Relative importance of tuning accuracy for fifths and major thirds

I've read that accuracy of tuning of the fifth is more critical than accuracy
of tuning of the major third for the overall harmoniousness etc. of the music.
It's beyond much doubt that in many ways the fifth is a more "fundamental"
consonance than the major third. It preceded the third historically and seems
to be more central in world music than the major third, which seems to have
become more important in western polyphonic music than elsewhere. I've had
the experience of playing and listening to recordings of music played on a
piano in three roughly equidistant tunings: 12 TET (fifths 2.0 cents flat,
thirds 13.6 cents sharp), sixth comma mean tone (MT) (fifths 3.6 cents flat,
thirds 7.2 cents sharp), and quarter comma MT (fifths 5.4 cents flat, thirds
just). My overall impression is that in 12 TET, the piano has a
characteristic "fine old burnished" sound - very lively with a lot of beating.
To me, the beating and shrillness considerably detract from the pleasantness
of the piano's sound. When the piano is tempered to sixth comma MT, the sound
is more relaxed. The major and minor triadic harmonies seem more natural
sounding and to me they sound better, warmer. Nevertheless, after having
become used to hearing the piano in quarter comma MT, the sixth comma MT music
still sounds somewhat strained and harsh to me, while it sounds to me much
more harmonious and pleasant in quarter comma MT.

When I listen closely to the piano's sound in these different tunings, I can
discern in both the 12 TET and sixth comma MT music a subtle "liveliness"
resulting from a lot of fast pulsatings in the sound. The effect is to give
the sound a roughness or harshness which takes away from the pleasantness and
somehow in a way the clarity of the music. By contrast, in the quarter comma
MT tuning, there seems to be an almost complete absence of a certain "brand"
of this "liveliness" which gives me a kind of sense of relief, as though the
air had been cleared of smoke or as though the fog on my windshield had
evaporated.

Now there's another kind of inharmoniousness which remains in the quarter
comma MT tuned piano. Now it's there and now it's not. Sometimes fourths in
the bass seem to be a bit unpleasant. The fifths are a little flatter than
would be ideal - not really nice and crisp. But by comparison with the sharp
thirds temperaments, the piano sounds absolutely beautiful. The triadic
harmonies are really sweet sounding while in 12 TET they sound strained and
labored and pinched, and the same holds to a somewhat lesser degree for the
sixth comma MT.

I've thought about this and tried to figure out why these temperaments affect
the sound of the piano I'm working with in this way.

I've just listened to some of Bill Sethares' fascinating music on a CD which I
just received from him. There is a delightful harmonious smooth feel to much
of the sound, and believe it or not, there's something about it reminiscent of
the way my piano sounded when it was in quarter comma MT - a very pleasant
freedom from roughness giving the music a feel of clarity and definition.

This makes me think that the beating and trembling which results from
mistuning plays a big part in marring the sound of mistuned music. It may be
that some kinds of beating actually enhance the music's sound. But the
tremulous, fast and I believe largely synchronous, coherent, and piercing
beating which mistuned thirds in particular give to the piano's sound seem to
me to be really detrimental to the sound.

If I listen for beating in the music of the piano tuned to quarter comma MT, I
can hear some waviness. Generally it is not prominent and doesn't impose
itself on my attention. It doesn't seem to give the sound any noticeable
"muddiness".

I thought about the beating phenomenon. Looking at a simple case, with
fifths, the third partial of the root will beat against the second partial of
the mistuned fifth above it and the beat rate will be proportional to the
product of the frequency of the root note, the number 3, and the degree of
mistuning in cents. In the case of mistuned thirds, the fifth partial of the
root will beat against the fourth partial of the mistuned third above it and
the beat rate will now be proportional to the product of the frequency of the
root note, the number 5, and the degree of mistuning in cents.

I've done some spectral analysis of piano string partials and I've found that
for notes in the vicinity of middle C, the partials are quite strong up past
the 10th or 20th or so. I believe that it is not totally missing the mark to
suggest that there will be a tendency for a mistuning of thirds by X cents to
cause beating with a beat frequency 5/3 times the beat frequency which results
from a mistuning of fifths by X cents.

I'm sure the matter is more complex than the above simplified picture would
imply, but it would be in line with my impressions that mistuning of thirds
seems to me to be more deleterious to piano music's sound than mistuning of
fifths.

Somewhat related to this is my sense that in quarter comma MT piano music, the
narrowness of the minor thirds (they are theoretically narrow by 5.4 cents in
this tuning) impresses itself on my attention somewhat more than the
narrowness of the fifths, also by 5.4 cents, in harmonic contexts. The
melodic quarter comma MT fifth sounds "tempered", but it still seems close
enough to me to have the feel of a fifth, which rises (sails) high above the
lower note. Still, in comparison with a true fifth, it falls short.

It's clear, and Bill Sethares' work is demonstrating this, that specifics of
an instrument's or of an ensemble's timbre will to a large extent determine
how different kinds of tuning or mistuning will affect its overall sound.

Dave Hill La Mesa, CA

On Sat, 20 Jun 1998, Ed & Alita Morrison wrote:
> A long time ago Paul Hahn wrote,
> > 10/9 --- 5/3 --- 5/4
> > / \ / \10/7 / \
> > / \ / \ / \
> > /14/9 \ / 7/6 \ / 7/4 \
> >16/9 --- 4/3 --- 1/1 --- 3/2 --- 9/8
> > \ 8/7 / \12/7 / \ 9/7 /
> > \ / \ / \ /
> > \ / 7/5 \ / \ /
> > 8/5 --- 6/5 --- 9/5
>
> I am not familiar with lattices of this type. Would you please explain
> it for me? The middle row is sequence of fifths but what is the meaning of
> the other rows and the fractions between them?

First, let me restore the brief explanation that originally accompanied
this diagram:

> >This is a projection of part of the 3D 7-limit lattice into a plane.
> >The triangles are in the 3-5 plane, and the ratios inside the triads are
> >either one layer above or below depending on whether they're inside
> >major or minor triads.

If that doesn't do it for you, I'll put it a little less tersely.

The basic unit of this lattice is this tetrahedron which represents a
4:5:6:7 tetrad in three-dimensional space:

5/4
/ \
/ \
/ 7/4 \
1/1 --- 3/2

(Imagine the 7/4 corner of the tetra sticking up out of the plane of the
screen towards you.)

The six edges of the tetra represent the six interval classes by which
the pitches of the tetrad are related. Now, if you look again at the
original diagram, you'll see that every pitch in it is related to the
others around it by those same intervals, represented by the same
relative spatial position.

For example, the 8/7, 4/3, and 8/5 are each exactly opposite the 7/4,
3/2, and 5/4 relative to the 1/1, and together with it they form a
Utonal tetrad which is the inverse of the original Otonal tetrad.
Similarly, the interval between the 7/4 and the 5/4 is a 7/5, and that
same spatial relationship tells us where to place the 7/5 and 10/7 in
the diagram.

Does that help?

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
-\-\-- o
NOTE: dehyphenate node to remove spamblock. <*>

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End of TUNING Digest 1452
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