JI approximations of 11-tET

> [Paul Erlich, TD 264.3]
> Unfortunately, without a recognizable 5:4 or 3:2, there's no
> way to clearly imply 16:15 in 11-tET.

Paul's observation about the lack of recognizable 3- and
5-identities in 11-tET is worth noting; it tends to obscure
the harmonic and melodic implications of other near-JI intervals
in a 'traditional' harmonic context.

> [Paul]
> Otherwise, 11-tET is highly irrational. In fact, Blackwood
> and I independently arrived at the conclusion that it is the
> best ET for random dissonance.

However, playing devil's advocate (Paul and I just can't seem
to avoid an argument, even when we agree!), I thought it would be
interesting to pursue the other near-JI approximations of 11-tET
for those who, like me, find other JI intervals to be consonant
chord-members in an 'extended JI' context, even without the use
of 5:4 and 3:2.

For starters, the specifics of the original intervallic comparison
are: 2^(1/11) = ~109 cents, or ~2.64 cents flatter than 16:15
(= ~112 cents). Admittedly, this approximation doesn't mean
much in practical terms because of the absence of 5:4 and 3:2.

Its 'octave'-complement, however, 2^(10/11) = ~1091 cents, which
is the same distance sharper than 15:8 (= ~1088 cents), makes
a nice approximation to the 5-limit JI 'major 7th'. This can
be clearly heard in a harmonic context if the chord is rich
enough in other near-JI chord-members (more on that below).

> [Paul]
> The only clear near-just interval in 11-tET is 436 cents,
> (9:7 is 435 cents).

2^(4/11) = ~436 cents, or ~1.28 cents sharper than 9:7.
Of course, its 'octave'-complement, 2^(7/11) = ~764 cents,
is the same distance flatter than 14:9 (= ~765 cents), the
complement of 9:7.

> [Paul]
> The only other recognizable interval is 327 cents, which
> is not too far from 6:5 at 316 cents.

2^(3/11) = ~327 cents, or ~11.63 cents sharper than 6:5.
Again, its complement of 2^(8/11) = ~873 cents is the same
distance flatter than 5:3 (= ~884 cents), the complement of 6:5.

Here are some other close approximations of 11-tET to JI:

2^(9/11) = ~982 cents, or ~5.28 cents sharper than 225:128
(= ~977 cents), the 5-limit JI 'bridge' which may be used
to imply 7:4 (= ~969 cents). The inverse is true of the
complements, altho I can't really see a recognizable compositional
use for that.

Likewise (doing double duty in addition to its very close
approximation of 14:9), 2^(7/11) = ~764 cents, or ~8.99 cents
flatter than 25:16 (= ~773 cents), the 5-limit 'augmented 5th';
again I see no real use for the complements.

It could certainly be argued that without a recognizable
implication of 5:4, this near-25:16 is also quite useless,
but I like the JI chord 16:25:30, which gives a 5-limit
'root : augmented 5th : major 7th' triad which can be quite
closely approximated in 11-tET by 2^(0/11) : 2^(7/11) : 2^(10/11).

11-tET also has decent approximations of the primary 11-limit
intervals. 2^(5/11) = ~545 cents, or ~5.86 cents flatter than
11:8 (= ~551 cents). The complement 2^(6/11) = ~655 cents is
the same distance sharper than 16:11 (= ~649 cents).

-monz

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