RE: Militant JI'ism

Denny says,

> For those of you who think that I'm not being militant
>enough on the side of just intonation, ...

I certainly think it's better not to be militant about just
intonation, or anything else, for that matter.

>I would like to point out something that Easly Blackwood once
>said: "There is no such thing as just intonation. All you
>can do is approximate a ratio by some standard of precision. It
>all depends on the tolorance that you are willing to accept". I
>think that the same thing could be said about equal
>temperaments, as well.

That may well be true for mechanical instruments, but it does not
seem to represent the situation for frequency dividers, and since
all modern electronic musical instruments use some form of
frequency division, it may be a distinction of some importance.

Take for example the humble fifth, which is 3/2 in JI and
1.498307077 in 12ET. To produce the JI fifth to an almost
unbelievable level of accuracy simply takes three flip-flops with
no gating, assuming they are the right kind of flip-flops (JK or
D). The circuit needs to be fed with a frequency that is 3 times
the frequency of the lower tone.

But to produce the 12ET fifth to the accuracy represented above
is quite complex, and since it is a non-repeating decimal, to
produce the 12ET fifth with the same accuracy with which we can
easily produce the JI fifth is quite impossible. Since
498307077, and 1000000000 have no common factors, we would have
to use two frequency dividers of 29 and 30 flip-flops
respectively, and a substantial amount of gating logic. We would
have to feed this circuit with a frequency 4.983070769E+17 times
higher than the lowest tone we wished to produce.

There is the alternative of using the "phase accumulator" method
of generating the frequency, as in FM synthesis or Wavetable
synthesis. That eliminates the need for the very high frequency,
but introduces phase jitter and does little or nothing for
circuit complexity.

With electronics, we can easily produce a very accurate JI fifth,
or we can spend thousands of times as much and get a much less
accurate approximation of an ET12 fifth. That is a principle
reason why I prefer exact JI.

Another reason is that the ear/brain system looks a lot more like
an electronic mechanism to me that a mechanical system. And
then, of course, there is ease of analysis.

Marion

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