Re: [tuning] What is Meantone (MT)? Third draft.

The following incorporates almost every suggestion made:

WHAT IS MEANTONE (MT)?

(Second 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 pefect fifth
in quarter-comma MT has a size of 696.6 cents. Such a fifth can be tuned by ear
by
initially setting a just major third (i.e. c'-e') and then, tempering the
intermediate fifths identically (octave reduced: c'-g'-d'-a'-e'). 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.

Variants of MT include:

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

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

Sixth-comma temperament, where an octave-reduced just augmented fourth (45:32,
590.2 cents)
is the sum of six fifths of 698.4 cents). Sixth-comma MT was associated
especially with organs of the late baroque and classical eras. (The theorist
Sorge explicitly criticized the organ builder Silbermann for this practice; on
the other hand, W.A. Mozart roundly praised Silbermann's instruments).

The process of distributing the comma can continue indefinitely or in fractional
variations, i.e. Zarlino's 2/7-comma temperament. When the comma is distributed
over
eleven fifths, the result is very close to 12tet (12tet actually is a
redistribution of the _pythagorean_ comma over 12 fifths, see 12TET).
Quarter-comma MT is closely approximated by 31tet, third-comma temperament by
19tet,
fifth-comma temperament by 43tet, and sixth comma temperament by 55tet.

In quarter-comma meantone, with a keyboard of 12 keys per octave, eight major
triads will have just major thirds, typically the triads on Ab through A or 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 intrinsically
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. Although more
frequently found in the early and middle meantone era, instruments with up to 16
tones in an octave are known to have been built and played throughout the life
of G.F.H�ndel.

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 standard keyboard tuning in the 16th, 17th, and 18th centuries. The
earliest recorded description of a MT tuning procedure is possibly that
attributed to
Pietro Aron in his _Toscanella_ (Venice, 1523). This attribution is, however,
controversial, while Zarlino's description of meantone in "Dimostrationi
harmoniche", Venezia 1571, is not controversial. 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 quarter-comma MT which exploit characteristic features of
the tuning.

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 (for Viennese classical music through Mozart, 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 (the MT augmented sixth is a good approximation of a 7:4).

More fundamental is the assumption in common practice harmony that motion by
mediant intervals, thirds and sixths, can also be heard as the sum of successive
perfect fifths and fourths. For example, in MT, the harmonic progression from a
C major triad to an a minor triad is extremely smooth due to the common pure
third c-e. It can then be complemented by a satisfactory return motion by triads
related by fifths, from a to d to G to C. Inasmuch as common practice tonality
can be characterized by such distinctive interaction between triads with roots
related by fifths and fourths and triads with roots
related by thirds or sixths, it was the compromise of MT that first provided an
intonational environment in which this interaction could be realized on
instruments of fixed pitch. There is astonishing agreement between the
properties of MT and the requirements of common practice harmony, but a
causality relationship has not been established.

DJW

First off, thank you Daniel Wolf for trying hard and being generous with your
time. I for one am appreciative of your efforts. Frankly, I'm not fully
sold on the FAQ idea. Perhaps it is my difficulty with signing on to
institutional definitions on this list when there is disagreement with the
definitions. I would prefer to explain myself rather than have others
explain and don't feel that Paul needs to feel an obligation to respond to
every question raised by a newbie.

To this end, I recognize that perceived criticism threatens to derail
important feedback. I'm sure no one on this list wants to be afraid to post,
or to simply register an opinion. Since we are not in school and we are not
publishing...this has nothing to do with any one person, particularly, I
would like to add an education perspective. I have been teaching people of
all ages, as many on this list may have. My only question on the informative
second draft by Daniel is if by the end of the first line, one must still
look up "temperament" and "syntonic" and comma" and "cents" and deal with
numbers, there must be a better way to organize difficult material.

Since we surely have no embarrassment between us, Daniel, I want only to try
something out. Now that all the FAQs are going to Paul directly, your
contribution may be the only one that allows for redirection, albeit with
your willingness. As I recall, you didn't want to change anything at the
top, but maybe you will reconsider for reasons that are good ones.

WHAT IS "MEANTONE"

Meantone is a tuning that was developed in the Renaissance to modulate on a
keyboard instrument. Theorists devised a way to split a major third into 2
equal parts (the "mean" whole tones) by flattening the pure fifth
consistently in every key.
This produces variants; quartercomma meantone splits a pure major third
(ratio 5/4 or 386 cents), while sixthcomma meantone was preferred by other
musicians in different times and places.

When the pure fifth (formed by the ratio of 3/2) is consistently flattened by
anywhere from 2 cents to 6 cents (with 1200 cents equal to an octave), a set
of identical diatonic keys are set up. In addition to the modern keyboard
setup of 12 keys per octave, meantone practitioners (like composer Michael
Praetorious, 1571-1621) often favored split-key instruments.

etc.

Thank your for your attention to this. Maybe it is too sappy above, but I
hope you think it was worth the try. I realize that there are difficult
terms that still need to be looked at elsewhere, perhaps on a different FAQ.
But giving help should be the overriding interest here.

Johnny Reinhard

Johnny Reinhard wrote:

> Thank your for your attention to this. Maybe it is too sappy above,
> but I hope you think it was worth the try. I realize that there are
> difficult terms that still need to be looked at elsewhere, perhaps on a
> different FAQ. But giving help should be the overriding interest here.

It looks ideal to me. I don't think newbies should be expected to know
any of the things you pointed out, or keep cross-referencing one FAQ to
another to find out. So long as the meat's still there, which if you're
building on Daniel's excellent foundation shouldn't be a problem.

Graham

--- In tuning@y..., "Daniel Wolf" <
djwolf1@m...> wrote:
>
> MT was the standard keyboard tuning in the 16th, 17th, and 18th centuries. The
> earliest recorded description of a MT tuning procedure is possibly that
> attributed to
> Pietro Aron in his _Toscanella_ (Venice, 1523).

How about a reference to the _physical
evidence_ of meantone in the late 15th
century (I believe Ibo reported this).

The rest of the FAQ is excellent! A+!

--- In tuning@y..., Afmmjr@a... wrote:

> Frankly, I'm not fully
> sold on the FAQ idea. Perhaps it is my difficulty with signing on to
> institutional definitions on this list when there is disagreement with the
> definitions.

Again, I don't see any attempt to sign on to
any institutional definitions and to
disenfranchise any alternative viewpoints.
Please express your viewpoints in a positive
manner and they are likely to get
incorporated into the FAQ.
> I
> would like to add an education perspective.

Excellent.

> I have been teaching people of
> all ages, as many on this list may have. My only question on the informative
> second draft by Daniel is if by the end of the first line, one must still
> look up "temperament" and "syntonic" and comma" and "cents" and deal with
> numbers, there must be a better way to organize difficult material.

Hopefully the FAQ will be in hypertext form
and one can simply click on the words in
question to get more info. However, I agree
that each FAQ entry should be as self-
contained as possible and that Daniel could
have presented these concepts in a less
intimidating way (I'm sure his intent was not
to be intimidating, but as I've learned (as
someone who likes to be as concise as
Daniel), that can often be the effect.)
>
> Since we surely have no embarrassment between us, Daniel, I want only to try
> something out. Now that all the FAQs are going to Paul directly,

Johnny, absolutely nothing is going to me
directly, at least not yet.

> your
> contribution may be the only one that allows for redirection, albeit with
> your willingness.

I don't know what this means.

As I recall, you didn't want to change
anything at the
> top, but maybe you will reconsider for reasons that are good ones.
>
> WHAT IS "MEANTONE"
>
> Meantone is a tuning that was developed in the Renaissance to modulate on a
> keyboard instrument.

Johnny, this is misleading for two reasons.

(1) Modulation, as understood today, is a
change of key; in the Renaissance the key
system did not yet exist.

(2) Meantone temperament is as necessary
for a piece of Renaissance music that sticks
to the natural notes ("white notes" on the
piano) as it is for music which uses altered
notes as well.

> Theorists devised a way to split a major third into 2
> equal parts (the "mean" whole tones) by flattening the pure fifth
> consistently in every key.

Again, the reference to "key" is confusing . . .
"flattening every pure fifth consistently"
would work better.

> Thank your for your attention to this. Maybe it is too sappy above, but I
> hope you think it was worth the try. I realize that there are difficult
> terms that still need to be looked at elsewhere, perhaps on a different FAQ.
> But giving help should be the overriding interest here.

Agreed -- and may you continue to supply
suggestions, and may we continue to
consider all of them regardless of
"institutional definitions"!

Paul, I am not writing a FAQ, I was asking for simplicity so that one does
not have to look up 3 words in the first sentence, then run to another FAQ
which also has 3 words to look up, and then has 3 new terms each to look up.

Modulation and meantone is debateable, but it is not the point I am making.
If Meantone begins in the Renaissance and is used for greater movement and
modulation, then I don't have trouble being general. Please do as you will.
It's just that Meantone is one of the easier subjects to write about. I
won't want to look at or read the more complex ones.

Basically, I am in the midst of much and can't write more than I have. In
fact, Margo has made good sense in her last posts re drifts. I do not
believe you, Paul, like drifts and you seem to see them as an evil. I do
not, and see them as a valuable effect on Renaissance recordings with good
intonation. In bad modern music there is sometimes drift, but this is not
the same thing as writing music where the drift is expected. Margo has
addressed this very well.

Re Benedetti considering drift a vice, rather than something noticeable to
him, please quote the appropriate sentence. I have not noticed it.

Paul, when I started on these topics I noticed some serious personal stuff
with you and some others. I was uninvolved. Now, you seem to address every
notice to me with something condescending about being positive. Please stop
this stuff.

As for whether you get FAQ submissions privately or not, I had the good
opportunity to express my opinions here and to Daniel Wolf, publicly and
privately. I am finished with this exploration. I am disapointed that
things are so difficult working on these fine issues through internet alone.
Splitting hairs may be a microtonalist's predilection, however. All good
wishes.

Johnny Reinhard

Dear Johnny,

> Re Benedetti considering drift a vice, rather than something
noticeable to
> him, please quote the appropriate sentence. I have not noticed it.

I believe Margo's last post addresses this fully. Benedetti was using
the following logical induction:

just intonation leads to drift; therefore (by the contrapositive)
no drift implies non-just-intonation (i.e. temperament); so
singers, who do not drift, sing in tempered intervals.

Now, this deduction is false -- since, as in Vicentino's method, one
can have vertical just intonation and no drift -- but it's the one
Benedetti uses.
>
> Now, you seem to address every
> notice to me with something condescending about being positive.
Please stop
> this stuff.

It's just a reaction to statements to the effect that the FAQ is
evolving in a biased or unfair way. I want you to feel as comfortable
as possible with the final contents of these articles, and to make as
substantial a contribution as possible in order to attain that end.
Otherwise, they will be invalid in my eyes.

>All good
> wishes.

Best wishes to you to, my friend.

-Paul