Yahoo Tuning Groups Ultimate Backup tuning meantone temperament question

----- Original Message -----
From: Dave Keenan <D.KEENAN@UQ.NET.AU>
To: <tuning@yahoogroups.com>
Sent: Sunday, August 05, 2001 8:44 PM
Subject: [tuning] Re: meantone temperament question

> --- In tuning@y..., "tim oram" <tim_oram@h...> wrote:
> > I've been looking into meantone temperaments a bit lately and have
> become aware that composers in meantone (normally?) didn't specify
> what meantone temperament they were writing for. Is this because the
> composer (unless the piece was only meant to be played on the
> instrument it was composed on) could never be sure of what meantone
> temperament instruments in different regions would be tuned to?
> >
> > I'd be really grateful if somebody could clear this up for me.
>
> Hi Tim,
>
> I hope someone more knowledgeable has already answered this for you by
> email, but just in case, I thought I'd have a go.
>
> My guess is that you are right. But also, they probably didn't care.
> Not many instruments would stay that accurately in tune and not many
> listeners can tell the difference between 1/3, 1/4, 1/5 comma
> meantones etc. anyway. All the composer would really care about is
> whether the wolf was in the right place. e.g. is it Eb to G# or Ab to
> C# etc. That requirement is obvious from the score.

Hi Tim and Dave,

I posted something on this a few weeks ago, in a response to
Herbert Anton Kellner, mainly quoting from Mark Lindley and
Roland Turner-Smith, and it seems relevant to repost it here:

> From: monz <joemonz@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, July 13, 2001 9:28 AM
> Subject: Re: [tuning] trias harmonica - for Margot Schulter
>
>
> Note also that 1/4-comma meantone, the type you describe
> here, where the "5th" is calculated as ( (3/2) / (81/80)^(1/4) )
> = ~696.5784285 cents (nearly identical to 31-EDO), was only
> one early solution. Its "major 3rd" is of course the 5:4
> ratio = ~386.3137139 cents. Its "whole-tone" is ~193.1568569
> cents, which is *precisely* the mean between the 9:8 and 10:9
> ratios. This is the only tuning which can be labeled *exactly*
> as "mesotonic".
>
> But while 1/4-comma meantone was indeed very popular during the
> 1500s and 1600s, it was far from being the exclusive "meantone"
> tuning employed.
>
> By the mid-1600s 1/4-comma had fallen out of general favor and was
> usually replaced by 1/6-comma meantone or its equivalent 55-EDO:
>
>> "Sauveur ... said the 55 division was "the one which ordinary
>> musicians use" [Sauveur 1711, p 315].
>> ...
>> Sauveur's remarks when taken together with other pertinent
>> evidence [from p 54, regarding 1/5-comma meantone; see
>> quote below] suggest that the 1/4-comma temperament is not
>> the most valid model for a menatone system of his day.
>> It had been the theorists' favourite model, however, during
>> the period between Zarlino's _Dimostrationi harmoniche_
>> (1570) and Mersenne' _Harmonie Universelle_ (1637), and
>> presumably it was used a good deal during that period, even
>> though one writer said in 1613 [Cerone 1613, p 1049] that
>> master organ builders used a meantone temperament with the
>> major 3rds larger than pure." [Lindly/Turner-Smith 1993, p 149]
>
> In 1/6-comma meantone the "5th" is calculated as
> ( (3/2) / (81/80)^(1/6) ) = ~698.3706193 cents
> (nearly identical to 55-EDO... see
> <http://www.ixpres.com/interval/monzo/55edo/55edo.htm>).
> The "major 3rd" of 1/6-comma meantone = ~393.4824771 cents,
> or ~7.168763199 cents wider than that of 1/4-comma meantone
> (which is the "pure" 5:4). Under certain conditions this
> could be an audible difference.
>
> The 55-EDO "5th" is 2^(32/55) = ~698.1818182 cents. Its
> "major 3rd" is 2^(18/55) = ~392.7272727 cents, and its
> "whole-tone" is 2^(9/55) = ~196.3636364 cents.

And a bit further on:

> But Zarlino himself advocated 2/7-comma meantone earlier than,
> and as superior to, 1/4-comma! I quote:
>
>> "... the oldest quantitatively coherent account of a meantone
>> temperament is Zarlino's description (1558) of a tuning in
>> which the major (and minor) 3rds are smaller than pure
>> by 1/7 of the syntonic comma, and so the 5ths are tempered
>> by 2/7 comma ...
>> In 1571 Zarlino published a scheme which several later
>> theorists took as their only model of meantone temperament
>> (although by the end of the 17th century, it may have been
>> used less than tunings with the major 3rds tempered larger
>> than pure) ...
>> But Zarlino through the 1570s and '80s still liked the sound
>> of his original 2/7-comma meantone temperament as well...
>> Near the other limit which we have determined for a meantone
>> temperament lies this possibility: [1/6-comma meantone].
>> And there is an obvious intermediate possibility [1/5-comma
>> meantone]... Sauveur preferred it to any other kind of system,
>> and Louliï¿½ in 1698 [Louliï¿½ 1698, p 28] said it was more in
>> use than any other." [Lindley/Turner-Smith 1993, p 52-54]

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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🔗monz <joemonz@yahoo.com>

🔗Paul Erlich <paul@stretch-music.com>

Hi Tim!

Others have answered your question well -- Monz and Dave Keenan in
particular -- I just wanted to provide some clarification on one
point, in case this wasn't clear. Historically, composers never
specified that they were writing "in meantone". "Meantone" itself is
a word that has come into use more recently than the actual meantone
era. In the 16th and most of the 17th centuries, and for non-keyboard
instruments for most of the 18th century, what we today call meantone
was simply called "correct intonation". One recognized feature of
this was that the notes that we today consider to be "enharmonically
equivalent" were supposed to be performed as different pitches. G# is
lower than Ab, D# is lower than Eb, etc. In fact, the
name "enharmonic" comes from the fact that in the Renaissance, many
theorists were interested in ancient Greek scales, including the
enharmonic scale which featured quartertones. In a typical
Renaissance tuning, all the pairs of "enharmonic equivalents" are in
fact a quartertone apart (e.g., 50.3 cents in 2/7-comma meantone; 48
cents in 50-tET). Many keyboards had split black keys or other
mechanisms to allow for the performance of a piece with both G#s and
Abs, for example. In most cases, though, keyboard works in this era
were written with no more than 12 distinct pitches, so it would be up
to the performer to look over the piece and tune the keyboard
accordingly. In some cases, an Ab might appear in passing even though
a G# was used prominently elsewhere in the piece -- in this case, the
keyboard would be typically tuned to include G# but not Ab (unless
there were split keys).

I'm rambling . . . let me know if you have any other questions.

-Paul

🔗jpehrson@rcn.com

🔗Paul Erlich <paul@stretch-music.com>

🔗tim oram <tim_oram@hotmail.com>

Dear all,

just a quick message to thank everyone who answered my question about meantone. Your answers have been very helpful and I really appreciate it.

Cheers,

Tim

----- Original Message -----
From: Paul Erlich
To: tuning@yahoogroups.com
Sent: Tuesday, August 07, 2001 12:30 AM
Subject: [tuning] Re: meantone temperament question

--- In tuning@y..., jpehrson@r... wrote:

> Ummm, I'm not sure this is just rambling... I never knew a keyboard
> performer would tune a 12-note per octave keyboard in meantone
> according to pieces in this way...
>
Oh yes . . . this is well-documented.

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🔗mschulter <MSCHULTER@VALUE.NET>

Hello, there, everyone, and I'd mainly like to reply to some questions
about Renaissance meantone temperaments and the range of options
discussed by writers in this era.

First of all, I would say that 1/4-comma meantone (with pure 5:4 major
thirds) is a fine choice for Elizabethan music, and likely to fit most
16th-century music quite nicely. Joe Pehrson, I would warmly encourage
you to continue using this tuning for your Shakespearean setting, with
the following discussion intended to place this choice in a certain
historical and theorical context.

One point recently made by Mark Lindley in a number of articles, but
not always adequately emphasized when specific fractions of a comma of
tempering are discussing, is that the process of temperament or
_participatio_ was likely in use for about a century before it got
mathematically defined by Zarlino (1558).

One might guess that the transition from Pythagorean tuning for
keyboards to meantone took place in many parts of Europe around
1450-1480; and various sources around 1496-1555 either describe the
practice of tuning fifths narrower than pure in general terms, or give
specific but not rigorously mathematical instructions for negotiating
this practice. It is Zarlino, however, who first describes tempering
or _participatio_ in terms of fractions of a syntonic comma -- his own
2/7-comma meantone in 1558, and also in 1571 the alternatives of
1/4-comma meantone and 1/3-comma meantone (pure minor thirds).

While Lindley correctly asserts that Pietro Aaron (1523) does not
specify 1/4-comma meantone, I would myself conclude that Aaron's
procedure may easily -- although not inevitably -- be read to achieve
the same result. He urges tuning the major third C-E "as just and
sonorous" as possible, and then tuning a chain four equally impure
fifths C-G-D-A-E.

Lindley ably argues that this tuning of the first five notes does not
_necessarily_ imply that other notes should be tuned in exactly the
same way; and he shows how Aaron's instructions _can_ cleverly be
interpreted to call for different degrees of temperament of certain
other fifths. However, Aaron's text, addressed to musical novices in
the Italian vernacular, at least suggests how a meantone with pure
major thirds might often have been tuned without a specific
mathematical focus on fractions of a syntonic comma.

At the same time, other theorists of the era describe either what may
be regular meantones with major thirds wider than 5:4 (and less
heavily tempered fifths), or what might be called "irregular" tunings
(e.g. Arnold Schlick, 1511).

Also, while Zarlino is the first to _define_ 2/7-comma, 1/4-comma, and
1/3-comma meantone in terms of fractions of a comma, certain other
theorists do describe tuning systems that imply temperaments at or
very close to such tunings, although defined in other ways.

Thus Vicentino (1555), in describing a tuning for his archicembalo or
superharpsichord where the octave is divided into 31 conceptually
equal "minor dieses" or fifthtones, suggests something very close to
1/4-comma meantone. The constraining factor here is that a chromatic
semitone is divided into two essentially equal diesis or fifthtone
steps, an effect featured in his experimental music.

In 1/4-comma meantone -- or, "a meantone with pure major thirds" --
a chromatic semitone of around 76 cents is divided into fifthtones of
around 41 cents and 35 cents, fitting the apparent intentions of his
music as well as his theoretical system.

Likewise Guillaume Costeley (1570) describes a system dividing the
octave into 19 equal "thirdtones," almost exactly equivalent to
1/3-comma meantone with pure 6:5 minor thirds.

Where more experimental music of the era relies on such larger gamuts,
the precise degree of meantone can become more constrained than for
usual music involving 12-16 notes per octave, where the choice may be
largely one of taste.

Thus Zarlino himself, while advocating 2/7-comma meantone, recognizes
that 1/4-comma is pleasant and not difficult to tune, and finds
1/3-comma somewhat "languid" (a kind of description he also applies to
the simple minor third or the division of the fifth with the minor
third below the major third).

Interestingly, while Zarlino's former student Vincenzo Galilei takes
many radical views on theory at odds with his teacher, he agrees that
2/7-comma meantone is the most pleasant tuning for keyboard
instruments.

In Spain, Cerone (1613) reports that an organ tuning with major thirds
wider than pure is the practice -- and Lindley recommends a
temperament somewhere around 1/5-comma meantone as pleasing for some
pieces around 1600.

Given all this variety, I might just offer a few practical
suggestions.

(1) Generally 1/4-comma meantone seems to me a delightful
starting point, and a quite acceptable "default," for
most 16th-century keyboard music -- not to exclude
other alternatives.

(2) For most "usual" pieces involving not more than 12-16
notes per octave, any meantone in the range of around
1/3-comma to 1/6-comma _might_ represent a period
keyboard realization in some region of Europe or
tuning tradition.

(3) For a few specialized or more experimental pieces,
considerations such as the division of a chromatic
semitone into two equal or quasi-equal fifthtones
(Vicentino), or of the octave into 19 equal
thirdtones (Costeley), point to a specific kind of
tuning model.

(4) As Mark Lindley has remarked, if one has easy access
to two or more degrees of meantone, then trying the
same piece in each tuning and seeing which "sounds
best" is about as good a method as any -- with
differences of taste not at all unlikely.

While there are vertical and melodic fine points which we can discuss
at length -- again subject to taste -- I find it not unlikely in
practice that the same organ piece, for example, might be performed in
1/4-comma somewhere in Naples; in 2/7-comma around Venice; and
possibly in 1/5-comma in Madrid. Within each region, also,
temperaments might have varied considerably, whether through
deliberate exercise of taste or assorted forms of imprecision, some
possibly quite artful.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗jpehrson@rcn.com

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

/tuning/topicId_26689.html#26848

> Hello, there, everyone, and I'd mainly like to reply to some
questions about Renaissance meantone temperaments and the range of
options discussed by writers in this era.
>
> First of all, I would say that 1/4-comma meantone (with pure 5:4
major thirds) is a fine choice for Elizabethan music, and likely to
fit most 16th-century music quite nicely. Joe Pehrson, I would warmly
encourage you to continue using this tuning for your Shakespearean
setting, with the following discussion intended to place this choice
in a certain historical and theorical context.
>

Thank you so *very* much, Margo, for this post, and this is, at the
moment, a very, very, very important one for me. (Did I say *very?*)

I printed it out immediately, and will include it with all my
materials for The Tempest...

According to this research, it *does* look like a lot of such
meantone tempering really came into play *before* Shakespeare's birth
in 1564...

So far, for this project, I am only using 1/4 comma meantone for the
entire, but I VERY MUCH appreciate your additional comments and
suggestions on the others!

If I continue doing these kind of settings for the period, I am
heartened that I could even do an "experimental" piece with 19 tones
per octave a la Vicentino! I doubt that would be a *vocal* one,
because of rehearsal problems, but such an "instrumental" addition
might make a VERY interesting "atmospheric" piece... and the fact
that it would also be *historically accurate* means a lot!

So far, I have been using lots of music from the period... in
addition to a *few* of my own indulgent efforts: music of William
Byrd, John Dowland and Thomas Morley (Byrd's pupil, of course) who
were all writing during Shakespeare's time...

Of course, I am subjecting these great masters to some "overlaying"
and "processing" techniques using "Cool Edit" that would not be
entirely "historical" but which bring this music into the 21st
Century for this project...

Of course, I am aware that the Medieval and Renaissance techniques of
temporal overlay through rhythmic augmentation and diminution were
popular and went back, I believe to (I think) the first multiphonic
settings of Gregorian chant...

These processes were not quite as "extreme" as we can affect today,
certainly, through modern digital technology and a little "warped"
thinking, on *my* part.... :)

The good news is that the director, and most *non musicians* can't
distinguish between 1/4 comma meantone and 12-tET anyway, so the
project is a "go," but for the "cognocenti" this makes for a *much*
more interesting endeavor!

Thanks so *very* (again) much for the help!

__________ ________ _____
Joseph Pehrson

🔗jpehrson@rcn.com

🔗mschulter <MSCHULTER@VALUE.NET>

Hello, there, Joe Pehrson, and just a quick aside that while 19-tET is
Costeley's tuning, with 1/3-comma meantone as described by Zarlino and
favored by Salinas almost identical, you can go very authentically
"wild" with Vicentino in 31-tET, almost identical to the 1/4-comma
temperament he may have used for his own archicembalo.

Since Vicentino's system divides each whole-tone into five equal
dieses, theorists starting at least as early as Lemme Rossi (1666)
have equated this system with 31-tET. In this approach, each
whole-tone (5/31 octave) is divided precisely into five diesis steps
of 1/31 octave, with semitones at 2/5-tone (chromatic) and 3/5-tone
(diatonic).

With 1/4-comma, we get almost the same result -- so I'd regard either
tuning as quite "authentic."

May I add that Vicentino himself considered 19 notes as a very useful
subset (Gb-B#) for his adaptive JI tuning -- if you go with 31-tET,
19-out-of-31 -- and that around 1600 in Naples, 19-note meantone
keyboards likely tuned in 1/4-comma or something very close were quite
popular, and provided the occasion for some "wild" music by Trabaci
and others. John Bull's famous piece based on hexachord transitions
might be played on such a keyboard in Cb-E#, as I recall -- tune the
usual B# key a diesis higher to Cb, or use a larger tuning with both
notes already included, and you have the basis for a fascinating
period interpretation.

Of course, with a larger subset of 31, or all 31, you can do
Vicentino's "enharmonic genus" with its fifthtone steps (here
precisely 1/31 octave).

An attraction of 31 (or 1/4-comma) which Vicentino himself favors in
the "proximate minor" or neutral third at around 11:9.

In my view, either 1/4-comma meantone or 31-tET could be described as
a "31-tone" or "fifthtone" system, and either provides a valid
realization of Vicentino's system. As I've often commented, the
variations which people tuning by ear would be likely to get in the
16th century could be larger than the differences between these two
mathematical models.

Please don't let this stop you from experimenting with 19 also; I just
wanted to clarify that for people who find n-tET's a convenient
reference point, Vicentino's system can nicely be realized in 31.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗jpehrson@rcn.com

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

/tuning/topicId_26689.html#26911

> Hello, there, Joe Pehrson, and just a quick aside that while 19-tET
is Costeley's tuning, with 1/3-comma meantone as described by Zarlino
and favored by Salinas almost identical, you can go very authentically
> "wild" with Vicentino in 31-tET, almost identical to the 1/4-comma
> temperament he may have used for his own archicembalo.
>
> Since Vicentino's system divides each whole-tone into five equal
> dieses, theorists starting at least as early as Lemme Rossi (1666)
> have equated this system with 31-tET. In this approach, each
> whole-tone (5/31 octave) is divided precisely into five diesis steps
> of 1/31 octave, with semitones at 2/5-tone (chromatic) and 3/5-tone
> (diatonic).
>

Thank you very much, Margo, for the help with these "classic"
xenharmonic systems. It seems as though, according to your post, one
could "authentically" compose in either 31-tET or 19-tET and still be
within something historically accurate for Renaissance music. That
gives a *lot* of latitude... Of course, to be absolutely
historically correct, one would need to closely study such features
as voice leading, etc., etc., but at least this gives a general idea
of the wide-ranging possibilities and a "modern" composer might wish
to try something a little "neo-Gothic" in these systems... not
necessarily sticking entirely with the historical. Certainly, there
is a lot to work with here!!!

Now the question I have concerns the fifth-tone system. If 31 is
divided by 5, one gets 6.2. So that means there must have been
6 "whole tones" in the Vicentino system, yes?? What were the note
names of those?? Also, there is a "remainder." Where does *that*
go.... (??) Thanks for any help...

________ _______ _______
Joseph Pehrson

meantone temperament question

🔗tim oram <tim_oram@hotmail.com>

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

🔗monz <joemonz@yahoo.com>

🔗Afmmjr@aol.com

🔗carl@lumma.org

🔗monz <joemonz@yahoo.com>

🔗Paul Erlich <paul@stretch-music.com>

🔗jpehrson@rcn.com

🔗Paul Erlich <paul@stretch-music.com>

🔗tim oram <tim_oram@hotmail.com>

🔗mschulter <MSCHULTER@VALUE.NET>

🔗jpehrson@rcn.com

🔗jpehrson@rcn.com

🔗mschulter <MSCHULTER@VALUE.NET>

🔗jpehrson@rcn.com

🔗Paul Erlich <paul@stretch-music.com>

🔗jpehrson@rcn.com