RE: [tuning] Is There Always an Optimal Note?

Mark Nowitzky wrote,

>I created another new web page:

> "Is There Always an Optimal Note?"

> http://nowitzky.hypermart.net/justint/optimal.htm

>As usual, I'm leaning toward 5-Limit Just Intonation. But the webpage
>touches on the idea that there may not exist a single optimal tuning for a
>piece.

Mark, I get the sense you haven't been following too closely the discussion
on adaptive tuning (including adaptively tuned Bach and Mozart) with John
deLaubenfels, but I encourage you to go through the archives and see what
you can get out of those discussions. The point is, there are ways to keep
all the harmonies in JI, avoid key drift, and _still_ greatly reduce the
maximum "shift" required of any pitch from one bar to the next. A full
comma shift, which you recommend in this webpage, is often disturbing to
some ears, particularly if it occurs in the melody line -- it disturbs the
flow of the music too much (although your particular examples have other
interruptions to that flow, typically that won't be the case).

A simple example of adaptive JI is Vicentino's second tuning of 1555. Here
we have not one but _two_ chains of 1/4-comma tempered fifths, one tuned 1/4
comma higher than the other. (You may enjoy using NNN for this tuning). The
rule is: in a passage of music using triadic harmony, the root always comes
from the lower chain, the fifth comes from the upper chain, if there is a
major third it comes from the lower chain, and if there is a minor third it
comes from the upper chain. In this way all triads will come out just. Also,
any music written in the framework of meantone temperament (your examples
certainly qualify) will not require any melodic shifts larger than
1/4-comma. I suggest you add Vicentino's adaptive JI scheme to your list of
alternatives for the musical examples you quote.

Since 1/4-comma is smaller than the just noticeable difference for _melodic
shifts_, while a full comma shift can be quite disturbing, I'd have to judge
your strict-JI approach, with their full comma shifts, as less than optimal
in many situations. Meanwhile, meantone is less than optimal because the
total lack of melodic shifts forces all fifths and minor thirds to be
out-of-tune by 1/4-comma, which is quite a bit _larger_ than the just
noticeable difference for _harmonic intervals_.

So, in general, for pieces compatible with meantone tuning and using 5-limit
triads as the consonant harmonies, Vicentino's second tuning of 1555 is a
much better candidate for "the optimal tuning" than any of the alternatives
you suggest, and, under a minimax melodic error criterion, with no tolerance
for harmonic error, will often be _the_ optimal tuning.

-Paul

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