Yahoo Tuning Groups Ultimate Backup tuning questions about those C's (was: Piano Tuning Schools)

hi Bill,

> From: "Bill Arnold" <billarnoldfla@yahoo.com>
> To: <tuning@yahoogroups.com>; <celestial-tuning@yahoogroups.com>
> Sent: Thursday, October 03, 2002 3:04 PM
> Subject: Re: [tuning] Re: Piano Tuning Schools
>
>
> Can someone explain why a piano centers (I think) on Middle C? How
> many C's octaves are there below? above? Why only those many? Why
> are they 8 notes apart and not 10? Why are they "harmonics" of each
> other? How many C notes could you actually have on a piano that the
> human ear could hear if the keyboard could be infinite? Why are there
> not more?

i'm perhaps not really answering any of your specific questions
here, but you'll probably find this useful:

in the original paper i put online about my tuning theories
(<http://makeashorterlink.com/?M6C022302> -- wait a moment
for the redirected link to work), there's a section discussing
a "reference pitch", which i call "C n^0" and suggest as
referring to either 1 Hz or 256 (= 2^8) Hz.

since my notation is based on prime-factoring, this makes
"middle-C" either 2^8 in the former case or n^0 in the latter.

i also state here: "The approximate range of human hearing is
from 20 to 20,000 Hz. Described as powers of 2, this is roughly
2^4 (=16) to 2^14 (= 16,384) Hz." if "middle C" at 256 Hz is
C n^0, this range is described as 2^(-4...+6).

they are "harmonics" of each other because they are all
powers of 2, and by definition, the interval which most
scales set up as the "interval of equivalence" is one whose
bounding pitches have the frequency ratio of 2:1.

this ratio usually goes by the more familiar musical name
of "octave" (abbreviated "8ve"), which is Latin for "eighth".

during the medieval period, the standard scale in use was the
heptatonic diatonic scale -- that is, 7 different pitches
("heptatonic" = "7-tone") within one "8ve", assumed to repeat
exactly in all other "8ves", and which have a mixture of both
"half-steps" and "whole-steps" as the intervals between the
degrees of the scale.

("diatonic" is Greek for "thru tones", because of the three
basic _genera_ in ancient Greek theory, this was the one
which had 2 tones in each "tetrachord"; the other two _genera_
had an interval which was bigger and two which were smaller.
see my "Tutorial on ancient Greek tetrachord-theory"
<http://makeashorterlink.com/?X65121302> for more info.)

when listening to a scale such as this, one is immediately
struck by how similar the 8th note (doesn't matter if it's
the 8th note below or the 8th note above the starting pitch)
sounds like the starting note.

thus, the "8ve" became the most basic interval involved in
scale construction as well as harmonic usage and analysis.

i suggest you peruse the following definitions from my
Tuning Dictionary:

/tuning/files/dict/harm.htm
/tuning/files/dict/harmser.htm
/tuning/files/dict/equivalenceinterval.htm
/tuning/files/dict/octave.htm
/tuning/files/dict/diatonic.htm
/tuning/files/dict/tetrachd.htm
/tuning/files/dict/semi.htm
/tuning/files/dict/wholetone.htm
/tuning/files/dict/degree.htm

later (c. 1200s to 1300s), mainly thru the process of "mutation",
the scale resources expanded to include what we would now call
the "flats" and "sharps", eventually leading to the "chromatic"
scale.

during the early development of Western music, from its origins
in the Frankish kingdom c. 700 up to the recognition of "3rds"
and "6ths" as "consonances" c. 1500, the "Pythagorean" (3-limit)
scale was the basic tuning.

after 1500, 5-limit "just-intonation" became a theoretical
paradigm but proved difficult to achieve in practice, leading
to the idea of "temperament". in roughly chronological order,
the leading families of temperaments have been "meantone",
"well-temperament", and then "equal-temperament".

for a long time it has been recognized that 12 different pitches
could form a nearly-closed circle, and by tempering, it could
become an actual closed circle. thus during the 1900s the
tuning standard became "12-tone equal-temperament", so that
the 2:1 ratio or "8ve" now contained 12 different notes, each
one spaced the same distance from those adjacent to it.

more definitions:

/tuning/files/dict/mutation.htm
/tuning/files/dict/chromati.htm
/tuning/files/dict/major3rd.htm
/tuning/files/dict/minor3rd.htm
/tuning/files/dict/consonance.htm
/tuning/files/dict/pythag.htm
/tuning/files/dict/limit.htm
/tuning/files/dict/just.htm
/tuning/files/dict/temp.htm
/tuning/files/dict/meantone.htm
/tuning/files/dict/well.htm
/tuning/files/dict/eqtemp.htm

the real question about the range of the piano keyboard
(a real piano has 88 keys, or a range of about 7&1/2 "8ves")
is not "why aren't there more notes?", but rather "why are
there that many?". the notes at either end of the keyboard
are so low or high that their pitch can barely be discerned,
and they're usually used only in combination with like notes
an "8ve" apart, to give and effect of increased depth or
brilliance respectively.

for an illustration and explanation of a keyboard that was
designed to use the 24-tone equal-temperament, commonly known
as the "quarter-tone scale", see my newly updated web version
(with Klaus Schmirler's English translation) of Willi
Mï¿½llendorff's _Musik mit Vierteltï¿½nen_ (Music With
Quarter Tones) at <http://makeashorterlink.com/?L67124302>.

the diagram of the keyboard is here:
<http://makeashorterlink.com/?C1A123302>.

-monz (this group's list-mom)
"all roads lead to n^0"

questions about those C's (was: Piano Tuning Schools)

🔗monz <monz@attglobal.net>

🔗monz <monz@attglobal.net>