Re: Well-temperament corrections -- 182:121 etc. (Paul Erlich)

Hello, there, and please let me correct some unfortunate errors which
led to me giving the correct measurement in cents for the wrong
ratios. Thanks to Paul Erlich for pointing out the problem, and giving
me the opportunity to clarify not only the numbers but the basic
concept involved.

[My earlier statement]

>> or as a tempering of 121:143:181 or 121:154:181,
>> with a pure 14:11 major third and 13:11 minor third joined to form a
>> fifth at 196:121, or about 706.72 cents (~4.76 cents wider than pure).

[Paul's response]

> Hmm . . . the outer interval of those chords is 181:121, or 697.18 cents,
> and 196:121 is 835.02 cents.

Here what I am describing is a sonority combining a pure 14:11 major
third (~417.51 cents) and a pure 13:11 minor third (~289.21 cents,
resulting in a fifth of 182:121, or ~706.72 cents.

As you can say, although I got the size of the 182:181 correct, I
mistyped the ratio itself no less than three times. I should have
given 121:143:182 (13:11 below 14:11) and 121:154:182 (14:11 below
13:11), but in both cases typed "182" instead of "181."

The third time I typed "196:121," a slip of the keyboard maybe
stemming from the distraction of a different ratio in the tuning, the
diminished fourth at 121:98 (~364.98 cents, close to 21:17), whose
octave complement is an augmented fifth at 196:121, indeed 835.02
cents just as Paul observes.

Now that I've clarified that we're dealing with the situation of a
sonority ideally combining a pure 13:11 and pure 14:11, and thus
producing a fifth not at 3:2 but at 182:121 (~4.76 cents wide), maybe
I can more clearly state the point I meant to raise.

Even with such techniques as adaptive tuning, there is no way to make
a 13:11 and a 14:11 add up to a pure 3:2. One might say that the laws
of mathematics themselves "temper" the fifth.

In my article on the 17-note well-temperament, I was suggesting that
the actual tuning of these intervals in the "nearer" 12-note range of
the temperament (Eb-G#) could be seen as a kind of variation on this
tuning in which the 13:11 is slightly compromised but the fifth is
actually brought closer to pure.

Over the range of Eb-G#, this tuning is identical what I term a
regular hypermeantone with pure 14:11 major thirds. In other words,
fifths are tuned so that four of them form a 14:11, calling for a
temperament of ~2.422 cents in the wide direction. Compare this with
the ~4.72 cents of "virtual temperament" in the 182:121 (as compared
to a pure 3:2) which the pure 13:11 and 14:11 together would produce.

The following diagram may help to show the difference:

All integer ratios (121:143:182) 14:11 hypermeantone

| B3 | B3
182:121 | 14:11 (~417.51) | 14:11 (~417.51)
~706.72 | G3 ~704.377 | G3
~+4.76 | 13:11 (~289.21) ~+2.42 | ~286.87 (-~2.34)
| E3 | E3

In the just tuning, the full burden of "virtual temperament" falls on
the fifth, maybe somewhat ironically in a musical context where fifths
are prime concords and thirds are unstable although _relatively_
"concordant." In the tempered version, the burden gets about equally
divided between the wide fifth (slightly wider than 3:2) and the minor
third (slightly narrower than 13:11).

Thus if one wishes to combine 3:2, 14:11, and 13:11 in a single
sonority -- and I'm not sure how widespread such a concept may be -- a
14:11 hypermeantone tuning just might be, in effect, one of the
closest approaches to "adaptive tuning," giving the inherent
incommensurability of these three ratios together.

Incidentally, I should add that my musical approach is guided mostly by
the "grammar" of Gothic sonorities and progressions; in such a setting,
the ratios 14:11 and 13:11 seem to me mathematically noble and also
musically beautiful and intriguing, attracting me to the idea of
sonorities uniting both -- or their close approximations.

P.S. As long as we're addressing corrections in my well-temperament
post, I should also note that my statement of the tempering of the
fifth as ~2.421 cents represents a truncation rather than a correct
rounding of the result given by GNU Emacs Calc, properly ~2.422
cents.

Most respectfully,

Margo Schulter
mschulter@value.net