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Description  |
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FIELD OF THE INVENTION
This invention relates to the tuning of musical instruments in just
intonation. More particularly, the invention relates to a just intonation
tuning system that can be applied to musical instruments in real time to
cause instruments to be dynamically retuned in just intonation, while
played in real time.
BACKGROUND OF THE INVENTION
It is generally known that the intervals of the equal tempered scale in
popular use today are slightly out of tune in relation to pure harmony.
Chords made from the intervals of this scale are disturbed by beats caused
by this inexact tuning, resulting in dissonance. In contrast, tones
derived from intervals of the just intonation scale form perfect
harmonies, when sounded together. When a cappella choral singers sing or
well trained chamber groups use unfretted instruments (violin, viola,
cello), the pure harmonies of just intonation are heard. The equally
tempered intervals were fixed in the seventeenth century to overcome
mechanical difficulties in changing keys in fixed tone instruments like
the piano, and fretted instruments like the guitar. In music dominated by
the equally tempered intervals of the piano and guitar, pure harmonies are
lost.
Just intonation intervals that create pure harmony can be defined by ratios
of whole numbers such as 1:1, 2:1, 3:2, 4:3, and 5:4. Strings divided into
these precise lengths give the same pure harmonies that singers had
discovered naturally by ear. However, the tones created by these intervals
are not entirely interchangeable when the key or chordal root of the music
changes. That is, when the frequency of the tonic or key tone changes, a
new musical scale is defined by the perfect ratios as applied to the new
key tone. If the singers modulate the key from a key tone A(1:1) up to the
key tone B(9:8 of key tone A) so as to define a new scale, some of the
tones in the original scale will be found in the new scale, but not all;
some tones of the new scale will be different. The D note played as a
Fourth (4:3) of the key tone A is not the same frequency as the D note
played as a Minor Third (6:5), of the key tone B. They are different
because in the first case D is 4/3 the frequency of A, whereas in the
second case D is 6/5 of 9/8 the frequency of A. These two values are
different by a small ratio: 81:80. Modern music makes them equal by
splitting the difference between both notes. This is only one example of
the errors of the equal tempered scale.
Staying in perfect tune while changing keys is not difficult for singers or
for players of instruments that allow any tone to be played, for example a
violin. But fixed-tone instruments like the organ, clavichord, harpsichord
and piano had to be altered or tempered in order to play in more than one
key.
In the seventeenth century, the scale of "equal temperament," was developed
fixing 12 equal intervals into an octave, thereby allowing all fixed tones
to be used in every key. In 1685 German organist and music theorist
Andreas Werckmeister, and Prussian musician Johann Neidhardt calculated
the equal intervals as the 12th roots of the powers of two (2.sup.1,
2.sup.2, 2.sup.3, 2.sup.4, 2.sup.5, 2.sup.6, 2.sup.7, 2.sup.8, 2.sup.9,
2.sup.10, 2.sup.11, 2.sup.12,). This solved the problem of easy modulation
for the pianos, but at the cost of throwing every interval out of pure
tune.
Mechanical solutions to the problem of key modulation in just intonation
were proposed by Hermann Helmholtz, Perronet Thompson, Henry Poole and
others, but were simply too cumbersome and too limited to offer complete
just intonation in all keys.
U.S. Pat. No. 3,821,460 to Motorola Inc. discloses an electronic keyboard
capable of being tuned to equal temperament and just intonation, using
programmable frequency dividers. The tuning, however, is not
instantaneous, and the instrument can not be used for playing while
allowing for modulation and chordal change in real time, but was rather
meant as a static instructional tool. Furthermore, the keyboard does not
realize true and complete just intonation scales.
U.S. Pat. No. 3,871,261 to Wells correctly pointed out that "the `equal
tempered` system has virtually gained universal acceptance . . . but does
not eliminate the beats" caused by notes "not perfectly in tune." His
invention proposes 12 frequency modifiers (12 potentiometers) for each
key, to render the pitch of each note adjustable, and a key selection
device to switch musical keys. Wells' scales are not truly just in all
cases, and the combination tones and overtones create disturbing beats.
Furthermore, there is no provision for changing chordal root within a
given key.
Electronic keyboard manufacturers began introducing various microtuning
features in 1985 using logarithmic cents as a micro tuning unit. Keyboards
and tone generators were produced with preset alternative scales including
so-called "Pure" scales in as many as 12 major and 12 minor diatonic
scales. To access one of these scales, the user has to step through many
menu choices, and therefore modulating to another key during a composition
is out of the question. Also, no provision is made for chordal root
changes.
U.S. Pat. No. 4,152,964 to Waage discloses an electronic system to
approximate just intonation by retaining "the tempered fourths and
fifths," and shifting "the pitch of certain notes to correct the larger
tuning errors of the scale." This invention was only an approximation of
just intonation.
U.S. Pat. No. 4,248,119 to Yamada is a pitch correction gate system that
attempts to detect chord structure and then alter tones from equal
temperament to just intonation as chords are being played. This approach
is impractical because the mixture of equal temperament and just
intonation is more dissonant than tempered tuning alone.
U.S. Pat. No. 4,434,696 to Conviser recognized 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."
The Conviser invention uses compound ratios to create the frequencies of
equal temperament and just intonation. Conviser uses the correct
just-intonation intervals from Ptolemy: 9/8, 5/4, 4/3, 3/2, 5/3, and 15/8,
but derives the other intervals by multiplying "by 16/15 to obtain the
flats . . . and by 25/24 to obtain the sharps." The resulting scale is not
a correct nor a complete just intonation scale. No truly just scale is
given, and there is no provision for the necessary tonal changes when
changing chordal root within a given key.
U.S. Pat. No. 4,498,363 to Shimada disclosed a "just intonation electronic
keyboard instrument". The keyboard comprised "a plurality of tonality
selection switches for selecting each key from among twenty-four just
intonation keys . . . " It noted that keyboard instruments which are tuned
according to equal temperament are unfit for use in teaching during chorus
practice. The patent describes 12 major diatonic scales, and twelve minor
diatonic scales, but not complete chromatic scales. The invention is
intended for choral practice, and there is no provision for changing the
tuning in real time nor is there any provision for chordal root changes.
U.S. Pat. No. 4,796,509 to Yamaha Corporation of Japan disclosed an
electronic tuning apparatus based on both equal temperament and just
intonation scales. This apparatus generates a scale based on a reference
signal, and displays a tone name for each frequency of the scale. The
tuner can accommodate a single just intonation scale, but does not provide
for chordal root changes as a composition is being played.
The Yamaha YMF262 FM Operator Type L3 chip was developed as a sound source
for computer musical keyboards and tone generators. It is also used on
many commercially available audio cards. This chip contains a frequency
modulation sound source which may be controlled by software. All functions
of the synthesizer are controlled by data written to its register array.
The function for sending the frequency requires that the frequency be
multiplied by 1.31072, rounded off to the next whole number, and then sent
to a 10 bit address on the chip. This rounding-off makes it impossible to
attain the simple fractions required for perfect just intonation
harmonies.
U.S. Pat. No. 4,860,624 to Dinnan attempted to solve overtone collision, or
dissonance. However, only some of the ratios given by Dinnan are correct
just intonation intervals. Others have no relationship to historically
used just scale intervals, and they create most unusual harmonies that
cannot be considered Just or Pure. The Dinnan invention makes no provision
for altering the scale when changing chordal root within a given key.
In view of the foregoing review of the prior art, and the failure of
previous proposals to solve the problem of pure intonation for fixed-tone
musical instruments, one of the objects of the present invention is to
create a just intonation system that overcomes the aforementioned
disadvantages and answers all the requirements of pure intonation
including ease of play and modulation of both key and chordal root while
playing.
The failure of the previously proposed solutions is that they are only
half-measures at best, and do not offer a comprehensive just intonation
system. To be practical for musicians a just intonation system must be
comprehensive and perfect for all chordal roots, all keys, all inversions
of chords, and in relation to all overtones and combination tones. It must
also allow dynamic play in real time with instantaneous switching of key
and root while playing the notes.
SUMMARY OF THE INVENTION
The present invention is an electronic just intonation tuning apparatus and
method that can be applied to musical instruments to create just
intonation so that the instruments can be played in real time, based on
any pitch, in all musical scales, using all musical scale intervals, in
all chordal roots, in all musical keys.
The invention is based in part on the discovery that within the same key,
when a chord changes, a new tuning of the musical scale is defined, based
on the frequency of the new chordal root, and the new tuning variables are
finite and can be identified by the selection of a key tonic and a chordal
root. A key is defined by a tonic, or keynote, which is the fundamental
note of a scale. The remaining notes of that scale are derived by the
application of appropriate ratios to the tonic. The chordal root is the
fundamental note of a chord within a given key. The present invention uses
3-dimensional (key, chordal root, and note) just intonation arrays based
on accurate just intonation intervals for all chordal roots in all keys.
The arrays may be implemented with an electronic logic circuit or by other
logic means, including a programmed computer, mechanical linkage,
hydraulics, pneumatics, or optics.
Each array defines n.sup.3 tone identifiers, (per octave), where n=number
of intervals (notes) of the scale (per octave). These are grouped in sets
of n tone identifiers for each of n roots for each of n musical keys. The
key tones of each of the n musical keys are related by a set of n ratios
of whole numbers. The chordal roots of each key are also defined by a set
(preferably the same set) of n ratios applied to each of the key tones.
The tone identifiers in turn are defined by a set (preferably the same
set) of n ratios applied to each of the chordal roots. In most, if not
all, implementations of the invention, including the preferred embodiment,
many of the tone identifiers will have the same value, greatly reducing
the total number of individual pitches that must be generated. And, for
particular embodiments, the number of tone identifiers can be further
reduced by eliminating the possibility of selecting certain keys or
certain chordal roots within the keys. Consequently, although the
theoretical number of pitches identified by tone identifiers is n.sup.3
actual embodiments may have a much smaller number.
The tone identifiers correspond to the pitches or intervals above a
reference which are representative of an individual musical tone to be
sounded when a note is selected by a musician. The tone identifiers can be
direct representations of frequency, such as 660 Hertz, an indirect
reference to a specific musical interval or tone, such as MU68, an
electronic circuit, such as a tone generator circuit which is directly
activated when the musician selects the key, the chordal root, and a note,
or any other means for generating the appropriate pitch.
In general, the invention provides a key and root selector as well as a
logic unit containing the array so as to maintain just intonation in all
roots in all keys while playing.
Means are provided for the selection of a key and a root within that
particular key before a musical composition is played or while it is being
played, and means are provided to communicate the selections to the logic
unit. If the instrument is a type that can receive a set of tone
identifiers to specify each pitch that should be sounded when each note is
selected by the musician, the set of tone identifiers corresponding to the
selected key and root are transmitted to the musical instrument to be
played. If not, the logic unit also receives note selections from the
musician and, based on the selected key, the selected root, and the
selected notes, causes the generation of appropriate pitches.
In one of its aspects, the invention is a method for adjusting the tuning
of a musical instrument including a means for receiving a selected key and
chordal root and a means for determining the just intonation tone to be
sounded upon receipt of a selected note.
In another aspect, the invention comprises an electrical circuit having one
or more inputs for receiving the selected key and the selected chordal
root within the key and having an output which specifies the just
intonation tones to be sounded. Either an entire set of tone identifiers
is communicated to a note selection receiving means which causes the
appropriate tone to be sounded when a note is selected by the musician, or
the electrical circuit also has an input for receiving selected notes and
the circuit in turn causes appropriate tones to be sounded.
In another aspect, the invention is computer software which causes a
computer to perform the method described above or to become an embodiment
of the apparatus described above.
In another aspect, the invention is a playable musical recording made by
the method described above.
In another aspect, the invention is a method for generating musical
recordings or output from musical data sequence recordings which were
originally created with unspecified tuning or equal tempered tuning (or
any tuning) by adding to the musical data sequence recording selections of
key and chordal root, allowing the recording to be played in just
intonation.
When a musician determines in advance the composition to be performed, the
musician may make a recording of the key selections and the chordal roots
selections desired by the musician. Then the musician plays the
composition while the recording of key and chordal root selections is
being played, eliminating the need for the musician to change the chordal
root specification during the performance. Consequently, in another
aspect, the invention is a recording of a sequence of selected keys and
chordal roots for performing the method described above.
In another of its aspects, the invention comprises apparatus for adjusting
the tuning of a musical instrument to play in just intonation while the
instrument is being played, comprising sounding means associated with the
musical instrument for producing musical tones, a logic unit for storing
n.sup.3 tone identifiers, where n=number of tones in one octave of a
scale, the tone identifiers being grouped in sets of n tone identifiers
for each of n chordal roots for each of n musical keys, and wherein each
tone identifier corresponds to a tone to be generated by the sounding
means of the instrument and wherein the set of tones corresponding to the
tone identifiers produce just intonation intervals, selection means
associated with the musical instrument for enabling a musician to select a
key and chordal root within a key in which tones of a composition are to
be played, a logic means associated with the musical instrument, means for
communicating the key and chordal root selected by the selection means to
the logic means, retrieving at least one tone identifier from the set of n
tone identifiers corresponding to the key and the chordal root selected by
said selection means and communicating to the sounding means said at least
one tone identifier.
In another of its aspects, the invention comprises a method of adjusting
the tuning of a musical instrument to play in just intonation while the
instrument is being played wherein selection means are associated with the
musical instrument for enabling a musician to select a key and a chordal
root, and memory means are associated with the musical instrument for
storing sets of n tone identifiers for each of n chordal roots for each of
n musical keys; and sounding means are associated with the musical
instrument for producing musical tones, comprising selecting a key and a
chordal root within a musical key, communicating said selected key and
chordal root to the said memory, retrieving from said memory at least one
tone identifier selected from the set of n tone identifiers corresponding
to the selected key and chordal root, communicating said at least one tone
identifier to said sounding means of said musical instrument whereby to
cause the sounding means to produce said at least one tone when the note
which calls for that tone is selected to be played by the musician.
The invention may be more fully appreciated by reference to the following
description of the preferred and alternative embodiments of the invention
and by reference to the drawings thereof and associated tables.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 illustrates the preferred embodiment of the invention in association
with an electronic keyboard.
FIG. 2 illustrates an alternative embodiment of the invention for
simultaneously controlling tuning in just intonation of several musical
instruments.
FIG. 3 is a flowchart of the software used in the preferred embodiment of
the invention.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
The preferred embodiment of the invention is illustrated in FIG. 1. A
standard digital electronic keyboard 10 is provided having instrument keys
12, hand wheels 14 and LCD displays 16. The keyboard 10 also includes a
MIDI OUT port 18.
A separate key and root selector unit 20 is provided. The selector unit 20
includes 12 key selectors 22, 12 root selectors 24, a numerical keypad 26,
a scale selection button 28, a pitch selection button 30, and two LCD
displays 32, 33. The selector unit has a MIDI IN port 34 and a MIDI OUT
port 36. The MIDI IN port 34 of the selector unit 20 is connected to MIDI
OUT port 18 of the keyboard by means of MIDI compatible cabling 38.
The MIDI IN port 40 of a tone generator 42 is connected to the MIDI OUT
port 36 of the selector unit 20. The tone generator 42 must be one that is
capable of being tuned. The tone generator 42 is connected to an amplifier
44 which is in turn connected to a speaker 46.
A CPU 48, a ROM chip 50 and a RAM chip 52 are provided on a circuit board
(not shown) within the housing of the selector unit 20.
The CPU 48 is provided with software to implement the invention. FIG. 5 is
a flowchart of the software of the preferred embodiment, although other
approaches might be used within the parameters of the invention.
The RAM chip 52 is used to store an array 54 of tone identifiers which are
used to adjust the tuning of the tone generator 42 as described below.
A just intonation musical scale is defined according to a set of ratios of
whole numbers which by convention and by empirical confirmation by the
inventors define just intonation scales. The preferred embodiment of the
invention uses the sets of ratios identified in Table I.
TABLE I
(Sets of Ratios Defining Scales in Just Intonation)
(a) 1:1, 16:15, 9:8, 6:5, 5:4, 4:3, 7:5, 3:2, 8:5, 5:3, 7:4, 15:8, (plus
Octaves)
(b) 1:1, 16:15, 9:8, 6:5, 5:4, 4:3, 7:5, 3:2, 8:5, 5:3, 9:5, 15:8, (plus
Octaves);
(c) 1:1, 16:15, 9:8, 6:5, 5:4, 4:3, 7:5, 3:2, 8:5, 5:3, 16:9, 15:8, (plus
Octaves);
(d) 1:1, 16:15, 9:8, 7:6, 5:4, 4:3, 7:5, 3:2, 8:5, 5:3, 7:4, 9:5, 11:6,
15:8, (plus Octaves);
(e) 1:1, 16:15, 9:8, 8:7, 7:6, 6:5, 5:4, 4:3, 7:5, 3:2, 8:5, 5:3, 7:4,
16:9, 9:5, 11:6, 15:8, (plus Octaves);
(f) 1:1, 16:15, 9:8, 6:5, 5:4, 4:3, 45:32, 3:2, 8:5, 5:3, 9:5, 15:8, (plus
Octaves);
(g) 1:1, 16:15, 9:8, 6:5, 5:4, 4:3, 45:32, 3:2, 8:5, 5:3, 16:9, 15:8, (plus
Octaves);
(h) 1:1, 9:8, 5:4, 3:2, 5:3, (plus Octaves);
(i) 1:1, 16:15, 9:8, 6:5, 5:4, 4:3, 45:32, 64:45, 3:2, 8:5, 5:3, 9:5, 15:8
(plus Octaves);
(j) 1:1, 16:15, 9:8, 6:5, 5:4, 4:3, 45:32, 64:45, 3:2, 8:5, 5:3, 16:9, 15:8
(plus Octaves).
The sets of ratios, such as those in Table I, are stored in the ROM chip
50.
A just intonation scale may be defined for any reference pitch. The
preferred embodiment of the invention uses a default pitch of A=440 Hz.
The invention allows for any calibration of pitch, for example as where a
musician wishes to sing a melody in a key that is half way between
standard A and B flat, at perhaps 455 Hz or 460 Hz, due to the
peculiarities of the song or the limitations of voice range. The reference
pitch is chosen by a musician by using the numerical keypad and the pitch
selection button of the selector unit. Any reference pitch may be chosen
so long as it is within the range of the tone generator.
The musician also selects the just intonation scale which is to be used
from the scales in Table I, using the numerical keypad 26 and the scale
selection button 28. The default selection of the preferred embodiment is
scale (c) of Table I representing a chromatic scale.
According to the invention, the just intonation array 54 is based on the
number of ratios in the set of ratios defining the just intonation scale.
In the case of scale (c) of Table I, n=12. The array will contain n.sup.3
(1728) addresses. When the musician selects a scale using the keypad, the
CPU reserves a block of RAM sufficient to contain an array of n.sup.3
addresses. Each address will contain a tone identifier.
The array is constructed by applying the set of n ratios to the reference
pitch to define n key tones. The key tones represent the tonic for each
musical key. The set of n ratios is applied to each of the n key tones to
define n chordal root tones for each key tone. This results in n.sup.2
chordal root tones. Chordal root tones will be referred to in this
specification and in the claims as "chordal roots". They represent the
tonic of any given chord. The set of n ratios is again applied to the n
chordal roots to define n tone identifiers for each of the n.sup.2 chordal
roots. The result is n.sup.3 tones. The tones are generally symbolic or
numerical representations of tones and are therefore referred to as tone
identifiers in this specification and in the claims, The calculation of
the array is accomplished by the CPU 48 which first retrieves from ROM 50
the set of ratios defining the selected scale and performing the necessary
calculations based on the selected reference pitch. The resulting array of
n.sup.3 tone identifiers is stored in the block of RAM 52 which was
reserved by the CPU 48.
The tone identifiers of the array are arranged in groups of musical keys,
chordal roots and individual tone identifiers. The tone identifiers may be
any direct or indirect representation of tones, including individual tone
generation circuits or other devices. In the preferred embodiment, this
representation is a binary representation of frequency in Hertz, to an
accuracy of at least four decimal places. The musical keys, chordal roots
and tones represented by the tone identifiers are each in just intonation
with respect to one another to define a flexible just intonation musical
scale.
Table II illustrates the array based on a reference pitch of 440 Hz and the
scale (c) of Table I.
TABLE II
__________________________________________________________________________
Array of tone identifiers for one octave
based on reference pitch of 440 Hertz and scale (c) of Table I
K = Key R = Chordal Root I = Tone Identifier
__________________________________________________________________________
K1 R1 K1 R2 K1 R3 I K1 R4 K1 R5 K1 R6
__________________________________________________________________________
440.0000
469.3333
495.0000
1 528.0000
550.0000
586.6667
469.3333
500.6222
528.0000
2 563.2000
586.6667
625.7778
495.0000
528.0000
556.8750
3 594.0000
618.7500
660.0000
528.0000
563.2000
594.0000
4 633.6000
660.0000
704.0000
550.0000
586.6667
618.7500
5 660.0000
687.5000
733.3333
586.6667
625.7778
660.0000
6 704.0000
733.3333
782.2222
616.0000
657.0667
693.0000
7 739.2000
770.0000
821.3333
660.0000
704.0000
742.5000
8 792.0000
825.0000
440.0000
704.0000
750.9333
792.0000
9 844.8000
440.0000
469.3333
733.3333
782.2222
825.0000
10 440.0000
458.3333
488.8889
782.2222
834.3704
440.0000
11 469.3333
488.8889
521.4815
825.0000
440.0000
464.0625
12 495.0000
515.6250
550.0000
__________________________________________________________________________
K1 R7 K1 R8 K1 R9 I K1 R10
K1 R11
K1 R12
__________________________________________________________________________
616.0000
660.0000
704.0000
1 733.3333
782.2222
825.0000
657.0667
704.0000
750.9333
2 782.2222
834.3704
440.0000
693.0000
742.5000
792.0000
3 825.0000
440.0000
464.0625
739.2000
792.0000
844.8000
4 440.0000
469.3333
495.0000
770.0000
825.0000
440.0000
5 458.3333
488.8889
515.6250
821.3333
440.0000
469.3333
6 488.8889
521.4815
550.0000
862.4000
462.0000
492.8000
7 513.3333
547.5556
577.5000
462.0000
495.0000
528.0000
8 550.0000
586.6667
618.7500
492.8000
528.0000
563.2000
9 586.6667
625.7778
660.0000
513.3333
550.0000
586.6667
10 611.1111
651.8519
687.5000
547.5556
586.6667
625.7778
11 651.8519
695.3086
733.3333
577.5000
618.7500
660.0000
12 687.5000
733.3333
773.4375
__________________________________________________________________________
K2 R1 K2 R2 K2 R3 I K2 R4 K2 R5 K2 R6
__________________________________________________________________________
469.3333
500.6222
528.0000
1 563.2000
586.6667
625.7778
500.6222
533.9970
563.2000
2 600.7467
625.7778
667.4963
528.0000
563.2000
594.0000
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