Yahoo Tuning Groups Ultimate Backup tuning Squirrel-like tunings: temperament extraordinaire

Hello, everyone, and I'd like to share my excitement on trying a
tuning for the first time -- with warm gratitude to Gene Ward Smith
for an article which led me to reflect on the creative expression of
squirrel-like qualities in music, and thus to tune this modified
meantone system.

As I'll explain more in a longer message, squirrels can for me stand
for a variety of qualities such as beauty, agility, grace,
incisiveness, playfulness, and a bit of unpredictability. Of course,
incisiveness ties in very nicely with my taste for large and expansive
major thirds (Pythagorean or wider), and compact semitones
(Pythagorean or smaller).

Thus it intuitively appeals to me to associate squirrel-like qualities
in a 12-note circulating system with major thirds larger than 400
cents -- or, I would say, more specifically such thirds substantially
larger than the Pythagorean 81:64 (~407.82 cents). This means that it
is squirrel-like to have some fifths tempered wider than pure; in
contrast, a conventional 12-note "well-temperament" has all fifths
either pure or tempered in the narrow direction.

A 12-note circulating temperament with some fifths wider than pure is
often known as a _temperament ordinaire_. This genre seems to have
developed in France around the middle to later 17th-century (although
Schlick's irregular organ temperament of 1511 might qualify), and has
had exponents ranging from Rameau and Alembert to George Secor, whose
system sometimes known as "Secor No. 3" is an outstanding example.

Last year I mentioned the idea of a system which Gene's articles
prompted me actually to tune -- with much excitement and delight.

In this "temperament extraordinaire," as I call it, the eight fifths
in the range F-C# are tempered as usual in 1/4-comma meantone, and the
other four are tempered by identical amounts in the _wide_ direction
to close the circle, about 4.888 cents each. The result is a tuning
circle in which all fifths are within 1/4-comma of pure, while major
thirds range from 4:5 (~386.31 cents) to 25:32 (~427.373 cents). That's
what I call "modal color."

Another way to describe this is to say that the just 25:32 major third
or diminished fourth at C#-F, already defined by the tuning of the
first nine notes in regular meantone, is divided into four equal
fifths C#-G#-D#-A#-E# or Db-Ab-Eb-Bb-F at about 706.843 cents, and
into two even-tones Db-Eb-F or C#-D#-E# at about 213.686 cents each.

Just as the usual meantone fifths are tempered narrow by 1/4 syntonic
comma (81:80, ~21.506 cents), or ~5.377 cents, so these wide fifths
are tempered by 1/4 of the "small comma" of 2048:2025 (~19.553 cents),
the difference between a 25:32 major third and a Pythagorean major
third formed from four pure fifths (~407.820 cents). This difference
is one meaning of the term "diaschisma," which can also refer to a
larger type of interval equal to half of a Pythagorean diatonic
semitone or limma at 256:243 (~90.224 cents), or about 45.112 cents.

Here's a Scala file:

! qcmte84a.scl
!
Temperament extraordinaire: F-C# 1/4-comma meantone, other 5ths ~4.888c wide
12
!
76.04900
193.15686
289.73529
5/4
503.42157
579.47057
696.57843
782.89214
889.73529
996.57843
1082.89214
2/1

A special delight of this tuning is that a chain of one usual narrow
fifth plus three wide fifths (e.g. F#-C#-G#-D#-A#) produces a major
third of ~417.108 cents, only ~0.40 cents from a pure 11:14 (~417.508
cents). Similarly, a chain of two wide fifths plus one narrow fifth
(e.g. Eb-Bb-F-C) produces a minor third of ~289.735 cents, only ~0.526
cents from a pure 11:13 (~289.210 cents).

A characteristic of this temperament extraordinaire is that interval
sizes within a given category (e.g. major thirds) jump in steps of
about 10.265 cents, or 1/4 of the 128:125 diesis (~41.059 cents)
defining the difference between the regular meantone diatonic semitone
at ~117.108 cents (e.g. C#-D) and chromatic semitone at ~76.049 cents
(e.g. C-C#). This large a jump in a circulating temperament could be
described as somewhat "squirrel-like," in contrast to the finer
gradations typical of many conventional 12-note well-temperaments.

For example, there are major thirds at around 386.314 cents (the usual
just 4:5); 396.578 cents (comparable to 1/8-comma meantone); 406.843
cents (close to Pythagorean); 417.108 cents (a virtually just 11:14);
and 427.373 cents (the meantone diminished fourth at a just 25:32)

The near-Pythagorean thirds Eb-G and B-D#, and especially the
near-11:14 thirds (Ab-C, Gb-Bb) and 25:32 third (Db-F) are ideal for a
neo-medieval kind of style where they tend to expand to stable fifths,
often together with major sixths expanding to octaves. These intervals
could indeed be described as optimal in such a context, just as the
usual meantone intervals are ideal for a typical Renaissance idiom.

There are actually two reasons I would give to leave the eight fifths
from F to C# in regular meantone: to maintain an uncompromising
meantone quality for lots of the most common Renaissance sonorities;
and at the same time to include at least one 25:32 diminished fourth
(a superb neo-Gothic major third as well as "special effects" interval
in some 16th-century music) and two excellent 76-cent semitones (C-C#
and F-F#, also spelled in neo-medieval contexts as C-Db and F-Gb).

We might define a temperament extraordinaire as a subcategory of
12-note temperament ordinaire or modified meantone system where at
least eight of the fifths are left in regular meantone, thus including
at least one standard diminished fourth (e.g. 25:32 in 1/4-comma).
To this I would add the ideal of a "reasonably smooth and even quality"
for all fifths: the wide fifths optimally should not be tempered
substantially more than the regular meantone fifths. Here they are
tempered a bit less (about 4.89 cents wide, vis-a-vis 5.38 cents
narrow).

We might consider it an especially characteristic procedure simply to
tune a chain of eight fifths in a regular meantone, with the other
four tempered equally wide. This approach could be especially
attractive in the general range from around 1/4-comma to Zarlino's
2/7-comma (regular fifths ~6.14 cents narrow, others ~6.42 cents
wide), with the latter tuning offering a diminished fourth (~433.517
cents) very close to a pure 7:9 (~435.084 cents).

While the 25:32 of our 1/4-comma temperament extraordinaire is not so
close to 7:9, I was delighted today to find this approximation of a
sonority to which Paul Erlich introduced me, the 4:6:7:9 chord:

F4 F#4
Db4 B3
Bb3 B3
Eb3 B2

Here I've used the first spelling that occurred to me: equivalents in
a 12-note circle are quite an adventure to me, with the 17-note
equivalents more familiar. The near-4:6:7:9 is around 0-707-986-1414
cents, generally rather less accurate than in Paul's beloved 22-EDO,
but a treat to find in a 12-note circle. As it happens, Bb-F-Ab-C
would have yielded about 0-707-986-1403 cents, closer in terms of
ratios above the lowest voice. In fairness, I should add that George
Secor's temperament ordinaire or modified meantone, although the
largest major third of ~418.573 cents (a near-11:14) is further from
7:9, has a position (Bb-F-Ab-C) which does better overall (around
0-702-985-1400 cents, including a pure 2:3 and a more accurate 4:7
above the lowest note).

Anyway, I find that my temperament extraordinaire has me fascinated
with the transpositions, modal colors, and "enharmonic equivalents"
possible in a 12-note circle. The system seems like a fusion of two
universes: a usual meantone universe (F-C#) merging into a realm more
like a 17-note circle, with the wide fifths tempered by an amount
recalling the nearer nine fifths (Ab-B) of George Secor's 17-tone
well-temperament (17-WT) at ~5.265 cents wide.

Thanks to Gene for that set of posts leading me to consider
squirrel-like 12-note circles, and also to George Secor for the
inspiring example of his temperament ordinaire or modified meantone
system.

Most appreciatively,

Margo Schulter
mschulter@value.net

Hello, there, Gene Ward Smith and Robert Wendell -- thank you both for
the inspiration you've lent to my "squirrel-like temperaments," with
special appreciation to Bob for discussing some "desiderata" and thus
getting Gene started on his "squirrel" ratings for 12-note circles and
the like.

Gene, your comment about a temperament turning into a scale suggests
to me this version of Eb Dorian (the medieval-Renaissance European
mode, that is, rather than the ancient Greek one) in my temperament
extraordinaire based on eight fifths (F-C#) in 1/4-comma, and the
others equally wide (~4.888 cents) so as to close the circle.

! tedorian.scl
!
Eb Dorian in temperament extraordinaire -- neo-medieval style
7
!
213.68628
289.73528
493.15685
706.84314
910.26471
986.31371
2/1

Here's a diagram of the steps, with conventional meantone spellings:

Eb4 F4 F#4 G#4 Bb4 C5 C#5 Eb5
0 214 290 493 707 910 986 1200
214 76 203 214 203 76 214

This scale, to which I am much drawn, is at once "squirrel-like" in
having those incisive near-11:14 and 25:32 major thirds and 76-cent
semitones from a neo-medieval point of view, and also a bit
"squirrely" in having a certain degree of charming asymmetry,
quirkiness, or unpredictability in comparison to a regular tuning.

Thus there are two sizes of whole-tones, at around 203 cents and 214
cents, reminding one of JI schemes with two such sizes. There,
however, one size of fifth is typically pure while the other is impure
by some comma equal to the difference of whole-tone sizes, for example
9:8 vis-a-vis 10:9 or 8:7; here the two fifths are impure by almost
identical amounts (-5.38 cents or +4.89 cents), but in opposite
directions. The two tetrachords Eb-F-F#-G# (214-76-203) and Bb-C-C#-Eb
(203-76-214) have the same interval sizes, but differently ordered,
for a bit of squirrel-like and squirrely asymmetry.

Here the near-11:14 thirds are formed from a 203-cent plus a 214-cent
step at F#-Bb, or vice versa at G#-C. Two of the large 214-cent steps
(actually ~213.686 cents) form the just 25:32 at C#-F.

Gene, when you talked about a temperament turning into a scale, this
might suggest looking especially at a curious affinity to a style of
JI/RI with "virtual temperament" that I've used in a neo-medieval
system. You mentioned the close approximations of 11:14 and 11:13,
which might make the question of brats (beat rates) a bit less
important for at least this portion of the tuning circle, but there's
a special squirrel-like (or squirrely?) quirk here: the tempered
interval sizes almost exactly mirror the 121:154:182 or 121:143:182
sonorities that occur in a "virtually tempered" system.

In other words, we have a sonority consisting of an 11:14 major third
below and an 11:13 minor third above (121:154:182), or the converse
(121:143:182), with a wide 121:182 fifth exceeding a pure 2:3 by the
11-13 schisma of 363:364 (~4.763 cents). Our tempered Dorian mode
includes very close approximations of both arrangements:

C#5 Bb4
(706.84, -0.125) (706.84, -0.125)
Bb4 F#4
(417.11, -0.40) (289.74, +0.535)
F#4 Eb4

~121:154:182 ~121:143:182

The charm of this temperament is that in addition to excellent
neo-medieval scales and modes like this, which invite either
drone-based melody or polyphony in a 13th-century European style (a
good opportunity for me to practice more remote transpositions), we
get an impeccable 1/4-comma meantone in the range of F-C#.

Such a scheme illustrates an exception to the rule that wide fifths in
a 12-note circle constitute "harmonic waste," an exception implied in
Owen Jorgensen's explanation of this rule applying to conventional
well-temperaments of this size. Generally one assumes that fifths are
meant to be tempered in the narrow direction, with major thirds
ideally near 4:5 and minor thirds near 5:6, so that unnecessary
"reverse tempering" needlessly increases the inaccuracy of a scheme.

Here, however, wide fifths are just as useful in producing ~11:14 or
25:32 major thirds and 76-cent semitones for use in a neo-medieval
context as narrow ones are in producing the usual Renaissance meantone
intervals.

Indeed, as I mentioned in my previous message, the wide fifths at
~4.888 cents larger than pure are very close to the size of the nearer
fifths in George Secor's 17-tone well-temperament at about 5.265 cents
wide. We have, as it were, a bit of neo-medieval forest or park in the
midst of a meantone metropolis -- or meantone countryside, as you like
your metaphors.

Anyway, Gene and Bob, thanks again for your ideas and encouragement!

Most appreciatively,

Margo Schulter
mschulter@value.net

Squirrel-like tunings: temperament extraordinaire

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Robert Wendell <rwendell@cangelic.org>

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗Joseph Pehrson <jpehrson@rcn.com>