The ideal comma pump

[Paul E, TD 495.20:]
>I can do better with chords tuned vertically just like those. I'll
>arbitrarily use the same starting pitches:
>
> C major:
> C +5.68
> E -8.01
> G +7.49 (change: +5.94 cents)
>
> A minor:
> C +8.67 (change: +2.99 cents)
> E -5.02 (change: +2.99 cents)
> A -6.84
>
> D minor:
> D -2.66
> F +12.86
> A -0.90 (change: +5.94 cents)
>
> G major:
> B -12.24
> D +3.28 (change: +5.94 cents)
> G +1.53
>
>Here the total squared shift is 3*(5.94)^2+2*(2.99)^2 = 123.7310

Well, Paul, you had me going for awhile trying to match or beat that
number! But it turns out it's a phantom value; the ideal comma pump
with truly exact vertical intervals would have a total squared shift of
132.15 (based upon theoretical best distribution: 2/7 comma, or 6.1447
cents, at the three points where only one note shifts, and 1/7 comma, or
3.0723 cents, where two notes shift). I see now that you began by
"taking" the slight variation from exact JI that I had produced. All
well and good, but we're splitting unimportant hairs here! And you
already agree that slight reduction from exact JI is desirable, so
what's the point?

JdL

John deLaubenfels wrote,

>I see now that you began by
>"taking" the slight variation from exact JI that I had produced. All
>well and good, but we're splitting unimportant hairs here! And you
>already agree that slight reduction from exact JI is desirable, so
>what's the point?

It sounds like you're contradicting yourself -- the slight variation from
exact JI that you produced is "splitting unimportant hairs," and yet it's
"desirable."

[I wrote:]
>>I see now that you began by "taking" the slight variation from exact
>>JI that I had produced. All well and good, but we're splitting
>>unimportant hairs here! And you already agree that slight reduction
>>from exact JI is desirable, so what's the point?

[Paul Erlich, TD 499.17:]
>It sounds like you're contradicting yourself -- the slight variation
>from exact JI that you produced is "splitting unimportant hairs," and
>yet it's "desirable."

Sorry, I'm not speaking clearly. We need two words for "small" or
"slight". The comma pump I originally put forth, TD 492.2, allows
enough relaxation from exact vertical JI to make a significant
improvement in horizontal motion: a mere 70.27 sum of squares. I'm
saying that we both think this type of motion recovery is worthwhile,
so it's splitting hairs to refine to the nth degree what would happen if
we DIDN'T take some relaxation of this sort.

JdL