Post: 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.

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It's certainly true, on my opinion and that of most others on this list,
that small whole-number ratios are the clearly basic navigational
guideposts in pitch space. But just because you find navigational bouys
useful, doesn't mean that you have to chart collision courses for them!

The character of how you MISS small whole-number ratios is every bit as
interesting-sounding as nailing them right on.



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Ray Tomes said:Numbers 24,30,36 and 40 form ratios of
4:5:6:8...shouldn't that 40 be 48?

Numbers 24,30,36,and 42 form the 4:5:6:7 ratios which are the heart and
core of bbshop harmony and therefore THE center of the harmonic
universe.It's so obvious but I never see it mentioned.



Ed Dotson
ldotson@sprynet.com

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By the way, Dave Hill recently did a semiformal study of preferences
between just intonation and equal-temperament, the subjects being a set of
junior-high and high-school band-campers at Florida State University. He
had more than a thousand respondents. Some of you might be interested in
asking him more details on his findings. They struck me as interesting,
and I took the same test as well.

Here's my short summary from reading his reports (although you should
probably hear it from Dave himself):
1. Across all responses, there was a strong preference for ET,
2. Among those who scored highest on the raw pitch-acuity section of the
test, there was strong preference for JI. (The pitch acuity section had
a series of pairs of tones separated by a comma, 13 cents, a quarter
comma, and a skhisma, and the respondents were asked to state whether the
second pitch was lower or higher than the first.)
3. Many of those who preferred JI seemed to have more specific reasons, and
just generally more thoughtful responses.

His study also included comments about the nature of their preference
either way - whether they preferred it, for example, for having a "richer
sound".

Dave Hill's address is "ascend11@aol.com".



SMTPOriginator: tuning@eartha.mills.edu
From: mr88cet@texas.net
Subject: Re: testing an experimental tuning program
PostedDate: 01-09-97 05:07:06
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