"raised pythagorean 'leading tone'", "flat chords" (was: Beethoven JI tuning experiment & 'lateral' harmonies)

[Paul Erlich, posting as Brett Barbaro, TD 206.3]
>
> George Kahrimanis, who has (questionably) backed up his theories
> with experimental observations on preferences in synthesized
> performances, analyzes the dominant in minor as a utonal chord,
> the dominant seventh being tuned 1/9:1/7:1/6:1/5 and with its
> lowest note on the regular dominant. That gives a highly raised
> leading tone much in accord with the modern practice of a typical
> string quartet playing Beethoven.

Fascinating - I've never encountered Kahrimanis or seen his work.
(how about a reference?)

The chord you describe (where 1/9 is the 'dominant' of the scale,
which makes 27/16 the utonal '1-identity' of the chord) can also
be expressed as:

1/9 : 1/7 : 1/6 : 1/5
= 3^1 : 3^3 * 7^-1 : 3^2 : 3^3 * 5^-1
= 3/2 : 27/14 : 9/8 : 27/20

The interval between 27/14 (= 1/7) and the 'tonic' 1/1, the
'highly raised leading tone much in accord with the modern
practice of a typical string quartet playing Beethoven', has
a size of ~1137.04 cents.

Without actually checking it, this does indeed seem to agree with
my perceptions of recordings of Beethoven string quartets.

So apparently a tuning experiment involving Beethoven's music
is amenable to the inclusion of 7-limit ratios, which I excluded.
Perhaps I'll expand my experiment to include them.

[Dale C. Carr, TD 206.6]
>
> [Excerpting:]
>
> """From: monz@juno.com
>
>>[...] The (culturally-conditioned?) preference for a sharpened
>> Pythagorean 'leading-tone' makes this chord sound flat, at
>> least in this piece, and to my ears. It sounds a lot better
>> using the Pythagorean pitch for the leading tone, and choosing
>> the 'root' and '5th' which make a nice 4:5:6 triad with it,
>> in the corresponding area of the harmonic lattice.[....]"""
>
> I wonder whether the term "sharpened Pythagorean 'leading-tone'"
> here isn't perhaps pleonastic and potentially confusing too?

OK, perhaps a few additional punctuation marks would have made
my statement more precise. How about:

"The (culturally-conditioned?) preference for a sharpened
(Pythagorean?) 'leading-tone' makes this chord sound flat"

[Carr]
> Can one actually speak of a Pythagorean leading tone?

Musicians, or tuning theorists anyway, routinely use the term
'Pythagorean leading tone' to mean an interval with the ratio
243:128 [= 2^-7 * 3^5 = ~1109.78 cents].

[Carr]
> The description of a flat sounding chord also has me wondering.
> monz might mean that the *third of the chord* is lower than he
> expected, but not likely that there's a flat sign in front of it.

I did indeed mean the former and not the latter.
(wasn't that clear in context?)

[Carr]
> But a *chord* might sound flat [= dull?] for some other reason
> than the tuning, <...snip>

This may or may not be true.

In fact, I've found that there's an intimate relationship between
tuning and timbral perception, and I mean this in a way that's
different from Bill Sethares's ideas.

In my analysis of Louis Armstrong's singing and playing, in
particular, I found that tiny microtonal differences in pitch
sounded like timbral differences; or perhaps it's the other way
around - it's impossible to tell by ear which is the cause and
which the effect.

[Carr]
> <...> so maybe he's not referring to the third at all.

But yes, I was referring to the fact that the '3rd', which in
the 'Dominant [V]' chord is the 'leading-tone' of the scale,
sounded too flat when tuned to 15/8, and to my ears, made the
entire chord sound too flat in the context of the overall
harmonic progression of the piece.

[Carr]
> As a relative novice to tuning matters, may I plead for more
> precise expression? The subject is too complex to tolerate
> unclear use of terms like sharp and flat.

Exactly why I started my online Tuning Dictionary.
(altho 'sharp' and 'flat' are not yet included)

Sorry that you found my use of terminology unclear.

What's unfortunate in this case is that 'sharp' and 'flat'
have more than one meaning: either specifically a semitone
(whatever that is! - that word also has multiple meanings,
depending on the tuning) higher or lower in pitch, as in
referring to the accidentals denoted by those names; or an
amount higher or lower which is either unspecified or is
not a semitone.

I used 'sharp' and 'flat' in the latter sense: lower and higher
in pitch than the 'target' by an unspecified amount, respectively.

The amount of discrepancy was indicated precisely by my use of
the terms 'Pythagorean' or '5-limit'. I considered including
the actual ratios in my post, but they're all on the webpage
already, along with a lattice diagram.

Did you look at the webpage yet?
Or listen to the experiment itself?

Perhaps I should have been more pedantic and descriptive,
for the benefit of those subscribers without web access.

Hope this clarified things for you.

(and *please*, call me monz - I *detest* excessive formality)

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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I wrote,

>> George Kahrimanis, who has (questionably) backed up his theories
>> with experimental observations on preferences in synthesized
>> performances, analyzes the dominant in minor as a utonal chord,
>> the dominant seventh being tuned 1/9:1/7:1/6:1/5 and with its
>> lowest note on the regular dominant. That gives a highly raised
>> leading tone much in accord with the modern practice of a typical
>> string quartet playing Beethoven.

Joe Monzo wrote,

>Fascinating - I've never encountered Kahrimanis or seen his work.
> (how about a reference?)

George was, I believe, on this list for a while, back when it
resided on the Mills server. He and I corresponded on e-mail
quite extensively. He sent me a huge paper which embodied years
of work, both theoretical and experimental, but which I can't
seem to locate right now. His theory is that chord progressions
in the music of the great common-practice composers strictly
follows the following rule: the root (in the Partchian sense)
always moves by a consonant interval (syntonic comma errors are
allowed here). He tested this theory by creating different
synthsized versions of "great" musical passages, which differed
microtonally in the details of the tuning, so that some versions
conformed to his theory while others did not. He claimed to find
statistically significant evidence that subjects preferred the
versions that conformed to his theory.