Wat leesvoer uti het addendum van OS 6.5 van de Virus C waarin het volledig gaat over pure tuning en de schaalverdelingen binnen oktaaf en de compromissen.
PURE TUNING
For centuries we have been playing musical instruments that are out of tune...
Ever since the emergence of polyphonic instruments, including all keyboards as well as fretted
instruments such as the lute and guitar, tuning has always been a compromise. Several attempts
from the 16th-18th centuries to standardise the temperament (tuning) of church
organs and virginals helped a little, but they were battling against the mighty laws of physics
– see the “Theory” section below.
First suggested in 1636, our modern “Equal Temperament” was only in common use from
the late 18th century onwards because it was considered much too much of a compromise at
the time. Despite it’s one advantage (the freedom to play in any key), Equal Temperament
simply dilutes the fundamental problem, spreading it across all the notes in the octave.
Experienced singers and string players use “just” intonation – they adapt to any keys and
modulations (key changes) because they have infinitely variable control over pitch. Within certain
limits, the pitch of wind instruments can also be varied by adjusting embouchure (lip
position/tension). A group of musicians instinctively approaches a common overtone structure,
minimizing the “friction” between all the voices in a chord. This results in the wonderfully
rich but compact sound of symphony orchestras or gospel choirs.
Unfortunately, realtime intonation was not a feasible proposition for makers of keyboard instrument.
Finding a usable method of performing fine adjustments to each and every note
seemed physically impossible, especially when playing polyphonically.
These days, digital musical instruments can automate this process. The Pure Tuning (aka. Hermode)
algorithm analyses chords and immediately adjusts the pitch of each note so
that the prominent harmonics line up. Especially for normal synthesizer sounds, the difference
between Equal Temperament and Pure Tuning may appear to be rather subtle at first (though
this difference can be accentuated.
...
Although tuning is also nominally 100% pure at this setting, the actual pitch of notes is
once again subject to the fluctuations, the natural anomalies responsible for a lot of the “life” in
acoustic instruments (as well as true analogue synthesizers).
Even if you only play octaves, the “Natural” setting often causes beating between the notes!
This effect is therefore independant of the chord structure, and can/should be accentuated via
oscillator Detune (or even Unison mode) to beef up the sound. That’s how the monumental
sound of a symphony orchestra arises: the richness of the sound is solely dependant upon the
number of musicians (oscillators), not upon harmonic complexity.
PureTuning works perfectly for major triads and dominant seventh chords. Due to the physics
involved (see below), minor chords are more of a problem – they don’t sound quite as pure.
However, PureTuning is also very effective here because the subharmonics are managed well.
Equal-temperament was a radical break from all the other “tempered” methods, where polyphonic
instruments could only be played in a few diatonic modes. Note that Johann Sebastian
Bach’s “The Well-Tempered Clavier” is thought to have been based on a variation of
“Werckmeister 3” tuning. Much more suitable for playing in any key, but not quite the same as
Equal Temperament!
THEORY
Interestingly, the feeling of harmony in a major chord is a phenomenon based upon physics,
not on psycho-acoustics alone. The notes in a perfectly tuned C-major triad (C, E, G) have exact
integer frequency ratios: From E to C is 5:4 (or 1:1.25) and from G to C is 3:2 (1: 1.5). The
same applies to any major chord. It is fairly common knowledge that the frequency ratio of
an octave is 2:1 – the upper note is exactly double the frequency of the lower one. Assuming a
C is exactly 1000 Hz, the E would be 1250 Hz and the G would be 1500 Hz. The next C would
be 2000 Hz. Very simple, very nice...
Reality isn’t quite that simple. We expect more from our (chromatic) keyboard instruments. We
demand that the frequency ratio of all semitone steps should be the same, that no particular
key has “priority” over any other. Chords in the key of B or C# should sound just as good as
they do in C. The truth is that they actually sound just as bad!
Rule Number One: An octave interval should be an octave, it should be exactly double/half the
frequency of the reference note. If you take this rule as the basis and work out a constant ratio
you can apply to all 12 semitones, you arrive at the 12th root of 2, i.e. 1.059463. Test this by
multiplying this value with itself 12 times, or take a pocket calculator with the “power” (^)
function and tap in 1.059463 ^ 12. The result is exactly 2 (or 1.9999 recurring). A perfect octave.
Sounds complicated, but it seems to work – so where’s the problem?
Let’s get back to our C-major chord and work out the frequency ratios using this “magic”
number. The interval C-E is four semitones, which means we should multiply the magic
number with itself four times (1.059463 ^ 4). Feel free to try it – your Virus is doing this kind
of thing all the time :-). The result is 1.2599. Fairly close to 1.25 (see above), but not near
enough: It is detuned all of 14 cents (14% of a semitone). A singer with perfect intonation
would sing the E exactly 14 cents lower than this!
The interval from C to G is seven semitones: 1.059463 to the power of 7 is 1.4983. Not bad,
but not quite the 1.5 me might have expected. This detuning is equivalent to only 2 cents, but
still... So that’s the big disadvantage of the tempered
scale, the standard for all modern (western) keyboard instruments. Why can’t the notes simply
be tuned “correctly” instead? Just intonation depends upon the current situation, upon
the context: An A-minor triad (A, C, E) also includes the notes E and C like in our C-major
chord, but their functions are now Fifth and minor Third (instead of major Third and Root). So
the frequency ratio should certainly not be 5:4 (1.25) for them to be perfectly in tune!
Is it actually possible to hear the disadvantages of Equal Temperament? Not immediately – we
have become too accustomed to it throughout our lives. However, we can certainly react to its
more unpleasant side-effects: Electric guitarrists avoid playing full chords when using very
distorted sounds. Instead, they often limit their playing to the so-called “power chords” which
leave out the Third (so they are neither major nor minor). Attempting to add the Third to a
power chord only results in a very rough and muddy sound.
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