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Description  |
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BACKGROUND OF THE INVENTION
The present invention relates to a new technique for eliminating overtone
collisions in musical scales, and to a novel interval system for tuning of
musical instruments wherein dissonances resulting from struck chords are
eliminated. More particularly, the invention relates to an electronic
musical instrument which reproduces musical scales so that normal struck
chords, such as major fifths, do not have such dissonances. The invention
is based on a wave system of communication which relies on a different
basis of periodicity in wave propagation.
One problem in human communication since the first attempts to use and
reproduce vibrations has been the development of a standard wave system of
communication such that production of a wave of a given definition does
not interfere with any other wave which occupies space. In the context of
music, such interference causes overtone collisions, or dissonances, which
can make individual notes, or melody lines in music difficult to identify
and to follow.
Throughout history, there have been a number of interval system whose goal
has been to minimize such dissonances. Four of the most widely-used such
systems in auditory octave tuning for Western music are the Pythagorean,
just intonation, meantone, and equal temperament intervals. These are
defined, for example, in The New Harvard Dictionary of Music, (1986,
Harvard Press). As described in that publication, pitch is calculated by a
ratio involving logarithms, cents, and string lengths, all of which
involve a measure of periodicity of waveforms based on .pi.=3.1416.
None of the above-mentioned interval systems suffices, by itself, to
prevent overtone collisions. Various attempts have been made to combine
various ones of these systems as appropriate in electronic musical
instruments, to minimize the degree of overtone collision present. Some of
these now will be discussed.
U.S. Pat. No. 4,152,964, which relates to a correction of the "larger
tuning errors of equal temperament as each interval or chord is played",
switches intervals between equal temperament and just intonation,
depending on whether a chord or a single note is to be played. Also in
that patent is a discussion of some of the inherent deficiencies in the
mathematical base from which the various then-known intervals were
derived. It is useful to consider that discussion here:
"No chromatic scale with tones of fixed pitch can yield perfectly tuned
chords and also allow complete freedom of modulation. A scale composed of
perfectly tuned chords must have notes whose frequencies form an
arithmetical progression, while if the scale is to allow complete freedom
of modulation, the notes must have frequencies that form a geometrical
progression. In the first case, although the frequency differences are all
congruent, the sizes of the various intervals, measured logarithmically,
are not congruent with respect to the octave or with one another because
the logarithms of simple interval ratios are irrational decimals. In the
second case the sizes of the intervals, measured logarithmically, are
congruent with one another and with the octave but now, since the interval
ratios are all expressed as fractional powers of two, and hence
irrational, all the intervals of such a scale except the octave are more
or less out of tune.
This dilemma which lies at the root of the difficulty of realizing just
intonation with scales of fixed pitch, can be resolved by converting the
present scale of equal temperament into a scale with tones of mutable
pitch. Thus, the modulational advantage of the present scale is preserved
by retaining the tempered fourths and fifths without alteration while the
harmonic potentialities are greatly enlarged by the use of a keyboard
controlled computer which automatically shifts the pitch of certain notes
to correct the larger tuning errors of the scale.
The technique of the above-discussed patent, then, does not represent a
systematic approach to elimination of overtone collision. The lack of a
systematic approach makes the scales reproduced by the electronic musical
instrument difficult to transpose into different keys. While equal
temperament facilitates such transposition, the problem of overtone
collision is serious.
Another illuminating discussion is found in U.S. Pat. No. 4,248,119,
relative to the incompatibility of freedom of modulation in one system
with chords of the same system, wherein it is stated:
Generally, an electronic musical instrument is tuned in an equally
tempered scale so that it is easy to modulate or transpose to other keys
or to make ensemble performance with other musical instruments. However,
when the electronic musical instrument is thus tuned with the equally
tempered scale, such chord tones as major triad chord tones are not
produced in perfect consonant intervals so that it constitutes one of the
factors that disturb harmony. For example, when major triad chord tones
are produced by a just intonation scale, the frequency ratio of the root
note tone to the major third note tone is just "4:5", and the frequency
ratio of the root tone to the perfect fifth note tone is just "2:3" and
accordingly "4:6". On the other hand, when the major triad chord tones are
produced with the equally tempered scale, the frequency ratio of the root
note to the major third note is "4:5.03984". Thus, the pitch of the major
note in the equally tempered scale become higher by 14 cents than that of
the major third note in the just intonation scale. Furthermore, when major
triad chord tones are produced in an equally tempered scale, the frequency
ratio of the root note to the perfect fifth note is "4:5.993228". Thus,
the pitch of the perfect fifth note in the equally tempered scale is lower
by 2 cents than that of the perfect fifth note, in a just intonation
scale. As a consequence, where chord tones are produced in a just
intonation scale, clear tones can be produced with consonant intervals. On
the other hand, where chord tones are produced in an equally tempered
scale, the tones become a bit unharmonic.
Thus, in both of the just-quoted patents, there is recognition that no
single interval system has been able to provide sufficient harmony for the
different situations in which both single notes and chords are struck.
In U.S. Pat. No. 4,498,363, it is stated:
"On the other hand, an instrument tuned according to the equal-temperament
cannot obtain perfect chords when compared to an instrument tuned
according to the temperament of just intonation. However, the instrument
tuned according to the temperament of equal-temperament is capable of
obtaining chords which sound substantially natural, and in addition, the
modulation operation is simple. For this reason, general electronic
keyboard instruments, piano, and the like were conventionally tuned
according to the temperament of equal-temperament. However, the chords
obtained from the keyboard instruments which are tuned according to the
temperament of equal-temperament are not perfect chords as described
before, and these instruments are unfit for use in teaching during chorus
practice, for example.
In the above-mentioned U.S. Pat. No. 4,152,964, there is discussion of some
of the disadvantages of both the just intonation and the equal temperament
scales.
The disadvantages of fixed scale systems will be evident from the
following description: It is well known that the just scale CDE.sub.1
FGA.sub.1 G.sub.1 C which is generated by the perfectly tuned chords
FA.sub.1 C, CE.sub.1 G and GB.sub.1 D, contains the imperfect minor chord
DFA.sub.1 in which the note D is a comma too sharp relative to the note
A.sub.1. On a fixed scale basis, a perfectly tuned chord D.sub.1 FA.sub.1
can be had only as the submediant triad in the key of F Major or as the
mediant triad in the key of B Flat Major, by momentarily turning on either
of these tonality stops. A further disadvantage of just intonation on a
fixed scale basis is that the same mis-tuned triad DFA which would also be
contained in the dominant ninth chord GB.sub.1 DFA.sub.1, renders that
chord even more dissonant than the same chord in equal temperament.
Thus, there has been clear recognition in the prior art that no fixed-scale
system has been known which eliminates overtone collision by itself. An
interesting summary of the problem is provided in U.S. Pat. No. 4,434,696,
wherein it is stated:
For centuries numerous scholars and critical listeners have argued that
the influences of fixed-pitch instruments have contributed to a loss of
correct pitch and have caused vocalists and instrumentalists not
constrained by fixed pitch to sing and play "out of tune" either for
equally tempered or "just" performance. Basic to this problem has been the
lack of technological development in instruments for either tempered
tuning or just intonation. An examination of the abundant literature on
the subject discloses that no fixed-pitch or keyboard instruments have
previously been proposed or built capable of approaching precisely equal
tempered intervals, nor any that could accurately produce just intonation
and all of its enharmonic notes or modulational pitch changes for either
instructional or performance use.
SUMMARY OF THE INVENTION
In view of the foregoing, one of the objects of the present invention is to
provide a novel and useful electronic keyboard instrument with a
retrievable system of stored frequencies within an octave without any
alteration of the present keyboard, in which the above described
disadvantages have been overcome.
The present invention accomplishes what previous efforts have failed to
achieve. According to the invention, there is provided a system of notes
in an octave which allows complete freedom of modulation and perfectly
tuned interval chords using a stored memory computer source as a signal
for a predetermined assigned frequency or frequencies. The novel interval
used is called Tru-Scale by the present inventors. With this novel
interval, all of the advantages of fixed-scale intervals, such as relative
ease of transposition, are retained, while the disadvantages, such as more
severe overtone collision, are eliminated.
Tru-Scale tuning involves new mathematical principles of a standard unit of
measurement, related to a new measure of periodicity of wave transmission.
When applied to the sound production components of an electronic
instrument, primary or secondary, or other wave producing equipment, this
tuning system can profoundly enhance the equipment's sound or performance.
The enhancement is accomplished by eliminating the amount of dissonance
caused by overtone collision by providing simultaneous frequencies with
independent time-space relationships.
In the present production of electronic sound, a controlled electric
impulse is sent to an oscillator, in which the impulse is turned into a
specific assigned frequency. It is important to note that the initial
impulse, which ultimately ends up as a predetermined frequency, is
determined by mathematical computations using logarithms based on the
present imperfect mathematical system. These various divisions, Equal
Temperament, Just Intonation, Meantone, Pythagorean, of sound represent
many prior attempts to divide sound into a non-dissonate interval system.
The Tru-Scale tuning system solves the problem of dissonance by using a new
mathematical base. The new base incorporates the curve imposed by nature
on all moving objects, including sound waves. Current mathematics, which
is used in all prior tunings, is calculated on a two dimensional plane.
Tru-Scale tuning is based on a three dimensional mathematical mode. (This
system takes into account the natural curve of wave travel). Therefore,
intervals between waves can be calculated to move in unison with no
dissonance or overtone collision. This cannot be done with current
mathematical theory due to improper calculations of wave movement. Such
improper calculations yield harmonic dissonance, as will be discussed
below.
The overall effect of Tru-Scale tuning creates a much cleaner and stronger
sounding interval system, which in turn creates better sounding chords.
The mathematical foundation behind the Tru-Scale tuning can also be used
to enhance all forms of wave production, transmission and reception.
BRIEF DESCRIPTION OF THE DRAWINGS AND PLATES
In the accompanying drawings and holographs:
FIG. 1A is a block diagram of an electronic musical instrument according to
the present invention, FIG. 1B is a block diagram of a tone generator
according to the present invention, connected through a suitable interface
to an electronic musical instrument, and FIG. 1C is a block diagram
showing the memory layout of a general construction of one embodiment of
an electronic musical instrument to which the present invention may be
applied;
FIGS. 2A-2C are holographs of a speaker at rest (2A), the same speaker
vibrating at 185 Hz (2B), and the same speaker vibrating at 220 Hz (2C);
FIG. 3 is a holograph of the speaker at rest (Groundstate);
FIG. 4 is a holograph of the speaker vibrating with a standard "C" chord
using the Equal Temperament scale; and
FIG. 5 is a holograph of the same speaker vibrating with a "C" chord using
the Tru-Scale scale.
DETAILED DESCRIPTION OF A PREFERRED EMBODIMENT
In FIG. 1A, an electronic musical instrument according to the present
invention includes a keyboard 110, which may contain any desired number of
keys. Typically, such keyboards may enable selection of keys from as many
as four different octaves. An analog-to-digital (A/D) converter 115
converts the keyboard input to digital form for input to a central
processing unit (CPU) 120, which preferably is a microprocessor.
A memory 130 stores Tru-Scale frequency values which are output in
accordance with particular keys being struck. When single keys are struck,
the memory 130 provides a single value which is reproduced through the
frequency reproduction circuitry 150, a digital-to-analog (D/A) converter
155, and a loudspeaker 160. For reproduction of struck chords, the memory
130 provides a value for each key being struck, and the CPU controls
timing of output of sound as a single chord, though the memory 130 may
output only one value at a time.
In FIG. 1B, a programmable tone generator 300 contains a memory 130' of
suitable construction, as is well known in the art, the memory 130'
storing the Tru-Scale algorithm. The tone generator 300 is connected
through a suitable interface 200, such a MIDI interface, to an electronic
musical instrument 100', enabling the musical instrument to reproduce
scales and chords using the Tru-Scale interval system, even though the
memory of the instrument 100' may not contain the Tru-Scale algorithm. The
MIDI interface is well known in the art; thus, a detailed description of
the MIDI interface is not necessary here to a full appreciation of the
invention.
FIG. 1C shows in block form an exemplary layout of the memory 130 for an
electronic musical instrument, such as a synthesizer, according to the
present invention. Synthesizers are used for reproducing many different
types of sound (voices), and so the memory 130 may contain data
corresponding to various musical instruments, such as a piano, an organ, a
guitar, a flute, etc. When the keyboard 110 (FIG. 1A) produces a pitch
determining voltage signal and keying signal in response to depression of
a selected key or keys, the CPU 120 instructs the retrieval of
predetermined signals from the memory 130.
The memory 130 may have stored therein data for 64 voice memories, 32
performance memories plus one (A,B,C) system setup memory and two (D)
user-defined micro tunings. The Tru-Scale sequence of frequencies is
programmed into the internal memory system for retrieval as desired, thus
avoiding the need for multi-scale devices to eliminate dissonance caused
by overtones or clashing frequencies.
Tru-Scale is a scale with tones of fixed pitch, yielding perfectly tuned
chords and allowing complete freedom of modulation. These frequency data
can be used for basic MIDI settings on all other instruments calling for
reception, storing, or transmission of Tru-Scale frequencies, as alluded
to with reference to FIG. 1B above. Further, these data may be stored for
use as a dependent or independent computer source on optical or magnetic
disks, in cartridges, or in semiconductor form (RAM or ROM). Data storage
in memories is well known in the art, and details of implementation are
not necessary here to a full appreciation of the invention.
Table I on the next page shows the algorithm for the Tru-Scale frequency,
and a comparison of the internal separation between the notes of Tru-Scale
and Equal Temperament. While only a few scales are shown, the pattern for
continuing the algorithm in either direction (toward a higher or lower
frequency) may be seen readily, and suggests applicability of the
Tru-Scale system to elimination of noise, interference, etc. in any range
of frequencies. The separations reflected in contemporary and of reported
historical divisions are based upon Pi .pi.=3.1416, which provides a base
that never closes, hence the continuous fractional parts, e.g. 130.81,
261.63. In contrast, the Tru-Scale separation provides a system of
time-space relationships that allows a frequency to be used with other
frequencies, which are compatible, and thus avoids the dissonance caused
by all other interval systems.
These mathematical data are reaffirmed in the reflection of the holographic
images of FIGS. 4 and 5. All audible octave interval frequencies are
stored on the memory source using the Tru-Scale octave interval system.
The relationships viewed in the presentation of the data in Table II, on
the page after Table I, in regard to the "C" chord of Equal Temperament
indicate the unbalanced association of the fractional extension of the
interval system used. The Tru-Scale "C" chord data reflect the balance of
the system and its integration of the parts to the whole which remains
constant in single or mutual multiple relationships.
TABLE I
______________________________________
Tru-
Equal Interval Interval Scale
Temperament Between Between Frequency*
Note Frequency(Hz)
Notes(Hz) Notes(Hz)
(Hz)
______________________________________
C 130.81 150.0
C.music-sharp.
138.91 8.10 7.5 157.5
D 146.83 7.92 10 167.5
D.music-sharp.
155.82 8.99 10 177.5
E 164.81 8.99 10 187.5
F 174.61 10.20 12.5 200.0
F.music-sharp.
185.26 10.65 12.5 212.5
G 196.00 10.74 12.5 225.0
G.music-sharp.
208.00 12.00 12.5 237.5
A 220.00 12.00 12.5 250.0
A.music-sharp.
233.97 13.97 12.5 262.5
B 246.94 12.97 17.5 280.0
C 261.63 14.69 20 300.0
C.music-sharp.
277.82 16.19 15 315.0
D 293.66 15.84 20 335.0
D.music-sharp.
311.64 17.98 20 355.0
E 329.62 17.98 20 375.0
F 349.22 20.40 25 400.0
F.music-sharp.
370.52 21.30 25 425.0
G 392.00 21.48 25 450.0
G.music-sharp.
416.00 24.00 25 475.0
A 440.0 24.00 25 500.0
A.music-sharp.
467.94 27.94 25 525.0
B 493.88 25.94 35 560.0
C 523.26 29.38 40 600.0
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*Modified to fit Western music.
TABLE II
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INTERVAL RELATIONSHIPS IN A CHORD
Equal Temperament Frequencies
(Hz) Tru-Scale Frequencies (Hz)
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"C" = 261.63 329.62 392.00
"C" = 300 375 450
(chord) (chord)
67.99 62.38 75 75
130.37 150
##STR1##
##STR2##
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The historical instability of the octave, relative to a standard wave
frequency, is reflected in Table III on the next page. The mean pitch
frequency of a from 1495 to 1812 ranges from a high of 506 Hz to a low of
394 Hz. The "Standard a'" of 440 Hz was agreed upon by the International
Organization of Standardization in 1955 (but some performers today may
adopt 442 or 443 Hz as their standard a"). (Harvard Dictionary of Music,
pp. 638-639.) Thus, the use of 500 Hz for a' in the Tru-Scale interval
system is within the range of frequencies which have been considered for
tuning purposes.
Looking now at some of the results yielded by employing the Tru-Scale
interval system, FIGS. 2A2C are the result of a process known as
stroboscopic holographic interferometry, in which, by stroboscopically
illuminating the surface of a vibrating subject, fringe systems can be
formed holographically which provide information not obtainable from
time-average fringes.
TABLE III
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Instability of the Octave
Frequency
Mean pitch of of a'(Hz)
Sample pitches, 1495-1812
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506 Halberstadt organ, 1495
489 Hamburg organ, 1688
15
German organs 1495-1716
487
14
Silbermann organs 1717-50
484
7 Austrian organs ca. 1550-1700
466
48
Venetian cornettos 16th-17th c.
466
33
German Mincks, 16th-175h c.
465
464 Stormthal organ, 1723
25
cornettos of unknown provenance,
461
16th-17th c. 455 Hamburg organ, 1749
454 Amati violins, high resonance, ca. 1650
454 London tuning fork, ca. 1720
7 English organs, 1665-1708
450
32
Italian (non-Venetian) cornettos,
16th-17th c. 448
440 Paris Conservatoire fork, 1812
435 Hamburg choir tuning fork, 1761
5 French cornets, 16th-17th c.
431
427 Sauveur's standard, 1713
426 Praetorius's Cammerton, 1619
=425
Padua pitch pipe, 1780
424 Amati violins, low resonance, ca. 1650
423 "Handel's" tuning fork, 1751
13
English and American organs,
1740-1843 421
415 Dresden choir tuning fork, ca. 1754-1824
6 German organs, 1693-1762
412
92
French oboes, ca. 1670-1750
411
408 Hamburg organ, 1762
405 Deslandes-Sauveur organ, 1704
13
French organs, 1601-1789
399
394 De Caus's standard, 1612
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To describe the holographic process briefly, real-time fringes are observed
by (1) forming a hologram of the static surface, (2) replacing the
processed hologram in its original forming position, (3) setting the
surface in vibration, and (4) illuminating the surface once each vibration
period with a short pulse of light. If the pulse is short enough, the
method is equivalent to real-time holographic interferometry of static
objects. However, by altering the phase at which the light flash appears,
the vibrating surface may be compared with the static image at any phase
in its vibration cycle. As with the previous real-time method, one can
vary the vibration frequency, in this case keeping the light pulse and
surface vibration in synchronism, and so examine the full range of
frequency response of the vibrating body. This enables narrowband
resonances to be studied easily.
Real-time methods allow observation of the evolution of a fringe pattern as
vibration amplitude is increased from zero or as the vibration frequency
is brought to the resonance value. From such observations one can count
fringes, follow their movement, and detect the positions of nodes. By
displacing the surface slightly in the direction of its normal and
observing the motion of the fringes, the relative sign of the vibration
over the mode pattern can be determined. One major advantage over the
time-average system, common to nearly all the stroboscopic techniques, is
that the fringes are of equal visibility independent of the amplitude of
vibration. For this reason larger vibration amplitudes can be studied
stroboscopically than with the time-average method.
FIG. 2A shows a loudspeaker at groundstate (i.e. not moving). FIGS. 2B and
2C show the same loudspeaker reproducing a sound of 185 Hz and 220 Hz,
respectively. As can be observed from FIGS. 2B and 2C, the space between
adjacent frequencies is clearly defined and is independent of any
interference. Also, it should be noted that, because only a single
frequency is being reproduced, the loudspeaker has only a single mode of
movement, as reflected by the relative clarity with which the structure of
the loudspeaker can be seen.
FIG. 3, a holograph, is the same speaker as in FIGS. 2A-2C. The speaker has
been put into a fixed position with a ruby-red strob-laser. The speaker is
at rest (groundstate), and shows no wave patterns at all.
FIG. 4, a holographic picture taken with the laser, provides a uniform
fringe visibility of an Equal-Temperament "C" chord, including C=261.62
Hz, E=329.62 Hz, G=392.00 Hz. It can be observed that about 40% of the
lower half of the speaker is involved in the production of the wave
pattern. The closeness of the fringes and the erratic line patterns are
observed when the three note chord is triggered at a constant volume.
Having only about 40% of the speaker in motion during playing of a chord
is clear indication of dissonance.
FIG. 5 is a holographic picture taken with the laser, using the exact same
volume as observed in FIG. 4 but reproducing a Tru-Scale "C" chord,
including C=300 Hz, E=375 Hz and G=450 Hz. There is shown a uniform fringe
visibility that involves 100% of the speaker in motion, not just 40%, as
was the case with the reproduction of the Equal-Temperament "C" chord.
Having virtually 100% of the speaker in motion indicates lack of
dissonance. As can be seen from FIG. 5, this Tru-Scale "C" chord has the
same characteristics as the single frequency patterns found in FIGS. 2B
and 2C. In addition to the well spaced waves, each frequency occupies its
own space without interfering with another's. Thus, the clarity of the
tone is apparent not only auditorially but also visually, depicting
harmony in motion as captured by a laser illuminating the surface of a
vibrating speaker.
As can be seen from the foregoing, according to the present invention, a
novel interval system employed in an electronic musical instrument enables
elimination of overtone collisions in struck chords, yielding a cleaner
sound while retaining the same ease of transposition of scales as was
possible with previous fixed scale interval systems.
While the present invention has been described in detail above with
reference to a preferred embodiment, numerous variations within the spirit
of the invention will be apparent to ordinarily skilled artisans. Thus,
the scope of the invention is limited only by the appended claims which
follow immediately.
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