Another "modified meantone" circle in 12 -- with 32:25

Hello, there, everyone, and thanks to the people who have contributed
to the recent thread on Chaumont, which has helped suggest to me
another 12-note temperament based in part on 1/4-comma meantone.

Before presenting a Scala file for this temperament, I should briefly
comment that Chaumont's tuning looks to me like the kind of scheme
known as a _temperament ordinaire_, popular in France during the later
17th and 18th centuries.

The characteristic feature of this type of 12-note temperament is that
some of the fifths (e.g. Eb-Bb and Bb-F) are tuned _wider_ than pure,
resulting in some major thirds wider than Pythagorean, or minor thirds
narrower than Pythagorean.

In some of these schemes, like Chaumont's in the version provided by
Manuel Op de Coul, there is still something of a "Wolf fifth," for
example at G#-Eb.

In others, however, all fifths are reasonably close to pure, as in
Bethisy's temperament ordinaire also included in the Scala archive,
where all fifths are no more than 1/4 syntonic comma from pure.

Mark Lindley has suggested that the temperament ordinaire may have
evolved during the 17th century from a kind of creative misreading of
the instructions by Mersenne for a usual 1/4-comma meantone tuning.

The basic idea is that these instructions call, when tuning up the
chain of fifths (e.g. C-G-D-A-E-B-F#-C#-G#) for the higher note of the
fifth to be somewhat lower than a pure 3:2.

However, when tuning fifths in a flatward or "downward" direction on
the chain, as with Bb-F and Eb-Bb, the lower note of the fifth should
be _raised_ or tuned "strong" to form a meantone fifth again narrower
than pure.

In the "creative misreading," some French musicians around the middle
of the 17th century may have instead tuned Bb-F and Eb-Bb "strong" in
the opposite meaning of "wider than pure," giving rise to the scheme
soon recognized and relished as a temperament ordinaire. By the time
of Couperin, Lindley suggests, music was being written with this kind
of temperament in mind.

Whether some versions of the temperament ordinaire qualify as 12-note
"well-temperaments" depends in part on the disposition of the fifths,
and also in part on how one chooses to define the limits of sizes for
"acceptable" thirds in a circulating scheme.

In the kind of well-temperament documented by Werckmeister and
followed by various 18th-century theorists and practical musicians,
all fifths are either pure or narrow of pure, and thus all major
thirds no larger than Pythagorean (81:64, ~407.82 cents). In other
words, they are all within a syntonic comma (81:80, ~21.51 cents) of a
pure 5:4 (~386.31 cents).

In a temperament ordinaire with a circle of 12 "playable" fifths,
ranging from around meantone to somewhat larger than pure, certain
remote major thirds will be larger than Pythagorean, for example up to
around 413.197 cents (Ab-C or G#-C) in Bethisy's tuning.

Such a major third (or diminished fourth) might be regarded as at
least "semi-interchangeable" with a more usual major third for the
period with a size closer to 5:4. If so, then we might speak in
historical terms of a kind of well-temperament.

Starting from the idea of a temperament ordinaire as a kind of
"modified meantone," I considered the problem of how many fifths in a
12-note tuning one can temper at 1/4-comma narrow while still
balancing out the circle to keep all fifths comfortably "playable,"
and preferably no more than 1/4-comma impure.

The solution to which I was drawn is a chain of 8 fifths (F-C#) in
1/4-comma meantone, with the remaining 4 fifths each tempered wide by
around 4.888 cents, or more precisely (2048/2025)^(1/4), to balance
out the 12-note circle.

The ratio of 2048:2025, a "diaschisma" in one definition of that term
(another is half of a Pythagorean diatonic semitone or limma at
256:243 or ~90.22 cents, giving an interval of ~45.11 cents), is equal
to the difference between a Pythagorean 81:64 major third and a 32:25
major third at ~427.37 cents, about 19.55 cents.

Thus we have 8 fifths tempered at 1/4-comma narrow, and the
remaining 4 fifths tempered at 1/4-diaschisma wide.

Here's a Scala file for this tuning:

! qcmqd8_4.scl
!
F-C# in 1/4-comma meantone, other 5ths ~4.888 cents wide or (2048/2025)^(1/4)
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

From a 17th-19th century historical perspective, this circle of 12
fifths is _not_ a "well-temperament," because the 32:25 major third or
diminished fourth at C#/Db-F is hardly interchangeable with a major
third at or reasonably near 5:4.

However, from the 21st-century perspective of a "mixed" style
combining Renaissance meantone with neo-Gothic sonorities and
progressions where anything from 81:64 to a bit beyond 9:7 (~435.08
cents) is routinely accepted as a "major third," this is a kind of
well-temperament with striking contrasts of "modal color."

In choosing a chain of 8 meantone fifths from F to C#, the idea is at
once to get an ideally smooth "16th-century" flavor for sonorities
within this range, and also to make the common thirds Bb-D and E-G#
not too far from 5:4, although they are inevitably somewhat
compromised by Renaissance standards.

The major third E-G# can be an especially delicate point for a 12-note
circle, since this third from around 1520 on is used prominently in
pieces in various modes, and often appears in closing sonorities in
the Phrygian mode (E-E).

In this scheme, E-G# and Bb-D are tuned at around 396.578 cents, or
about 10.265 cents wide of 5:4 -- a bit less than 1/2 syntonic comma.

By comparison, in various well-temperaments of the late 17th-19th
centuries, E-G# tends to be somewhat wider, often larger than in
12-EDO at 400 cents.

Once we move out of the Bb-G# region, of course, things move quickly
toward Pythagorean and beyond. The thirds Eb-G and B-D#/Eb are close
to Pythagorean at ~406.843 cents; Ab/G#-C and F#-Bb/A# are at ~407.108
cents, very close to a just 14:11 (~417.508 cents); and C#/Db-F is at
a just 32:25.

There are also two minor thirds (C-Eb, Eb/D#-F#) at ~289.735 cents,
very close to 13:11 (~289.210 cents), and two thirds (F-G#/Ab,
Bb-C#/Db) at ~279.471 cents, rather close to 27:23 (~277.591 cents).

These "Monzian thirds" are a special adornment to a tuning. This term
originated from a famous exploit of Joe Monzo, the "Monz," in which he
started out with 7:6 (~266.87 cents) as a possible size for an
interval in a composition, but found by ear that around 279 cents
sounded like the right size, eventually deciding on a rational ratio
of 75:64 (~274.58 cents).

Thus any third from around 274 to 280 cents can attract the pleasant
adjective "Monzian."

A curious feature of this "8_4" tuning, in contrast to more
conventional well-temperaments, is that interval sizes within a given
category jump in steps of around 10.265 cents, the difference in size
between the narrow fifths at 1/4 comma (~5.377 cents) smaller than pure
and the wide fifths at 1/4 diaschisma (~4.888 cents) larger than pure.

Since the tuning uses only these two sizes of fifths, the 10.265-cent
steps result from the replacement of a narrow meantone fifth by a wide
one. With major thirds, for example, we have these possibilities with
"N" showing a narrow fifth and "W" a wide one:

---------------------------------------------------------------
Interval Cents Ratio Chain of fifths
---------------------------------------------------------------
A-C# ~386.31 5:4 A E B F# C#
just N N N N
...............................................................
E-G# ~396.58 5:4 E B F# C# G#
+~10.27 N N N W
...............................................................
B-D#/Eb ~406.84 81:64 B F# C# G# D#
-~0.98 N N W W
...............................................................
F#-A#/Bb ~417.11 14:11 F# C# G# D# A#
-~0.40 N W W W
...............................................................
C#/Db-F ~427.37 32:25 Db Ab Eb Bb F
just W W W W
---------------------------------------------------------------

This scheme, to conclude, is like some versions of the temperament
ordinaire in combining narrow and wide fifths to form a 12-note circle
where all of these intervals are reasonably close to pure; but it
differs in including a major third or diminished fourth as large as
32:25, identical to the diminished fourths of 1/4-comma meantone.

Since in the kind of neo-medieval style favored for the more remote
transpositions, a 32:25 major third is routine, a special kind of
"well-temperament" results in this musical context.

Most appreciatively,

Margo Schulter
mschulter@value.net

hello Margo,

thanks very much for this post! indeed, Chaumont's tuning
is a form of "temperament ordinaire" -- and based on the
information you provided here, my opinion is that this
term should be included in the Tuning Dictionary as a
separate family of tunings distinct from "well-temperaments".

the interesting feature that relates them both together
is that the 5th which is notated as the "wolf" turns out
to be quite close in size to the "regular" 5ths, which
has the effect of closing the scale at 12 tones.

in a well-temperament, all 5ths are within a few cents
of being the same size (pretty close to Pythagorean or 12edo),
whereas in a _temperament ordinaire_ most 5ths are derived
from a meantone (quite a bit smaller than Pythagorean) and
a few (usually two) are quite large.

i agree with Lindley that a misreading may have been the
original stimulus in creating _temperament ordinaire_,
only because it makes a lot of sense and this kind of
thing has happened before -- the association of Greek
names with the modes which are in use today is the result
of a medieval misreading in which Frankish scribes took
the ancient Greek descriptions of tone and semitone patterns
to apply to an ascending scale, whereas the Greeks had
applied them to descending scales. the result is that
the old Greek names and the old modes are both still used,
but the associations connecting the names and modes is
completely different than that given by the Greeks.

the fact that this plausible misreading of meantone instructions
results in a tuning which closes at 12 tones -- rather than
being left open as in meantone -- would be enough to ensure
some measure of popularity on keyboard instruments with the
conventional Halberstadt design.

if i get some time, i hope to explore your new _temperament
ordinaire_ more fully, and perhaps even make a lattice diagram
and webpage about it. i have been making quite a few lattices
of well-temperaments and meantones, actually cutting and taping
them physically to create cylinders which show the closure of
the system in one dimension.

and thanks also for naming a family of intervals after me!!

-monz
"all roads lead to n^0"

----- Original Message -----
From: "M. Schulter" <MSCHULTER@VALUE.NET>
To: <tuning@yahoogroups.com>
Sent: Thursday, September 05, 2002 4:47 PM
Subject: [tuning] Another "modified meantone" circle in 12 -- with 32:25

> Hello, there, everyone, and thanks to the people who have contributed
> to the recent thread on Chaumont, which has helped suggest to me
> another 12-note temperament based in part on 1/4-comma meantone.
>
> Before presenting a Scala file for this temperament, I should briefly
> comment that Chaumont's tuning looks to me like the kind of scheme
> known as a _temperament ordinaire_, popular in France during the later
> 17th and 18th centuries.
>
> The characteristic feature of this type of 12-note temperament is that
> some of the fifths (e.g. Eb-Bb and Bb-F) are tuned _wider_ than pure,
> resulting in some major thirds wider than Pythagorean, or minor thirds
> narrower than Pythagorean.
>
> In some of these schemes, like Chaumont's in the version provided by
> Manuel Op de Coul, there is still something of a "Wolf fifth," for
> example at G#-Eb.
>
> In others, however, all fifths are reasonably close to pure, as in
> Bethisy's temperament ordinaire also included in the Scala archive,
> where all fifths are no more than 1/4 syntonic comma from pure.
>
> Mark Lindley has suggested that the temperament ordinaire may have
> evolved during the 17th century from a kind of creative misreading of
> the instructions by Mersenne for a usual 1/4-comma meantone tuning.
>
> The basic idea is that these instructions call, when tuning up the
> chain of fifths (e.g. C-G-D-A-E-B-F#-C#-G#) for the higher note of the
> fifth to be somewhat lower than a pure 3:2.
>
> However, when tuning fifths in a flatward or "downward" direction on
> the chain, as with Bb-F and Eb-Bb, the lower note of the fifth should
> be _raised_ or tuned "strong" to form a meantone fifth again narrower
> than pure.
>
> In the "creative misreading," some French musicians around the middle
> of the 17th century may have instead tuned Bb-F and Eb-Bb "strong" in
> the opposite meaning of "wider than pure," giving rise to the scheme
> soon recognized and relished as a temperament ordinaire. By the time
> of Couperin, Lindley suggests, music was being written with this kind
> of temperament in mind.
>
> Whether some versions of the temperament ordinaire qualify as 12-note
> "well-temperaments" depends in part on the disposition of the fifths,
> and also in part on how one chooses to define the limits of sizes for
> "acceptable" thirds in a circulating scheme.
>
> In the kind of well-temperament documented by Werckmeister and
> followed by various 18th-century theorists and practical musicians,
> all fifths are either pure or narrow of pure, and thus all major
> thirds no larger than Pythagorean (81:64, ~407.82 cents). In other
> words, they are all within a syntonic comma (81:80, ~21.51 cents) of a
> pure 5:4 (~386.31 cents).
>
> In a temperament ordinaire with a circle of 12 "playable" fifths,
> ranging from around meantone to somewhat larger than pure, certain
> remote major thirds will be larger than Pythagorean, for example up to
> around 413.197 cents (Ab-C or G#-C) in Bethisy's tuning.
>
> Such a major third (or diminished fourth) might be regarded as at
> least "semi-interchangeable" with a more usual major third for the
> period with a size closer to 5:4. If so, then we might speak in
> historical terms of a kind of well-temperament.
>
> Starting from the idea of a temperament ordinaire as a kind of
> "modified meantone," I considered the problem of how many fifths in a
> 12-note tuning one can temper at 1/4-comma narrow while still
> balancing out the circle to keep all fifths comfortably "playable,"
> and preferably no more than 1/4-comma impure.
>
> The solution to which I was drawn is a chain of 8 fifths (F-C#) in
> 1/4-comma meantone, with the remaining 4 fifths each tempered wide by
> around 4.888 cents, or more precisely (2048/2025)^(1/4), to balance
> out the 12-note circle.
>
> The ratio of 2048:2025, a "diaschisma" in one definition of that term
> (another is half of a Pythagorean diatonic semitone or limma at
> 256:243 or ~90.22 cents, giving an interval of ~45.11 cents), is equal
> to the difference between a Pythagorean 81:64 major third and a 32:25
> major third at ~427.37 cents, about 19.55 cents.
>
> Thus we have 8 fifths tempered at 1/4-comma narrow, and the
> remaining 4 fifths tempered at 1/4-diaschisma wide.
>
> Here's a Scala file for this tuning:
>
> ! qcmqd8_4.scl
> !
> F-C# in 1/4-comma meantone, other 5ths ~4.888 cents wide or
(2048/2025)^(1/4)
> 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
>
>
> From a 17th-19th century historical perspective, this circle of 12
> fifths is _not_ a "well-temperament," because the 32:25 major third or
> diminished fourth at C#/Db-F is hardly interchangeable with a major
> third at or reasonably near 5:4.
>
> However, from the 21st-century perspective of a "mixed" style
> combining Renaissance meantone with neo-Gothic sonorities and
> progressions where anything from 81:64 to a bit beyond 9:7 (~435.08
> cents) is routinely accepted as a "major third," this is a kind of
> well-temperament with striking contrasts of "modal color."
>
> In choosing a chain of 8 meantone fifths from F to C#, the idea is at
> once to get an ideally smooth "16th-century" flavor for sonorities
> within this range, and also to make the common thirds Bb-D and E-G#
> not too far from 5:4, although they are inevitably somewhat
> compromised by Renaissance standards.
>
> The major third E-G# can be an especially delicate point for a 12-note
> circle, since this third from around 1520 on is used prominently in
> pieces in various modes, and often appears in closing sonorities in
> the Phrygian mode (E-E).
>
> In this scheme, E-G# and Bb-D are tuned at around 396.578 cents, or
> about 10.265 cents wide of 5:4 -- a bit less than 1/2 syntonic comma.
>
> By comparison, in various well-temperaments of the late 17th-19th
> centuries, E-G# tends to be somewhat wider, often larger than in
> 12-EDO at 400 cents.
>
> Once we move out of the Bb-G# region, of course, things move quickly
> toward Pythagorean and beyond. The thirds Eb-G and B-D#/Eb are close
> to Pythagorean at ~406.843 cents; Ab/G#-C and F#-Bb/A# are at ~407.108
> cents, very close to a just 14:11 (~417.508 cents); and C#/Db-F is at
> a just 32:25.
>
> There are also two minor thirds (C-Eb, Eb/D#-F#) at ~289.735 cents,
> very close to 13:11 (~289.210 cents), and two thirds (F-G#/Ab,
> Bb-C#/Db) at ~279.471 cents, rather close to 27:23 (~277.591 cents).
>
> These "Monzian thirds" are a special adornment to a tuning. This term
> originated from a famous exploit of Joe Monzo, the "Monz," in which he
> started out with 7:6 (~266.87 cents) as a possible size for an
> interval in a composition, but found by ear that around 279 cents
> sounded like the right size, eventually deciding on a rational ratio
> of 75:64 (~274.58 cents).
>
> Thus any third from around 274 to 280 cents can attract the pleasant
> adjective "Monzian."
>
> A curious feature of this "8_4" tuning, in contrast to more
> conventional well-temperaments, is that interval sizes within a given
> category jump in steps of around 10.265 cents, the difference in size
> between the narrow fifths at 1/4 comma (~5.377 cents) smaller than pure
> and the wide fifths at 1/4 diaschisma (~4.888 cents) larger than pure.
>
> Since the tuning uses only these two sizes of fifths, the 10.265-cent
> steps result from the replacement of a narrow meantone fifth by a wide
> one. With major thirds, for example, we have these possibilities with
> "N" showing a narrow fifth and "W" a wide one:
>
> ---------------------------------------------------------------
> Interval Cents Ratio Chain of fifths
> ---------------------------------------------------------------
> A-C# ~386.31 5:4 A E B F# C#
> just N N N N
> ...............................................................
> E-G# ~396.58 5:4 E B F# C# G#
> +~10.27 N N N W
> ...............................................................
> B-D#/Eb ~406.84 81:64 B F# C# G# D#
> -~0.98 N N W W
> ...............................................................
> F#-A#/Bb ~417.11 14:11 F# C# G# D# A#
> -~0.40 N W W W
> ...............................................................
> C#/Db-F ~427.37 32:25 Db Ab Eb Bb F
> just W W W W
> ---------------------------------------------------------------
>
> This scheme, to conclude, is like some versions of the temperament
> ordinaire in combining narrow and wide fifths to form a 12-note circle
> where all of these intervals are reasonably close to pure; but it
> differs in including a major third or diminished fourth as large as
> 32:25, identical to the diminished fourths of 1/4-comma meantone.
>
> Since in the kind of neo-medieval style favored for the more remote
> transpositions, a 32:25 major third is routine, a special kind of
> "well-temperament" results in this musical context.
>
> Most appreciatively,
>
> Margo Schulter
> mschulter@value.net

--- In tuning@y..., "monz" <monz@a...> wrote [#38930]:
> hello Margo,
>
> thanks very much for this post! indeed, Chaumont's tuning
> is a form of "temperament ordinaire" -- and based on the
> information you provided here, my opinion is that this
> term should be included in the Tuning Dictionary as a
> separate family of tunings distinct from "well-temperaments".
>
> the interesting feature that relates them both together
> is that the 5th which is notated as the "wolf" turns out
> to be quite close in size to the "regular" 5ths, which
> has the effect of closing the scale at 12 tones.
>
> in a well-temperament, all 5ths are within a few cents
> of being the same size (pretty close to Pythagorean or 12edo),
> whereas in a _temperament ordinaire_ most 5ths are derived
> from a meantone (quite a bit smaller than Pythagorean) and
> a few (usually two) are quite large.
>
> i agree with Lindley that a misreading may have been the
> original stimulus in creating _temperament ordinaire_ ...
>
> the fact that this plausible misreading of meantone instructions
> results in a tuning which closes at 12 tones -- rather than
> being left open as in meantone -- would be enough to ensure
> some measure of popularity on keyboard instruments with the
> conventional Halberstadt design.

Monz,

This sounds very much like a temperament I came up with in the
early '70s (it appeared in _Xenharmonikon_ #5), and it's nice to see
that there's a name for this particular class of temperaments, even
if the term "ordinaire" is not really very descriptive. I call
it "Secor No. 3", and I happened to email it in a message to Margo
last fall, because I thought that she might be interested in it.

Essentially, it's an alternative to a 12-note meantone tuning (Eb to
G#) that eliminates the wolf fifth, while keeping the intonation of
the eight usable meantone triads significantly better than
Pythagorean -- five of the eight have intonation comparable to
meantone, and the other three are comparable to 12-ET.

A technical description from the Xenharmonikon article follows:

<< Secor No. 3 – meantone fifths (1/4 comma narrow) between C, G, D,
A, E, and B; just fifths between F#, C#, and G# and between Bb and F;
fifths 1/4 comma wide between Ab, Eb, and Bb; fifths narrowed by the
sum of 1/8 comma and 1/2 skhisma between F and C and between B and
F#. >>

This is for a Scala file:

! secor12_3.scl
!
George Secor's closed 12-tone temperament #3 with 5 meantone, 3 just,
and 2 wide fifths
12
!
83.13700
193.15686
292.42357
5/4
501.71015
729/1024
581.18200
785.09200
889.73529
999.75514
1082.89214
2/1

I also published a well temperament in the same issue of
Xenharmonikon. A short excerpt follows:

<< The Secor No. 2 is a 12-tone well temperament that meets all 35 of
the requirements set down by Owen Jorgensen in his specifications for
a "well temperament." It is intended to be usable for diatonic music
in all keys, having no triads more dissonant than Pythagorean triads
(of which there are five). The major triads on F, C, and G have
intonation comparable to meantone temperament, and those on D, A, Bb,
and Eb are comparable in intonation to 12-ET.

Secor No. 2 – meantone fifths (1/4 comma narrow) between G, D, A, and
E, and B; just fifths between B, F#, C#, G#=Ab, Eb, Bb, F, and C;
fifths narrowed by the sum of 1/8 comma and 1/2 skhisma between C and
G and between E and B. >>

And this is for a Scala file:

! secor12_2.scl
!
George Secor's closed 12-tone temperament #2, a well-temperament with
7 just fifths
12
!
256/243
194.86828
32/27
388.02514
4/3
1024/729
698.28985
128/81
891.44671
16/9
4096/2187
2/1

With these two tunings my intention was to provide optimal solutions
for a well-temperament and temperament ordinaire. (Should we
abbreviate these 12-WT and 12-TO, respectively? That would allow me
to do away with the numbers 2 and 3.)

I had my piano in the Secor No. 2 tuning for a number of years. I
played all sorts of things on it and found that I preferred the well
temperament to 12-ET for just about everything.

Just to let you know that things like this do happen intentionally.

--George

> This is for a Scala file:

My previous message had a couple of errors in the Scala file listing
for my "temperament ordinaire." It should have been:

! secor12_3.scl
!
George Secor's closed 12-tone temperament #3 with 5 meantone, 3 just,
and 2 wide fifths
12
!
83.13700
193.15686
292.42357
5/4
501.71015
581.18200
696.57843
785.09200
889.73529
999.75514
1082.89214
2/1

--George

hi George,

> From: "gdsecor" <gdsecor@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, September 10, 2002 6:40 AM
> Subject: [tuning] Yet another "modified meantone" circle in 12
>

> --- In tuning@y..., "monz" <monz@a...> wrote [#38930]:
> >
> > <snip>
> >
> > i agree with Lindley that a misreading may have been the
> > original stimulus in creating _temperament ordinaire_ ...
> >
> > the fact that this plausible misreading of meantone instructions
> > results in a tuning which closes at 12 tones -- rather than
> > being left open as in meantone -- would be enough to ensure
> > some measure of popularity on keyboard instruments with the
> > conventional Halberstadt design.
>
>
> Monz,
>
> This sounds very much like a temperament I came up with in the
> early '70s (it appeared in _Xenharmonikon_ #5), and it's nice to see
> that there's a name for this particular class of temperaments, even
> if the term "ordinaire" is not really very descriptive. I call
> it "Secor No. 3", and I happened to email it in a message to Margo
> last fall, because I thought that she might be interested in it.
>
> <snip>
>
> With these two tunings my intention was to provide optimal solutions
> for a well-temperament and temperament ordinaire. (Should we
> abbreviate these 12-WT and 12-TO, respectively? That would allow me
> to do away with the numbers 2 and 3.)
>
> I had my piano in the Secor No. 2 tuning for a number of years. I
> played all sorts of things on it and found that I preferred the well
> temperament to 12-ET for just about everything.
>
> Just to let you know that things like this do happen intentionally.

hmmm... OK, sure, i do know that all kinds of tunings have been
devised intentionally. i was simply agreeing with the plausiblity
of Lindley's speculation that it was a fortuitous accident as a
result of a mistake, but i didn't say anything about the
possibilities of a _temperament ordinaire_ being designed
deliberately, so thanks for adding your comments.

i've been using "MT" as an abbreviation for "meantone" and
"WT" for "well-temperament" for a few years now, and see no
reason why "TO" couldn't be used for "temperament ordinaire".

-monz
"all roads lead to n^0"

George Secor wrote:

>My previous message had a couple of errors in the Scala file listing
>for my "temperament ordinaire." It should have been:

With the latest version it's quite easy to calculate these
temperaments. It's also checked which tones are just.
For your #3 you enter this:

pyth/temp
12
2
3/2
$k
1/4
1/4 (or default is the previous size)
1/4
1/4
1/4
x
0
0
-1/4
-1/4
0
x

Manuel

--- In tuning@y..., manuel.op.de.coul@e... wrote:
> With the latest version it's quite easy to calculate these
> temperaments. It's also checked which tones are just.
> For your #3 you enter this: ...

Thanks for this information.

After the better part of a year I still haven't tried Scala -- too
busy with ongoing projects -- but I did download it and install the
latest version a few weeks ago. Maybe soon.

(And Paul Erlich, if you're out there: I've printed out the latest
version of your 22-tone paper and have started reading it. It will
take some time to digest everything, but I'll be getting back to you
with my comments, hopefully soon.)

--George

George Secor wrote:
>After the better part of a year I still haven't tried Scala -- too
>busy with ongoing projects -- but I did download it and install the
>latest version a few weeks ago. Maybe soon.

Good, when you do, please don't miss
Edit->Options...->Notation system->Select...

Manuel