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

--- In tuning@yahoogroups.com, "M. Schulter" <MSCHULTER@V...> wrote:

> ! 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

This is fascinating, Margo--a temperament caught in the act of turning

into a scale. Of course with all the 14/11 and 13/11 intervals the

beat ratio question seems no longer very relevant, but you've given me

some other tunings to look at beyond what Paul thought relevant in

temperament ordinaire.

> 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.

Don't forget Robert Wendell!

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "M. Schulter" <MSCHULTER@V...> wrote:

>

> > ! 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

>

> This is fascinating, Margo--a temperament caught in the act of

turning

> into a scale. Of course with all the 14/11 and 13/11 intervals the

> beat ratio question seems no longer very relevant, but you've given

me

> some other tunings to look at beyond what Paul thought relevant in

> temperament ordinaire.

>

> > 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.

>

> Don't forget Robert Wendell!

Bob:

Thank you, Gene! I'm very sincerely honored.

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

--- In tuning@yahoogroups.com, "M. Schulter" <MSCHULTER@V...> wrote:

/tuning/topicId_42903.html#42932

>

> 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.

>

***Could someone please explain to me a bit about this "temperament

turning into a scale...?" I seem to have gotten a bit lost in this

thread... But, I'd like to know more about this concept...

Thanks!

J. Pehrson