"true" meantone?

> > as it turns out, modern Euro-centric harmonic theory
> > evolved within a meantone context, and the particular
> > variety of meantone was generally assumed to be
> > 1/4-comma (which is the "true" meantone tuning -- i.e.,
> > it really does have a "tone" [major-2nd] which is
> > exactly midway in pitch between the two JI major-2nds
> > of ratios 10/9 and 9/8; other "meantones" are approximately
> > mean but not exactly).

I have to speak up about an issue of nomenclature here.......

There's an infinite set of regular and exact meantone layouts available to 12-note keyboards. 1/4 syntonic comma merely happens to be the interesting case where a *pure* major third is the thing being bisected, by geometric mean. SQRT(5/4). The "mean" happening here (for the nomenclature) doesn't have anything to do with splitting 10/9 and 9/8. D is the geometric mean between C and E, exactly. It's a "mean tone", two of which make up a major third, whatever size that major third happens to be. Any interval can be bisected, the same way that any angle on paper can be bisected. [Whether that point is also mean between two JI positions is irrelevant, since those JI are not in the scale!] All we're deriving here is the square root of whatever interval is our major third by definition.

Demonstration on a harpsichord: set up any C-E of your choosing (as long as it still sounds like a major third!). Tune C down to G as a pure 4:3. Tune E down to A as a pure 3:2. Tune D (diatonic step between the original C-E) from both G and A, such that the beat rate of the fifth G-D is exactly 2/3 the beat rate of the fourth A-D. D is therefore constructed exactly mean between C and E, geometrically, being a mean tone--half the distance from C to E.

That D can then be used in turn to construct the proper G and A above the existing notes. Put the G so the C-G fifth beats 2/3 as fast as the D-G fourth. Put the A so the D-A fifth beats 2/3 as fast as the E-A fourth. Retune our previous (temporary) G and A to match these, as pure octaves. Now we have C-G-D-A-E all in a regular meantone layout (exactly) defined by whatever size the original C-E third happened to be.

That beat-rate technique always works (listening to the simplest partials of the tone, as we harpsichord tuners do). [On harpsichord the inharmonicity of the strings is negligible this low in the harmonic series.] Wherever x, y, and z are three notes in positions 1, 2, and 5 of a diatonic scale: when x to z beats exactly 2/3 as fast as the beating of y to z, the fifths x to z to y are equal in size (geometrically). That is, all three notes are elements of some meantone set.

So, knowing the positions of any two of the three notes, one can derive the third note by moving it until the beat relationship is correct. For example, if we have already established the fifth A and E in whatever meantone we're setting on the keyboard, we can get the B between them by moving it until the fourth B-E beats 3/2 as fast as the fifth A-E. Then we can use this B and the E a fifth below it to set up our F#, and etc etc etc etc as far around the circle as we want to go...all generated by the original size of our C-E major third plus this single operation (the sesquialtera beat rate) to bisect it. All these are regular and exact "meantone" tunings: every note exactly splits the correctly spelled major third around it, and all the fifths are the same size as one another.

All the regular meantone tunings also have a similar musical character to one another: the difference being of degree, not shape. How fast do the resulting major thirds beat (might be 0, or might be wide, or might be narrow)? If our "major third" happens to something ludicrously large such as 420 cents or as small as 350 (or whatever), making the whole thing laughably bad for tonal music, an exact meantone layout can still be derived. C-E doesn't have to start at 386.3137138648 (or whatever) cents. It could be any rational or irrational value anywhere in the range "sounds like a major third" [which I haven't bothered to calculate....]...obviously an infinite set there. And that's apart from considerations such as choosing G# instead of Ab, making a bigger infinity!

Back to my quip above about the JI 9/8 and 10/9 not being in the scale. Start with some random C-E, for example 390 cents (*). One can tune D as a 9/8 from C, but then D to E is not 10/9; or one can tune the D as 10/9 from C but then D to E is not 9/8. They don't both fit, because we didn't start with 5/4. (Truisms here, all.) So, if we stick the meantone D of 195 cents in there, it matters not one whit that it happens not to bisect 10/9 and 9/8, and not to be half a syntonic comma away from either of them, for that matter...they can't both be in that same room at the same time anyway!

=====

To boil all this down and recap: "meantone" doesn't mean "half a syntonic comma away from 9:8 and/or 10:9"; but only that there's no such thing as a major tone and a minor tone available anymore, since we've split the major third (whatever size it is) by square root (geometric mean) instead of anything to with JI superparticular ratios. Instead of those two other expected tones with some claims to purity, we have a mean tone.

Where our major third is not 5:4, and we're deciding where to put the intervening note, the tones either have to be a major tone plus garbage, or a minor tone plus garbage, or two carefully regulated and equal pieces of garbage, or two sloppy pieces of garbage. "Meantone" charts that third course there, controlling the impurity and turning it into the melodic virtue of smoothness. Do-Re and Re-Mi become the same size. So do Fa-Sol, Sol-La, and La-Ti along with those. Therefore it becomes very easy for music to modulate (or "mutate" to use the older term from hexachord parlance) by moving from one diatonic scale to a different one, where the intervals will be the same sizes. No bumpiness, no foreign exchange rates. Need to transpose the whole piece? Merely move it over, and then make sure we don't go off the edge into the wrong enharmonic set of accidentals.

=====

(*) Major third of 390 happens to be between the major thirds of 1/5 syntonic comma and 1/5 Pythagorean comma meantone, but whatever it is, it can be split. There's nothing magical about simple fractions of any particular comma; it's all a continuum. Numbers are merely measurements. There's nothing magical about cents, either. All the regular meantones can be set up very easily by ear, using the procedure above. Establish one major third of whatever size one's taste says sounds good; doesn't even need to be measured by any machine, necessarily. Then whack it into equally-sized four fifths in between, and keep going with that same size of fifth until there are twelve different notes. All the resulting major thirds (spelled correctly) will be the same size as the original one; all the tones will be mean within their major thirds; a regular meantone tuning. There's no need to *count* beats to any specified standard of time; merely to observe that they are 3-against-2 of some other beat rate nearby. Triplets against duplets, like in music.

Regular 1/3, 1/4, 1/5, 1/6, 1/7, 1000/7001, 0.237563, etc etc comma meantone (whichever comma) are all manifestations of the same shape, varying only by intensity of the same character. Merely pick your size of generating fifth, or your size of generating major third, and everything else follows automatically. Two notes automatically become twelve, or more if your instrument has split keys. The lighter the fifth-tempering gets, the more one can start cheating by playing enharmonically wrong notes and having them serve more or less decently as one another.

=====

Anyway, such a debate about the proper use of the term "meantone" has been going on for centuries already, and will probably continue to do so after we're all gone. I'm in the camp that define it as I've described above. SQRT(5/4) is merely one special case within an infinite set...namely the case where the tone happens to bisect 9:8 and 10:9 also, as coincidence, but not the case that merits the restrictive name "meantone".

Meantone, to me, is a shape; not only that single point. The splitting of major thirds equally is the main thing. If it happens to come into eclipse with the splitting of just-intonation intervals (a completely different game) at one coincidental point, so be it; but I don't see why we should use the word "meantone" to mean only that eclipse point, when the special case's nomenclature "1/4 syntonic comma meantone" describes that single point so nicely within the whole family of possibilities, and presents the whole family at once. The whole family sounds similar, varying only by intensity.

If somebody wants to invent a special keyboard that is able to play all three of the equally spaced notes 9:8, SQRT(5/4), 10:9 above a given tonic, fine, but that has nothing to do with the diatonic music I care about or can play on my harpsichord. If I've tuned a meantone layout onto it, my G is exactly between F and A, being a mean tone between them. I needn't tell anyone what size the interval F-A is; it's still a meantone tuning. D is mean between C and E. It's mean within the diminished fifth B-[D]-F. D also happens to be mean within the augmented fifth Bb-[D]-F#. D is mean within the minor seventh A-[D]-G, and within the major ninth G-[D]-A, and so forth; and within the diminished third C#-[D]-Eb. Meantone-tuned notes are exactly halfway between other notes that are also available to be played; that's why it's called "meantone".

Brad Lehman

hi Brad,

--- In tuning@yahoogroups.com, Brad Lehman <bpl@u...> wrote:
>
> > > as it turns out, modern Euro-centric harmonic theory
> > > evolved within a meantone context, and the particular
> > > variety of meantone was generally assumed to be
> > > 1/4-comma (which is the "true" meantone tuning -- i.e.,
> > > it really does have a "tone" [major-2nd] which is
> > > exactly midway in pitch between the two JI major-2nds
> > > of ratios 10/9 and 9/8; other "meantones" are approximately
> > > mean but not exactly).
>
> I have to speak up about an issue of nomenclature here.......
>
> There's an infinite set of regular and exact meantone
> layouts available to 12-note keyboards. 1/4 syntonic comma
> merely happens to be the interesting case where a *pure*
> major third is the thing being bisected, by geometric
> mean. SQRT(5/4). The "mean" happening here (for the
> nomenclature) doesn't have anything to do with splitting
> 10/9 and 9/8. D is the geometric mean between C and E,
> exactly. It's a "mean tone", two of which make up a major
> third, whatever size that major third happens to be.
> Any interval can be bisected, the same way that any angle
> on paper can be bisected. [Whether that point is also
> mean between two JI positions is irrelevant, since those
> JI are not in the scale!] All we're deriving here is the
> square root of whatever interval is our major third by
> definition.
>
> <big snip>

good summary. thanks for straightening out something
that was misleading the way i wrote it.

-monz

Brad Lehman wrote:
>There's an infinite set of regular and exact meantone layouts available to

>12-note keyboards. 1/4 syntonic comma merely happens to be the
interesting
>case where a *pure* major third is the thing being bisected, by geometric
>mean. SQRT(5/4). The "mean" happening here (for the nomenclature)
doesn't
>have anything to do with splitting 10/9 and 9/8.

I disagree. Mathematically it's exactly the same. And theorists like
Salinas write that the difference between 10/9 and 9/8 (tonus minor and
tonus maior), the syntonic comma, is split in two, hence the meantone
(tonus aequalis) is the middle between 10/9 and 9/8.

Manuel

--- In tuning@yahoogroups.com, "Manuel Op de Coul" <manuel.op.de.
coul@e...> wrote:
>
> Brad Lehman wrote:
> >There's an infinite set of regular and exact meantone layouts
available to
>
> >12-note keyboards. 1/4 syntonic comma merely happens to be the
> interesting
> >case where a *pure* major third is the thing being bisected, by
geometric
> >mean. SQRT(5/4). The "mean" happening here (for the nomenclature)
> doesn't
> >have anything to do with splitting 10/9 and 9/8.
>
> I disagree. Mathematically it's exactly the same. And theorists like
> Salinas write that the difference between 10/9 and 9/8 (tonus minor
and
> tonus maior), the syntonic comma, is split in two, hence the
meantone
> (tonus aequalis) is the middle between 10/9 and 9/8.
>
> Manuel

As I pointed out, but maybe didn't say very well: 1/4 SC is the only
meantone where the splitting (such as placing D between C and E)
happens to be a split of both the major third and the 10/9 to 9/8.
Sort of an "eclipse" of two systems, interlocked. In *all* the
regular layouts, D is placed exactly between C and E, no matter what
distance that original C to E was. 10/9 and 9/8 come into the
picture
(from this perspective) only if that distance C to E was 5/4.

In all the other regular systems, the ones I refer to (maybe loosely)
as "meantone" since their tones are mean within whatever major third
is available, since 5/4 isn't available either, the 10/9 and 9/8 are
moot.

Yes, it's true that when we split the 81/80 (i.e. syntonic comma) we
get its square root which happens to be mean between 10/9 and 9/8.
But, since both 10/9 and 9/8 do not make an appearance on a regular
1/4 comma keyboard, it's moot (to me) whether they're being split or
not. That's from some other orbit of JI intervals and the
relationships among them.

If it happens to make an eclipse (intersect) with the other set of
all
regular layouts, at that one particular size of fifths generated from
the 5/4 third, so be it; but in most of the practical keyboard music
from c1500-c1900 we care only about the regular layouts themselves,
not about the JI placements.

The set of JI frequencies (rational numbers) is so irregularly
spaced,
anyway! At what point do we decide that they stop being interesting,
and therefore stop including them in our enumeration of the set?
When
their numerators or denominators have more than two digits in them?

Brad Lehman

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:
> --- In tuning@yahoogroups.com, "Manuel Op de Coul" <manuel.op.de.
> coul@e...> wrote:
> >
> > Brad Lehman wrote:
> >
> > > There's an infinite set of regular and exact meantone
> > > layouts available to 12-note keyboards. 1/4 syntonic
> > > comma merely happens to be the interesting case where
> > > a *pure* major third is the thing being bisected, by
> > > geometric mean. SQRT(5/4). The "mean" happening here
> > > (for the nomenclature) doesn't have anything to do with
> > > splitting 10/9 and 9/8.
> >
> > I disagree. Mathematically it's exactly the same. And
> > theorists like Salinas write that the difference between
> > 10/9 and 9/8 (tonus minor and tonus maior), the syntonic comma,
> > is split in two, hence the meantone (tonus aequalis) is
> > the middle between 10/9 and 9/8.
> >
> > Manuel
>
>
> As I pointed out, but maybe didn't say very well: 1/4 SC is
> the only meantone where the splitting (such as placing D
> between C and E) happens to be a split of both the major third
> and the 10/9 to 9/8. Sort of an "eclipse" of two systems,
> interlocked. In *all* the regular layouts, D is placed
> exactly between C and E, no matter what distance that original
> C to E was. 10/9 and 9/8 come into the picture (from this
> perspective) only if that distance C to E was 5/4.
>
> In all the other regular systems, the ones I refer to
> (maybe loosely) as "meantone" since their tones are mean
> within whatever major third is available, since 5/4 isn't
> available either, the 10/9 and 9/8 are moot.
>
> Yes, it's true that when we split the 81/80 (i.e. syntonic comma)
> we get its square root which happens to be mean between 10/9
> and 9/8. But, since both 10/9 and 9/8 do not make an appearance
> on a regular 1/4 comma keyboard, it's moot (to me) whether
> they're being split or not. That's from some other orbit
> of JI intervals and the relationships among them.
>
> If it happens to make an eclipse (intersect) with the other
> set of all regular layouts, at that one particular size of
> fifths generated from the 5/4 third, so be it; but in most
> of the practical keyboard music from c1500-c1900 we care only
> about the regular layouts themselves, not about the JI
> placements.
>
> The set of JI frequencies (rational numbers) is so
> irregularly spaced, anyway! At what point do we decide
> that they stop being interesting, and therefore stop
> including them in our enumeration of the set? When
> their numerators or denominators have more than two
> digits in them?
>
>
> Brad Lehman

during the meantone era, the size of the "major-2nd"
(diatonic "whole-tone") in actual practice could range
anywhere from ~182.4037121 cents (exactly the JI
"small tone" with ratio 10:9) in 1/2-comma meantone,
to 200 cents in 12-et.

but in theoretical treatises, nearly every theorist
continued to define the "tone" as 9:8, as had been
done already for millennia.

this is merely my vague observation ... if anyone can
cite examples where theorists *do* actually define the
"tone" as a meantone interval i'd welcome seeing it.

the meantones described during the 1500-1600s were
always seen as practical solutions to a tuning problem
where 5-limit JI was the theoretical ideal, and
meantone necessarily tempered the size of those JI
intervals for practical reasons (whether of drift,
keyboard manufacture, whatever).

as i've pointed out, the way standard musical notation
works, with a chain extending from Gbb ... Ax, we
get a 31-note system, which aruges strongly in favor
of seeing 1/4-comma meantone as a paradigm, whether
or not it was the tuning used in actual practice.

Brad has a point that "meantone" does refer physically
to every member of the meantone family, if you consider
the mean to be that between the two notes bounding
the "major-3rd".

but in terms of historical usage, i'd say that by far
the evidence points to "meantone" referring specifically
to 1/4-comma meantone.

i think it's really only within the last few years
that we here have extended the word "meantone" to
cover the whole family of temperaments in which the
syntonic-comma vanishes. theorists of the 1500-1600s
generally just called them "temperaments" or "participato"
(or something like that).

the point at which JI pitches stop being included in
the set, is determined by which unison-vectors are
tempered out (i'm calling them "vapros").

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> i think it's really only within the last few years
> that we here have extended the word "meantone" to
> cover the whole family of temperaments in which the
> syntonic-comma vanishes.

Someone with a copy of Barbour might tell us his usage; I thought he
called eg 2/7-comma "meantone".

In any case to make sense of the theory you need to distinguish what
is tempered out from the detail of precise tunings.

Brad,

That's all true, but I reacted because you mentioned that
the nomenclature of meantone, or maybe you want me to spell it
as mean tone, has nothing to do with being the mean of 10/9
and 9/8, and I said that historically speaking, it does.

Manuel

--- In tuning@yahoogroups.com, "Manuel Op de Coul" <manuel.op.de.
coul@e...> wrote:
>
> Brad,
>
> That's all true, but I reacted because you mentioned that
> the nomenclature of meantone, or maybe you want me to spell it
> as mean tone, has nothing to do with being the mean of 10/9
> and 9/8, and I said that historically speaking, it does.

Right! Fair enough! As I pointed out at the beginning of this
thread:
/tuning/topicId_55723.html#55723
(...) "Anyway, such a debate about the proper use of the term
'meantone' has been going on for centuries already, and will probably
continue to do so after we're all gone." (...)

My own objection, obviously, was to the notion that 1/4 syntonic
comma is the only "true" meantone. To some people, with one usage of
the word "meantone", the statement is true; to others of us who
understand that term differently, it's false because there's an
infinity of meantone layouts.

I set up my Italian virginal in regular 2/7 syntonic comma to check
out Gene's question about Bull's "Ut re mi fa sol la". I chose the
usual Eb-Bb...C#-F#-G# disposition. The piece sounds like a total
mess, except of course in the outer diatonic sections of the piece
around the hexachord of G. The diminished fourths B-Eb, F#-Bb, G#-C,
and C#-F are hideous and make the music sound randomly raucous as
soon as the misspelled notes come in; I don't see any way that Bull
could have expected this to be a positive feature. (Even if, as Gene
mentioned, those diminished fourths somewhat resemble 9/7, they still
stick out egregiously, being *so* different from the slightly tight
major thirds.)

Then I moved the Bb and Eb to be A# and D#, in effect moving the wolf
around to A#-F. All this accomplishes is to delay the disasters for
about 15 seconds further into the piece. As soon as we run into
C#-F, G#-C, D#-G, and A#-D, it's as bad as before.

So, then I tried the thing I mentioned yesterday: putting C#/Db mean
between A and F, G#/Ab mean between E and C, D#/Eb mean between B and
G; and D-F#-Bb-D as 400s. Still awful: a bunch of little wolves in
F-Bb and F#-B and elsewhere.

Next thing to try was (still keeping all the naturals where they are
in 2/7 comma "meantone")...B-F#-C# and F-Bb-Eb slightly wide, and
then the mean G#/Ab between the C# and Eb. That is, the wolf is
chopped into many small bits and nothing's horrible anywhere. The
four brightest triads are of course still the misspelled B-Eb-F#,
F#-Bb-C#, C#-F-G#, and G#-C-Eb as in all the regular ("meantone")
temps, by this point having a character somewhat resembling their
counterparts in regular 1/5 or 1/6. Eb and E major are also both
quite bright, being on the outskirts of the bad neighborhood. The
whole piece can be played without disaster, and there's still a very
strong "center of gravity" when we hit C, F, or G major. All the
other major triads have increasingly colorful beating in them.

The bigger problem, though, is that some of our hexachords (so nicely
demonstrated in all twelve possible positions) are quite bumpy,
melodically. If we were going to end up with our wolf-puppy major
triads at the four classic positions anyway, letting the music more
or less circulate through all keys, why not just do the whole keyboard
in 1/5 or 1/6 to begin with instead of messing around with something
as tight as 2/7 on the naturals....? The hexachords end up sounding
smoother, the major triads of A, E, Eb, and Bb relax a bit, and the
whole thing is less startling overall.

Not that this exercise has proven anything new. In regular
temperaments, the sharper the generating major third is (while less
than 400), the better the accidentals are able to proxy for one
another in misspelled situations; that's a truism. Why not just do a
nice 1/5 or 1/6, optionally stick a mean G#/Ab into it to slice the
slight remaining wolf, and be done? With or without that final step
of the G#/Ab, these are all "meantone" *in character* and they all
have the same shape, varying only in intensity, when playing tonal
music.

So I went ahead and raised G, D, A, E, and B, and lowered F...while
not touching the accidentals from the previous step. Voila, we're
back in something very close to regular 1/5 or 1/6, and the only
remaining harshness is in the classic places of the misspelled major
triads and some canine derivative in the Eb-x-C# kennel.

Run the whole experiment backwards from 1/6 or 1/5 comma regular.
Leave C where it is, and also as many of the naturals and early
sharps as you can get away with; dink everything else around to make
the fifths purer at some expense of the major thirds. That's what
"Werckmeister III" is! Starting from a "meantone" (i.e. regular)
organ, which by 1690 meant it could be anywhere between 1/4 and 1/6
or even lighter, you can leave C and three or four other notes of
each octave essentially where they are (within 2.5 cents), and not
have to rebuild the whole instrument!

1/6 syntonic: leave C, C#, F#, G, and B.
1/6 Pythagorean: leave C, E, F#, G, and B.
1/5 syntonic: leave C, E, F#, G.
1/5 Pythagorean: leave C, D, E, G.
1/4 syntonic: leave C, D, G, A.

"Werckmeister III" is therefore a remarkably clever way to renovate a
regular organ without having to rebuild all of it, but merely 7 or 8
of the notes in each octave (and mostly in the direction of making
the notes flatter, i.e. adding pipe material, rather than cutting into
the precious legacy instruments). It's brilliant, really, under such
conditions of converting existing instruments. It still does not
solve the basic problem of improving those same four classic major
triads from the outer darkness; it merely takes some of the edge off
them. It doesn't really make chromatic music more playable, but only
less egregiously offensive in the situations of enharmonically
misspelled notes...making the parishioners wince less intensely at
such moments, and therefore giving them some appreciable return for
the investment of the conversion.

"Werckmeister IV" gives an alternate way to give regular 1/6 Pyth
comma a facelift, if the goal is to stick more closely to diatonic
music in the commonest keys. "Werckmeister V" is an alteration of
regular 1/8 Pyth comma.

These three are good for organ conversions, to save expense. They're
inferior, though, if the goal is to start from scratch and do
something more musically flexible and subtle. As with any job,
there's a big difference in results between settling for a half-assed
compromise (trying for minimum rework) and seriously rethinking the
basic design in other ways.

Brad Lehman
(not really a fan of Werckmeister III's sound....)

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

> I set up my Italian virginal in regular 2/7 syntonic comma to check
> out Gene's question about Bull's "Ut re mi fa sol la". I chose the
> usual Eb-Bb...C#-F#-G# disposition. The piece sounds like a total
> mess, except of course in the outer diatonic sections of the piece
> around the hexachord of G. The diminished fourths B-Eb, F#-Bb, G#-C,
> and C#-F are hideous and make the music sound randomly raucous as
> soon as the misspelled notes come in; I don't see any way that Bull
> could have expected this to be a positive feature. (Even if, as Gene
> mentioned, those diminished fourths somewhat resemble 9/7, they still
> stick out egregiously, being *so* different from the slightly tight
> major thirds.)

They don't "somewhat resemble" 9/7s, they closely approximate 9/7s,
being flat by only 1.567 cents. What I'm wondering is why, if they
stick out egregiously and make a raucous mess, they don't seem to in
my version. I've listened to a lot of music now where 9/7s are mixed
in, and often it is much more apparent than it seemed to be in this piece.

Anyway, "hideous" is an opinion about supermajor triads I don't share;
they *do* sound quite distinctly different than 1-5/4-3/2 triads, and
one may well question if Bull would have found them acceptable, but
without any evidence that he used a circulating temperament, or that
slightly tempered meantone was in use at the time, there's a prima
faciae case that he did.

> Not that this exercise has proven anything new. In regular
> temperaments, the sharper the generating major third is (while less
> than 400), the better the accidentals are able to proxy for one
> another in misspelled situations; that's a truism. Why not just do a
> nice 1/5 or 1/6, optionally stick a mean G#/Ab into it to slice the
> slight remaining wolf, and be done?

Where's the documentary evidence anyone did in Bull's time?

> Run the whole experiment backwards from 1/6 or 1/5 comma regular.
> Leave C where it is, and also as many of the naturals and early
> sharps as you can get away with; dink everything else around to make
> the fifths purer at some expense of the major thirds. That's what
> "Werckmeister III" is!

Andreas Werckmeister was born in 1645 and John Bull died in 1628, so
assuming Werckmeister is a historically apt tuning is a big leap, and
one you clearly don't buy yourself. Of course, one can simply look at
the numbers and conclude it would make *sense* as a tuning, but that
leaves us with the question of what the heck Bull was doing with his
tuning fork.

> "Werckmeister III" is therefore a remarkably clever way to renovate a
> regular organ without having to rebuild all of it, but merely 7 or 8
> of the notes in each octave (and mostly in the direction of making
> the notes flatter, i.e. adding pipe material, rather than cutting into
> the precious legacy instruments).

Yes, if I remember correctly this was also mentioned here by Ibo
Ortgies.

Perhaps you can quickly assess how many of the 36 tones on
the Archicembalo Bull's piece would need.

Manuel

--- In tuning@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:

> Perhaps you can quickly assess how many of the 36 tones on
> the Archicembalo Bull's piece would need.

The library here has several copies of the Fitzwilliam Virginal Book,
so I'm going to see about making an extended meantone version.

Brad, (and Gene)

Musically, Brad's argument makes sense, but in absence
of any historical documentation that Bull was concocting
proto-well-temperaments, I think I have to side with Gene that Bull may have
been making perverse use of a who-knows-which standard meantone tuning. Bull
was known to be not only a great virtuoso, but a real interesting
character...who knows, maybe he *was* trying to make people go through hell
and back in this piece !!!

I'm not quite sure that it would have been 2/7 comma though. Is there any
evidence for this choice?

Other possibilities for a live acoustic realization: two harpsichords, using
1/3 comma extended meantone. Or one of those Neopolitan 19-tone harpsichords:
like the one Christopher Stembridge uses on his disk 'Consonante
Stravagante'.

Interesting that this thread has come up now. I've been of the mind to finish
my midi realization of this very piece in 19 equal. I posted about this last
year sometime, remember?

It's fantastic in 19-equal IMO.

Cheers,
Aaron. (not a WerckmeisterIII fan much myself)

Aaron Krister Johnson
http://www.dividebypi.com
http://www.akjmusic.com

On Tuesday 24 August 2004 02:32 pm, Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:
> > I set up my Italian virginal in regular 2/7 syntonic comma to check
> > out Gene's question about Bull's "Ut re mi fa sol la". I chose the
> > usual Eb-Bb...C#-F#-G# disposition. The piece sounds like a total
> > mess, except of course in the outer diatonic sections of the piece
> > around the hexachord of G. The diminished fourths B-Eb, F#-Bb, G#-C,
> > and C#-F are hideous and make the music sound randomly raucous as
> > soon as the misspelled notes come in; I don't see any way that Bull
> > could have expected this to be a positive feature. (Even if, as Gene
> > mentioned, those diminished fourths somewhat resemble 9/7, they still
> > stick out egregiously, being *so* different from the slightly tight
> > major thirds.)
>
> They don't "somewhat resemble" 9/7s, they closely approximate 9/7s,
> being flat by only 1.567 cents. What I'm wondering is why, if they
> stick out egregiously and make a raucous mess, they don't seem to in
> my version. I've listened to a lot of music now where 9/7s are mixed
> in, and often it is much more apparent than it seemed to be in this piece.
>
> Anyway, "hideous" is an opinion about supermajor triads I don't share;
> they *do* sound quite distinctly different than 1-5/4-3/2 triads, and
> one may well question if Bull would have found them acceptable, but
> without any evidence that he used a circulating temperament, or that
> slightly tempered meantone was in use at the time, there's a prima
> faciae case that he did.
>
> > Not that this exercise has proven anything new. In regular
> > temperaments, the sharper the generating major third is (while less
> > than 400), the better the accidentals are able to proxy for one
> > another in misspelled situations; that's a truism. Why not just do a
> > nice 1/5 or 1/6, optionally stick a mean G#/Ab into it to slice the
> > slight remaining wolf, and be done?
>
> Where's the documentary evidence anyone did in Bull's time?
>
> > Run the whole experiment backwards from 1/6 or 1/5 comma regular.
> > Leave C where it is, and also as many of the naturals and early
> > sharps as you can get away with; dink everything else around to make
> > the fifths purer at some expense of the major thirds. That's what
> > "Werckmeister III" is!
>
> Andreas Werckmeister was born in 1645 and John Bull died in 1628, so
> assuming Werckmeister is a historically apt tuning is a big leap, and
> one you clearly don't buy yourself. Of course, one can simply look at
> the numbers and conclude it would make *sense* as a tuning, but that
> leaves us with the question of what the heck Bull was doing with his
> tuning fork.

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...> wrote:

> I'm not quite sure that it would have been 2/7 comma though. Is
there any
> evidence for this choice?

It was discussed during the period Bull was composing, which seems to
have been before 1/4-comma had established itself as the meantone of
choice. Given that 1/3-comma, and even more so 2/7-comma, were
discussed, it seems possible that this is because a rather flat
meantone was in actual use at the time. These flat systems all have
supermajor thirds well enough in tune to make the identification
reasonable; 2/7 comma has it 1.57 cents flat, 1/3 comma 6.63 cents
sharp, and the rest in between.

> Other possibilities for a live acoustic realization: two
harpsichords, using
> 1/3 comma extended meantone.

But 1/3 comma seems to have not recieved the attention 2/7 comma has.
The most logical reason to favor it it would have been a 19-tone
harpsichord, which perhaps you are suggesting, but meanwhile Manuel is
talking about an archicembalo.

> Interesting that this thread has come up now. I've been of the mind
to finish
> my midi realization of this very piece in 19 equal. I posted about
this last
> year sometime, remember?

I remember it coming up and trying to find a midi realization then,
which I've only now managed to do.

> It's fantastic in 19-equal IMO.

I presume you mean all 19 notes, not 12 out of 19? Yes, that would be
interesting to hear!

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
wrote:

[re: Bull's "Ut re me fa sol la"]

> Other possibilities for a live acoustic realization:
> two harpsichords, using 1/3 comma extended meantone.
> Or one of those Neopolitan 19-tone harpsichords:
> like the one Christopher Stembridge uses on his disk
> 'Consonante Stravagante'.
>
> Interesting that this thread has come up now. I've been
> of the mind to finish my midi realization of this very
> piece in 19 equal. I posted about this last year sometime,
> remember?
>
> It's fantastic in 19-equal IMO.

interesting to me that you juxtaposed these two tunings
one after the other. i'd bet all the money that i can
find that there's not a single person who could audibly
tell the difference between 19-equal and 1/3-comma meantone.

see my webpage to see just how close they are:

http://tonalsoft.com/enc/index2.htm?19edo.htm

looking forward to hearing your version too, Aaron!
i really love Gene's ... have been listening to it a lot.

-monz

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

> But 1/3 comma seems to have not recieved the attention
> 2/7 comma has.

1/3-comma was really only favored very early on.
as soon as Zarlino published his account of 2/7-comma,
that one seems to have gained a good deal of interest.

> The most logical reason to favor it it would have
> been a 19-tone harpsichord, which perhaps you are
> suggesting, but meanwhile Manuel is talking about
> an archicembalo.

Gene, sounds like you're not all that familiar with
Vicentino.

"cembalo" is the Italian word for "harpsichord" ...
actually, i think it just means "keyboard" but in
those days it generally referred to a harpsichord.

Vicentino's "archicembalo" is a word that means something
like "super-harpsichord". he devised it to play his
adaptive JI scheme, which i've written about here:

http://tonalsoft.com/enc/index2.htm?../monzo/vicentino/vicentino.htm

-monz

--- In tuning@yahoogroups.com, "Manuel Op de Coul" <manuel.op.de.
coul@e...> wrote:
> > "Werckmeister III" is therefore a remarkably clever way to
renovate a
> > regular organ without having to rebuild all of it, but merely 7
or
8
> > of the notes in each octave (and mostly in the direction of making
> > the notes flatter, i.e. adding pipe material, rather than cutting
into
> > the precious legacy instruments).
>
> Yes, if I remember correctly this was also mentioned here by Ibo
> Ortgies.

In reference to converting *only* 1/4 comma organs, it's well known
as
a remark from Lindley's _New Grove_ articles. My point yesterday was
that Werckmeister III also converts 1/5 and 1/6 comma organs, either
comma: a much more important conclusion. It's a conclusion of my
own,
from recent study of charts of these layouts, comparing frequencies.

Did Ibo mention those other strains of meantone, in reference to this
conversion business? (I haven't seen his message.)

Brad Lehman

> > Perhaps you can quickly assess how many of the 36 tones on
> > the Archicembalo Bull's piece would need.
>
> The library here has several copies of the Fitzwilliam Virginal
Book,
> so I'm going to see about making an extended meantone version.

Good idea. THEN (if you stick with 2/7) you'll hear the 9/7s you've
been asking about, and will notice how egregiously they stick out.
Your current version doesn't have them. As I mentioned, there's some
comma-cheating going on in your current one, since it uses wrong
enharmonics....

Brad Lehman

"Strike that/reverse it!" You'll hear your 9/7s only if you do NOT
use extended meantone, but instead stick to exactly 12 notes even if
they're misspelled on the page.

Furthermore, you'll hear things as John Bull did (and be in a
position to venture any aesthetic judgment of results) only by
discarding MIDI altogether and setting this up on a plucked-string
keyboard (i.e. harpsichord or virginals), with the complex
interactions of the upper harmonics coming from the plucked strings.
Optionally, try it also on pipe organ....

Whether the *fundamentals* make nearly-pure 9/7 is IRRELEVANT. Beats
come from the interaction of upper partials, the simplest ones. On
harpsichord, the diminished fourths of 2/7 comma meantone sound like
garbage...as I reported yesterday. The beats of those nearly-9/7
intervals are so fast as to be a mere blur, a discordant bit of
vinegar. Paper theory, and practice on MIDI simulations, does not
tell us the sound Bull would have heard on stringed keyboards, if
indeed (as I doubt) 2/7 comma had anything to do with it anyway.

And as I pointed out, the dodge of using split-key keyboards for this
piece also doesn't work; at the place where Bull spells a triad as
A-Db-E (coming out of an A-major section and an E-major dominant) and
then ties the Db forward into the next chord of Gb-Bb-Db-Gb, one MUST
pick either C# or Db; and one or the other is going to be wildly out
of tune, in one context or the other. There's no way to slip it a
clandestine comma during the sustain of the Db! An archicembalo does
not solve that problem.

For this piece to work out on a 12-key harpsichord/virginals, it has
to be very light tempering of the fifths all the way around (if we
want to use a regular temperament, a "meantone"), or an irregular
system where some of the fifths or thirds are different from others.
There's really no substitute here for setting up various systems on
acoustic instruments, and playing the piece straight out.... :)

Brad Lehman

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:
> > > Perhaps you can quickly assess how many of the 36 tones on
> > > the Archicembalo Bull's piece would need.
> >
> > The library here has several copies of the Fitzwilliam Virginal
> Book,
> > so I'm going to see about making an extended meantone version.
>
> Good idea. THEN (if you stick with 2/7) you'll hear the 9/7s
you've
> been asking about, and will notice how egregiously they stick out.
> Your current version doesn't have them. As I mentioned, there's
some
> comma-cheating going on in your current one, since it uses wrong
> enharmonics....
>
> Brad Lehman

> > Not that this exercise has proven anything new. In regular
> > temperaments, the sharper the generating major third is (while
less
> > than 400), the better the accidentals are able to proxy for one
> > another in misspelled situations; that's a truism. Why not just
do a
> > nice 1/5 or 1/6, optionally stick a mean G#/Ab into it to slice
the
> > slight remaining wolf, and be done?
>
> Where's the documentary evidence anyone did in Bull's time?

Chapter 7 of Barbour: Grammateus (1518: 50 years before Bull's
birth), Artusi (1603: when Bull was 40), et al.

And Arnolt Schlick asked for a mean compromised G#/Ab (halfway
between G and A) as early as 1512; see Barbour, p137ff.

See also Lindley's _Lutes, Viols, and Temperaments_....

Brad Lehman

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:
> > > Perhaps you can quickly assess how many of the 36 tones on
> > > the Archicembalo Bull's piece would need.
> >
> > The library here has several copies of the Fitzwilliam Virginal
> Book,
> > so I'm going to see about making an extended meantone version.
>
> Good idea. THEN (if you stick with 2/7) you'll hear the 9/7s you've
> been asking about, and will notice how egregiously they stick out.
> Your current version doesn't have them.

This is backwards; in extended meantone I *won't* have them. The
current version does. At least, it has the SysEx tuning dumps, and the
pitch-bend version sounded the same. If the supermajor triads fail to
sound egregious in my version, that is not evidence of
"comma-cheating", whatever that means (adaptive tuning?) It could be
taken as evidence that my thesis is possible.

As I mentioned, there's some
> comma-cheating going on in your current one, since it uses wrong
> enharmonics....

I don't see how this claim makes much sense; I used only twelve notes
of 2/7 comma meantone, which means the only choice I get is where to
start the circle of fifths. There's nothing it is possible to fudge.

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

> "Strike that/reverse it!" You'll hear your 9/7s only if you do NOT
> use extended meantone, but instead stick to exactly 12 notes even if
> they're misspelled on the page.

Right; we are on the same page here.

> Furthermore, you'll hear things as John Bull did (and be in a
> position to venture any aesthetic judgment of results) only by
> discarding MIDI altogether and setting this up on a plucked-string
> keyboard (i.e. harpsichord or virginals), with the complex
> interactions of the upper harmonics coming from the plucked strings.
> Optionally, try it also on pipe organ....

Do you mean "use a harpsichord font" or "use an actual harpsichord"?
The latter is not an option for me.

> Whether the *fundamentals* make nearly-pure 9/7 is IRRELEVANT.

Hardly! If the fundamentals make a nearly pure 9/7, you get an
interval which can, to some degree, be audibly appreciated.

Beats
> come from the interaction of upper partials, the simplest ones.

What makes you think beats are what is most important for evaluating
consonances?

It seems to me you are a hard man to please--pure 5/4 is too
consonant, and pure 9/7 too dissonant.

On
> harpsichord, the diminished fourths of 2/7 comma meantone sound like
> garbage...as I reported yesterday.

And yet you've also several times indicated that those exact same
diminished fourths don't sound like garbage to you in the version I
put up.

The beats of those nearly-9/7
> intervals are so fast as to be a mere blur, a discordant bit of
> vinegar. Paper theory, and practice on MIDI simulations, does not
> tell us the sound Bull would have heard on stringed keyboards, if
> indeed (as I doubt) 2/7 comma had anything to do with it anyway.

My theory is speculative, but so is yours--very speculative, in fact,
given the lack of any evidence beyond the score.

> And as I pointed out, the dodge of using split-key keyboards for this
> piece also doesn't work; at the place where Bull spells a triad as
> A-Db-E (coming out of an A-major section and an E-major dominant) and
> then ties the Db forward into the next chord of Gb-Bb-Db-Gb, one MUST
> pick either C# or Db; and one or the other is going to be wildly out
> of tune, in one context or the other.

What I was thinking of doing was simply accepting Bull's spelling as
what he intended and seeing what resulted.

> For this piece to work out on a 12-key harpsichord/virginals, it has
> to be very light tempering of the fifths all the way around (if we
> want to use a regular temperament, a "meantone"), or an irregular
> system where some of the fifths or thirds are different from others.

By your definition of "work". Given that no one has so far produced
any evidence that Bull knew about either irregular tuning systems or
light meantones, concluding he used them is very speculative.

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

> > Where's the documentary evidence anyone did in Bull's time?
>
> Chapter 7 of Barbour: Grammateus (1518: 50 years before Bull's
> birth), Artusi (1603: when Bull was 40), et al.

Can you give tuning for these? It would be interesting to have them.

Point taken, Gene; you're right in the comments below. (And I'd
already corrected myself five minutes after sending, at
/tuning/topicId_55723.html#55840
)....

The almost pure 9/7s are in your current version (I've noticed them
now, on a much closer listen to your results) but they don't stick
out, while they're awful on the harpsichord in 2/7 comma. Why?

I think I've figured it out now; I suspect it has very much to do
with
your choice of timbre in the orchestration! Those
pseudo-clarinets/chalumeaux in there have only half the overtone
series in them, skipping the odd partials as clarinets do.
Therefore,
they're dodging each other in the places where beats would occur in
harpsichord tone, which has all the partials. The 7th and 9th
partials aren't in your tone, and therefore there's no battle that
they have to do up there at their common multiple frequency!

Try a re-orchestration to some timbre rich in the odd partials, and
you'll hear those 9/7s really bash.

Again, this would matter to John Bull if he had some keyboard
instrument on which he could omit all the odd partials from the tone..
..

Brad Lehman

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:
> > > > Perhaps you can quickly assess how many of the 36 tones on
> > > > the Archicembalo Bull's piece would need.
> > >
> > > The library here has several copies of the Fitzwilliam Virginal
> > Book,
> > > so I'm going to see about making an extended meantone version.
> >
> > Good idea. THEN (if you stick with 2/7) you'll hear the 9/7s
you've
> > been asking about, and will notice how egregiously they stick
out.

> > Your current version doesn't have them.
>
> This is backwards; in extended meantone I *won't* have them. The
> current version does. At least, it has the SysEx tuning dumps, and
the
> pitch-bend version sounded the same. If the supermajor triads fail
to
> sound egregious in my version, that is not evidence of
> "comma-cheating", whatever that means (adaptive tuning?) It could be
> taken as evidence that my thesis is possible.
>
> As I mentioned, there's some
> > comma-cheating going on in your current one, since it uses wrong
> > enharmonics....
>
> I don't see how this claim makes much sense; I used only twelve
notes
> of 2/7 comma meantone, which means the only choice I get is where to
> start the circle of fifths. There's nothing it is possible to fudge.

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

> --- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:
>
> > > Where's the documentary evidence anyone did in Bull's time?
> >
> > Chapter 7 of Barbour: Grammateus (1518: 50 years before
> > Bull's birth), Artusi (1603: when Bull was 40), et al.
>
> Can you give tuning for these? It would be interesting to
> have them.

i don't have time to do Artusi's ... here's Grammateus's ...

on page 139 of the 1951 edition of Barbour, he says:

>> "Grammateus tuned the diatonic notes of his monochord
>> according to the Pythagorean ratios. But when it came
>> to the black keys, the "minor semitones", he followed
>> a different prodedure. There were formed by dividing
>> each tone into two equal semitones by the Euclidean method
>> for finding a geometric mean proportional."

and in Table 118 on p 140, he gives Grammateus's monochord
tuning as follows (i reformatted it to fit better here):

name pyth-com cents
C .... 0 .... 1200
B .... 0 .... 1110
Bb .. +1/2 .. 1008
A .... 0 ..... 906
G# .. -1/2 ... 804
G .... 0 ..... 702
F# .. -1/2 ... 600
F .... 0 ..... 498
E .... 0 ..... 408
Eb .. +1/2 ... 306
D# .. -1/2 ... 306
D .... 0 ..... 204
C# .. -1/2 ... 102
C .... 0 ....... 0

the monzos are as follows, with more accurate cents:

. 2,3-monzo ....... ~cents

[ 1 .... 0 > .... 1200.00000000
[-7 .... 5 > .... 1109.77500433
[-11/2 . 4 > .... 1007.82000346
[-4 .... 3 > ..... 905.86500260
[-5/2 .. 2 > ..... 803.91000173
[-1 .... 1 > ..... 701.95500087
[ 1/2 .. 0 > ..... 600.00000000
[ 2 ... -1 > ..... 498.04499913
[-6 .... 4 > ..... 407.82000346
[-9/2 .. 3 > ..... 305.86500260
[-3 .... 2 > ..... 203.91000173
[-3/2 .. 1 > ..... 101.95500087
[ 0 .... 0 > ....... 0.00000000

-monz

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

> I think I've figured it out now; I suspect it has very much to do
> with
> your choice of timbre in the orchestration! Those
> pseudo-clarinets/chalumeaux in there have only half the overtone
> series in them, skipping the odd partials as clarinets do.

This can't very well be it; clarinets skip the even partial tones. You
*can* here the 9/7s in there, and it seems to me it adds a certain
flavorful quality. It's much less strange sounding than a lot of other
music I've retuned to this section of the meantone spectrum, however,
and I still don't know why. I also made a harpsichord version when I
did this one, and it didn't seem whacked-out to me. I'll put that up
also, as I originally intended, though how much like a harpsichord it
actually sounds is another question.

> Do you mean "use a harpsichord font" or "use an actual harpsichord"?
> The latter is not an option for me.

Yes, acoustic instruments.

> > Whether the *fundamentals* make nearly-pure 9/7 is IRRELEVANT.
>
> Hardly! If the fundamentals make a nearly pure 9/7, you get an
> interval which can, to some degree, be audibly appreciated.

Point taken; but consider this: on acoustic instruments, an in-tune
pure interval sounds pure and so characteristic BECAUSE the upper
partials coincide exactly and reinforce the resonance of the whole
thing...not only the fundamentals. The purity of your near-9/7 is
going to sound increasingly compromised by various beats, the more
partials you allow into your timbre above the basic fundamental sine
wave...and especially the odd partials.

> Beats
> > come from the interaction of upper partials, the simplest ones.
>
> What makes you think beats are what is most important for evaluating
> consonances?

Being a professional harpsichord player/tuner. :) The close
listening to those beats is the way we do our job. That's EXACTLY
how we evaluate consonances and differentiate them from
near-consonances.

> What I was thinking of doing was simply accepting Bull's spelling as
> what he intended and seeing what resulted.

Yes, that would sound nearly consonant at most spots, except at the
change-over points between sharps and flats, where melodically you're
going to have some comma jumps up or down.

> > For this piece to work out on a 12-key harpsichord/virginals, it
has
> > to be very light tempering of the fifths all the way around (if
we
> > want to use a regular temperament, a "meantone"), or an irregular
> > system where some of the fifths or thirds are different from
others.
>
> By your definition of "work". Given that no one has so far produced
> any evidence that Bull knew about either irregular tuning systems or
> light meantones, concluding he used them is very speculative.

The opposite conclusion by anyone that musicians did not know about
light meantones is also very speculative. How clueless are
harpsichordists and organists supposed to be, especially at such a
high level of achievement as Doctor Bull? Anybody who has ever sat
at a harpsichord with a tuning hammer, trying to set 1/4 comma fifths
but accidentally making them too lightly tempered (i.e. getting too
slow a beat rate), will hear the resulting wide major third and
notice what it sounds like. And, at least once or twice, he or she
will continue onto the entire instrument to see what happens, and try
some repertoire on it. I don't see how John Bull or any other
harpsichord player then or now would not know this, unless they've
had their tuning tasks hired out 100% of the time, or unless they had
tin ears insensitive to pitch control.

Ditto for the dinking around in the wolf area, trying to make things
more usable by nudging notes higher or lower to taste...this is
Harpsichord Tuning "101" there, from a practical point of view. The
chimera is that practical players EVER stuck as closely to regular
layouts as the theorists (then and now) would have us believe.
Regular layouts are too restrictive, both melodically and
harmonically, to get the job done in playing any substantially large
body of repertoire.

Sure, different regular strains could be deployed on different
occasions, or one could pick different dispositions for different
pieces (quickly retuning all the Eb to D#, for example) as Barbour
would have it in his speculations. But that "bag of golf clubs"
approach DOES NOT work for organs or for fretted clavichords, where
the alteration of temperament takes quite a bit more work than
turning a tuning pin. One must either set up a system where proxy
enharmonics sound decent, or live with a bunch of occasional and
random-sounding dissonances in music that looks perfectly diatonic on
the page, or stick to a very restricted set of keys and modulations.
This is where practice meets theory and overrules it. It's theory's
duty to catch up, not to squelch practice.

"Well-tempered" tunings are not an invention of the 1690s; it's
merely a theoretical codification of the continuum that had already
been happening in "ordinary" tuning for at least a hundred years
already...the tasteful nudging up of some or all of the sharps, and
nudging down of some or all the flats, until they're somewhat
passable as one another. And, that's starting from whatever strain
of regular meantone best suits the milieu and the music, i.e.
sensitivities of musical taste; and not necessarily 1/4 syntonic
comma. Start with something mostly regular, dink it around carefully
(by taste and experience and training) until it sounds even better in
the music to be played, and there's your ordinary tuning.

The alternative is to believe that musicianship ever moved in
stair-step jumps: that everybody used regular 1/4 comma, until
Werckmeister (or whoever) realized suddenly one day that things must
be reformed drastically. But the 16th and 17th century music itself
argues against such a view; the exotic enharmonic notes are in there.
Why would composers write notes that deliberately make the instrument
sound unresonant, or tune the instrument in such ways that make the
music sound randomly strident, for generations? Did they all have no
taste at all? Isn't it much simpler to conclude, from the music,
that they had practical ways of tuning so things sounded fine, and
just didn't bother to write those methods down because they were so
common, everybody knew them? Practice is not restricted only to the
things that theorists make up and square off with mathematical
nicety, or have bothered to write down at all!

Brad Lehman

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:
>
> > I think I've figured it out now; I suspect it has very much to do
> > with
> > your choice of timbre in the orchestration! Those
> > pseudo-clarinets/chalumeaux in there have only half the overtone
> > series in them, skipping the odd partials as clarinets do.
>
> This can't very well be it; clarinets skip the even partial tones.

Only on the distinction of starting the counting at 1 vs 0. Partials,
overtones, whatever; the half of them that would matter most here are
missing.

Brad Lehman

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:
>
> > I think I've figured it out now; I suspect it has very much to do
> > with
> > your choice of timbre in the orchestration! Those
> > pseudo-clarinets/chalumeaux in there have only half the overtone
> > series in them, skipping the odd partials as clarinets do.
>
> This can't very well be it; clarinets skip the even partial tones.

That's probably what you meant, though. Skipping the even partials and
superimposing a 7 and a 9 seems nice in clarinet terms.

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

> Those
> pseudo-clarinets/chalumeaux in there have only half the overtone
> series in them, skipping the odd partials as clarinets do.
> Therefore,
> they're dodging each other in the places where beats would occur in
> harpsichord tone, which has all the partials. The 7th and 9th
> partials aren't in your tone, and therefore there's no battle that
> they have to do up there at their common multiple frequency!

the 7th and 9th partials are very strong in the chalumeaux register
of the clarinet. 9/7 is easy to tune with two clarinets.

> Try a re-orchestration to some timbre rich in the odd partials, and
> you'll hear those 9/7s really bash.

the clarinet does have mainly odd partials.

Brad,

You are so right on about timbre and its effect on dissonance, I was going to
post a response about this last night but you beat me to it.

I was listening to an excellent William Byrd recording by Davitt Moroney--

http://www.hyperion-records.co.uk/details/66551.asp

--at my weekly listening session tonight, and I was thinking about our thread
re:Bull 'Ut Re Mi Fa Sol La'. The extensive liner notes (200 pages or so!)
include a contention that 1/4 comma meantone was *not* common in England
before the late 1600's, and it was more a Continental European phenomenon.
Instead, he claims they (the Brits of Elizabeth's time) used modified
Pythagorean tunings. For the recording on harpsichord and virginals, he used
the following tuning (reconstructed here from the description he gives in the
liner notes):

0: 1/1 0.000 unison, perfect prime
1: 256/243 90.225 limma, Pythagorean minor second
2: 193.157 cents 193.157
3: 32/27 294.135 Pythagorean minor third
4: 5/4 386.314 major third
5: 4/3 498.045 perfect fourth
6: 45/32 590.224 diatonic tritone
7: 696.578 cents 696.578
8: 128/81 792.180 Pythagorean minor sixth
9: 889.735 cents 889.735
10: 16/9 996.090 Pythagorean minor seventh
11: 15/8 1088.269 classic major seventh
12: 2/1 1200.000 octave

Although the liner notes don't mention it, it turns out that this is
Kirnberger, what we commonly think of as a not-very subtle, but usable,
Baroque tuning !!! In effect, he claims that before Kirnberger put this to
paper, it was going on in English Elizabethan music as a practical tuning
*before* meantone became a standard in England, and that it allows for the
strange modulations we see in the British Isles virginal music from time to
time.

We do know that the written documentation of meantones were Continental, so he
has a point. Obviously, Zarlino in Italy talked about 2/7 comma, Pietro Aron
talked about 1/4 comma, Salinas talked about 1/3 comma, and composers in
Germany like Praetorius were talking about it as a standard tuning, and how
one should go about laying the bearings of it.....but apparently not in
England, which after all was somewhat isolated. The question remains--how
isolated was England--was it enough that practical tuning ideas amongst
Continental and island musicians were *not* shared. This is Moroney's claim.
(However, it wasn't so isolated that Rome didn't want to squash Henry VIII and
Elizabeth, after all--they knew about the Anglican church!!)

Perhaps he (Moroney) didn't want to appear ignorant (he's a great player and
the disk is really amazing, BTW) or controversial, but in his liner notes he
didn't directly allude to Kirnberger, (if he knows Kirnberger, he would have
known that the knee-jerk reaction of people 'in the know' would be to scoff)
-- but stated (which me be a fact--can someone else verify this) that English
musicians in the 1600's and even late 1500's were doing a proto-Kirnberger
(nay, an actual Kirnberger!!) but calling it an 'modified Pythagorean'.

Someone care to tune Bull to Kirnberger?

Best,
Aaron.

Aaron Krister Johnson
http://www.dividebypi.com
http://www.akjmusic.com

I agree 100% with everything you say here Brad....right on!

Aaron.

On Wednesday 25 August 2004 03:33 pm, Brad Lehman wrote:

> "Well-tempered" tunings are not an invention of the 1690s; it's
> merely a theoretical codification of the continuum that had already
> been happening in "ordinary" tuning for at least a hundred years
> already...the tasteful nudging up of some or all of the sharps, and
> nudging down of some or all the flats, until they're somewhat
> passable as one another. And, that's starting from whatever strain
> of regular meantone best suits the milieu and the music, i.e.
> sensitivities of musical taste; and not necessarily 1/4 syntonic
> comma. Start with something mostly regular, dink it around carefully
> (by taste and experience and training) until it sounds even better in
> the music to be played, and there's your ordinary tuning.
>
> The alternative is to believe that musicianship ever moved in
> stair-step jumps: that everybody used regular 1/4 comma, until
> Werckmeister (or whoever) realized suddenly one day that things must
> be reformed drastically. But the 16th and 17th century music itself
> argues against such a view; the exotic enharmonic notes are in there.
> Why would composers write notes that deliberately make the instrument
> sound unresonant, or tune the instrument in such ways that make the
> music sound randomly strident, for generations? Did they all have no
> taste at all? Isn't it much simpler to conclude, from the music,
> that they had practical ways of tuning so things sounded fine, and
> just didn't bother to write those methods down because they were so
> common, everybody knew them? Practice is not restricted only to the
> things that theorists make up and square off with mathematical
> nicety, or have bothered to write down at all!
>
>
> Brad Lehman

--
Aaron Krister Johnson
http://www.dividebypi.com
http://www.akjmusic.com

hi Brad and Gene,

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

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> > --- In tuning@yahoogroups.com, "Brad Lehman" <bpl@u...> wrote:
> >
> > > I think I've figured it out now; I suspect it has
> > > very much to do with your choice of timbre in the
> > > orchestration! Those pseudo-clarinets/chalumeaux
> > > in there have only half the overtone series in them,
> > > skipping the odd partials as clarinets do.
> >
> > This can't very well be it; clarinets skip the even
> > partial tones.
>
>
> Only on the distinction of starting the counting at
> 1 vs 0. Partials, overtones, whatever; the half of them
> that would matter most here are missing.

Brad, it has nothing at all to do with 1 vs. 0.
neither partials nor overtones are ever counted starting
with zero -- both of them start with 1.

the distinction is that counting partials begins with
the fundamental, while overtones begin with the next
partial (2:1) above that -- so the fundamental is the
1st partial, the 1st overtone is the 2nd partial, etc.

Gene was precise in saying that the clarinet timbre is
lacking in even partials. (and note that this is most
true is the lowest "chalumeau" register ... even-number
partials become more and more prominent the higher the
clarinet's note.)

so the 9th and 7th partials are definitely present.

-monz

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
wrote:
> Perhaps he (Moroney) didn't want to appear ignorant (he's a great
player and
> the disk is really amazing, BTW) or controversial, but in his liner
notes he
> didn't directly allude to Kirnberger, (if he knows Kirnberger, he
would have
> known that the knee-jerk reaction of people 'in the know' would be
to scoff)
> -- but stated (which me be a fact--can someone else verify this)
that English
> musicians in the 1600's and even late 1500's were doing a
proto-Kirnberger
> (nay, an actual Kirnberger!!) but calling it an 'modified
Pythagorean'.
>
> Someone care to tune Bull to Kirnberger?

Yes; me. Way back in 1983 when I was first learning to tune and play
harpsichord, and reading through the Fitzwilliam Virginal Book for
fun
and at my lessons, I simply tuned it in "Kirnberger III" all the time
because I didn't know any better, and because it sounded good to me;
I
didn't know any of the regular meantones yet.

It doesn't surprise me at all that musicians could have used
"Kirnberger III" a century or more before Kirnberger wrote it down
(1770s); I've been saying that same thing on HPSCHD-L for quite a
while now already. It's trivially easy for novices to tune, or to
discover from scratch! Furthermore, Sorge wrote down something very
similar in the 1740s, but with Pythagorean comma rather than syntonic
in those four fifths C-G-D-A-E.

Brad Lehman

Brad wrote:
>Did Ibo mention those other strains of meantone, in reference to this
>conversion business? (I haven't seen his message.)

Can't find it. What I probably remembered was the remark
in Paul Poletti's paper that retuning an organ to
Werckmeister III would save a considerable amount of
work compared to retuning all the pipes.

Manuel

>> Where's the documentary evidence anyone did in Bull's time?

>Chapter 7 of Barbour: Grammateus (1518: 50 years before Bull's
>birth), Artusi (1603: when Bull was 40), et al.

>And Arnolt Schlick asked for a mean compromised G#/Ab (halfway
>between G and A) as early as 1512; see Barbour, p137ff.

Even older is Boulliau's temperament: 1373.

Manuel

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

/tuning/topicId_55723.html#55782

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > i think it's really only within the last few years
> > that we here have extended the word "meantone" to
> > cover the whole family of temperaments in which the
> > syntonic-comma vanishes.
>
> Someone with a copy of Barbour might tell us his usage; I thought he
> called eg 2/7-comma "meantone".
>
> In any case to make sense of the theory you need to distinguish what
> is tempered out from the detail of precise tunings.

***Barbour calls 2/7 comma "meantone..."

Pg. 32:

"The first regular temperament to be advocated after the description
of the ordinary meantone temperament was that described by Zarlino in
which "each fifth remains diminished and imperfect by 2/7 comma."

Then, there is a table of this temperament on page 33, distinctly in
the "meantone" chapter...

J. Pehrson

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> /tuning/topicId_55723.html#55782
>
> > --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> >
> > > i think it's really only within the last few years
> > > that we here have extended the word "meantone" to
> > > cover the whole family of temperaments in which the
> > > syntonic-comma vanishes.
> >
> > Someone with a copy of Barbour might tell us his usage;
> > I thought he called eg 2/7-comma "meantone".
> >
> > In any case to make sense of the theory you need to
> > distinguish what is tempered out from the detail of
> > precise tunings.
>
>
> ***Barbour calls 2/7 comma "meantone..."
>
> Pg. 32:
>
> "The first regular temperament to be advocated after
> the description of the ordinary meantone temperament was
> that described by Zarlino in which "each fifth remains
> diminished and imperfect by 2/7 comma."
>
> Then, there is a table of this temperament on page 33,
> distinctly in the "meantone" chapter...
>
> J. Pehrson

thanks, Joe. i should have responded to this when
it first came up.

i really only meant "within the last few years" in
my statement in a vague sort of way, of course knowing
that Barbour had already placed a lot of other tunings
into the "meantone" category and called them by that
name.

my point was that *at the time the older treatises
were written* (1500s-1600s) the theorists were not
calling anything other than 1/4-comma by the "meantone"
name.

as i said, i'm not 100% sure about this and would have
to go back and do research that i don't have time for
now. but i'm fairly certain that the older theorists
who wrote about tuning we would now call "meantones"
but which use a division of the comma other than 1/4
(Zarlino, Salinas, Smith, etc.), did not use the
"meantone" moniker when discussing them. it was only
in the 1900s that that began ... i think.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> i really only meant "within the last few years" in
> my statement in a vague sort of way, of course knowing
> that Barbour had already placed a lot of other tunings
> into the "meantone" category and called them by that
> name.
>
> my point was that *at the time the older treatises
> were written* (1500s-1600s) the theorists were not
> calling anything other than 1/4-comma by the "meantone"
> name.
>
> as i said, i'm not 100% sure about this and would have
> to go back and do research that i don't have time for
> now. but i'm fairly certain that the older theorists
> who wrote about tuning we would now call "meantones"
> but which use a division of the comma other than 1/4
> (Zarlino, Salinas, Smith, etc.), did not use the
> "meantone" moniker when discussing them. it was only
> in the 1900s that that began ... i think.

for those who care to read Zarlino and Salinas to find
out what terms they used:

Salinas, Franciscus. 1577. _De musica, liber tertius_.
(Latin only)
http://www.music.indiana.edu/tml/16th/SALMUS3_TEXT.html

Zarlino, Giuseppe. 1558. _ Le institutione harmoniche_, part 2
(original Italian and English translation)
http://tonalsoft.com/monzo/zarlino/1558/zarlino1558-2.htm

-monz

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

/tuning/topicId_55723.html#55857

>
> "Well-tempered" tunings are not an invention of the 1690s; it's
> merely a theoretical codification of the continuum that had already
> been happening in "ordinary" tuning for at least a hundred years
> already...the tasteful nudging up of some or all of the sharps, and
> nudging down of some or all the flats, until they're somewhat
> passable as one another. And, that's starting from whatever strain
> of regular meantone best suits the milieu and the music, i.e.
> sensitivities of musical taste; and not necessarily 1/4 syntonic
> comma. Start with something mostly regular, dink it around
carefully
> (by taste and experience and training) until it sounds even better
in
> the music to be played, and there's your ordinary tuning.
>

***This is really interesting. I've never seen or heard this
viepoint of how "Well Temperaments" came about either on this list or
any other place... Is this a commonly accepted view??

Thanks!

J. Pehrson

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

>
> ***This is really interesting. I've never seen or heard this
> viepoint of how "Well Temperaments" came about either on this list or
> any other place... Is this a commonly accepted view??
>

This is the same opinion as Daniel Wolf's. But he got so much bad
reaction for it on this list in making a FAQ on meantone that he gave up.

Gabor

Jorgensen on p9 of _Tuning_ ascribes it back to Grammateus (aka
Heinrich Schreiber), 1518, in the sense that I presented it last week
at the beginning of this thread. (For what that's worth!) In
Schreiber/Grammateus' layout, "all the chromatic notes were exact
meantones between their two natural neighboring notes, and the
natural
notes D, G, and A were also exact meantones between their two
chromatic neighboring notes. All the semitones in the Schreiber
tempered Pythagorean were exactly the same size except for EF and BC
which were smaller. Therefore, the Schreiber Pythagorean Temperament
has been called meantone temperament and also equal temperament. It
is an example of the equal type of temperaments that were used on
lutes and viols in the sixteenth century. In conclusion, the terms
'Pythagorean tuning', 'meantone temperament', and 'equal temperament'
have commonly been scrambled together in history."

But, just ahead of that on page 8, Jorgensen gave us: "The term
'meantone temperament' is a relatively new term that was invented by
theorists long after the era of meantone temperament supremacy had
passed. Before the eighteenth century, meantone temperament was
simply called 'keyboard tuning' by musicians. It was the common
temperament. (...)" [See the book for the fuller context....]

Brad Lehman

> > > > i think it's really only within the last few years
> > > > that we here have extended the word "meantone" to
> > > > cover the whole family of temperaments in which the
> > > > syntonic-comma vanishes.
> > >
> > > Someone with a copy of Barbour might tell us his usage;
> > > I thought he called eg 2/7-comma "meantone".
> > >
> > > In any case to make sense of the theory you need to
> > > distinguish what is tempered out from the detail of
> > > precise tunings.
> >
> > ***Barbour calls 2/7 comma "meantone..."
> >
> > Pg. 32:
> >
> > "The first regular temperament to be advocated after
> > the description of the ordinary meantone temperament was
> > that described by Zarlino in which "each fifth remains
> > diminished and imperfect by 2/7 comma."
> >
> > Then, there is a table of this temperament on page 33,
> > distinctly in the "meantone" chapter...
> >
> thanks, Joe. i should have responded to this when
> it first came up.
>
> i really only meant "within the last few years" in
> my statement in a vague sort of way, of course knowing
> that Barbour had already placed a lot of other tunings
> into the "meantone" category and called them by that
> name.
>
> my point was that *at the time the older treatises
> were written* (1500s-1600s) the theorists were not
> calling anything other than 1/4-comma by the "meantone"
> name.
>
> as i said, i'm not 100% sure about this and would have
> to go back and do research that i don't have time for
> now. but i'm fairly certain that the older theorists
> who wrote about tuning we would now call "meantones"
> but which use a division of the comma other than 1/4
> (Zarlino, Salinas, Smith, etc.), did not use the
> "meantone" moniker when discussing them. it was only
> in the 1900s that that began ... i think.

hi Brad and everyone,

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

> But, just ahead of that on page 8, Jorgensen gave us:
> "The term 'meantone temperament' is a relatively new term
> that was invented by theorists long after the era of
> meantone temperament supremacy had passed. Before the
> eighteenth century, meantone temperament was simply
> called 'keyboard tuning' by musicians. It was the common
> temperament. (...)" [See the book for the fuller context....]

and *in* the 18th century "keyboard tuning" meant some
type of circulating (well) temperament, whereas
"correct tuning" for orchestral players still meant meantone.

you can find stuff about this on my "55edo" page:

http://tonalsoft.com/enc/index2.htm?../monzo/55edo/55edo.htm

Paul Erlich contacted me offlist to tell me essentially
the same thing i quoted from you above. i'd really have
to go back and read treatises from the 1500s-1600s to
get to the bottom of this, and i don't have time now.

-monz

Brad Lehman wrote:

> Jorgensen on p9 of _Tuning_ ascribes it back to Grammateus (aka > Heinrich Schreiber), 1518, in the sense that I presented it last week > at the beginning of this thread. (For what that's worth!) In > Schreiber/Grammateus' layout, "all the chromatic notes were exact > meantones between their two natural neighboring notes, and the
> natural > notes D, G, and A were also exact meantones between their two > chromatic neighboring notes. Hmm, that sounds like "double diatonic" from Graham Breed's catalog!
http://x31eq.com/catalog.htm#doubleneg

Herman Miller wrote:
> Brad Lehman wrote:
> > >>Jorgensen on p9 of _Tuning_ ascribes it back to Grammateus (aka >>Heinrich Schreiber), 1518, in the sense that I presented it last week >>at the beginning of this thread. (For what that's worth!) In >>Schreiber/Grammateus' layout, "all the chromatic notes were exact >>meantones between their two natural neighboring notes, and the
>>natural >>notes D, G, and A were also exact meantones between their two >>chromatic neighboring notes. > > > Hmm, that sounds like "double diatonic" from Graham Breed's catalog!
> http://x31eq.com/catalog.htm#doubleneg

I also found this one in the Scala archive:

artusi.scl

Lute tuning of Giovanni Maria Artusi (1603). 1/4-comma w. acc. 1/2-way naturals
|
0: 1/1 0.000 unison, perfect prime
1: 96.578 cents 96.578
2: 193.157 cents 193.157
3: 289.735 cents 289.735
4: 5/4 386.314 major third
5: 503.422 cents 503.422
6: 600.000 cents 600.000
7: 696.578 cents 696.578
8: 793.157 cents 793.157
9: 889.735 cents 889.735
10: 986.314 cents 986.314
11: 1082.892 cents 1082.892
12: 2/1 1200.000 octave

Considered as a 7-limit linear temperament, this is the one we've been calling "injera" (named after a kind of Ethiopian bread), which repeats at the half octave. It has a tuning map of [<2, 3, 4, 5|, <0, 1, 4, 4|]. The 7-limit TOP tuning for this temperament is 93.610 cents for the generator and 600.889 cents for the period. It's interesting that the same pattern shows up in a 486 year old tuning!