Re: 225/224

Hello -

The discussions have been intriguing recently. I'd like to
say more but will make an observation about differences
between a 1.5 cent wide 7/4 and an 8 cent wide 7/4 I've
noted or believe I've noted on the piano. The 8 cent wide
7/4 is a lot closer than the 12 TET 31 cent wide 7/4 or the
"wide" 1/4 comma MT 7th which is 38 cents wide of 7/4.
In intended 4:5:6:7 chords is still has some slight harshness
or feeling of being a bit off, while the 7/4 which is only 1.5
cents wide just seems to have a subtle more powerful
appeal and emotionality. I had my piano recently tuned to
a JI scheme with three rows of slightly stretched from just
fifths (1 cent wide) and slightly stretched 5/4 and 6/5 thirds
1/2 cent wide. With this tuning, the G# would be 7.8 cents
plus 3 cents or 10.8 cents wide from a just 7/4 on Bb, but
I had that single pitch class - G# - tuned lower, sacrificing
the C# - G# fifth and the E - G# major thirds in order to
have nearly just 7/4 on Bb and a 7/6 on F wide by only .5
cents.

In 1/4 comma MT tuning, there are two very close
approximations to the 4:5:6:7 chords and these chords have a
very smooth consonant sound. Still, to my hearing, the
considerably closer approximation I presently have seems
to have a more emotionally intense character - which pervades
the different inversions of the chord, ninth chord harmonies - here, in the
4:5:6:7:9 chord all the thirds are wide by .5 cents.

I might speculate that having the 7 really close to just
tends to intensify a Chopinesque character to music played
in something like his style using nearly just harmonies on
the piano.

I had the intervals tuned slightly wide from just in an effort to
achieve a harmonious sound best suited to the piano's
slightly wide-stretched partials. I also had the unisons
detuned more than usual - 3 cents separation highest and
lowest pitch of a trichord unison for eight of the pitches and
4.5 cents separation for the B, F#, C#, and G# which usually
form major thirds (or for C# and G#, app. 7/4 sevenths also)
on other notes. This is an experiment. Maybe I detuned the
unisons too much, but in some ways this seems to have
improved the piano's sound, depending on what music is played.

An added comment regarding the piano - in my opinion, the
piano can sound really beautiful when it is tuned otherwise
than to 12 TET and for many of us, until very recently we
never heard a piano tuned to anything but 12 TET and I had
some kind of thought that if a piano were tuned differently
than to 12 TET, it might explode or suffer some similar
horrendous damage (exaggerating slightly). For my ears,
putting the piano into a harmonious tuning suddenly turns
the instrument into a HARMONIOUS instrument as well as
a smooth sounding complex subtly impressionistic almost
harmonious musical instrument - difficult to get precisely
the right shade of expression for the 12 TET piano's falling
short of really sounding beautifully harmonious. Of course,
even when tuned to a just intonation pattern, the piano is
probably not as harmonious sounding as a really skilled and
careful vocal group could be because of the inharmonicity of
its strings' partials, but I agree with Carl Lumma that with
some changes, including more notes per octave, and with
a change in the way its tuning is thought of - a questioning
of the satisfactoriness of 12 TET - the piano may have more
life ahead than it has had up to the present time.

Ascend11@aol.com wrote:

> The discussions have been intriguing recently. I'd like to
>say more but will make an observation about differences
>between a 1.5 cent wide 7/4 and an 8 cent wide 7/4 I've
>noted or believe I've noted on the piano. The 8 cent wide
>7/4 is a lot closer than the 12 TET 31 cent wide 7/4 or the
>"wide" 1/4 comma MT 7th which is 38 cents wide of 7/4.
>In intended 4:5:6:7 chords is still has some slight harshness
>or feeling of being a bit off, while the 7/4 which is only 1.5
>cents wide just seems to have a subtle more powerful
>appeal and emotionality. I had my piano recently tuned to
>a JI scheme with three rows of slightly stretched from just
>fifths (1 cent wide) and slightly stretched 5/4 and 6/5 thirds
>1/2 cent wide. With this tuning, the G# would be 7.8 cents
>plus 3 cents or 10.8 cents wide from a just 7/4 on Bb, but
>I had that single pitch class - G# - tuned lower, sacrificing
>the C# - G# fifth and the E - G# major thirds in order to
>have nearly just 7/4 on Bb and a 7/6 on F wide by only .5
>cents.

Ignoring the (uneven?) stretch for now, you seem to be describing this 7-limit lattice:

B---F#--C#
/ \ / \ / \
G---D---A---E
/ \ /G#\/ \ /
Eb--Bb--F---C

If you want to preserve the fifth C#-G# and use both of these pitch classes as good 7/4's as well as good thirds, the trick is to make the 6/5's Just, but make the 5/4's and 3/2's 1.9 cents narrow. This will result in Just 7/4's, and 7/6's and 7/5's that are also 1.9 cents narrow. Then apply your stretch (evenly?) on top of that.

Regards,
-- Dave Keenan
http://dkeenan.com