Fabrice Mogini
2.0 PARTIALS AND TUNING SYSTEMS.
2.1 CONSONANT PARTIALS FOR ALTERNATIVE EQUAL TEMPERAMENTS.
We have seen in the section about harmonic series how certain partial distributions
can confer musicality to sounds. The partial's distribution, to prevail its
harmonic functionc, has to be transformed when using alternative tuning system
so we can use musical sounds in the context of any particular tuning.
A simple harmonic sound1 may be dissonant in an alternative equal tuning.
This is because the partial's frequencies of each note considered have little
to do with the frequencies of the other notes we can choose from in this tuning.
This will be the case if we decide to use for instance a tuning with seven
notes per octave with an harmonic sound.
Some partials contained in the sound like fifth or third will not have any
corresponding frequencies within the set of frequencies available in the tuning.
A good solution is to tune the partials so they can work harmonically with
the chosen temperament. In that case, partial frequencies will be multiples
of the frequencies found in the set and will fulfil their harmonic function.
With this technique, we use certain notes from the tuning as partials. These
partials are contained in each sound and are in tune with any note belonging
to the tuning.
Partial's distribution:
• After the root, the first partial will still be the octave, which
exists in any equal temperament of base two.
• Then we will have to find which note of the temperament to use as
the second partial.
• We must also be able to calculate this partial from any of the notes
that exist in the tuning with a single equation instead of doing it case by
case.
The next section describes a way to find the right partials depending on the
number of equal steps per octave and starting from a fundamental frequency.
2.2. CALCULATIONS OF CONSONANT PARTIALS FOR EQUAL TEMPERAMENTS.
t is number of steps per octave (temperament);
p is the wanted partial frequency;
f is the fundamental frequency;
Note: log.2 is not a base two logarithm but a normal log. It can be calculated
beforehand because this value will always remain the same in the equation
if we keep the octave as the basic unit from which the other intervals are
deduced.
For a verification of the formula, we can start calculating the first eight
partials for sounds in the usual temperament (twelve equal tones per octave).
Partials for 12 note equal temperament
0 = fundamental
12 = octave
(it is our first partial or the twelfth note of the series of tones in this
temperament when the fundamental is 0).
19.0195 = equal tempered fifth
(nineteenth note appearing in the series of tones).
23.9999 = octave.
27.8632 = equal-tempered major third.
31.0195
33.6883
36
38.039... (more partials could be calculated).
After a rounding of the values we have finally found that these partials are
those of the harmonic series with a small deviation from just ratios because
we are in an equal-tempered system.
We can explore then about partials for other equal temperaments (with a different
number of equal steps per octave).
Partials for 14 note equal temperament
0
14 = fourteenth note = octave.
22.1894
27.9999
32.507
36.1894
39.303
42
44.3789Now we need to use these results with our formula and find each partial
frequency starting from a certain fundamental.
14 equal notes Partials with a fundamental of 440Hz:
It would be fastidious to calculate each partial frequency, especially when
we now that we will have in a musical context several different fundamentals
(melodic line for instance).
Nevertheless, the partials given as a step number in the series above remain
the same for this temperament.
In a musical context we will build an array of these values and ask the computer
to apply the formula for each element of the array and this for each fundamental
frequency to be used in the music.
Finally, each complex-tone will be the result of all these frequencies played
together with different amplitudes.
We have to keep the first value, zero, to represent the fundamental itself.
2.3. EXAMPLE OF THE CALCULATION IN A MUSICAL CONTEXT.
Here is an example of partial's calculation in a musical context with 14 equal
notes per octave.
(i ) is the index that defines a position in the array of "partialfreqs".
It enables the computer to apply the whole calculation to each member of this
array.
• With this formula, we just need to call our series of notes "fundamental
freq".
• Each sound of the melodic line will be a complex tone with its partials
being harmonic with the tuning.
• It is even possible of course to change temperament during the music
piece. In that case, 14 (above) would be replaced by another value while the
array "partialfreqs" would have to be changed to the relevant one.
2.4. APPLICATIONS OF EQUAL TEMPERED PARTIALS.
With the calculations described in the last section, we have created some
harmonic partials deduced from equal temperaments. This means that these partials
are multiples from the fundamental frequencies available in the chosen temperament.
Sethares (1988) comes to similar results using a different technique.
These complex sounds, even though they are by nature inharmonic, share some
common points with the acoustic sounds with partials derived from the common
harmonic series.
They are the most harmonic sounds (or simple harmonically) that can be used
in the context of a particular tuning system.
For this reason a complex inharmonic sound gets here the status of harmonic
sound.
A major consequence is the relative justification for the presence of inharmonic
sounds in a musical context. We wanted to use equal temperament with wholeness
by
creating equal-tempered partials but we have also achieved a way of using
inharmonic sounds with integrity.
Complex sounds with different partial tunings and amplitudes, can be directly
created to suit better a particular tuning. The same principle can even be
used to deduce the most harmonic complex sound to use in a special scale of
a tuning.
Starting from an existing complex sound, we can predict in which tuning it
will be the most consonant.
2.5. SPECTRAL TUNING.
Spectralist techniques are directly connected to the analysis of an inharmonic
sound's spectrum to create an adequate tuning. Spectralists base their tuning
on the spectral analysis of an existing or preconceived sound or instrument,
and are using to this effect unequal divisions of the octave. Not only partial's
frequencies are analysed but also partial's amplitudes, ring times and detune
(slow beating).
Grisey (1975), in Partiels, starts from the trombone natural harmonics to
deduce the octave divisions and instruments to use accordingly to these partials.
Murail (1985), in Time and Again, uses a DX7 synthesizer to
trigger the creation of a set of artificial spectra played by the
orchestra. The DX7 plays the partials contained in the timbre
of its own spectrum (inharmonic) while the other orchestral
instruments enrich this spectrum with partials belonging
either to this spectrum, or to the potential fundamentals
which would be composed by the DX7.
With additive-synthesis, a composer can create and control the spectrum of
computer-generated sounds and base the octave divisions on such a spectrum
(Risset 1989).
For the spectralists, each different spectrum can generate a particular composition
with a specific tuning system.
In other words, there are as many tunings as there are complex sounds.
2.6. SPECTRAL TUNING AND EQUAL TEMPERED PARTIALS.
Spectral analysis can indeed use most complex sounds in an adapted tuning
context. Equal tempered partials, on the other hand, can help modulation,
polyphony even though they are restricted to the use of certain complex sounds
only.
Unequal partials limit the composer to a mapping of the harmonies to the chosen
spectrum, the harmonies are rich but it is more difficult to influence the
music development on a compositional level.
The configuration of the partials's structure (intervals and order) remain
the same whatever the fundamental chosen. Finally, each partial frequency
does not correspond anymore to the fundamental frequencies available in the
pitch system.
The partials can be considered as a chord made out of unequal intervals. Transposing
this whole structure to one of these intervals will not necessarily distribute
it along the other type of organisation which as a major role in the tuning
system: the set of available fundamentals.
There are limited homogenous interactions possible between horizontal and
vertical music with these unequal partials.
With this system, horizontal music as a sequence of fundamentals is opposed
to the vertical notion of music: polyphony, counterpoint, partials analysed
as a chord.
With equal-tempered partials, sounds are made out of harmonics that are all
found within the tuning system: each harmonic corresponds to a fundamental
available in the tuning. Each step chosen will generate a series of partials
created from some of the steps that exist in the temperament.
This is why equal-tempered partials help the exploration of the musical language
and allow more freedom for composing.
3.0 TUNINGS NOT BASED ON THE OCTAVE OR ON A SINGLE
STEP SIZE.
3.1. BEYOND THE OCTAVE.
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