Re: [tuning] Digest Number 2236

>Thus, precisely, what is the "lowest C on the piano" and what is "the
> very *highest* C on the piano"? Can you also give their mathematical
> equivalencies? Is not Middle C at 256 Hz? By the way, no one has
> told me yet how many zeroes are after the 256?

Greetings,
Braid-White gives the lowest ET C a freq. of 32.703 Hz and the highest
C = 4186.009 Hz. Middle C is listed as 261.626 Hz. These numbers are only
approx. The tuner that uses less stretch will have different figures for the
extremes. The piano scales that produce higher inharmonicity will also
"expand" these figures.
I also believe that it makes no sense to associate the pianos range with
C's, since the earlier keyboards didn't necessarily begin and end with
C,(today's don't either, they ususally have A as the lowest note). Indeed,
before the 1880's, pianos usually only went to A7 with the 85 note keyboards.
The Romantic era of music, and perhaps marketing, caused a shift upwards to
finalize the scale at C.
The top note is C because of several reasons. One is that it approaches
the human limits of hearing and physical arm length. Another is that it
represents a logical musical end point, (it wouldn't make sense to add a C#
as a final note, since musically, C# didn't usually represent an oft-used
tonic and ending on a raised key would have been mechanically awkward).
Another is the point of diminishing returns, the string length for C8 is
approx. 1 7/8" on ALL pianos, be they concert grands or spinets, and it
becomes physically difficult to create a shorter scale and still leave room
for soundboard under it as well as finding a way for the hammer to contact
it.
The bottom of the keyboard reached A0 and ended there because we can't
really hear much under 27 Hz. The extended Bosendorfer scales are intended
to add tension beyond the last playable note (A0) in order to improve the
response of that string. (the last string on a bridge often suffers tonal
deficincies due to mechanical reasons).

>How many C's make up what you call "a circle of fifths"?
The same number of C's as there are in an octave, One for sure, two if
you want to "close" it.

>And, as a novice, I really would like to know why you call it "a
> circle" of notes, and not a sphere or a linear series or a triangle
> or some other "gon"--say a hexagon or other?

The difference between the circle and the "gon" is a matter of degree of
observation,(from sufficient distance, any regular polygon of enough sides
may appear as a circle). Obviously, no shape made from straight lines can be
a "circle", but the verb form of "circle" does describe the results of a
string of fifths; they "circle" back to an octave equivalent of the
beginning point, so the "circle" represents the return to the starting point
in an easily describable manner.
Regards,
Ed Foote RPT