a sum of squares variation from just
>> of around 8.92 cents,

> You mean 8.92 cents _squared_, not 8.92 cents, right? If the sum-of-
> squares is 8.92 cents squared, and six intervals are being compared
> here, then the mean-square is 1.49 cents squared, and the root-mean-
> squared variation from just is, enfin, 1.22 cents.

Just to clarify, since you picked up on this in your next remarks: the
sum of squares of the temperings in cents is indeed 8.92, with 3:2
counted twice, and each of three other interval categories once: 7:6,
9:7, and 7:4.

As you also very correctly suggest, I would have gotten the same
result if I had counted all six intervals, weighting accordingly.

>> 5. Who should enter the room but Dan Stearns, who does his own quick
>> test and excitedly comments: "This tuning has a very accurate
>> emulation of the 20-tET approximation of 28:27." He also notes that
>> this near-28:27 is tempered

> "Relative to 20-tET, or relative to JI?"

Relative to JI -- and thanks for calling this unintended ambiguity to
my attention.

>> by twice as much as the best approximation
>> of 9:7,

> In 20-tET, or in this tuning?

In this tuning -- again, thanks for this helpful query.

> Hmm . . . I'm pretty sure the rest of your clues should be enough
> information to solve the puzzle . . . I'll leave this alone for now
> so others may jump in . . . how'm I doing so far, Margo?

Very well, I'd say -- both in solving the puzzle, and in pointing out
some places where my language may have been more "puzzling" than
intended.

Most appreciatively,

Margo Schulter
mschulter@value.net