Fabrice Mogini
1.0 A PLURALITY OF TUNING SYSTEMS.
1.1 PYTHAGOREAN TUNING.
This is an ancient way of tuning instruments used during the Middle Ages.
It is based on a series of pure fifths that gives very consonant results if
thirds are avoided.
Major thirds are larger and minor ones are smaller in that system.
Nevertheless, most of the medieval music, written in pythagorean tuning, was
vocal. Singers could adjust their intonation to reach pure thirds. These intervals
were also
used scarcely since unison, octave, fourths and fifths were considered as
the most consonant intervals (Texier 1999).
These perfect fifths added to each other would extend the range beyond an
octave. This is why a transposition to the lower octave was done to obtain
the Pythagorean diatonic
scale but there was a problem in the first place: after twelve fifths added
to each other, One would not obtain a perfect octave of the first note.
Diagram 2. Series of pure fifths and Pythagorean comma.(available soon)
One fifth had to be diminished which had for consequence the deviation of
one of the semitones.
The Pythagorean comma1 is the difference between this larger semitone and
the former type of semitone used for the other notes. This could be a reason
why many
compositions still avoid the last two notes in the sequence of pure fifths
(pentatonic scale).
1.2 UNEQUAL TEMPERAMENTS, JUST TUNING.
The Pythagorean system privileged fifths and substracted a Pythagorean comma
to thirds to obtain in the end a pure
octave. This imperfect third seemed too dissonant compared to the other intervals
but was rarely used.
The third had to be modified to its just acoustic value1 from the XVIth century,
with the appearence of the common triad. Diagram 3. The common triad.(available
soon)
Gioseffo Zarlino gave his name to this unequal tuning in 1558.
Only certain major and minor chords could be purely consonant in this tuning:
C, F and G.
The aim was to give priority to certain harmonies and achieve consonance but
modulation to a different key was not possible without having to retune the
instrument.
From that period and through the XVIIth century, several types of unequal
temperaments were created which provided a consonant quality to certain chords
or intervals but gave a different intervalic sound to each scale (Texier 1999).
1.3 EQUAL TEMPERAMENT.
Andreas Werkmeister (1686), organist in Weimar Germany, described in "Musicalishe
Temperatur" a new tuning system: the equal temperament. Bach was to be
the first to write successfully a cycle of pieces in all the keys using this
system: 'The Well-Tempered Keyboard' (1722, first book: bwv 846-849, second
book: bwv 870-893).
The Equal-Temperament is obtained with root calculations.
It does not give precedence to any interval (apart from the octave that has
the function of frame).
There is no pure or just interval, harmonically, in this system.
On the other hand, all the intervals are experienced as acoustically correct
because they are equally out of tune1, which maintains a certain unity.
1.4 EQUAL TEMPERAMENT: CONSEQUENCES ON MODERN MUSIC.
Instead of using pure fifths and having small thirds so fifths and third can
coexist within the octave, Equal-Temperament
reduces slightly and equally each interval in the octave.
The use of each interval becomes an option available to composers using this
tuning system.
Equal-Temperament was soon to become a standard of Western music. It is adapted
to harmonic changes because modulation to many different keys becomes possible.
This aspect has the particularity to favour developments of the musical language
for the composer who whishes to explore the interactions between several keys.
Because counterpoint and modulation were liberated, more contrast was possible
betwen consonance and dissonance.
Before Equal-Temperament, a dissonant chord was any chord containing one of
the imperfect interval. Such an imperfect interval was an exception to the
system and could easily be noticed.
With Equal-Temperament, each interval is equally imperfect and a chord's consonnance
or dissonance started to depend more on their place within a musical development.
Consonance and dissonance depend then on the harmonic system from a compositional
point of view.
1.5 CALCULATION OF THE PITCHES OF A TWELVE NOTES
EQUAL TEMPERAMENT.
This formula is borrowed from Moore (1990).
r is the reference frequency or fundamental;
f is a wanted frequency;
n is the step number or the number of semitones between f and the frequency
wanted;
2 represents the range of an octave.
12 is the number of steps per octave.
If the reference frequency is 440, and we want to find the fourth step from
that pitch in the twelve notes equal temperament, we obtain:
1.6 CALCULATION OF THE PITCHES OF ALTERNATIVE
EQUAL TEMPERAMENTS.
It is possible from the same formula to calculate tones of any other equal-tempered
scales by changing the value that will divide n. The equation is then written
down as such for an eight-notes equal temperament (still looking for the fourth
note):
Equal-tempered intervals are easy to perceive but more difficult to tune correctly
by ear on an instrument.
This is why electronics and the computer are better tools to tune them quickly
and accurately. It is also possible to program the computer to change from
one tuning to another one in real time, during the course of a same composition.
Generally, pitch bend messages1 are used to detune frequencies already conceived
for the twelve-tone system but it is sometimes also possible to program the
frequencies of a special tuning beforehand.
The computer has the capacity to allow the composer to experiment freely with
tuning. Unfortunately tuning is rarely a challenged notion.
Twelve-tone equal temperament is still the only tuning most synthesisers are
equipped with. Timbres, envelopes, types of soundwaves are generally implemented
with most synthesisers but not tuning (as a system rather than for a single
note).
Composing with "tuning" as a parameter is then still reserved to
those who can have access to special musical instruments or who can write
computer programs.
1.7 EQUAL TEMPERED TUNINGS WITH A LARGE NUMBER OF NOTES PER OCTAVE.
James McCartney (1996), has written on the advantages of the 72 tone equal
temperament and has compared each note to those made with simple just ratios.
Advantages of the 72 tone equal temperament :
• This temperament closely approximates the first 12 harmonics of the
harmonic series.
• It is a superset of 12 tone equal temperament, which means that the
existing Western repertoire can be played unchanged.
• 72 can be evenly divided by many numbers:
36, 24, 18, 12, 9, 8, 6, 4, 3, 2
Table of just approximations of small integer ratios in 72 equal notes per
octave .
RATIO
STEPS
ERROR
(cents)
COMMENT
1/1 0 0 unison
16/15 7 +5 minor second
10/9 11 -1 minor tone
9/8 12 -4 major tone,
9th harm.
8/7 14 +2 septimal super major
tone
7/6 16 0 septimal sub minor third
6/5 19 +1 minor third
5/4 23 -3 major third, 5th harm.
9/7 26 -2 septimal super major third
4/3 30 +2 perfect fourth
11/8 33 -1 11th harm.
7/5 35 +1 septimal tritone
10/7 37 -1
16/11 39 +1
3/2 42 _2 perfect fifth, 3rd harm.
14/9 46 +2 septimal sub minor sixth
8/5 49 +3 minor sixth
5/3 53 -1 major sixth
12/7 56 0 septimal super major sixth
7/4 58 -2 7th harm.
16/9 60 +4
9/5 61 +1 minor seventh
15/8 65 -5 major seventh
2/1 0 octave
1.8 EQUAL TEMPERED TUNINGS WITH A SMALL NUMBER OF NOTES PER OCTAVE.
When the smallest interval of a tuning system is larger than a semitone, we
obtain frequencies that are easily distinguished by the ear. The amount of
dissonance is limited and these tunings generate harmonies of a modal quality.
_The eight-tone equal temperament was used by Pierce (1966) in Eight-tone
canon. This tuning has many of his tones in common with the usual twelve tone
system. It has some of the harmonic qualities of the diminished seventh scale.
_Tunings such as the five or seven notes equal temperaments are part of ancient
Greek or oriental cultures (Javanese). These intervals offer a limited number
of possible combinations but have the advantage of being easily heard which
is not always the case with large microtonal systems.
We will see later how it is possible to use several simple tunings together
to have access to more combinations and still favour simplicity.
1.9 TWELVE NOTES PER OCTAVE SUBDIVIDED IN SMALLER INTERVALS.
The 72 note tuning discussed earlier seems to fall in this category of 'twelve-note
expanded tunings' but is in fact designed for other goals. Even though the
twelve notes can be used with this large system, they are not meant to be
the base of the tuning because we can explore any other intervals without
having to come back to the twelve notes.
On the other hand, with the twelve notes subdivided, the small intervals are
just ornamental and used in the melody as grace notes or modulations near
from a tuned vibrato effect. The main tuning remains then a twelve-note system.
The Hindustany and Karnatic systems of India are based on 22 unequal notes
per octave. In reality there are twelve notes and small variations of intonation.
These variations were measured and discussed by Danielou (1969).
A similar effect is obtained in some of the quarter tone tuning used in contemporary
music. Quarter-tones truly extend the musical language if the twelve tone
system is not anymore sensed as central: this depends finally on a composer's
inclination.
1.10 HARMONIC SERIES.
Pure sounds with a simple and single soundwave are rare in nature. Most musical
sounds contain partials that are multiples of the fundamental frequency. We
tend to only hear this single frequency but with a particular timbre which
is a single complex-sound.
Sometimes the fundamental is not even present but the brain can deduce it
by analysing the partials's relationships. This happens, for example, with
portable radios and telephones that generally have small loudspeakers that
cannot reproduce lower frequencies (Plomp 1976).
The notion of musicality of sounds is subjective but generally refers to sounds
with partials that are multiples of the fundamental frequency. The first natural
harmonics on most instruments are those of the just or perfect intervals.
The harmonic series from a fundamental (1/1) is:
2/1 3/2 5/4 7/4 9/8 13/8 15/8
We can observe in the example above that some intervals such as octave, fifth,
third, major second and seventh are found in this series, even though usual
tunings will not include (7/4) and (13/8).
Harmonic tuning could in theory include such intervals because they are naturally
justified but in reality, ratios with prime numbers larger than five are generally
omitted.
_Some electronic organs that are tuned with the twelve-tone equal temperament
while based on sounds containing partials derived from the harmonic series,
demonstrate an application of such natural series.
In that case of course, some of the partials cannot be in tune with the fundamental
(and partials) of certain other notes of the keyboard.
This ambiguity, creates some interesting beating effects, even though there
is not a perfect homogeneity between the temperament and the harmonic partials.
1.11 RATIOS SYSTEMS.
Using ratios to build a musical system means that we use several dimensions
or proportions within a given space. The octave (2/1) is the standard space
that can be deconstructed in different proportions. This is different from
the equal temperament system where we have steps of the same size. Adding
twice the size of the smallest interval, the semitone gives us the next interval,
in size, in this system which is equal to one tone(in the case of the twelve
notes per octave temperament).
Before they relate to each other, the ratios tend to have a perfect vibrational
relationship with our basic unit: the octave. While the adepts of just intonation
would choose to favour a consonant interval in the construction of a scale,
the ratio-based system accepts a gradual natural dissonance when including
ratios with prime or higher-integer numbers.
_Clarence Barlow (1992) came up with the notion of indigestibility of numbers.
Digestible numbers are divisible. Harmonic intervals are then not only small
but divisible numbers. This explains why intervals that contain prime numbers
larger than five1 are not found in Western music.
The most harmonic intervals, depending on a limit threshold, can be compared
to the intervals of an equal temperament system. Prime numbers ratios can
then be used if they are near enough from the next harmonic ratio.
A chosen proportion derived from the smallest interval can set the limits
of an area where an interval is near enough from a simple-ratio interval and
can then borrow its harmonic function in the tuning system.
Diagram 4. A seventeen equal scale (black dots), the nearest ratios (rectangles)and
the level of harmonicity.
_Harry Partch was aware of the limitation of a classical tuning system based
on ratios with numbers not exceeding five. To extend his tuning system, he
had to face the problems of interpretation by performers as well as acoustic
realities and started designing real instruments on which his new ratios could
be played accurately. For him the smallest interval that one can perceive
measures 14 cents for a median register. Partch (1974) has experimented with
these perfect ratios and built a system where simple as well as complex ratios
coexist. One of them is his famous 43- note ratio tuning.
Critical bandwitch.
The register1 is an important factor in the perception of consonance. A low-number
interval (a third for instance) can sound dissonant in the lowest register
of a piano.
The critical bandwitch (Scharf 1970) is an area of closeness for two frequencies.
This bandwitch can help determine when some beating will occur between two
frequencies.
The critical bandwitch changes size depending on the register observed. The
ear's physiology is equiped for logarithmic distributions (volume and pitch)
because this organ is supposed to analyse in a single system low and high
pitch, partials for speech recognition, very loud and quiet sounds.
1.12 FAREY SYSTEMS OF MUSICAL INTONATION.
Rudolf A. Rasch (1987), questioned whether musical intervals used in alternative
tunings (such as equal temperaments) can be considered as consonant. The idea
of consonance here is different from the idea of tonal consonance in music
theory. It could then be called "sensory consonance".
Intervals are represented in the form of ratios. Simple ratios with low integer
numbers represent the most consonant intervals such as unison (1:1), octave
(1:2), perfect fifth (2:3).
When ratios are defined by higher integer numbers, the intervals become less
consonant. Some tuning systems (the equal temperament system for instance)
have intervals that show a small deviation from the exact ratios (as the tempered
third that has a ratio slightly different from the perfect third ratio).
For Rasch (1987), high-number ratios can be consonant if tuned accurately.
Rasch used the Farey series of fractions to define new sets of intervals.
John Farey (1766-1826) was a British geologist who created a new way of classifying
fractions.
A Farey series is a set of non-reducible fractions where ratio numbers do
not exceed a certain integer of order N. The magnitude of the ratios's quotients
defines the order of appearance in the set.
When N = 4, the Farey series is: 1/2, 2/1, 1/1
This series represents the frequency ratios with numbers not exceeding 2,
from unison to octave and ordered in function of their quotient. The most
consonant interval is the unison at the end of the series, while ratios at
the beginning of the series are less consonant.
A Farey interval plane can be used to represent the intervals as points on
this plane. The magnitude of the ratio numbers is then a dimension on this
two-dimensional plane.
This way of classifying intervals helps us understand the notion of space:
low-number intervals have more space (on the plane) around them than high-number
intervals. This space represents the interval faculty of having fast or slow
growing beat frequencies when mistuned.
Electronic equipment can allow the composer to use high-number intervals accurately
but these intervals require more tuning and intonation effort when using the
voice, or acoustic instruments. Rasch showed that high-number intervals can
be used as stable and consonant for the hear in alternative tuning systems
as long as they are in tune.
2.0 PARTIALS AND TUNING SYSTEMS.
2.1 CONSONANT PARTIALS FOR ALTERNATIVE EQUAL TEMPERAMENTS.
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