Yahoo Tuning Groups Ultimate Backup tuning enharmonics in meantone

> (...) in 1/4-comma
> meantone, Cbb does indeed come very close to one of
> the other notes already in the scale ... it's not
> Bb, but it's Ax.
>
> anyway, the "awkward" double-sharps and double-flats
> occur near the points where the basic diatonic scale
> only has semitones instead of whole-tones, i.e.,
> between E-F and B-C.
>
> so, if i put those "awkward" double-sharps and
> double-flats in brackets, the whole system looks
> like this:
>
> A - B - C - D - E - F - G
> A# - B# - C# - D# - E# - F# - G#
> Ab - Bb - Cb - Db - Eb - Fb - Gb
> Ax - [Bx] - Cx - Dx - [Ex] - Fx - Gx
> Abb - Bbb - [Cbb] - Dbb - Ebb - [Fbb] - Gbb

But from about the middle of the 17th century (yes, 17th), 1/4 comma and 1/5 comma meantone had yielded in common practice to 1/6 (having tritone at or near 45/32). There is nothing awkward about Bx; it's a Pythagorean comma higher than C#, which in turn is a Pyth comma higher than Db. What's the problem? :)

The "awkwardness" here comes from thinking inside the paradigm of 1/4 comma instead of the lighter meantone schemes. [Been there, done that; I had a very long period of 1/4 comma enthusiasm myself, in both its pure and many modified forms, until its vicissitudes finally annoyed me enough to switch.]

Tosi (c1646-1732), writing in 1723 near the end of his long career, instructed singers to preserve that comma distinction in pitch; this was nothing new. It's normal to 1/6 syntonic comma meantone; each time around the circle, we gain or lose a Pyth comma. No big whoop, once the sound is in the ears, and it's sitting right there on any ordinary 1/6 comma meantone keyboard. That's why he and some contemporaries found it easiest to present the 55edo model, as a way of explaining this already common-practice stuff to the mathematically less astute. A tone is built of 9 pieces of 55edo; a chromatic semitone is 4 pieces and a diatonic semitone is the other 5. It's 1/6 comma meantone, plain and simple, in terms that people could understand without knowing fractions. And those little pieces themselves are essentially the comma anyway, so it all boils down to differentiating enharmonics by one comma each, with only the simplest integer arithmetic. If the music says A# in the vocal part, sing that note one comma below the keyboard's Bb. That puts your A# at 392 cents against F#: the normal size of major thirds in common everyday meantone tuning (all the correctly-spelled major thirds on the keyboard having that everyday sound of 392 cents). [The sensitive keyboardist doesn't insist on playing a competing Bb there against the proper A#, being off by a comma! F#-Bb is still a wolf diminished fourth.]

I've also seen references to 45/32 in an Italian keyboard document from 1676, but I can't say more for sure until I've got better copies from the microfilm to be sure I'm seeing what I think I'm seeing in it; suffice it to say that 1/6 comma was already a norm. It sounds perfectly fine, really wonderful, as long as one sticks to notes that are spelled correctly!

Werckmeister's shtick c1690 was to offer ways to convert the old-fashioned and restricted 1/6 comma meantone into more circulating methods: keeping the axis C and F# where they were (near 45/32) and moving the other notes up or down. "Werckmeister III" keeps four notes very near their 1/6 comma positions (C, F#, and slightly raised B and lowered G). His IV keeps five notes (Bb, C, D, E, F#) of 1/6 comma and fusses around with the other seven. His V is a squared-off version of 1/8 comma meantone, de-regularizing it. All three of these were low-budget methods to convert existing organs out of everyday meantone, making minimal changes to the pipes (and mostly flatward: i.e. adding material to existing pipes if necessary, to lower the pitch a little bit, but not cutting into existing pipes to shorten them). [The addition of material is a somewhat reversible procedure, but cutting isn't....]

Werckmeister's musical choices in IV and especially in III have done so much unfortunate violence to tonal music, both from 1690-c1775 and c1960 to now...all (at the root of it) from keeping the F# as low as 45/32! So many of the notes, frankly, are grossly flat...sort of like the sound of unsupported singing by a weak choir. Instead of seeking to overturn the classic meantone restrictions for 12-key keyboards, with anything more ideal for music, he merely duct-taped over the most obvious problems without really solving them. Then those who have emulated his work (because it was so widely influential and fashionable) fell into the same silly traps, not perceiving the underlying shapes and issues well enough.

It's been even worse with the taking of his 1/4 comma prescriptions (in III) as any strong theoretical endorsement of 1/4 C per se, instead of recognizing it as a bastardized version of 1/6. 1/4 was already out of the picture by that point. "Werckmeister III" is basically "squaring off your 1/6 C meantone with as many pure fifths as possible, For Dummies and for churches unwilling to invest in a complete overhaul"...without really extending the range of usable keys at all, but merely watering down their offensiveness in some musical situations. In W-III and its offshoots one is still stuck within the Eb to G# set of 12 notes; all he did was to make the sorties outside that range a bit less garish. Ab, Db, Gb, and Cb are the big losers, but D# and A# and E# are pretty rotten, too.

As Werckmeister himself cleared up later, 12edo and its sound-alikes are better for music than those stopgap conversion-solutions are; but some threads of music history got too stuck on [misinterpretations of] his publications of the 1690s. And the three temperaments of his colleague Bendeler, also from c1690, are musically more advantageous anyway, as for playing music that circulates into the weirder enharmonics. Bendeler just hasn't had the entourage and household-name status that big W has. (Anybody ever seen an electronic tuner that has the excellent "Bendeler III" pre-programmed into it?)

I have more to say about this, sometime, but this isn't the time and place for it.

Brad Lehman

🔗Brad Lehman <bpl@umich.edu>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Brad Lehman <bpl@umich.edu>

> > The "awkwardness" here comes from thinking inside the paradigm of
> 1/4 comma
> > instead of the lighter meantone schemes. [Been there, done that;
I
> had a
> > very long period of 1/4 comma enthusiasm myself, in both its pure
> and many
> > modified forms, until its vicissitudes finally annoyed me enough
to
> switch.]
>
> What do you regard as the vicissitudes? Some problems can be reduced
> by sharpening the fifth, and some by flattening it. Sometimes these
> are the same problems, eg the wolf fifth and sharp major thirds cum
> diminished fourths in the remote keys. Sharpening the fifth, which
> moves the meantone closer to 12-equal, is one way to do it. Moving
> your thinking to higher prime limits and your fifth to 2/7-comma or
> 50-equal is another.

What annoys me most about 1/4 comma meantone?

- The just 5/4 thirds, while very attractive in themselves, are
static and eventually dull. They make the music stop and stop and
stop, instead of continuing to move forward.

- Melodic bumpiness. 117 cents is too wide for a diatonic semitone
in melody; again it generates stasis instead of moving the music
forward, plus it simply feels wrong to sing (even after using them
for 20 years, regularly). Sure, there's attractiveness in having the
strong contrast with the chromatic semitone of 76 cents; their
character is obviously different. But again, all this is so bumpy in
melody, and especially if we ever run into any notes that are spelled
incorrectly. Musical expression is reduced to the subtlety of a
blowtorch. The temperament draws too much attention to itself.

It sounds a bit more interesting in the French and Italian offshoots
of it. And of course it sounds fine in most of the Fitzwilliam
Virginal Book and other repertoire that early; but once we get into
the 17th century any appreciable distance, in its pure form it just
doesn't work so well anymore.

Brad Lehman

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:

> What annoys me most about 1/4 comma meantone?
>
> - The just 5/4 thirds, while very attractive in themselves, are
> static and eventually dull. They make the music stop and stop and
> stop, instead of continuing to move forward.

Normally you hear the pure 5/4 as an element of a chord, and as an
element of a chord it does not seem to me to possess the
characteristic quality of pure JI; it does impart to the chords a
certain quality, just as a pure fifth with a tempered third will, but
it isn't a JI quality.

> - Melodic bumpiness. 117 cents is too wide for a diatonic semitone
> in melody; again it generates stasis instead of moving the music
> forward, plus it simply feels wrong to sing (even after using them
> for 20 years, regularly).

It seems to me you might with more force complain that 100 cent
semitones are melodically bland beyond endurance, giving 25/24 and
16/15 the same size. 1/6 comma meantone has gone a long way in this
flabby direction, with a 16/15 of 108 cents and a 25/24 of 89 cents.
In comparison, the JI values are 112 cents and 71 cents. Melodically
you might draw the line at a pure 16/15, which is exactly 1/5-comma
meantone, and which 43-equal does a good job for.

Sure, there's attractiveness in having the
> strong contrast with the chromatic semitone of 76 cents; their
> character is obviously different. But again, all this is so bumpy in
> melody, and especially if we ever run into any notes that are spelled
> incorrectly. Musical expression is reduced to the subtlety of a
> blowtorch. The temperament draws too much attention to itself.

I've listened to things in meantones a lot flatter than 1/4-comma
without noticing any "bumpiness", though clearly the melodic character
changes. Even 19-equal, which makes 16/15 twice as large as 25/24,
doesn't strike me as "bumpy". From a diatonic scale point of view, the
shoe is on the other foot anyway; you are comparing steps whose size
is a tone and a semitone, and as the fifths get flatter, these become
more alike, not less. The diatonic scale in 19-et, where the tone is a
mere 50% larger than the semitone, is really the opposite of bumpy;
and even if we are comparing 12-note MOS, which is in effect what we
started out doing, the 1 and 2 step scale sizes in 19 are just like
what a diatonic scale looks like in 12--is that bumpy?

> It sounds a bit more interesting in the French and Italian offshoots
> of it. And of course it sounds fine in most of the Fitzwilliam
> Virginal Book and other repertoire that early; but once we get into
> the 17th century any appreciable distance, in its pure form it just
> doesn't work so well anymore.

Funny, since that's mostly what they wrote it for.

🔗Brad Lehman <bpl@umich.edu>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:
>
> > What annoys me most about 1/4 comma meantone?
> >
> > - The just 5/4 thirds, while very attractive in themselves, are
> > static and eventually dull. They make the music stop and stop
and
> > stop, instead of continuing to move forward.
>
> Normally you hear the pure 5/4 as an element of a chord, and as an
> element of a chord it does not seem to me to possess the
> characteristic quality of pure JI; it does impart to the chords a
> certain quality, just as a pure fifth with a tempered third will,
but
> it isn't a JI quality.

I agree; in triads the fifth livens things up nicely. But, for me,
it
still doesn't overcome the stasis of the thirds....

>
> > - Melodic bumpiness. 117 cents is too wide for a diatonic
semitone
> > in melody; again it generates stasis instead of moving the music
> > forward, plus it simply feels wrong to sing (even after using them
> > for 20 years, regularly).
>
> It seems to me you might with more force complain that 100 cent
> semitones are melodically bland beyond endurance, giving 25/24 and
> 16/15 the same size. 1/6 comma meantone has gone a long way in this
> flabby direction, with a 16/15 of 108 cents and a 25/24 of 89 cents.
> In comparison, the JI values are 112 cents and 71 cents. Melodically
> you might draw the line at a pure 16/15, which is exactly 1/5-comma
> meantone, and which 43-equal does a good job for.

I agree that 100c semitones are bland beyond endurance; they've lost
all functionality since there's no distinction between diatonic and
chromatic anymore.

But, I feel the broader issue here is: too much of ANY consistency (i.
e. complete regularity of the scale) becomes quickly bland and dull,
even if it's a harmonious set of attractive intervals. Without a
slight irritant of irregularity, oysters don't make pearls.

Brad Lehman

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:

> They wrote it for regular 1/4 comma meantone? That's the party-line
> view, due (I suspect) in large part to Barbour and some of his
> misconceptions that have been carried forward (and into most
> recordings of that repertoire, from the past 30 years or so). But
> it's not so. The lighter regular systems and the irregular ones
> (i.e. the role of tasteful adjustment to regularity) were much more
> common in the 17th century than general historiography has led us to
> believe.

I'm hardly able to argue history with anyone, but I would like to nail
this down--actual instances of irregular 17th century tunings would be
nice. Monz pointed out recently that Huygens called 1/4-comma
"ordinary temperament", or simply "temperament", and that suggests
that in his acutely mathematical mind 1/4-comma was firmly established
as the temperament musicians actually used.

The repertoire itself makes that clear. Play through all
> of Froberger's music, especially, and John Bull's "Ut re mi fa sol
> la" (piece #51 in the Fitzwilliam).

I think arguing from what chords seem to be used to what the tuning
must have been is inherently speculative. As I remarked, there is more
than one way of treating the extreme keys of meantone, and that
Zarlino and Salinas discussed flat systems suggests that at one time
(late 16th century, anyway, and that's really Bull territory also)
these might have been in common use. This I find interesting because
2/7 comma has some very interesting properities when you get to the
remote keys in twelve notes. I'd like to know if these properties ever
came into play in actual music.

It's been suggested to me before that the Bull piece would be
interesting for me to work with, but I couldn't find a midi for it;
this time I found one playing as background music for looking at
paintings, so I'll take a look at it, though of course I'm hardly
going to play it through.

🔗Manuel Op de Coul <manuel.op.de.coul@eon-benelux.com>

🔗Brad Lehman <bpl@umich.edu>

🔗Manuel Op de Coul <manuel.op.de.coul@eon-benelux.com>

🔗Brad Lehman <bpl@umich.edu>

> > No. Bendeler III is:
>
> Thanks. Curiously, it's close to one of Werckmeister's
> temperaments! No. 5, transposed by a whole tone.

No, it's not. Werckmeister V has five 1/4 Pythagorean comma fifths,
and one wide fifth, and six pure. Furthermore, the pattern's not
quite the same as Bendeler III anyway, even if we straighten out the
spot where the narrow and wide fifths are neighboring.

Bendeler III (1690):
Ab -1/4 Eb 0 Bb 0 F 0 C -1/4 G -1/4 D 0 A 0 E -1/4 B 0 F# 0 C# 0 G#

Werckmeister V (1691):
Ab +1/4 Eb 0 Bb 0 F -1/4 C 0 G 0 D -1/4 A -1/4 E 0 B 0 F# -1/4 C#
-1/4 G#

If we straighten out C# 0 G# 0 Eb there at the ends instead of having
the G# lowered, and shift over by a whole tone, we still don't have a
mapping onto the Bendeler. There are different numbers of pure
fifths in sequence.

=====

There is a Werckmeister/Bendeler connection elsewhere, though;
"Werckmeister III" is a shaved version of "Bendeler I" using 1/4
comma instead of 1/3. (And "Kellner" fabricated in 1975 is a 1/5
re-shaving of Werck III, and "Barnes" in 1979 is a 1/6 re-shaving of
Kellner...all having a similar layout with one tempered fifth in B-F#
and the rest in sequence in the naturals around C. Barnes had also
fooled around with an entirely different one, some years earlier.) Of
course Bendeler and Werckmeister knew one another; Bendeler was the
cantor at Quedlinburg while Werckmeister was the organist.

Bendeler I:
Ab 0 Eb 0 Bb 0 F 0 C -1/3 G -1/3 D 0 A 0 E 0 B -1/3 F# 0 C# 0 G#

Werckmeister III:
Ab 0 Eb 0 Bb 0 F 0 C -1/4 G -1/4 D -1/4 A 0 E 0 B -1/4 F# 0 C# 0 G#

Kellner:
Ab 0 Eb 0 Bb 0 F 0 C -1/5 G -1/5 D -1/5 A -1/5 E 0 B -1/5 F# 0 C# 0 G#

Barnes:
Ab 0 Eb 0 Bb 0 F -1/6 C -1/6 G -1/6 D -1/6 A -1/6 E 0 B -1/6 F# 0 C#
0 G#

Brad Lehman

🔗Manuel Op de Coul <manuel.op.de.coul@eon-benelux.com>

🔗Brad Lehman <bpl@umich.edu>

> >No, it's not. Werckmeister V has five 1/4 Pythagorean comma
fifths,
> >and one wide fifth, and six pure.
>
> A substantial difference indeed. Still if you transpose one of
> them by a tone, then 10 tones are exactly the same, and 2 differ
> by 1/4 P. comma.

True. But that operation of tranposition by a tone makes a huge
difference in *effect* vis-a-vis tonal music, reorganizing the
relationships of the keys and any Affekt they might contribute to;
and
it moves almost all the placements of notes for everybody else
(non-keyboard) who is trying to play and sing with it.

Granted, Bendeler III in Chorton would sound pretty much like
Werckmeister V in Cammerton, much of the time. But, string players
would mutiny at the placement of tight 1/4 comma fifths in their
D-A-E. It makes their use of open strings unreliable.

And, in the operation mentioned above (comparing Bendeler III in D
with Werckmeister V in C), Werckmeister has done two things typical
for him: ruin the note Ab to give a calmer G# (thereby making Ab-C
Pythagorean), and ruin E# to give a calmer F-A (thereby making C#-E#
Pythagorean). These are not improvements for the behavior of
circulating tonal music; they're steps backward in quality for music
that uses enharmonics.

It may not seem like such a big deal to crank F up a 1/4 comma, and
G#
down a 1/4 comma; but it is, in practical situations. Every such
move
of dinking a single pitch up or down has repercussions in multiple
other places, both in melody and harmony, because it's a closed
system
and all the pitches are doing multiple duty within complex
relationships. Each little move creates some equal and opposite
reaction somewhere else. Here, the motion of both F and G# creates a
fourth (especially small) size of semitone where Bendeler had only
three; and a fourth (especially large) size of tone where Bendeler
had
only three. That stuff "sticks out like sore thumbs" on close
hearing
in melodic contexts, and in the playing of Db major and Ab major
triads. Eww.

Brad Lehman

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@tonalsoft.com>

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

> --- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:
>
> > But, string players would mutiny at the placement
> > of tight 1/4 comma fifths in their D-A-E. It makes their
> > use of open strings unreliable.
>
> Does anyone know if string players have always tuned
> pure fifths? It doesn't make a lot of sense to do that
> during the meantone era. When Leopold Mozart discussed
> meantone violin method, for instance, what was said,
> if anything, about open string tunings?

that's a question that i've had for a long, long time.

it would make sense for string players to tune their
open strings to tempered 5ths during the "meantone era"
... but AFAIK the history of string instruments argues
strongly against it, where open strings are almost always
"pure" "4ths" (4:3s) or "5ths" (3:2s), with the occasional
"3rd" thrown in as on the modern standard guitar tuning.

part of the problem guitar players still have today is
that to be in tune with the frets, the open strings have
to be tuned to tempered "4ths" and a tempered "3rd".
it's easy to tune the "4ths" by ear as 4:3s, but then
the open strings are not in tune with the fretted versions
of those same notes.

-monz

🔗Carl Lumma <ekin@lumma.org>

🔗Aaron K. Johnson <akjmicro@comcast.net>

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@...> wrote:

> Kellner:
>Ab 0 Eb 0 Bb 0 F 0 C -1/5 G -1/5 D -1/5 A -1/5 E 0 B -1/5 F# 0 C# 0 G#

Kellner claims his patented chimera as
http://plaza.ufl.edu/wnb/baroque_temperament.htm
allegations in TUs (1/720 part degree of the PC):
Ab Eb Bb F C -144 G -144 D -144 A -144 E B -144 F# C# G#

in competing with Brad's wild speculations:
Ab -60 Eb -60 Bb +60 F-120 C-120 G-120 D-120 A-120 E B F# C# -60 G#

Procedure:
Add both of them and cut the result into half.
That yields the middle inbetween the two above obsolete forerunners:

Ab
-30 := (0 -60)/2 ; PC^(-1/24)
Eb
-30 := (0 -60)/2 ; PC^(-1/24)
Bb
+30 := (0 +30)/2 ; PC^(+1/24)
F
-60 := (0 -120)/2 ; PC^(-1/12)
C
-132 := (-144 -120)/2 ; PC^(-11/60)
G
-132 := (-144 -120)/2 ; PC^(-11/60)
D
-132 := (-144 -120)/2 ; PC^(-11/60)
A
-132 := (-144 -120)/2 ; PC^(-11/60)
E
B
-72 := (-144 -0)/2 ; PC^(-1/10)
F#
C#
-30 := (0 -60)/2 ; PC^(-1/24)
G#

with PC:= 3^12/2^19 = 531441/524288 , the Pythagorean-Comma

Disclaimer:
All that three tunings mentioned above,
consist altogehter in modern phantasms,
without the slightest historically relevance
for any imputed Baroque composer,
alike the insustainable
wishful thinking and
the unduly-ado about that, as in:

http://www.gcmusiccenter.org/php/facility/special.features/organ.php
Quote:
"This organ is the first since the 18th-century to use a proposed
reconstruction of Johann Sebastian Bach's tuning.....

There is no "evidence" that JSB tuned that way.

Awake from yours selfproclaimed smoking organ-pipe-dream nightmare.

"...Several other people had suggested previously that the drawing
means something else non-decorative,...."
In deed:
Probably JSB meant the smoky-curly-squiggles of Cantata BWV515a:
http://www.cs.ualberta.ca/~wfb/cantatas/515a.html
http://de.wikipedia.org/wiki/Knaster
" Erbauliche Gedanken eines Tobackrauchers [Bearbeiten]

aus "Notenbüchlein für Anna Magdalena Bach" (1725)

So oft ich meine Tobacks-Pfeife,
mit gutem Knaster angefüllt,
zur Lust und Zeitvertreib ergreife,
so gibt sie mir ein Trauerbild
und füget diese Lehre bei,
daß ich derselben ähnlich sei...."

http://www.tobis-notenarchiv.de/bach/22-Sammlungen/Anna-Magdalena2/20c.htm

Partially bilingua version (german/english) at the end of page:
http://fujipub.com/ooops/bsoct95.html
"5. DOTTLE-
Odds and ends to amuse
Erbauliche Gedanken eines Tobackrauchers
Constructive thoughts of a Tobacco smoker
author unknown....................."

bye
A.S.