FWD: Tuning thirds in triads

Paul asked me to repost this:

Date: Mon, 5 May 1997 07:16:11 -0700
From: PAULE
To: Multiple recipients of list <tuning@ella.mills.edu>
Subject: Tuning thirds in triads

The reason I differ with Paul Rapoport when he defines the m3 as simply
"left over" from the P5 and the M3 is that in tuning a major triad, both
thirds are important to my ears. Some of the experiments I discussed with
John Chalmers provide evidence for this belief. For example, John (in
Tuning
Digest 1008) and I both heard that a chord tuned 0 400 720 was clearly
"better" than a chord tuned 0 372 720, even though the M3 and P5 are
equally
false in both chords. (To refute an explanation based on a preference for
stretched intervals, use the first inversion of these chords.) Therefore
it
matters whether you're using the best possible m3 in a particular tuning;
just using the best M3 and the best P5 does not guarantee the best major
triad. For example, try 64-tet. There the best perfect fifth is 37 steps,
and the best major third is 21 steps. Changing either of these by 1 to
give
a better minor third of 17 steps improves the major triad. 64-tet is
actually an interesting tuning; it is the simplest equal temperament that
contains all "tonal" systems (defined according to my paper on 22-tET, the
systems are pentatonic, diatonic, and a new system first found in 22-tET
and
described at length in that paper).

The above assumes that the "best" tuning of a major triad is 4:5:6, a
statement which causes little serious controversy. Unfortunately things
are
not so simple for the minor triad. The 10:12:15 (or 1/6:1/5:1/4) tuning is
lowest in roughness, but for a typical root-position voicing,
(4:8:12:)16:19:24 will sound more stable. Gary Morrison has pointed out
that
there can be two different types of consonance judgments operating at the
same time; I believe that one is based on roughness while the other is
based
on various factors having to do with how well the tones fit into a single
harmonic series. 16:19:24 is typically still within the "dip" in the
roughness function whose local minimum is at 10:12:15, so 16:19:24 is
still
less rough than just about any triad besides the major. To me a
root-position minor triad sounds out-of-tune as 10:12:15, although I think
12:15:20 (1/5:1/4:1/3) is a very sweet first-inversion minor triad.

So it is possible that the best thirds for the major triad are different
that the best thirds for the minor triad. It destroys a lot of the
mathematical beauty of 5-limit theory by bringing in the 19th harmonic,
but
I have to let my ears be the final arbiter, and in this case they tell be
that the 19th harmonic is important.

A much-ignored study by Pierce and Roberts (I think; it's from the book
_Harmony and Tonality_) found that untrained listeners generally fall into
two categories, "pure" and "rich". The pure listeners liked just 4:5:6 and
3:5:7 triads better than versions of those chords where the middle tone
was
displaced by +/-15 or +/-30 cents, while the rich listeners like the
15-cents-off versions of the chords best, with no particular preference
between a +15 and a -15 cent "error" However, for 10:12:15 (or
1/6:1/5:1/4)
chords, both classes of listeners preferred the version where the middle
tone was lowered by 15 cents. The authors of the article did not provide
much of an explanation of this phenomenon. Note that although the 15-cent
shift has about the same effect on roughness for both major and minor
triads, the listeners may have been motivated not so much by the small
differences in roughness but by a preference for harmonic series. The
preferred version of the minor triad is very close to 16:19:24, which has
a
note octave-equivalent to the fundamental in the bottom voice. Thus in a
sense it better resembles a harmonic series than the "just" or 10:12:15
minor triad. The rich listeners may have also been attracted to the
beating,
which is on the order of 7 times per second for the +/- 15 cent versions
of
all of these chords.

--- In tuning@y..., manuel.op.de.coul@e... wrote:
> Paul asked me to repost this:

thanks manuel. i want to call attention in particular to the last
paragraph.

>
> Date: Mon, 5 May 1997 07:16:11 -0700
> From: PAULE
> To: Multiple recipients of list <tuning@e...>
> Subject: Tuning thirds in triads
>
> The reason I differ with Paul Rapoport when he defines the m3 as
simply
> "left over" from the P5 and the M3 is that in tuning a major triad,
both
> thirds are important to my ears. Some of the experiments I
discussed with
> John Chalmers provide evidence for this belief. For example, John
(in
> Tuning
> Digest 1008) and I both heard that a chord tuned 0 400 720 was
clearly
> "better" than a chord tuned 0 372 720, even though the M3 and P5
are
> equally
> false in both chords. (To refute an explanation based on a
preference for
> stretched intervals, use the first inversion of these chords.)
Therefore
> it
> matters whether you're using the best possible m3 in a particular
tuning;
> just using the best M3 and the best P5 does not guarantee the best
major
> triad. For example, try 64-tet. There the best perfect fifth is 37
steps,
> and the best major third is 21 steps. Changing either of these by 1
to
> give
> a better minor third of 17 steps improves the major triad. 64-tet is
> actually an interesting tuning; it is the simplest equal
temperament that
> contains all "tonal" systems (defined according to my paper on 22-
tET, the
> systems are pentatonic, diatonic, and a new system first found in
22-tET
> and
> described at length in that paper).
>
> The above assumes that the "best" tuning of a major triad is 4:5:6,
a
> statement which causes little serious controversy. Unfortunately
things
> are
> not so simple for the minor triad. The 10:12:15 (or 1/6:1/5:1/4)
tuning is
> lowest in roughness, but for a typical root-position voicing,
> (4:8:12:)16:19:24 will sound more stable. Gary Morrison has pointed
out
> that
> there can be two different types of consonance judgments operating
at the
> same time; I believe that one is based on roughness while the other
is
> based
> on various factors having to do with how well the tones fit into a
single
> harmonic series. 16:19:24 is typically still within the "dip" in the
> roughness function whose local minimum is at 10:12:15, so 16:19:24
is
> still
> less rough than just about any triad besides the major. To me a
> root-position minor triad sounds out-of-tune as 10:12:15, although
I think
> 12:15:20 (1/5:1/4:1/3) is a very sweet first-inversion minor triad.
>
> So it is possible that the best thirds for the major triad are
different
> that the best thirds for the minor triad. It destroys a lot of the
> mathematical beauty of 5-limit theory by bringing in the 19th
harmonic,
> but
> I have to let my ears be the final arbiter, and in this case they
tell be
> that the 19th harmonic is important.
>
> A much-ignored study by Pierce and Roberts (I think; it's from the
book
> _Harmony and Tonality_) found that untrained listeners generally
fall into
> two categories, "pure" and "rich". The pure listeners liked just
4:5:6 and
> 3:5:7 triads better than versions of those chords where the middle
tone
> was
> displaced by +/-15 or +/-30 cents, while the rich listeners like the
> 15-cents-off versions of the chords best, with no particular
preference
> between a +15 and a -15 cent "error" However, for 10:12:15 (or
> 1/6:1/5:1/4)
> chords, both classes of listeners preferred the version where the
middle
> tone was lowered by 15 cents. The authors of the article did not
provide
> much of an explanation of this phenomenon. Note that although the
15-cent
> shift has about the same effect on roughness for both major and
minor
> triads, the listeners may have been motivated not so much by the
small
> differences in roughness but by a preference for harmonic series.
The
> preferred version of the minor triad is very close to 16:19:24,
which has
> a
> note octave-equivalent to the fundamental in the bottom voice. Thus
in a
> sense it better resembles a harmonic series than the "just" or
10:12:15
> minor triad. The rich listeners may have also been attracted to the
> beating,
> which is on the order of 7 times per second for the +/- 15 cent
versions
> of
> all of these chords.