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BACKGROUND OF THE INVENTION
1. Field of the Invention
This invention relates to keyboard musical instruments and more
particularly to a keyboard controlled just intonation computer which
automatically corrects the larger tuning errors of the equal tempered
scale as each interval or chord is played.
2. Description of the Prior Art
The theoretical basis for all prior art systems of just intonation for
keyboard instruments is the assumption that all the harmonic resources
required for most types of music can be derived from just scales of fixed
pitch, one for each chosen tonality. As all prior art systems were scale
determined they could not be fully automatic because scale selection
requires additional mental and manual effort on the part of the player of
the instrument. There were two approaches to the problem of scale
selection.
One method was to have all the required notes playable by means of separate
keys on a complex keyboard constructed in the form of a matrix so that
harmonically related notes would lie within easily fingered rows of keys.
Greater versatility with this approach could be achieved only at the
expense of greater complexity of the keyboard. Typical of many such
keyboard designs is the one disclosed in U.S. Pat. No. 2,232,600 to Arthur
Fickensher.
The other method made it possible to simplify the keyboards or to retain
the conventional keyboard by providing tonality stops which would cause
the entire instrument to be retuned to any one of several just scales, one
for each chosen tonality. For each modulation into a new key or for each
new transposed chord, another tonality stop would have to be turned on.
Typical of the tonality stop systems are U.S. Pat. No. 2,293,499 to Sidney
T. Fisher and U.S. Pat. No. 2,525,524 to A. J. Chase.
The disadvantages of fixed scale systems will be evident from the following
description: It is well known that the just scale C D E.sub.1 F G A.sub.1
B.sub.1 C which is generated by the perfectly tuned chords F A.sub.1 C, C
E.sub.1 G and G B.sub.1 D, contains the imperfect minor chord D F A.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 F A.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 D F A.sub.1 which would also be
contained in the dominant ninth chord G B.sub.1 D F A.sub.1, renders that
chord even more dissonant than the same chord in equal temperament. A
correctly tuned chord G B.sub.1 D F.sub.2 A is available only by
temporarily turning on the tonality stop for the key of G Major, if this
scale has an additional note F.sub.2 about two commas lower than the
normal note F. Therefore, the use of at least three tonality stops would
be required to render in just intonation even the simplest music based
upon the seven notes of the diatonic scale. In order to play music of
greater harmonic complexity, in which chords are used for coloration of
the melodic line as well as for definition of tonality, too much attention
to tonality stops would be required and such music would be difficult, if
not impossible, to play upon a multidigital keyboard. A useful description
of typical present day organs may be found in a book entitled "Organ
Builder's Guide" 1976, 3rd Edition by Roy L. DeVault and published by
Devtronix Organ Products, 5872 Amapola Drive, San Jose, Calif. 95129.
SUMMARY OF THE INVENTION
The present invention and the inventor's pioneer disclosure entitled
"Musical Instrument", U.S. Pat. No. 2,422,940, differ from all prior and
subsequent systems of just intonation for keyboard instruments in that the
tone producing elements are initially tuned to the scale of equal
temperament and that keyboard controlled means are provided whereby all
the thirds and sixths, alone or combined in chords, are rendered in almost
perfect intonation automatically. Also, the minor sevenths are rendered as
harmonic sevenths when they occur in dominant seventh or dominant ninth
chords. As these automatic shifts of pitch are determined solely by the
note combinations played, this automatic just intonation system may be
referred to as an interval determined system of just intonation.
The classical art of singing a cappella is generally considered as
conducive to the achievement of just intonation. Actually, this can be
true only if the music is sung with moderate tempo and with a minimum of
vibrato. Excessive vibrato on the part of all the members of a choir has
the ffect of blurring distinctions of pitch so that the tonal spectrum is
almost continuous and in such a tonal environment it is impossible for
each member of the choir to sing with accurate intonation. However, some
choirs and instrumental ensembles do make an effort to achieve more
accurate intonation. This is especially true of quartettes because within
a more intimate tonal environment it is easier to achieve accurate
intonation by mutual pitch accomodation. The precedent for a more direct
approach to just intonation by pitch variation on an interval by interval
basis, as afforded by this invention, has therefore already been
established by the performances of the best vocal and instrumental
ensembles.
The chief object of this invention is to provide a novel system of
automatic just intonation which can be easily incorporated in existing
instruments having one or more conventional keyboards and having tone
producing elements initially tuned to the twelve tone scale of equal
temperament.
More specifically, an object of this invention is to provide a logic
circuit which receives signals from the keying voltages of the keyboards
through a diode branch circuit. The data thus gathered from simultaneously
played keys is processed in terms of their content of thirds and sixths,
whereupon outputs of the said logic circuit energize pitch shifting
devices associated with the tone producing elements of the instrument for
correcting the intonation of the notes played.
Another object is to dispense with the additional multicontact switches for
operating the logic circuit as was disclosed in the inventor's prior U.S.
patent. Instead, advantage can be taken of the direct current keying
systems used in electronic organs whereby a very small portion of the
current used for keying may also be used to operate the logic circuit.
Another object is to substitute solid state devices for the electromagnetic
relays used in the inventor's prior invention.
Another object is to provide pitch shifting means controlled by the logic
circuit for producing two degrees of pitch shift; a shift of about one
seventh of a semitone for the thirds and sixths, and a shift of about
three tenths of a semitone for the minor sevenths of dominant chords. The
inventor's prior invention provided for only one degree of pitch shift for
both classes of intervals.
Another object is to resolve anomalies arising from the novel combination
of the diatonic and the septimal systems of harmony which this invention
makes possible.
Another object, the attainment of which contributes greatly to the
attainment of the foregoing objects, is to incorporate this invention in
an electronic organ having printed circuit modules or large scale
integrated circuit modules. With very little added expense, the circuit
components for carrying out the invention could be added to the keying,
tone generating and tone modifying circuits normally used in such modules.
The advantage accruing therefrom would be that all the input and output
connections for the added circuit components of this invention would be a
built-in part of the other circuitry within the module and no additional
outside wiring would be required.
Further objects will appear from the detailed description taken in
connection with the drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a diagram drawn to a logarithmic scale showing the relative
pitches of all the notes in an octave of the equal tempered scale together
with one set of notes in the upper scale which form just major thirds with
notes in the middle scale and another set of notes forming harmonic
sevenths with notes in the middle scale.
FIG. 2 is an array composed of most of the notes of FIG. 1 expanded to form
three series of fifths such that harmonically related notes are positioned
closely together.
FIG. 3 is a schematic diagram of the complete logic circuit of this
invention.
FIG. 4 is a partial block diagram showing one form of master oscillator and
its associated frequency dividers together with one form of adjustable
pitch shifting circuit controlled by the logic circuit whereby the
frequency of the master oscillator may be altered by one comma or by two
commas.
FIG. 5 is a partial block diagram illustrating two of the twelve diode
branch circuits for interfacing the logic circuit with a plurality of
keyboards.
FIG. 6 is a block diagram illustrating the arrangement of circuit elements
in one of six identical modules, a circuit design which would minimize the
complexity of wiring between large single function units in an organ.
FIG. 7 is a lineage chart of a large variety of chords in the key of C
Major formed by successive additions of notes to a few basic triads and
showing by subscripts how each chord is intoned by the just intonation
computer.
FIGS. 8A and 8B is a split schematic of any one of the six basic plug-in
modules.
FIGS. 8C and 8D is a split schematic of the basic common circuit which
receives the six plug-in modules one of which is illustrated in FIGS. 8A
and 8B described above.
MUSIC THEORY UNDERLYING THE INVENTION
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.
In equal temperament the fourths are too wide and the fifths are too narrow
by only 1.955 cents or one fiftieth of a semitone. With such a small
tuning error they are practically indistinguishable from just fourths and
fifths. They have the great advantage of allowing continuous modulations
without the occurrence of overlapping notes as is the case with just
fourths and fifths. The equal tempered major third is too wide and its
inversion, the minor sixth, is too narrow, by 13.69 cents. Conversely, the
minor third is too narrow, and its inversion, the major sixth, is too wide
by 15.64 cents. On the average, therefore, the major thirds and the major
sixths are too wide and their inversions, the minor sixths and the minor
thirds, are too narrow by 14.66 cents, or about one seventh of a semitone.
The intonation computer of this invention automatically corrects these
mistuned intervals, singly or combined in chords, by lowering the pitch of
the top notes of major thirds or major sixths by a comma of one seventh of
a semitone or, conversely, by lowering the pitch of the bottom notes of
minor thirds or minor sixths by the same amount. For example, when B is
played together with G and/or D, B is lowered in pitch by one seventh of a
semitone to yield the combinations GB.sub.1, B.sub.1 D or GB.sub.1 D.
Another mistuned interval in equal temperament is the minor seventh in
dominant seventh and dominant ninth chords. When used in this harmonic
context it is too wide by 31.17 cents, or about three tenths of a
semitone. Accordingly, the top note of the minor seventh is automatically
lowered in pitch by three tenths of a semitone, or slightly more than two
of the above commas, when it is a part of a dominant chord. For example,
when F is played together with GB and/or BD, the pitch of F is
automatically shifted to F.sub.2 simultaneously with the shift of B to
B.sub.1, yielding the combinations GB.sub.1 F.sub.2, B.sub.1 DF.sub.2 or
GB.sub.1 DF.sub.2.
The minor seventh also occurs in two other harmonic contexts. No pitch
shift occurs in the normal minor seventh GF or its inversion, the major
tone FG. However, when the same minor seventh is played together with a
third note B.sup..music-flat., the note G shifts to G.sub.1 by virtue of
forming a just minor third with B.sup..music-flat.. The chord is therefore
rendered as G.sub.1 B.sup..music-flat. F. A slightly wider minor seventh
G.sub.1 F results which is the inversion of the minor tone FG.sub.1.
As each note in the equal tempered scale may be shifted downward in pitch
by one comma or by two commas, as described above, there will be 36
available pitches per octave consisting of three equal tempered scales.
FIG. 1 illustrates the three scales drawn to a logarithmic scale to show
the relative pitches of the notes. The normal pitches are represented by
the middle scale. When an interval or chord is played, one or more notes
will be shifted in pitch, a comma lower as in the upper scale or two
commas lower as in the lower scale depending upon the structure of the
chord. The upper and lower scales are therefore not distinct scales but
represent the required pitch deviations from notes in the middle row when
an interval or chord is rendered in just intonation. For the purpose of
comparison, the fourth octave of the harmonic series from the eighth to
the sixteenth harmonics has been inserted to show that the pitches of all
the harmonics except the eleventh and the thirteenth are closely
approximated by notes from the three scales.
In FIG. 2 notes from the three scales are arranged to form three series of
equal tempered fifths in which the upper row contains notes that are a
just major third above notes in the middle row and the lower row contains
notes that are a harmonic seventh above notes in the middle row. This
schematic arrangement clarifies the concept of close versus distant
harmonic relationships between notes of the scale. Thus, the notes
FA.sub.1 CE.sub.1 GB.sub.1 D from the just diatonic scale of C Major are a
more compact group than the notes FACEGBD from the equal tempered scale of
C Major found in the middle row. Likewise the harmonic relationships
between notes of the dominant ninth chord GB.sub.1 DF.sub.2 A are closer
than those between the notes GBDFA found in the middle row.
Among the many microtonic temperaments advocated from time to time over
many years by music theorists, the most important have been the cyclic
meantone scale of 31 equal divisions of the octave and the cyclic
pythagorean scale of 53 equal divisions of the octave. Many attempts have
been made to construct keyboard instruments for each of these temperaments
with keyboards having a corresponding number of digitals per octave. The
31 division system provides very good approximations to the major third
and the harmonic seventh but the fifths are quite noticably flat. On the
other hand, the 53 division system provides very good thirds, somewhat
poorer harmonic sevenths and extremely good fifths all at the expense of
an even more complex keyboard. The 84 division system, which is a
synthesis (31+53=84) of these two systems has the best characteristics of
both. It combines in one system the advantage of providing nearly perfect
fifths, thirds and harmonic sevenths. Because it consists of seven
distinct twelve tone scales, it is perfectly adapted for use with the
conventional keyboard. The present invention, however, uses only three of
the possible seven equal tempered scales with the further modification
that, in order to obtain perfect harmonic sevenths, one equal tempered
scale is slightly lower in pitch than it would be in the 84 division
system. A further distinction to be made is that this invention does not
utilize three distinct scales per octave as such but a scale of only
twelve initial tones per octave, each one of which may vary in pitch as
required to provide perfectly tuned intervals and chords.
A primary function of the logic circuit of this invention is to avoid a
difficulty that arises because of a difference in structure of normal
diatonic minor chords as compared with the structure of their septimal
counterparts contained in dominant chords. Thus, if the inharmonious minor
chord DFA.sub.1 contained in the just scale of C Major were perfectly
tuned it would be rendered as D.sub.1 FA.sub.1. The difference between the
chord D.sub.1 FA.sub.1 and the chord DF.sub.2 A is clearly shown in FIG.
2. The ideal frequency ratios between notes of the chord D.sub.1 FA.sub.1
are 10:12:15. When this chord is part of a dominant ninth chord the logic
circuit causes the note F to shift three tenths of a semitone lower to
F.sub.2 while the pitches of the notes D and A are unchanged so that the
full chord is rendered as GB.sub.1 DF.sub.2 A. The ideal frequency ratios
between notes of the chord DF.sub.2 A are 6:7:9. The requirements for
perfect diatonic harmony by which this chord is rendered as D.sub.1
FA.sub.1 and for perfect septimal harmony by which this chord is rendered
as DF.sub.2 A can both be met if a portion of the signal due to the
playing of the note B is routed into gates that inhibit the notes D and A
from each shifting a comma lower in pitch to D.sub.1 and A.sub.1. Thus,
the addition of the note B to the notes D, F, and A will cause the chord
D.sub.1 FA.sub.1 to change to the chord B.sub.1 DF.sub.2 A.
The various operations of the logic circuit thus far described largely in
musical terms may be summarized by the following logic equations for notes
in the tonality of C Major. Analogous equations may also be written for
chords in other tonalities. The note symbols on the left side of each
equation represent input signals to the logic circuit of FIG. 3, while
those on the right side represent output signals. Under each equation is
written an equivalent equation in which the note symbols are replaced by
the corresponding number designations of the energized input and output
busses in the logic circuit. All OR functions are understood to be
inclusive OR functions:
______________________________________
EQUATION NO. (1)
(G + D)B = B.sub.1
(7 + 2)11 = 23
EQUATION NO. (2)
(G + D)B . F = B.sub.1 . F.sub.2 . -- D.sub.1
(7 + 2)11 . 5 = 23 . 29 . -- 14
EQUATION NO. (3)
(D + A)F = D.sub.1 + A.sub.1
(2 + 9)5 = 14 + 21
EQUATION NO. (4)
(G + D)B . (D + A)F = B.sub.1 . F.sub.2 . -- D.sub.1 .
-- A.sub.1
(7 + 2)11 . (2 + 9)5 = 23 . 29 . -- 14 . -- 21
______________________________________
The foregoing equations are applicable to all note combinations contained
in the dominant ninth chord GB.sub.1 DF.sub.2 A in the key of C Major.
When the circuit of FIG. 3 is analyzed in greater detail, it will be
apparent that it has complete circular symmetry and therefore Mod 12
concepts, well known in the art, are applicable to it. Therefore, each of
the above equations represents only one of twelve equations that can be
written for each of the twelve transpositions of any given chord. If the
corrections of intonation that occur for any given chord are known, then
by analogy, the same corrections of intonation will also be known for all
of the twelve transpositions of the same chord.
There are two other well known chords which, for the purpose of this
invention, will be regarded merely as altered dominant seventh chords
although the same chords may be derived by altering other chords. One
chord is the diminished seventh chord such as G.sup..music-sharp. BDF
which is derived from the chord GBDF and the other is the augmented sixth
chord D.sup..music-flat. FGB which is derived from the chord DFGB.
Strictly speaking, the first chord consists of two diminished fifths while
the second chord consists of two augmented fourths or two tritones.
The chord G.sup..music-sharp. BDF may resolve into any one of four
different major and four different minor keys and likewise the chord
D.sup..music-flat. FGB may have a variety of resolutions. In view of this
ambiguity, and also because of the inherent dissonance of both chords, the
logic circuit has been designed to render both chords in equal
temperament, thus avoiding the following contradictory responses. For
example, the possibility exists that each note of the chord
G.sup..music-sharp. BDF will be lowered in pitch by either one comma or by
two commas. This chord contains four triads one of which would be rendered
as B.sub.1 DF.sub.2 in accordance with equation (2). By analogy, the other
three triads contained in this chord, if properly spelled, would be
rendered as A.sub.2.sup..music-flat. D.sub.1 F, A.sup..music-flat.
C.sub.2.sup..music-flat. F.sub.1, and G.sup..music-sharp..sub.1 BD.sub.2.
Thus, a note such as F might have any one of three pitches, F, F.sub.1, or
F.sub.2, and similarly the other notes of the diminished seventh chord may
each have any one of three different pitches. This confusion of responses
is resolved by the expedient of providing an inhibit function whereby the
signals F.sub.1 and F.sub.2 mutually cancel each other. As this provision
applies equally and simultaneously to all four notes, this diminished
seventh chord will be rendered in its equal tempered version as
G.sup..music-sharp. BDF.
In accordance with equation (2) the notes F, G and B contained in the
augmented sixth chord D.sup..music-flat. FGB would ordinarily be rendered
as F.sub.2 GB.sub.1. By analogy, the notes D.sup..music-flat., F and B
would be rendered as D.sup..music-flat. F.sub.1 B.sub.2. Here again,
inhibit gates simultaneously receiving the signals F.sub.1 and F.sub.2,
function to restore the original pitch F and other inhibit gates receiving
the signals B.sub.1 and B.sub.2, function to restore the original pitch B.
The above augmented sixth chord is therefore rendered in its equal
tempered version as D.sup..music-flat. FGB. Having thus set forth the
chief functions of the logic circuit largely in musical terms, its
electrical operation in a preferred embodiment will now be described.
DESCRIPTION OF THE PREFERRED EMBODIMENT
It will be understood that the present invention falls within a board class
of control systems performing three functions, namely, data gathering,
data processing, and control. A major aspect of this invention is the use
of a diode branch circuit for gathering data as to the content of major
and minor thirds or any inversion thereof in chords and the use of a data
processing unit or logic circuit for selecting which of the played notes
will be altered in pitch to render the chord more consonant. On the other
hand, the control function can be performed by a large variety of pitch
shifting devices each appropriate to the kind of keyboard instrument in
which they are used. Therefore, the scope of the invention includes many
types of keyboard instruments.
This invention is preferably embodied in an electronic organ of the type
having twelve master oscillators in the top octave which drive chains of
frequency dividers in the lower octaves. In such organs any note together
with all its octavely related notes are simultaneously shifted in pitch
merely by changing the frequency of the corresponding master oscillator.
It will be understood, however, that this invention can also be embodied
in an electronic organ with independently tuned tone generators if means
are provided whereby the frequencies of octavely related tone generators
may be simultaneously shifted to the same degree.
In the present state of the art it is possible to dispense with separate
master oscillators in a divider type organ. This may be done by means of
an integrated circuit that synthesizes the frequencies of the equal
tempered scale for the top octave from a single 4 megahertz clock. This
expedient would seem quite attactive due to reduced manufacturing costs,
ease of transposition to any absolute pitch without altering the interval
ratios and, if the tuning errors of equal temperament are not taken into
account, such an organ can never "get out of tune". Although the present
invention requires the use of twelve master oscillators of high frequency
stability, and provides a keyboard controlled logic circuit for varying
those frequencies, the ability of such an organ to yield perfectly tuned
intervals and chords is a significant advance in the art which cannot be
realized with an instrument in which the interval ratios are unalterably
fixed.
FIG. 4 shows a preferred two input pitch shifting means for one of the
master oscillators, MO1, in this case the one for the note F. The
preferred master oscillator shown, exemplary of any other type of
oscillator, is the highly stable 555 integrated circuit timer together
with an external network consisting of resistor RO1, variable resistor RO2
and capacitor CO1. The initial frequency or absolute pitch of the
oscillator is set by adjusting variable resistor RO2. The frequency can be
further varied either by changing variable resistor RO2 or by changing
capacitor CO1, but in the preferred embodiment advantage can be taken of
the fact that type 555 integrated circuit can also function as a voltage
controlled oscillator so that the frequency can more effectively be
lowered by applying a positive voltage to terminal 41. Since the logic
circuit LC1 consists preferably of CMOS integrated circuits, the logic
levels of outputs 17 and 29 thereof are either at zero voltage or the same
positive voltage as the power supply when high. The voltage level at
control voltage terminal 41 can be adjusted by means of potential divider
RO3 so that when output 17 goes high the pitch of master oscillator MO1
will change, for example, from F to F.sub.1. Likewise, the voltage level
at terminal 41 can be adjusted by means of potential divider RO4 so that
when output 29 goes high, the pitch of the master oscillator will be
lowered by two commas from F to F.sub.2. Isolation diodes DO1 and DO2 not
only insure that the potential dividers will not bias the internal trigger
level of the oscillator toward ground, but also that the setting of one
potential divider will not affect the voltage output from the other
potential divider. As was mentioned hereinbefore and which will be
described in greater detail, output terminals 17 and 29 cannot go high
simultaneously and therefore only two degrees of pitch shift can occur.
The output of master oscillator MO1 is connected to the input of frequency
divider FD1.
Direct current keying systems afford a convenient means of operating the
logic circuit without the necessity of providing additional keyboard
switches for this purpose. However, diode branch circuits will be required
for interfacing all key switches of one or more keyboards with the twelve
inputs of the logic circuit. With reference to FIG. 3, the logic circuit
has twelve input busses corresponding to the twelve notes of the chromatic
scale from C up to B, which are also labelled with the numerals 0 through
11, respectively. As shown in the partial block diagram of FIG. 5, each
input of a given nomenclature, for example input bus 0 for the note C, is
connected to all of the keyboard switches for the note C through a diode
branch circuit so that the playing of any one C digital or combination of
C digitals is equally effective to energize the single input 0 of the
logic circuit. For the sake of simplicity, only two of the required twelve
diode branch circuits are shown, one for the note C and the other for the
note E. As shown in FIG. 5, the twelve diode branch circuits encompass a
plurality of manuals and include the pedal keyboard as well. For certain
music replete with grace notes and arpeggios it may be more advantageous
to disconnect from the logic circuit the manual on which such passages are
played and allow another manual on which the accompanying harmony is
played to control the intonation. For this purpose a twelve pole switch
SW1 is provided. Another valuable feature of this invention is that a
choice may be made between just intonation or equal temperament by
switching on or off the operating current for the logic circuit with
switch SW2. The ease with which the sound of tempered versus justly
intoned intervals and chords may be thus compared would be of value of
teachers of harmony and music theory.
In the design of the logic circuit, an economy of circuit interconnections
is realized by grouping together the logic gates associated with each
major triad and its counterpart transposed by a tritone. Thus, at the left
side of FIG. 3, the logic gates required to correct the intonations of the
chord CEG and those required to correct the intonation of the chord
G.sup..music-flat. B.sup..music-flat. D.sup..music-flat. are contained
within the same integrated circuit packages. The entire circuit,
therefore, consists of six smaller circuits referred to as "basic
circuits". The fact that the logic circuit can be easily subdivided into
six such basic circuits has an application which will be described in
connection with FIG. 6.
All six basic circuits are identical in form such that one of the inputs to
NAND gates Q through Q11 are associated with the third of each major triad
and pairs of diodes connected to the other inputs of those NAND gates are
associated with the primes and fifths of each major triad. This may be
verified for each chord in the following circle of just major triads:
G.sup..music-flat. B.sub.1.sup..music-flat. D.sup..music-flat.,
D.sup..music-flat. F.sub.1 A.sup..music-flat., A.sup..music-flat. C.sub.1
E.sup..music-flat., E.sup..music-flat. G.sub.1 B.sup..music-flat.,
B.sup..music-flat. D.sub.1 F, FA.sub.1 C, CE.sub.1 G, GB.sub.1 D,
DF.sub.1.sup..music-sharp. A, AC.sub.1.sup..music-sharp. E,
EG.sub.1.sup..music-sharp. B and
BD.sub.1.sup..music-sharp.F.sup..music-sharp.. This list brings out the
fact, well known to musicians, that in most flat key signatures the
flattened notes serve as primes, fourths, fifths and ninths of each major
scale, while in most sharp key signatures the sharped notes serve as
thirds, sixths and leading tones of each major scale.
Only when both inputs of a NAND gate Q are high or logic 1 can its output
be low or logic 0. With both their inputs connected together, NAND gates R
function only as inverters. NOR gates S and T function both as inverters
and as inhibit gates. For the sake of simplicity, the required resistors
between all inputs of gates Q and ground are not shown. A detailed
description of the operation of the logic circuit will now be given in
connection with the logic equations (1) through (4).
The analysis of the logic states existing in various parts of the logic
circuit when certain note combinations are played, will be facilitated by
the use of the following notation: The logic states of the inputs of a
gate will be written in parentheses to the left of the letter designation
of the gate and the resulting logic state of the output will be written in
parentheses to the right, as for example, (0) (1)T23(0). The same notation
will be used to indicate the logic state of a bus, as for example, 2(0).
The "dont care" state will be designated by X which represents either
state 1 or state 0. Thus (0) (1)T23(0) or (1) (1)T23(0) can be written as
(X) (1)T23(0).
Equation (1) states that (G+D)B=B.sub.1. When these notes are played,
inputs 2 and/or 7 together with input 11 are all logic 1 so that the logic
levels for NAND gate Q11 are expressed by (1) (1)Q11(0). Hence one of the
inputs to NOR gate T23 is logic 0. The other input to NOR gate T23 is also
logic 0 because unused input busses 1 and 8 are low or logic 0. Therefore
the result is (0) (0)T23(1). Output bus 23 now being high or positive, the
tuning means for note B acts to lower its pitch to B.sub.1. This may be
concisely expressed as (0) (0)T23(1)=B.sub.1. A portion of the logic 0
output of NAND gate Q11 also enters one of the inputs of NOR gate S29.
Since unused input busses 1,5, and 8 are also logic 0, the result is (0)
(0)Q5(1) which, by inversion by gate R29, is (1)R29(0) so that (0)
(0)S29(1)=F.sub.2. This means that, although note F is not being played,
the pitch of the mast | | |
