quoted 72 lines From: "Medium Graham" <medium_graham@yahoo.co.uk>
>From: "Medium Graham" <medium_graham@yahoo.co.uk>
>To: <ian@webice.net>, "Investigative Data Mining" <idm@hyperreal.org>
>Subject: RE: [idm] music 101/tunings
>Date: Sun, 20 Aug 2000 12:59:21 +0100
>
>Actually, Josh is nearly right: In the equal temperament system, it's done
>by frequency...
>
>Each note has a particular frequency associated with it, for example the
>note A below middle C has a frequency of 440Hz. To get the note that's one
>octave above (A above middle C), you multiply the frequency by 2, so that
>becomes 880Hz. Now, each octave has twelve semitone steps in it, so it
>would
>seem sensible to mathematically divide each octave into twelve equal steps.
>So the frequency of the B below middle C, should be 513.333Hz. This is not
>how it works.
>
>If you think about it, go up another octave (to 1760Hz) the number of Hertz
>between the bottom note and the top note of the octave is now bigger than
>before. Instead of *adding* on a constant number of Hertz to the frequency,
>we *multiply* it by a constant factor, this being the 12th root of 2
>(1.059463). So, in fact the frequency of B below middle C is 466.164Hz.
>
>Of course, when tuning a piano or whatever, the guy usually uses a 440Hz
>tuning fork to fix the A, and does the rest of the piano by ear. Watch
>someone doing it, it's pretty interesting.
>
>Hope this is useful.
>
>
>G-love.
>
>www.gram.org.uk
>
>
> > Equal Temperament is how instruments are tuned today in the
> > Western world.
> > It means that every note is equidistant from the on before and
> > the one after
> > it. Basically, the octave is divided up into 12 equal half steps, thus
> > making the perfect 5th a bit flat and the Major 3rd a bit sharp
> > from their
> > natural values. This differs from how instruments were tuned
> > during, say ,
> > the baroque era, when they were tuned in Just Intonation, in which the
> > perfect 5th is made slightly flat but the "pure 3rd" between the
> > 4th and 5th
> > partial of the overtone series is preserved. This tuning was
> > used because
> > most music during that time was based on harmony of thirds. So,
> > back in the
> > day of JS Bach, some keys actually did sound "sadder" or "happier" to a
> > certain extent, whereas there are no real differnces between one
> > minor key
> > and another minor key in Equal Temperament. Even older is the
> > Pythagorean
> > method of tuning, which is tuning based on "Perfect" 5ths, which
> > facilitates
> > music based on harmonies of 5ths but leaves 3rds badly out of
> > tune. Equal
> > Temperament equally allows 5ths and 3rds to be used harmonically,
> > at least
> > to our young western ears.
> >
> > Geez, I hope this clarifies things.
> >
> > take care, have fun with tuning!!
> >
> > Joshua
> > \telefon tel aviv
> > \benelli design labs
> >
I probably should have been more specific in my explanation. Each note in
an Equally Tempered scale is equidistant in PITCH from the one before and
the one after it, NOT in frequency. Before the days of frequency measuring
devices, pitches were tuned by ear to be equidistant from one another on the
basis of preserving the perfect octave, or what G is referring to by means
of mathematics as the 2:1 ratio of say, C4 and C5. This is of course
achievable only by taking the frequency of one note and multiplying that
number by G's number 1.059463 (wow, that's tight, I just do it by ear!) to
get the pitch of the next note. It is in this case that the real distinction
between *pitch* and *frequency* seems to be drawn. What to our ears
represents an octave divided into 12 equal PITCH INTERVALS of a minor 2nd is
a frequency that is multiplied by an EQUAL or same amount (1.059463 pretty
much), that when multiplied with an exponent of 12, preserves the 2:1 aspect
ratio of the perfect octave.
Oh-I think that 466.164 is Bb above middle C, but don't quote me on that. B
below middle C should be around 247 hz, I think.
respect to G for the l33t math skills. Thanks for the clarification.
peace
_J_
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