Download Investigate the mathematics behind the tuning systems of Wendy

Investigate the mathematics behind the tuning
systems of Wendy Carlos (15)
Student number: 200666371
Thursday 10th November 2011
Introduction - The twelve-tone equal temperament has been the dominant tuning system used in Western music for centuries. However, this has not always
been the case. There exists a large variety of different tunings, more or less used
depending on time and place, from the Pythagorean scale to meantone temperament, or just intonation. Today, computers and synthesizers allow musicians to
invent even more tunings because there are no material restrictions anymore: it
is not necessary to construct a new instrument in order to use a new scale. Thus,
electronic music is very well adapted to experiments in musical tuning. Wendy
Carlos is an electronic musician and composer very interested in exotic sounds and
alternative tunings. She is famous for having composed the original soundtrack of
Kubrick’s film A Clockwork Orange. She experimented several alternative scales,
in particular in her album The Beauty in the Beast. Some of these scales were her
own creations: three equal-tempered scales called alpha, beta and gamma, and
a “super-just” scale called the harmonic scale. We will explain on which mathematical basis are constructed these scales, and in what extent this makes them
sound good.
All musical scales try to approximate simple integer ratios for the frequency ratios, in order to have the most consonant tuning. In equal-tempered scales, the
frequency ratio of two consecutive notes is constant, so the degree (distance between consecutive pitches) should be chosen in order to get as close as possible to
these ratios. Most existing equal-tempered tunings are based on an interval with
frequency ratio 2:1, the octave, which is divided into an equal number of steps:
12, 19, 24, 31, etc.
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The alpha, beta and gamma scales are equal-tempered scales, but their particularity comes from the fact that the octave is not divided by a whole number.
Ignoring the octave, equal temperament can obtain better approximations for the
other simple integer frequency ratios, such as the perfect fifth 3:2 or the major
third 5:4. In [4], Carlos said that she “discovered” these three scales while trying
. A method on how to
to find good approximations for the ratios 32 , 54 , 65 , 74 , and 11
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find frequency ratios close to these simple integer ratios is explained in [1]. Here,
we will only consider the perfect fifth (ratio 3:2) and the major third (ratio 5:4),
but we will actually find also a good approximation for the minor third (ratio
6:5) as 65 = 23 ÷ 45 . We need to find a scale degree x for which there exist some
integers m and n such that the perfect fifth occur at approximately m steps, and
the major third at approximately n steps, that is:
n × x × log2
5
3
≈ m × x × log2
2
4
In other words, the ratio log2 32 ÷ log2 45 should approximate an integer ratio,
and small integers would be better, in order not to get too much notes within
an octave. The best way to find a rational approximation of this ratio is using
continued fractions:
log2 32
1
5 = 1+
log2 4
1 + 4+ 1 1
2+
1
1
6+ 1+...
The alpha scale is based on the approximation 1 + 1+1 1 = 95 for this ratio. So with
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our previous notation, we have that m = 9 and n = 5, and as a consequence, 4
steps approximate the 6:5 minor third. In order to find the best x for our scale,
we have to minimise the mean square deviation: this is a function describing the
difference between the model we want to follow and the reality. This corresponds
to:
5
6
3
(9x − log2 )2 + (5x − log2 )2 + (4x − log2 )2
2
4
5
(where the unit is the octave). The minimum value for this function is obtained
when the derivative with respect to x equals zero, that is:
9 log2 23 + 5 log2 54 + 4 log2
x=
92 + 52 + 42
6
5
≈ 0.06497
and we find x = 0.06497... octave. A better unit to express this value is the cent,
where the octave is divided into 1200 cents. So this scale is of degree 78 cents,
and there are approximately 15.4 steps per octave.
The steps for beta and gamma scales are found with the same calculations,
but with different approximations for the ratio log2 23 ÷ log2 54 . For beta we choose
2
1+
1
1+ 15
=
11
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found by rounding up. For gamma, we take 1 +
1
1+
1
4+ 1
2
=
20
.
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We
find that beta has 18.8 steps to the octave, and gamma 34.2 (respectively 63.8
and 35.1 cents per step).
Figure 1: α, β and γ scales compared to exact ratios, from [1]
As you can see, these scales have better approximations for the simple ratios
than the 12-tone equal-tempered scale, where there are 300 cents for the minor
third, 400 cents for the major third and 700 cents for the fifth. Alpha and beta
have very similar properties, but the sevenths are a little more in tune in beta.
Gamma is closer to just tuning for minor thirds, major thirds and fifths, which
are almost perfect; this scale wasn’t used on the album The Beauty in the Beast.
As we said earlier, consonance occurs when frequencies of notes are related
by simple integer ratios: two pitches with a small integer ratio between their
frequencies are in perfect harmony. A possible tuning would be to construct notes
with frequencies following exactly these ratios. Such tuning systems are called
just intonation. The main problem of these scales is that all the pitches are
constructed in relation to a single note, so they are in harmony while we play in
that key, but we can’t modulate (meaning start to play in another key) because
some pitches will be dissonant with the new tonic. So with the classic 12 notes
per octave, we can play in one key only.
Wendy Carlos invented a ’super-just’ scale in the sense that it extends just
intonation, beyond 5-limit. In other words, the exact ratios of the frequencies
use multiples of primes other than 2, 3 and 5. This is the harmonic scale. She
described this in [2] and in [3]. This scale has 144 different pitches per octave.
This corresponds to 12 keys × 12 notes in a chromatic scale. The principle is to
consider a “root note” or key, and to construct a just scale related to this note.
This scale will be composed of harmonics of a low fundamental, which means notes
with a multiple of its frequency. For more clarity, let’s construct this scale with
the root C for example. The 12 notes of the octave related to C are harmonics of
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a low C, called the fundamental. Their wavelength ratio (from the fundamental)
are terms of the harmonic series.
Figure 2: Harmonics of C, from [2]
The first C in our octave will be the 16th harmonic of this fundamental. Then,
C] is the 17th harmonic, D the 18th, E[ the 19th, E the 20th, F the 21st, F] the
22nd, G the 24th, A[ the 26th, A the 27th, B[ the 28th and B the 30th. Finally,
C is the 32nd harmonic. We can then work out the frequency ratios of each note
of the octave from C, and they are simple integer ratios:
Figure 3: Harmonic scale on C table, from [2]
You can notice that E is the perfect major third, G is the perfect fifth. All the
other ratios are simple too, so all the notes will sound good with C. But if we use
only this table, transpositions from one key to another are not possible. Indeed,
16
in the key of D[, the fifth would be A[, but then the ratio is 17
× 13
= 26
, which
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17
is not perfect anymore. That is why, for her harmonic scale, Carlos established
such tuning tables for every note in the 12-tone equal-tempered scale: each one
plays the role of the root, and we will thus obtain a 12-tone harmonic scale for
each of them, hence the 144 pitches per octave. A harmonic scale for a certain
root is away from the previous one by 100 cents, that is one step in 12-tone equal
temperament. With all these pitches, we can now modulate between the distinct
keys, and the just intonation restriction of playing in only one key disappears.
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While playing music with the harmonic scale, these transpositions are controlled
by the appropriate note on a one-octave music keyboard, which automatically
retunes all the notes of the playing keyboard according to the corresponding tuning
table.
One interesting phenomenon of this scale is that if you play several notes of
one tuning table, you can ear a low note: the fundamental. As the frequencies
of the notes are multiples of the fundamental, when we add them we obtain a
periodic wave, with the same period as the fundamental wave.
Conclusion - The mathematics permit to invent new tuning systems, more consonant, and electronic music provide the technical tools to put them into practice.
It is of course easier to tune instruments in 12-tone equal-tempered scale than in
Carlos’ scales, because in alpha, beta and gamma there is no octave, and in the
harmonic scale we have 144 notes per octave! But when music is played using
synthesizers, there is no reason to keep this scale as a standard anymore. And in
this case, the musical scales invented by Wendy Carlos are very good alternatives
because the frequency ratios of their notes are closer to simple ratios than in 12tone equal temperament. However, our ears are not used to these tunings, so we
might find them a bit strange when listening to The Beauty in the Beast for the
first time...
References
[1] David J. Benson. Music: A Mathematical Offering. Cambridge University
Press, 2007.
[2] Wendy Carlos. Tuning: At the crossroads. Computer Music Journal, Spring
1987.
[3] Dominic Milano. A many-colored jungle of exotic tunings. Keyboard, November 1986.
[4] Wendy Carlos official website. http://www.wendycarlos.com.
[5] Wikipedia. http://en.wikipedia.org.
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