Temperament Question, Just Int.

[Richard Moody]

>Question: which temperament has the most just
> intervals?



It would be interesting to establish a temperament that might be called Just
I or that which has the most pure sounding intervals.   It is of course
impossible to have all the intervals on the piano in Just Int. which I take
to mean pure, ie no beats.   If you attempt to tune 3 pure thirds on top of
each other to make an octave, one of the 3rds cannot be pure.   To
understand this  one
needs to know about the ratios of the intervals.  To hear it,  one needs
only a tuning hammer and the experience of  tuning say C--E, E--G#, then
since G#--C is not a third, (at least in music theory class), you are
looking at Ab--C, but Ab has already been tuned as G#.   This will also give
insight into why G# is not the same as Ab.
    An interesting experiment is to try to tune the C diatonic scale in
Just.  These intervals are pure-----C--G, C--F, C--E, E--B.  Here you will
notice that G--B not tuned directly, does sound as a pure 3rd. Now if A is
tuned pure to F, and D is tuned pure to G then you will hear all the
intervals pure in the I--IV--V progression starting on C.  However this D
will not make a good 5th with A.
Or if D is tuned pure to G and A is tuned pure to D, A will not make a pure
3rd with F nor a pure 4th with E.   Hence the need for temperament on a
piano.   The black keys present the same problems.  You can tune them as
pure 3rds to a white key, or pure 5ths to another white key. Sooner or later
glaring faults, (wolves) show up.  So compromises must be made.
    Way back when, someone suggested that D might be tuned as a "mean tone"
between C and E.  Again the understanding of this involves knowing the
arithmetic of the  ratios of the intervals.  As just intervals C--D is 9/8
and D--E is 10/9.   This however makes C--E a wide 3rd.  The arithmetic
proves it. Since ratios are multiplied by each other to get a larger
interval, 9/8 times 10/9 = 90/81  but a pure 3rd (5/4) is 90/80.
Now the mean of two ratios is the square root of their  product.   This is a
number that means nothing to the tuner, because how can you get D "exactly"
between C and E with only a tuning hammer?  The theorists could on paper and
with the monochord.  They attempted this with lengths of organ pipes but
that didn't work either.  The precise tuning needed to be done by ear.
    The temperaments with pure 3rds but narrow 5ths are based on 1/4 comma
Meantone.  The "ancients" knew that a circle of 4 pure
fifths would produce a sharp 3rd (from the starting note)  Therefore if
these
fifths were each flattened, the 3rd would come out pure. So they tuned flat
5ths in a circle all the way from C to G#, then down from C to Eb.  Now the
next 5th after Eb is Ab.  But this has already been tuned as
G#.   So all 12 notes of the octave have been tuned, but the interval Eb--G#
sounds horrible, and this they called the "wolf".  Also since only 2 pure
3rds can be in an octave, the third 3rd is way off.  There are 4 of these.
    The tuning with the most pure intervals is probably Pythagorean.  Here a
circle of 5ths is tuned pure until the last one is reached.  This makes a
wolf with the starting note.  However strange as it may seem, G#--C even
though a d4 (dim 4th),actually sounds like a pure 3rd.  Now if pure 5ths are
tuned from C to G# and C to Eb the wolf is again between Eb and G#.    A d4
exists,  F#--Bb, which should sound pure.    Try a Pythagorean tuning and
you
will be amazed to hear these pure sounding "3rds" and also try the augmented
2nds, these sound as minor 3rds. (Bb--C# for example or
Eb--F#). Thus the 4 "diminished 4ths", F#--Bb, G#--C, B--Eb, and C#--F when
played as a major triad sound pure.  Wait till you hear the minor triads and
how they progress.   Talk about tonality.  How differently cadences and
progressions sound in these keys than in the ones with the so called
Pythagorean 3rds. The execption is G#--C--Eb, but what color! ; ).   ---ric

> There has been alot of interesting discussion about temperaments lately,
>I have a customer who
>is very interested  in Just Intonation.  I'd like to tune her piano in a
temperament in
> which she will be able to hear as much just intonation as possible. I have
> Jorgensen's book, and have been looking at the Pietro Aaron temperament,
> which has a lot of just thirds, but no just fifths; and the Kirnberger
> which has some just triads. Question: which temperament has the most just
> intervals,
> Thanks, Keith Jones
>