Tuned bodies, just intonation



TO: internet:harp-l@xxxxxxxxxxxxxx

TUNED BODIES

There's something in acoustics called forced resonance. If you
take a tuning fork vibrating at a frequency to which it is tuned,
and stand its stem on the bottom surface of an inverted dinner
plate, the plate will pick up and amplify the vibrations. Not
because the plate is tuned to any particular frequency - it's
just a passive conductor, spreading the vibration over a wider
area.

If you substitute reed for tuning fork and comb (with reedplates
and covers) for the dinner plate, it's pretty obvious that
there's not much of a tuned resonance going on in a harmonica.
After all, the comb is nothing but a slab of hard material of a
predetermined size with slots cut into it to accomodate the
reeds. I'm not even sure if they use differently-cut combs for
different keys of harp. And given the complex shape, how could
you tune it anyway? Everything described so far can easily be
explained by forced resonance. The harps that vibrate more may
simply have a better seal between the reedplate and the comb.

JUST and EQUAL TEMPERAMENT

Rick Barker has asked me both publicly and privately for
definitions for just and equal temperament. I take the Harvard
Dictionary of Music (1969 edition) as my source. This stuff gets
very intricate, and I won't go into great detail here. But I can
at least outline the differences.

Let's start with the overtone or harmonic series, the
naturally-occuring series of tones generated as byproducts of
notes played on most acoustic instruments. The harmonic series is
based on simple ratios (starting from the bottom):


16/1 (4 octaves)         C
15/1 (3 octaves +  7th)  B
14/1 (3 octaves + ~b7)  ~Bb
13/1 (3 octaves + ~6th) ~A
12/1 (3 octaves + 5th)   G
11/1 (3 octaves + ~#4)  ~F#
10/1 (3 octaves + 3rd)
9/1  (3 octaves + 2nd)   D
8/1  (3 octaves)         C
7/1  (2 octaves + ~b7)  ~Bb
6/1  (2 octaves + 5th)   G
5/1  (2 octaves + 3rd)   E
4/1  (2 octaves)         C
3/1  (octave + 5th)      G
2/1  (octave)            C
1/1  (fundamental)       C

The only intervals in the harmonic series matching equal
temperament are the octaves - nothing else matches up.

However, the octave, second, third, fifth and major seventh
(compressing the octave spread) all are in accordance with just
intonation, while the quasi-minor seventh, quasi-raised fourth
and quasi-major sixth are all out of tune with every standard
form of temperament.

The notes in the harmonic series that match up with just
temperament give us major triads for C (I) and G (V) that are
naturally in tune. The ratios used to generate these can be
boiled down to just two (if we ignore the 2/1 octave ratio):

3/2 (fifth)
5/4 (major third)

Just intonation uses these ratios to construct a scale in which
all major chords are in tune. The missing notes F (4th) and A
(6th) can be generated to complete the scale. F is 4/3
(essentially treating C as the 5th of F) and A is 5/3 (the 3rd of
F).

This is obviously desirable for a diatonic instrument that plays
a lot of major chords, like the diatonic harmonica. However, if
we take it much beyond those three chords, we get conflicts. For
instance, the fifth between D and A is out of tune. And we have
two different sizes of major second - C-D is 9/8, while G-A is
10/9. Even the simplest relations between the degrees of the
scale can create conflicts when refigured with the simplest
ratios. This is before we even consider chromatic tones.

Pythagoras (and unnamed others in China possibly before him) took
a stab at solving this problem by basing everything on the 3/2
ratio (the perfect fifth). This ratio is applied to C to get G,
to G to get D, to D to get A, and so on. However, this made major
thirds extremely sharp- even more so than in equal temperament.
Sharp notes like F# and C# are also out of tune when used in
chords or arpeggios of chords like D and A. And by the time we
get back around the chromatic cycle to C (B## at that point in
the Pythagorean system). This system used little corrections,
called commas, to get everything back in synch - very
complicated.

For centuries, the solution was to tune an instrument to either
just or Pythagorean intonation IN ONE KEY and stick to it, or use
a compromise system like mean-tone (look it up if you like).
Modulation to even a closely related key would sound bad. As for
choral singers, they had to punt - a literal adherence to just
intonation would cause the pitch to sink, while Pythagorean
tuning was unnatural to the ear, which rules vocal intonation.

Sometime in the 16th century, someone came up with the bright
idea of dividing the octave into 1200 cents, with 100 cents being
the size of each successive semitone. This created a simple,
sturdy structure that would allow for all keys to be treated
equally. It didn't really catch on until the 19th century, when
the need for such a vehicle became truly pressing as music became
more and more chromatic and ranged wider harmonically, modulating
through many keys. Of course it put every single interval out of
tune (see scale below), and singers and violinists still fudge
(Bb will be very flat, while A# will be very sharp - in mean-tone
tuning this discrepancy is called "the wolf").

Here is a three way comparison between the just, Pythagorean, and
equal tempered scales for a C major scale, expressed in cents:

      C  D   E   F   G   A   B    C

JUST  0  204 385 498 702 885 1088 1200

PYTH  0  204 408 498 702 906 1110 1200

EQUAL 0  200 400 500 700 900 1100 1200

Remember, this is expressed in cents, not in Hertz. By the way,
cents are not an equal division of any number of Hertz by 12.
Rather, It's logarithmic. The pitch of each successive semitone
(100 cents) is derived from multiplying the pitch of the previous
semitone by 1.05946. Thus if A=440, then A# is 440 x 1.05946 =
446.1624. B is 446.1624 x 1.05946 = 493.88, and so on. Want to
know more? Look it up.

What this means for harmonica players is that we can tune to
equal temperament and have our chords sound bad, or to just
intonation and have our melodies sound bad. Or we can fudge -
hence Steve Baker's compromise tuning as proposed in his Harp
handbook.

By the way, The Harvard Dictionary of Music is an excellent
reference work. Entries used in writing this little summary
included Acoustics; Intervals, calculation of; Just intonation;
Pythagorean scale; Temperament.

Winslow







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