margo on Meantone (real long)

[Richard Moody]

> Greetings,
>      Several have expressed interest in some further delving into the
history
> of temperaments, and Margo Schulter, an awesome author and authority on
the
> subject of real early tuning posted the following to the "Tuning"
list,(this
> is a truncated repost,  there is more!).   I hope it may be of some
interest.
> Regards,
> Ed Foote RPT
>






>>"
>Usual Pythagorean
> traits such as active thirds and sixths inviting efficient resolutions
> to stable 3-limit concords, and narrow diatonic semitones for
> expressive melody and incisive cadential action, are heightened in a
> form of intonational mannerism.
<<"

What is a 3-limit concord.?




>
> By describing these regular temperaments beyond Pythgorean as
> "neo-Gothic meantones," however, I elicited some predictable and
> constructive controversy. In a prompt response, Paul Erlich[2]
> questioned whether tunings with fifths wider than pure could be
> "meantones" in even "the most inclusive sense."

Any temperament or one using at least 4 "equal" fifths (in a row) will be a
meantone.   ET is a meantone.  Even Pythagorean is a meantone.
If one defines Meantone by D being the mean(tone) between C and E, this
happens in all temperaments.   (Starting from C)  It is an inescapible
mathematical phenomenon that results from the succession of 4 equally tuned
5ths whether tempered or pure.  Before boring everybody with the arithmetic
to prove it, I would like to bore everyone with my pet theory that Meantone
or at least the concept originated very early in the diatonic scaled organs.
    Now I have no proof of diatonic organs only that it seems natural that
an organ maker would make the diatonic scale first, then add the
"accidentals".  But be what ever may have been, the tuning of a diatonic
scale presents the perplexion of why temperament is necessary go make a
diatonic scale on the organ sound musical.  If the notes are tuned Just, all
make good 5ths and 3rds with each other execpt D.  To make a long story
short my guess is they decided "temper" D, so why not make it equal between
C and E or the mean ratio of C and E?   D as Just is 9/8 from C and 10/9
from E.  With E being 5/4 from C, the mean between C and E then is the mean
of nine to eight and ten to nine, or the square root of their products.
Well this product is 90/72 and that reduces to 5/4 (ya duh) (18 is the
reducer).  So we really need the square root of 5/4 to get the mean between
C and E.  Now this the ancients had various means to compute and one of them
was geometrical.  Organ pipes are geometrical,  so the geometrical length of
D as a mean between the length of C pipe and E pipe was figured.  Great
theory but practical problems, the least of which was the fact that two
identical organ pipes rarely sounded close enough to be called in tune.
Since organ makers had figured out since day two how to tune pipes they were
back to square one on how to tune a diatonic scale so that D would make both
a good 4th with G and 5th with A.  If they could only figure out how to
tuneD as a meantone they could solve the problem, at least that is what I
think they thought.
    Now this goes back a long time because we know of the names ascribed to
Guido (c1030)of the C major diatonic scale, and around 1530 we have Pietro
Aaron describing what today we call 1/4 comma meantone for a 12 note scale.
And this could have already been 100 years old as Margot suggests.  Aaron
did not say what his temperament was called, Mersenne in 1635 I don't
believe used the word "meantone".

    Since all meantone temperaments are constructed from the a fifth of
specific size they could be considered equal.  This though clashes with the
modern notion of Equal Temperament where ALL the intervals are equal.  Even
though Pythagorean and meantones  use equal fifths, some un-equal ones
result.  So it seems simpler to refer to  (x)tet, as just that, equal
temperaments, leaving meantone in its historic context to refer to temps
formed by fifths derived by the various fractions of the syntonic comma for
the purpose of purer 3rds and 6ths. ---ric





>
>I would like to propose
> the term "hypermeantone" to describe regular temperaments with fifths
> larger than pure.
>....................
> If we adopt this definition of "hypermeantone," then the term
> "hypomeantone" might apply to regular tunings with fifths_smaller_
> than in historical meantones, with 1/3-comma meantone or 19-tet as a
> possible line of demarcation.