From: john@jlc.mv.com (John Leslie)
Subject: Re: Fundamental Frequencies of the Musical Notes
Organization: John Leslie Consulting, Milford NH
Date: Sat, 02 Jan 1993 18:17:51 GMT

In article <TED.93Jan1173824@lole.nmsu.edu> ted@nmsu.edu writes:
>
> i am sure that there are quite a number
> of people out there who can correct me in the places that i go wrong.
> please feel free to do so, if you know better than i.
>
>>   Could you tell me, is there agreement on the absoulute frequencies?
>
> 440 A and 256 C define slightly different scales that are used in
> slightly different situations.  i think that most pianos are usually tuned
> with 256 = C, while orchestras (in my limited experience) invariably
> use 440 = A

   Sorry, wrong on both counts.  Pianos, on the average, are tuned close
to A-440.  Orchestras like to tune sharp of that -- A-444 was fairly common
the last I checked.  Of course, many individual pianos have been tuned well
flat of A-440.  A lot were designed for 435, and may break strings if tuned
up to 440.  Others are simply in poor shape, and may be tuned most anywhere.
Pianos rise and fall with the seasons, so piano tuners tend to tune higher
in summer and lower in winter for the same piano.

   Also, early-music instruments are often tuned to the historically
correct pitch, which is as much as two semitones higher or lower.

>>   Also, what is a "well tempered scale" and where can I read more on
>>   this subject?
>
> originally, scales were defined by going around the `circle of fifths'
> (or the essentially equivalent circle of fourths).

   Well, not quite "originally".  Historical evidence is that tunings
started out "just" (pronounced like Justice), meaning geometrically exact.
In a just tuning, the fifth (C-G) is exactly 3:2, and the third (C-E) is
exactly 5:4.  Clearly, there is no "just" tuning that satisfies all the
keys.  The "circle of fifths" was invented to allow playing in all keys,
but it generated a "pythagorean" fifth with a ratio of 81:64, which the
musicians really hated.  The musicians settled on "meantone" tunings,
which made the third exactly 5:4 by making the fifths quite flat.  The
battle between pure thirds and all keys playable continued for hundreds
of years.

> there are several problems with this.  first, C an octave up isn't
> right (it is 1.4% off, which is plenty enough to sound terrible).

   Absolutely true.  The octave (2:1) is the interval most sensitive
to tuning errors.  Piano-tuners tend to stretch some octaves by one or 
two hundredths of a semitone (because the second harmonic of a piano
string is *not* exactly an octave above the fundamental), and that's 
close to the limit of what sounds OK.  Two hundredths of a semitone 
is 1.001559..., ten times smaller than the pythagorean comma.

> secondly, chords other than the fifths used to construct the scale
> sound off.

   Well, they do, but that's not really the point.  The point is that
the fifth (3:2) can be detuned a lot more than the octave before it
sounds wrong.  To get ahead of the story, the equal-tempered fifth
comes out at 1.498307..., which is less than two hundredths of a
semitone low.  Piano-tuners hear the difference, but few others do.

> for example F# and the C above middle C have a frequency
> ratio of 1.42 instead of the desired 1.5

   Er... well... No.  C to F# is a tritone, which is supposed to
sound wrong in any tuning.  1.5 is the desired ratio for C to G.
I think Ted means to refer to the "wolf" fifth, which is properly
called E# to C in the tuning he gave.  (E# is erroneously called
F in *many* music textbooks.)  This interval comes out 1.4798...,
which is 1.3% low, and does indeed sound gross.

   I think it's important to note that anything we say about
historical tunings *must* be only approximate, limited by the
capabilities of the human ear to measure differences.  We generally
talk of "cents" -- hundredths of a semitone (calculated as the
1200th root of 2).  One "cent" is considered to be the smallest
pitch difference the human ear can resolve.  Thus *no* interval
in an historic tuning should be considered to be closer than one
cent, and many were undoubtedly three or more cents away from
where we claim they were.

John Leslie <john@jlc.mv.com>
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