Basic Tuning Theory

    INTRODUCTION TO PROPERLY TUNING CHORDS USING GENERAL MIDI

                      John A. deLaubenfels
                        <jdl@adaptune.com>
                            01-03-01


Though most musicians don't realize it, the chords produced with the 
12-tone equal tempered scale are not well in tune.  Everyone grows
up hearing badly formed chords until our ears are "burned in" on 
dissonance.  With modern sound cards and synthesizers, however, it is 
possible to hear chords properly tuned.  The sensation is quite a 
surprise!

The foundation chord in Western music is the major triad, represented
by the notes C, E, and G (of course, there are 11 other major triads
in 11 other keys, but the same considerations apply to all).

The absolute tuning in general use today has A above middle C tuned at 
440 beats per second.  For ease of computation, however, consider a 
tuning in which "C" is exactly 400 beats per second, and all other notes
follow the 12-tone convention.

The math for calculating equal-tempered frequencies is as follows: 
going up 12 notes (an octave) causes a doubling of frequency.  In our
example, the C above is at exactly 800 beats/sec.  In between, the
ratio of adjacent frequencies is the twelfth root of 2, or approximately
1.059463.  From this ratio, we can calculate 12-tone frequencies for
our entire scale as follows (note that labeling of Eb instead of 
D#, etc., is completely arbitrary in 12-tone):

   C   400.000
   C#  423.785
   D   448.985
   Eb  475.683
   E   503.968
   F   533.936
   F#  565.685
   G   599.323
   G#  634.960
   A   672.717
   Bb  712.719
   B   755.099
   C   800.000

Here's the problem: a properly tuned major triad, with the root note
at 400 beats/sec, should have its other notes tuned at 500.000 and
600.000.  What we get instead is 503.968 and 599.323.  Of these, the
interval of a fifth, from C to G, is actually very good (599.323 is
very close to 600.000), closer than the ear can easily hear.  The
interval of a major third, however, from C to E, is NOT good: 503.968
is sufficiently far from 500 to produce an ugly audible beat frequency.

What's so special about the ratio 400 to 500 and 500 to 600?  They are
aligned with the natural overtones found within each individual note
(aside: some natural sounds, and many computer-generated sounds, don't
have so-called "harmonic overtones" like this, but string and wind
instruments do).  If, for example, we play a low C of 100 beats/sec, 
that one note will probably include overtones at 200, 300, 400, 500, 
600, ... beats/sec.  When we play another note at 503.968 beats/sec, it 
audibly conflicts with the overtone at 500 beats/sec, giving a "fuzzy" 
sound.  Why we humans prefer a steady sound to a fuzzy sound is not 
clear, but is apparently inborn.

It is possible to hear well tuned chords by applying a MIDI pitch-bend
message.  Consider the default case in General MIDI, where the range
of bending for each instrument is +/- 2 semitones.  That's a total range
of 4 semitones, or 400 cents (a cent is defined as 1/100 semitone, or 
1/1200 octave).  There are 16384 bend values in the range Ex 00 00 to 
Ex 7F 7F (x is the channel), which works out to 40.96 bv/cent (where 
bv == bend value).  In our above scale, an unbent E (with the pitch 
wheel at its center position of E1 00 40) sounds at 503.968; we wish to 
bend it down slightly to exactly 500.000.  How many cents is that?  The 
math is a bit esoteric: log(500.000/503.968)/log(2) gives us the number 
of octaves: -0.011404.  Multiplying by 1200 cents/octave, we get -13.685
cents.  Multiplying by 40.96 bv/cent, we get -560.53 bend values, or 
about -561 bv.  Changing this to hex and applying to the center bend, we
get Ex 4F 3B.

The magnitude of the correction is not dependent upon our choice of an
example tuned at 400 beats/sec; it applies to all 12-tone tunings and
to every key within them.

To play a properly bent major triad, we play its notes on separate
channels, for example:

E0 00 40   (no bend)
E1 4F 3B   (bend down by 561 bv, or about 13.685 cents)
E2 50 40   (bend up by 80 bv, or about 1.955 cents)

90 3C 40   (C, without bending)
91 40 40   (E, with -13.685 cents bending)
92 43 40   (G, with  +1.955 cents bending)

The correction for the "G" note can be omitted (just play the G on
the same channel as C) without great loss of chordal quality.

The tuning difference is best heard with non-decaying voices, such as 
horns, with any effects that add quavering turned OFF.

Minor triads, ex: C, Eb, G, contain the same intervals as major chords,
but in the opposite order.  The frequency of the Eb therefore needs to 
be RAISED to tune it correctly against C and G.  We could use E3 01 45 
to achieve this, for example:

E0 00 40   (no bend)
E3 01 45   (bend up by 641 bv, or about +15.640 cents)
E2 50 40   (bend up by 80 bv, or about 1.955 cents)

90 3C 40   (C, without bending)
93 3F 40   (Eb, with +15.640 cents bending)
92 43 40   (G, with  +1.955 cents bending)

Seventh chords, ex: C, E, G, Bb, require even MORE correction to the
seventh: it should sound at 700 beats/sec in our scale above (aside:
this is if we tune in "7-limit", which I assume here), but in 12-tone it
sounds at 712.719.  The correction needed is -31.174 cents, or -1277 bv,
represented by E4 03 36.

E0 00 40   (no bend)
E1 4F 3B   (bend down by 561 bv, or about 13.685 cents)
E2 50 40   (bend up by 80 bv, or about 1.955 cents)
E4 03 36   (bend down by 1277 bv, or about 31.174 cents)

90 3C 40   (C, without bending)
91 40 40   (E, with -13.685 cents bending)
92 43 40   (G, with  +1.955 cents bending)
94 46 40   (Bb, with  -31.174 cents bending)

Doing this kind of correction does not increase the number of notes
playing, but DOES increase the number of MIDI channels needed to play
them.  A sequence file containing many instruments sounding
simultaneously would run out of channels, and therefore could not be
re-tuned.  A sequence with no more than 3 or 4 instruments, however, can
be corrected without exceeding the 16-channel limit.

Unfortunately, not all chords can be correctly tuned.  Consider the 
chord C, D, F, G.  On the one hand, the notes are all connected via 
fifths: F C G D, requiring D to be bent UP relative to F.  On the other,
D and F form a minor third, requiring that D be bent DOWN relative to F.
It is not possible to bend a note two ways at once, so there is no 
perfect way to tune chords that contain four or more notes connected by 
successive fifths.  Other chords that are difficult include full 
diminished (ex: C#, E, G, Bb) and augmented (ex: C, E, G#).  Chords such
as these occur relatively rarely, however.  When they do, one way to
deal with them is simply to leave them in equal tempered tuning.

What does the future hold?  Twelve-tone tuning is probably under no 
threat in the immediate future.  Every Western instrument supports it,
our ears are used to it, there are no annoying notational specifications
for micro-tuning, etc.  And, on first hearing, properly tuned chords 
seem "strange."  Once appreciated, however, they are far superior to the
fuzzy mess that we've been stuck with for centuries.  At least now we 
have a choice, which makes this a good time to be alive!