1/7th comma meantone

[Richard Moody]

While "meantone"  is supposed to refer to  the ratio of the tone
between the tonic and the major 3rd, there is little in the theory(s)
of meantone to suggest this was actually used to construct the scales.
It was/is a result rather than a method.  The important words in the
name "quarter comma meantone" are "comma"   Here comma refers to the
comma of Didymus or the syntonic comma.   This comma results when four
successive 5ths form a wide major third of the ratio 81/64.   Now a
pure major 3rd has the ratio of 5/4. The difference between 81/64 and
5/4 (or 80/64) is the syntonic comma.   Since the differences between
ratios are determined by division rather than simple subtraction this
value is 81/80.   because 81/64 divided by 81/64 gives 81/80.
(21.506)
    Remember,  to see what ratio is formed by four successive 5ths,
the ratio of the 5th (3/2) is multiplied together 3 times, or
3/2*3/2*3/2*3/2   or 3/2^4.   If you compute this you get 81/16.  This
is an intgerval two octaves higher than a 3rd, and to get an interval
an octave down you divide by 2, or multiply by 1/2 so two octaves down
you multiply by 1/2 again which gives 81/84.    Since musical
intervals are detemined always by multiplicatio or divison, never
added or subtracted, a quarter of the comma in this case is the fourth
root of 81/80, or (81/80)^1/4.
    Why do we want one quarter of the comma of Didymus? Because a
series of four 5ths reduced by this amount will give a pure 3rd,
rather than the sharp 3rd of four pure 5ths sometimes called the
Pythagorean 3rd.  The problem of tempering seems to have been
approached through the succession or circle of 5th reduced by a
certain amount.  Meantone does convey a tempering that gives pure
3rds.  So when one encounters the term "1/7 meantone"  that means the
5th has been tempered or narrowed by 1/7 of the syntonic comma.  This
will give a wider 3rd than pure of course but narrower than the
Pythagorean 3rd.  Remember that the circle of twelve 5ths narrowed by
1/4 comma will end up with a tonic flatter than a circle of pure 5ths.
This gives the "last 5th" very much out of tune or the so called "wolf
fifth".   Perhaps they thought with 1/7 comma the last 5th might not
be so objectionable.  Whether this was actually used is a matter of
historical research. But whatever ratio is used to "divide" the comma,
the tone between the tonic and 3rd will always be a mathematical mean.
In that regard ET is also a meantone.
    In the article "Temperament" in New Groves several fractions are
given for types of meantone.   1/7th, 2/7th, 1/9, 2/9 if memory
serves.  Also associated with these various meantones were  scales
with more than 12 notes which the microtonalists have studied and
archived in great detail.
    When cents are used simple division will suffice.  Thus if we see
that the syntonic comma is 21.5 cents, then 1/4 of that is 5.375
cents.   1/7th of that is 3.07 cents, 2/9 is 4.7 while 1/9th is 2.3
cents.  ---ric




. I
| must confess to some mystification also to the idea of 1/7th. comma
| meantone. Surely there can be no such thing, as meantone has always
meant
| the division of the just major third into two (mean)tones rather
than the
| major tone with a ratio of 8:9 and the minor tone ratio of 9:10
which added
| together comprise the just major third. Perhaps someone can be kind
enough
| to enlighten me.
|
| Ted Sambell
|