Re Definition of Just Intonation

David Keenan: At the time I wrote the quoted definition of JI, it agreed
with the general usage in the music theory literature. Some earlier
writers have even defined JI as the C mode of Ptolemy's Intense
(syntonic) diatonic. Others admitted ratios of 7 and higher and there
was an unspoken or unwritten assumption that JI should sound different
from 12-tet.

It is only recently that theorists have tried to confuse the issue by
deliberately inventing tuning systems which are aurally
indistinguishable from 12-tet and calling them JI. Systems such as
Hammond's gear ratios and Ellis's ratio approximation to 12-tet have
been considered by most writers to be merely approximations of equal
temperament and not true JI. Since irrational intervals have to be
approximated by rational fractions and because no physical system is
either exactly in tune or completely stable, I think these quibbles are
rather pointless. One can be clear enough by using a practical or
effective definition -- JI is a tuning system whose intervals closely
approximate ratios of small integers and which does not sound like an
approximation to a tuning based on irrational frequency relationships.

Ivor Darreg faced similar definitional problems when he introduced the
term xenharmonic. He was aware that one can write music in certain
non-12-tet tuning systems which still sounds as if it were in 12-tet.
Such music is NOT xenharmonic. Brian McLaren generated some musical
examples (as midi files) and retuned them by computer while keeping all
other musical parameters constant. In some cases, it was very difficult
for most listeners to tell the tuning systems apart. In other cases,
such as David Hills's examples, the distinction between meantone and
12-tet was quite clear. The conclusion I draw from this is that one must
take musical context as well as mathematics and psychophysics into
account when describing tuning systems and their audible properties.

--John