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> [Patrick K. Mullen, Tue Aug 22, 2000 4:33pm]
http://www.egroups.com/message/tuning/11701
>
> 12tET has spawned harmonic extensions that sound really cool
> in 12tET, but not so good when tuned to just intervals. As if
> such a thing could even be done with some chords. Try tuning
> the beats out of a G13#11! But it sounds very nice in 12tET.

Hello, Patrick, and welcome to the Tuning List!

Go to:

http://www.egroups.com/files/tuning/monz/G13#11.mid

to hear 3 examples of me 'tuning the beats out of a G13#11' chord,
presenting it in both 12-tET and 13-limit just-intonation.

(The quality of what you hear in MIDI depends on your sound source.)

The pitches are registered as in a typical jazz voicing,
using the minimum number of notes (= 5) that would identify
the specific set of harmonic extensions you indicated
(this 'minimum' type of usage is typical of jazz, which you
indicated as your background):

12-tET note cents ~cents to --> harm(JI) ~cents
------ ---- ----- --------- -------- -----

13th E 900 -59 13 841
3rd B 400 -14 5 386
7th F 1000 -31 7 969
#11 C# 600 -49 11 551
root G 0 0 1 0

1st, a piano timbre (no sustain or vibrato) plays the chord
in each of the two tunings.

2nd, a harmonium timbre (no vibrato) plays the chord in 12-tET
and sustains the notes as the 4 upper voices slide into their
respective rational slots.

3rd, a violin timbre (loads of vibrato) does the same.

In the voicing I used, the 12-tET tuning has the proportions:
2^(0/12) : 2^(6/12) : 2^(10/12) : 2^(16/12) : 2^(21/12)
= 8 : ~11.31 : ~14.25 : ~20.16 : ~26.91

The JI tuning has the proportions 8 : 11 : 14 : 20 : 26,
which doesn't entirely 'tune out the beats', but does reduce
them to a mathematical (but not necessarily acoustical!) minimum.

Of course, there are infinitely many other ratios to which I
could have tuned these harmonic extensions; this particular
attempt results in what Paul Erlich calls a 'nice periodicity buzz'.
He used this term to characterize the sound of the JI version
of my piece _Invisible Haircut_:

http://www.ixpres.com/interval/monzo/haircut/haircut.htm

Perhaps the fact that I'm already used to working with chords
like this, tuned in JI, conditions my easier acceptance of them.

To my ears, and in contradiction to your opinion, the JI version
of this chord is much richer and more sensual than the 12-tET version.

Notice also that each of the upper voices had to go flatter (while
none had to go sharper) in order to attain the rational frequencies
which describe the nearest harmonic 'fit'.

Recent psychoacoustical research has shown that people tend to hear
the most consonant intervals as wider (= upper note sharper) than
the small-integer ratios. The ubiquitousness of 12-tET for several
generations now may be what accounts for this (if it is indeed
caused by cultural conditioning, which it may not be).

I do agree with what you're saying to some extent, but (as with
Paul Erlich) I would have chosen as an example a different kind
of chord, most likely a major6/9.

This is a sonority that simply cannot be translated into
small-integer JI because of the commatic 'punning' involved.
See the Tuning List archives for more on this chord and terminology.

And always be willing to stretch your ears and your mind, the two
things we use when we listen to music!

-monz
http://www.ixpres.com/interval/monzo/homepage.html