Re: [tuning] Draft: What is meantone (MT)?

WHAT IS MEANTONE (MT)?

(First draft of a FAQ entry)

Meantone (MT) is a temperament where the syntonic comma (81:80; 21.5 cents) is
distributed equally among a fixed number of successive fifths. The standard, or
_quarter-comma MT_, distributes the comma among four fifths, so that their
octave-reduced sum is a just major third (5:4, 386.3 cents). The fifth in
quarter-comma MT has a size of 696.6 cents. This can be tuned by ear by first
setting a just major third and then tempering the intermediate fifths. The name
MT is derived from the size of the wholetone (193.15 cents), which divides the
just major third equally and falls between between the just major (9:8) and
minor (10:9) wholetones.

The major variants of MT include:

Third-comma MT, where an octave-reduced just major sixth (5:3, 884.4 cents) is
the sum of three fifths of 693.3 cents),

Fifth-comma, where an octave-reduced just major seventh (15:8, 1088.3 cents) is
the sum of five fifths of 697.6 cents),

Sixth-comma, where an octave-reduced just augmented fourth (45:32, 590.2 cents)
is the sum of six fifths of 698.4 cents). After quarter-comma MT, sixth-tone
temperament was the most widely used, especially in organs of the late baroque
and classical eras.

The process of distributing the comma can continue indefinitely or in fractional
variations, i.e. Zarlino's 2/7-comma MT. When the comma is distributed over
eleven fifths, the result is equivalent to 12tet. Quarter-comma meantone is
closely approximated by 31tet, third-comma by 19tet.

In quarter-comma meantone, with a keyboard of 12 keys per octave, eight major
triads will have just major thirds, typically the triads on Eb through E. It is
essential to note that although these tunings were chiefly used on keyboard
instruments with finite numbers of keys per octave, MTs are not tunings with
fixed numbers of pitches. The series of MT-fifths can be continued indefinitely,
with each additional tone adding an additional available tonality. MT
instruments with more than 12 keys per octave were not unknown, and G.F.H�ndel
owned instruments with 14 and 16 keys per octave.

Music in MT is notated with the standard pythagorean scheme: seven nominals or
staff positions without accidentals are modified by sharps (#) as one ascends by
fifths and flats (b) as one descends. This process continues indefinitely. Due
to the smaller size of fifth the chromatic semitone will be smaller than the
diatonic. Thus in MT c# is lower in pitch than db, the opposite relationship is
heard in pythagorean tuning.

MT was the pre-eminent keyboard tuning in the 17th and 18th centuries. The
earliest recorded description of a MT tuning procedure is usually attributed to
Pietro Aron in his _Toscanella_ (Venice, 1523). Common usage of MT or MT
variants continued well into the 19th century with its final replacement by
various well temperaments and 12tet occuring definitively only around 1850. MT
has been widely revived for performances of early music; modern tracker organs
in MT are not uncommon. Contemporary composers Gy�rgy Ligeti and Douglas Leedy
have composed works in MT.

Given the pre-eminence of MT in the era when common practice tonality developed,
it is useful to consider which qualities of MT were assumed by composers and
positively reflected in musical repertoire. These qualities included the purity
of the major third and a good major triad; a preference for major over minor
tonality (and when minor, a preference for g); a limited range of usable
tonalities (typically Eb to A); a leading tone significantly lower than that of
pythagorean or 12tet; a dissonant minor seventh, requiring resolution; an
augmented sixth intonationally distinct from the minor seventh (indeed, the MT
augmented sixth is a good approximation of a 7:4). More fundamental, however, is
the assumption in harmonic practice that motion by mediant intervals, thirds and
sixths, can also be heard as the sum of successive perfect fifths and fourths.
Inasmuch as common practice tonality can be characterized by a distinctive
interaction between triads with roots related by fifths and fourths and triads
with roots related by thirds or sixths, it was MT that provided the intonational
compromise to realize this interaction.

DJW

--- In tuning@y..., "Daniel Wolf" <djwolf1@m...> wrote:
> WHAT IS MEANTONE (MT)?
>
..............

This information, and the like that will surely follow, will be
extremely helpful for the neophytes like me. Thanks.
Regards,
Haresh.

Daniel wrote,

>MT was the pre-eminent keyboard tuning in the 17th and 18th centuries.

It seems, from all of Margo's posts, that the 16th would be included as
well, though the mathematics of it was not well-established until at least
1523.

My overall comment is: Start more slowly, as if explaining to a child, and
then give much more in the way of details, examples, and alternate
formulations for this:

>More fundamental, however, is
>the assumption in harmonic practice that motion by mediant intervals,
thirds and
>sixths, can also be heard as the sum of successive perfect fifths and
fourths.
>Inasmuch as common practice tonality can be characterized by a distinctive
>interaction between triads with roots related by fifths and fourths and
triads
>with roots related by thirds or sixths, it was MT that provided the
intonational
>compromise to realize this interaction.

May I make some suggestions about this text? They're interpolated & marked off w/ ###.

I'm sure glad I didn't have to make the draft - - it's much easier to take potshots at somebody
else's :-)

�ale C. Carr

Message: 24
Date: Wed, 21 Feb 2001 09:44:26 +0100
From: "Daniel Wolf" <djwolf1@matavnet.hu>
Subject: Re: Draft: What is meantone (MT)?

WHAT IS MEANTONE (MT)?

(First draft of a FAQ entry)

Meantone (MT) is a temperament where the syntonic comma (81:80; 21.5
cents) is
distributed equally among a fixed number of successive fifths.

###The term "successive fifths" might be misleading to a real tyro, or to one used to the
nomenclature of harmony: c-g, d-a, e-b are surely 'successive fifths', but just as surely not
what you mean. Maybe somewhere else in the FAQ the term 'successive fifths' is defined as meaning
'along the circumference of the circle of 5ths? Maybe this is too nitpicky?###

The
standard, or
_quarter-comma MT_, distributes the comma among four fifths, so that
their
octave-reduced sum is a just major third (5:4, 386.3 cents). The
fifth in
quarter-comma MT has a size of 696.6 cents. This can be tuned by ear
by first
setting a just major third and then tempering the intermediate fifths.

###Again, 'intermediate' might be unclear to someone looking at a keyboard rather than envisioning
the circle of 5ths.###

The name
MT is derived from the size of the wholetone (193.15 cents), which
divides the [...]

[...] the sum of five fifths of 697.6 cents),

Sixth-comma, where an octave-reduced just augmented fourth (45:32,
590.2 cents)
is the sum of six fifths of 698.4 cents). After quarter-comma MT,
sixth-tone
temperament was the most widely used, especially in organs of the late
baroque
and classical eras.

###I seriously doubt that there is any documentary evidence for this broad historical statement.
If there is, I'd love to be referred to it.###

The process of distributing the comma can continue indefinitely or in
fractional
variations, i.e. Zarlino's 2/7-comma MT. When the comma is distributed
over [...]

[...]The series of MT-fifths can be continued
indefinitely,
with each additional tone adding an additional available tonality. MT
instruments with more than 12 keys per octave were not unknown, and
G.F.H�ndel
owned instruments with 14 and 16 keys per octave.

###It would surely be more to the point to refer to split key instruments from the heyday of MT,
the 16th and 17th centuries. H�ndel's instrument was exceptional rather than even slightly
representative.###

Music in MT is notated with the standard pythagorean scheme: seven
nominals or [...]

MT was the pre-eminent keyboard tuning in the 17th and 18th centuries.

###Again, this historical statement is unjustifiably broad. A historical atlas of the spread of
MT in Europe would be fascinating; so far as I am aware, nobody has yet attempted it. In Italy,
England, & France it may indeed have been common *for organs* until the late 18th C., but - even
in these countries - other more easily tunable instruments will have been commonly tuned in
something between MT & 12-tet. In the area where Bach performed his organ music, MT would have
been impossible without split keys, which are not documented from that area & period.###

The
earliest recorded description of a MT tuning procedure is usually
attributed to
Pietro Aron in his _Toscanella_ (Venice, 1523). Common usage of MT or
MT
variants continued well into the 19th century with its final
replacement by
various well temperaments and 12tet occuring definitively only around
1850. MT
has been widely revived for performances of early music; modern
tracker organs
in MT are not uncommon.

###I wish it were so! By far the most non-ET modern organs are tuned in something between MT &
12-tet. Even historical organs when 'historically' restored are far too often tuned to a
temperament which makes possible performance of Bach's organ works, whether this repertoire is
relevant to the style of the instrument or not.###

Contemporary composers Gy�rgy Ligeti and
Douglas Leedy
have composed works in MT.

Given the pre-eminence of MT in the era when common practice tonality
developed,
it is useful to consider which qualities of MT were assumed by
composers and
positively reflected in musical repertoire. These qualities included
the purity
of the major third and a good major triad; a preference for major over
minor
tonality (and when minor, a preference for g);

###Again I think this is historically unjustifiable, quite apart from the fact that the repertoire
was not conceived in a 'tonality' but rather in a mode, for most of the period in question at
least. [If you're referring to final chords, that's another matter; but your reference to the
'preference for g' then makes no sense.] Probably as many pieces were composed in what we would
now anachronistically call 'minor keys' as in major.###

a limited range of
usable
tonalities (typically Eb to A);

###Here again the reference to tonalities is misleading. A d# in the key of C major will sound as
badly out of tune as it will in B major - unless of course the eb key has been tuned,
exceptionally, down to d#; the difference is that the d# doesn't occur as often in C as in B.
But it's irrelevant: the point is simply the available tones i.e. spellings, not the keys i.e.
tonalities.###

a leading tone significantly lower
than that of
pythagorean or 12tet; a dissonant minor seventh, requiring resolution;

###Isn't this a function of harmonic practice rather than tuning practice? Maybe it's the term
'dissonant' that's out of place here. A sticky issue: you can't very well substitute 'out of
tune minor seventh'. And even if you could, its being out of tune would require not resolution
but tuning. Hmmmm.###

an
augmented sixth intonationally distinct from the minor seventh
(indeed, the MT
augmented sixth is a good approximation of a 7:4). More fundamental,
however, is
the assumption in harmonic practice that motion by mediant intervals,
thirds and
sixths, can also be heard as the sum of successive perfect fifths and
fourths.
Inasmuch as common practice tonality can be characterized by a
distinctive
interaction between triads with roots related by fifths and fourths
and triads
with roots related by thirds or sixths, it was MT that provided the
intonational
compromise to realize this interaction.

###This seems pretty difficult for a beginner to grasp. By "motion by mediant intervals" do you
mean something like 'harmonic progressions by mediant intervals'? But harmonic motion by mediant
intervals was pretty much the exception, again, in the heyday of MT; and MT was pretty much
forgotten by the time harmonic motion by mediant intervals became anything other than a good
harmonic surprise.###

I hope this will help to refine some issues!

�ale

List !
I agree with this in general and believe we should remember that not
everyone on this list is as rigorous at mathematics as others, We are on
a music list and FAQ should be written in a music based language and not
a math except when unavoidable. when unavoidable should be explained in
layman terms.

"Paul H. Erlich" wrote:

> My overall comment is: Start more slowly, as if explaining to a child,
> and
> then give much more in the way of details, examples, and alternate
> formulations for this:

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

On Wed, 21 Feb 2001 09:44:26 +0100, "Daniel Wolf" <djwolf1@matavnet.hu>
wrote:

>WHAT IS MEANTONE (MT)?
>
>(First draft of a FAQ entry)
>
>Meantone (MT) is a temperament where the syntonic comma (81:80; 21.5 cents) is
>distributed equally among a fixed number of successive fifths.

This is fine for historical meantone tunings, but what about irrational
meantones? I think any tuning based on a series of tempered fifths of
identical size (assuming octave equivalence and exact 2/1 octaves), between
a certain range of acceptable fifths, ought to qualify as meantone.
Probably around 691 - 702 cents would be the extreme range of fifths that
might be acceptable: fifths outside of that range would result in whole
steps smaller than 10/9 or larger than 9/8. Actually, 700 cents would be a
reasonable upper limit, since fifths larger than 700 cents would result in
sharps being higher than flats.

Stretched octave meantone tunings, with octaves greater than 1200 cents,
are also useful in certain contexts, although strictly speaking they're not
really meantone scales. The benefit of tempering the octave is that the
fifths don't need to be tempered as much if the octaves are tuned slightly
sharp.

>The process of distributing the comma can continue indefinitely or in fractional
>variations, i.e. Zarlino's 2/7-comma MT. When the comma is distributed over
>eleven fifths, the result is equivalent to 12tet.

Only approximately equivalent (12-tet fifths are tempered by 1/11 of a
Pythagorean comma, not a syntonic comma), but close enough.

> Quarter-comma meantone is
>closely approximated by 31tet, third-comma by 19tet.

And 1/5-comma by 43-TET.

--
see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--
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@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

dcc carr wrote,

###This seems pretty difficult for a beginner to grasp. By "motion by
mediant intervals" do you
mean something like 'harmonic progressions by mediant intervals'? But
harmonic motion by mediant
intervals was pretty much the exception, again, in the heyday of MT; and MT
was pretty much
forgotten by the time harmonic motion by mediant intervals became anything
other than a good
harmonic surprise.###

Dear Dale,

(1) Progressions like I-vi-IV-V-I and I-vi-ii-V-I include motion by mediant
intervals and are typical of tonal practice in the meantone era.

(2) Daniel Wolf was not referring to pre-tonal music in his post, since the
interaction with "COMMON PRACTICE" was what was meant by the title. A whole
other FAQ could be written on "MEANTONE TUNING AND PRE-TONAL WESTERN MUSIC"