tuning Middle C; the "octave"

[Drew Skyfyre:]

> For 12-TET the std. is A 440Hz,which puts C at
> 261.6255653006 Hz (Graphing Calc. at work->1 more
> reason to love a Mac).

A-440 is not just a tuning standard for 12-tET.
It's the most universal tuning standard that's ever
existed in the world, and, largely thanks to modern
electronic tuning equipment being calibrated to it,
is today used as a basis by lots of musicians
everywhere.

> Now what I need to know is, what do those of
> you who have been at this for a while use as a
> fundamental.Does anyone use 261.625......? ,it
> appears to me to be a poor choice.A whole no.
> would be easier to deal with.Do some of you
> use 'arbitrary' freqs ?.

Many composers who work in JI make their Middle C
the 5-limit just major 6th, or 5/3, below A-440,
which makes it 264 Hz.

I use a different tuning myself, with two alternative
numerical representations. I felt that the simplest
way to handle this was:

n^0 = 1/1 = 1 Hz

This makes the lowest recognizable pitch somewhere
around 2^5 = 32 Hz, and gives a "Middle C" of 2^8 = 256 Hz.

Thus, the ratios of the octave "C"s in my system describe
simultaneously the exact frequencies.

However, I use a slighly different nomenclature more often:

Because the range of musically useful pitches is roughly
between 2^4 and 2^12, or 8 "octaves", I usually prefer
to call my Middle C of 256 Hz the 2^0, giving the lower
octave Cs (from the bottom up) as 2^-4, 2^-3, 2^-2, and
2^-1, and giving the octave Cs above Middle C as 2^1,
2^2, 2^3, and 2^4 -- a nice symmetrical layout of my
whole pitch-range, centered on good ol' Middle C.

La Monte Young, who frequently writes droning electronic
pieces which consist of one chord played continuously
for hours, days, or weeks at a time, came up with a very
reasonable solution:

All electrical current in the USA flows thru the lines at
60 cycles per second, which produces the hum you
hear in amplifiers, refrigerators, etc., and whose frequency
happens to be (let me get my calculator...) 60 Hz.
So Young tunes his 1/1 to 60 Hz and calculates all other
ratios from that, thus sublimating the hum into his piece
by masking it.

(This 60 cycle hum is a 15/8 "major 7th" in my tuning.)

> Also,since the term Octave is related to 12-TET,

Actually, "octave" literally means "8th tone" in Latin and
comes from the old Pythagorean diatonic minor scale.
ETs are based on dividing the 2/1, or "octave", but
the name is really much older.

Though the ancient Chinese and Greeks recognized
the similarities of the 2/1 from the earliest theoretical
writings that still exist, the first theorist to recognize
the 2/1 in any really meaningful way as far as scale
structure is concerned is the Graeco-Roman Ptolemy
(200s AD). Before that, theory was all tetrachords
or pentatonic scales.

The Franks, beginning around 800, based their theory
(at least in concept) on the ancient Greek, and gave
the 2/1 the name "octave" because it was the 8th tone
in the diatonic scale. Our modern Eurocentric music
theory follows in an unbroken line from the Franks.

> is there a more approprite word to use ?
> This is just semantics,but it may help think along a different plane .

For this very reason, Partch chose to call this basic
interval simply 2/1.

In my own work, manipulating prime factors and
exponents, I don't really find it necessary to use any
term for this interval. Usually 2^n works fine. If I must
refer to it in the abstract, I just call it "octave", always
with quotes, to distinguish it from the unquoted octave
of regular theory.

Joseph L. Monzo
monz@juno.com

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