Re: [tuning] Meantone FAQ

Some responses:

(1) I stand by "modern tracker organs in MT are not uncommon". The adjective was
very carefully chosen after doing a survey of all historical organ builders who
have web sites. I think I turned up some 50 new or restored organs in MT, not
including small continuo organs and the like.

(2) I have qualified the reference to the distribution of 1/6 comma meantone to
specify a late era and specifically the association with the maker Silbermann
(whose organs were so roundly praised by Mozart). The H�ndel reference has been
made more modest (I've not got any reference materials here in Budapest).

(3) To Hermann Miller: I'm aware of many other variants, but I'd hesitate to
include them in this FAQ item. In fact, I think I pushed the envelope enough
already by including the historical nth-comma variants, which are debatably not
MT, as they no longer have wholetones at the mean of a just major third. If i
were writing DJW's tuning dictionary, instead of a FAQ, I would have done it
differently, i.e. given more weight to fifth-comma temperament, which I used in
my opera, THE WHITE CANOE (shameless plug).

(4) Setting a meantone: do you start with a just third or with the fifths? I
learned with the third first and can't imagine not having that reference point
to work with. But the order is immaterial for the purposes of a FAQ.

(5) Preference for g minor, this has been qualified to refer to Viennese
classicism. My reference is Kelletat.

So, here's the second draft:

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. This 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.

The major historical 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 perhaps most widely used, especially in the organs of the late
baroque
and classical eras built by Silbermann.

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 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 MT by 19tet,
fifth-comma MT by 43tet, sixth comma 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 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
played on instruments with up to 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 standard keyboard tuning in the 16th, 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 quarter-comma 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 (for Viennese classical music, 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.
For example, in MT, the harmonic progression from a C major triad to an a minor
triad is extremely smooth due to the pure third c-e. It can then be complemented
by a satisfactory return motion 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 compromist of MT that provided the
intonational environment in which this interaction was first realized on
instruments of fixed pitch.

DJW

Daniel Wolf wrote,

>G.F.Händel
>played on instruments with up to 16 keys per octave.

up to 16 tones per octave, with enharmonic variants selected via organ stops
rather than split keys.