Field of the Invention
[0001] This invention relates to algorithms and devices for use in producing music. It is
disclosed in the context of an instrument including a keyboard, but is believed to
have utility for any other polyphonic instrument or in other applications as well.
Related Applications
[0002] This application is based upon U. S. S. N. 60/106,150 filed October 29, 1998. The
disclosure of U. S. S. N. 60/106,150 is incorporated herein by reference.
Background of the Invention
[0003] For centuries musicians and mathematicians attempted to find a way of scaling a limited
number of notes so that natural harmonics could be preserved, while melodies and harmonies
were pitched at different levels, i.e., played in different keys. Many ways of tuning
12-note scales (12 notes per octave) were tried. All produced annoying dissonances
and/or severely limited the keys (pitches) in which a piece could be played and the
harmonic intervals which could be used. 13-note scales were tried (D# and Eb were
different notes) to provide more consonant intervals. Fourteen-note scales were also
tried, and Handel even invented an instrument with a 70-note scale but could find
noone who could play it. Finally, the compromise twelve tone, equal tempered scale
was adopted. In this scale, all intervals except the octaves are dissonant, but music
played in different keys retains the same interval relationships because the scale
is a geometric progression. Even though this scale has now been in use for over two
centuries, many musicians still find the dissonances produced by the scale to be annoying.
String quartets eliminate some dissonances by tuning to each other, and find it difficult
to play with pianos, which are generally tuned in equal tempered tuning. Likewise,
the voices of barbershop quartets tune to each other, but almost always perform unaccompanied.
[0004] Several methods and apparatus are known which modify the equal tempered musical scale.
There are, for example, the methods and apparatus described in U. S. Patents: 4,152,964;
4,248,119; 5,501,130; and, 5,736,661.
Disclosure of the Invention
[0005] According an aspect of the invention, an instrument is provided which retunes itself
in response to the chord being sustained and the way that chord is voiced.
[0006] According to another aspect of the invention, an instrument is provided which retunes
itself in response to the chord being sustained and the separation of the notes in
the chord.
[0007] According to another aspect of the invention, an instrument is provided which blends
the notes of a chord the instrument is playing in view of the chord being sustained
and its voicing.
[0008] According to another aspect of the invention, an instrument is provided which blends
the notes of a chord the instrument is playing in view of the chord being sustained
and the separation of the notes in the chord.
[0009] According to another aspect of the invention, an instrument is provided which retunes
itself in view of the chord being sustained and the way talented musicians in ensembles
tune to each other.
[0010] According to another aspect of the invention, a method is provided to develop alternative
methods to retune the notes of a keyboard in view of the harmonics of contention,
that is, harmonics that are separated typically by more than about one and one-half
cents and less than about thirty-five cents apart, produced by the notes of the chord.
[0011] According to another aspect of the invention, a method is provided to produce consonant
harmonics on a keyboard with equal tempered stretch tuning.
[0012] According to another aspect of the invention, a method is provided to obtain the
consensus of experts as to the most desirable strategies for tuning different styles
of music.
[0013] According to another aspect of the invention, a method is provided to obtain the
consensus of experts as to the most desirable strategies for blending notes of a chord.
[0014] According to another aspect of the invention, a method is provided to obtain the
consensus of experts as to the most desirable strategies for tuning in view of the
kind(s) of ensemble(s) which is (are) performing (a) musical composition(s).
[0015] According to one aspect of the invention, a method of retuning a keyboard-type instrument
starts from, and returns to, equal tempered stretch tuning based on the type of chord
being sustained and the voicing of the chord.
[0016] According to another aspect of the invention, a method for generating harmonics for
stretched tuning preserves consonance of harmonics.
[0017] According to yet another aspect of the invention, a method for retuning a keyboard
type instrument is based on the chord type being played and the way the chord is voiced.
[0018] According to yet another aspect of the invention, a method is provided for determining
which notes should be tuned as a sustained chord and which notes should be treated
as passing notes.
[0019] According to yet another aspect of the invention, a method is provided for implementing
options for how sustained chords can be retuned to eliminate dissonances and generate
enhanced overtones.
[0020] According to yet another aspect of the invention, a method is provided for permitting
musicians to select tuning strategies from combinations of options.
[0021] According to yet another aspect of the invention, a method is provided for retuning
based on the chords, for example, 2-note chords, 3-note chords, 4-note chords, 5-note
chords, created by the sustained notes.
[0022] According to yet another aspect of the invention, a method is provided for retuning
based on the history of sustained notes.
[0023] According to yet another aspect of the invention, a method is provided for retuning
based on tuning options as indicated by the setting of switches.
[0024] According to yet another aspect of the invention, a method is provided for tuning
based on the length of time notes have been sustained and the interval positions they
serve.
[0025] According to yet another aspect of the invention, a method is provided for starting
from, and returning to equal tempered tuning based on the chord type being sustained,
the voicing of the chord, and choices among options that have been made by experts.
[0026] According to yet another aspect of the invention, a method is provided for blending
sustained chords so that no note stands out.
[0027] According to yet another aspect of the invention, a method is provided for retuning
instruments so that they will closely approximate the way musicians and ensembles
typically tune to each other.
[0028] According to an other aspect of the invention, a musical instrument includes a first
switch having a first position in which the instrument is capable of producing tones,
the intervals between which are equal tempered intervals of a twelve note octave.
The first switch has a second position in which the instrument is capable of producing
tones, the intervals between at least some of which are determined by identifying
at least selected ones of the notes the instrument is being commanded to produce.
The instrument also includes a processor including a map by which the identified notes
are mapped to a chord type. The processor identifies a note in that chord type and
substitutes a frequency closer to a harmonic of the identified note for the frequency
of at least one harmonic of at least one other note the instrument is being commanded
to produce.
[0029] Illustratively according to this aspect of the invention, the instrument includes
a second switch. The processor includes at least two different maps. The second switch
has a position for each map, permitting selection of one of the at least two different
maps by which the instrument maps the identified intervals to a chord type.
[0030] Further illustratively according to this aspect of the invention, the instrument
includes a third switch. The processor includes at least two different chord type
decision engines. The third switch has a position for each chord type decision engine,
permitting selection of one of the at least two different decision engines by which
the instrument identifies a note of the chord type.
[0031] Additionally illustratively according to this aspect of the invention, the processor
is a processor for substituting a frequency within a predetermined range of a harmonic
of the identified note for the frequency of at least one harmonic of at least one
other note the instrument is being commanded to produce.
[0032] Illustratively according to this aspect of the invention, the processor is a processor
for substituting frequencies closer to at least two harmonics of the identified note
for the frequencies of harmonics of at least two other notes the instrument is being
commanded to produce.
[0033] Further illustratively according to this aspect of the invention, the processor is
a processor for substituting frequencies closer to at least two harmonics of the identified
note for the frequencies of at least two harmonics of at least one other note the
instrument is being commanded to produce.
[0034] Additionally illustratively according to this aspect of the invention, the processor
is a processor for permitting mapping of the identified notes to at least one of:
a major triad; a minor triad; a triad suspended by a second; a triad suspended by
a fourth; a major sixth; a minor sixth; a major seventh; a minor major seventh; a
dominant seventh; a minor dominant seventh; a half diminished chord; a full diminished
chord; and, an augmented chord.
[0035] Illustratively according to this aspect of the invention, the processor is a processor
for resolving contention among competing ones of: a major triad; a minor triad; a
triad suspended by a second; a triad suspended by a fourth; a major sixth; a minor
sixth; a major seventh; a minor major seventh; a dominant seventh; a minor dominant
seventh; a half diminished chord; a full diminished chord; and, an augmented chord,
and mapping according to the contention resolution.
[0036] Further illustratively according to this aspect of the invention, the instrument
includes a second switch. The processor includes at least two different chord type
contention resolutions. The second switch has a position for each chord type contention
resolution, permitting selection of one of the at least two different chord type contention
resolutions by which the instrument identifies the chord type.
[0037] Additionally illustratively according to this aspect of the invention, the the processor
is a processor for permitting mapping of the identified notes to an inversion of the
chord.
[0038] Illustratively according to this aspect of the invention, the instrument includes
a second switch. The processor includes a substitution decision engine. The second
switch has a position in which the substitution decision engine is disabled and a
position in which the substitution decision engine is enabled.
[0039] Further illustratively according to this aspect of the invention, the substitution
decision engine has as an input at least one of: how long the instrument is commanded
to sustain one of the twelve notes; the history of accumulated time of uninterrupted
sustainment of a sustained note; the position a sustained note occupies in a chord;
the position a sustained note occupied in a chord on at least one prior occasion;
and, how much the note's current assigned frequency varies from equal-tempered tuning.
[0040] Additionally illustratively according to this aspect of the invention, the the processor
includes a lookup table by which the identified notes are mapped to a chord type,
by which a note of the chord type is identified, and/or by which a frequency closer
to a harmonic of the identified note is substituted for the frequency of at least
one harmonic of at least one other note the instrument is being commanded to produce.
[0041] Illustratively according to this aspect of the invention, the instrument includes
a keyboard having multiple keys for producing tones which are octaves of the at least
one harmonic of the at least one other note the instrument is being commanded to produce.
The processor substitutes octaves of the frequency closer to a harmonic of the identified
note for the octaves of the frequency of at least one harmonic of the at least one
other note the instrument is being commanded to produce.
[0042] Further illustratively according to this aspect of the invention, the processor includes
a substitution decision engine having as an input how long the instrument is commanded
to sustain one of the twelve notes. The processor reassigns the keys to producing
tones which are octaves of the at least one harmonic of the at least one other note
the instrument is being commanded to produce when the instrument is no longer commanded
to sustain one of the twelve notes.
[0043] Additionally illustratively according to this aspect of the invention, the the processor
is a processor for adjusting the amplitude of the frequency closer to a harmonic of
the identified note which is substituted for the frequency of at least one harmonic
of at least one other note the instrument is being commanded to produce.
[0044] Illustratively according to this aspect of the invention, the processor is a processor
for adjusting the amplitudes of more than one of the tones the instrument produces
in response to the commands to produce.
[0045] Further illustratively according to this aspect of the invention, the instrument
includes a second switch. The processor includes at least two different amplitude
decision engines. The second switch has a position for each amplitude decision engine,
permitting selection of one of the at least two different amplitude engines by which
the instrument adjusts the amplitudes of the tones.
[0046] According to another aspect of the invention, a musical instrument includes a first
switch having a first position in which the instrument is capable of producing tones,
the amplitudes of which are determined by identifying at least selected ones of the
notes the instrument is being commanded to produce. The instrument further includes
a processor including a map by which the identified notes are mapped to a chord type.
The processor identifies a note in that chord type, and adjusts the amplitude of at
least one of the tones the instrument produces in response to the commands to produce
in response to the identified note.
[0047] Illustratively according to this aspect of the invention, the first switch has a
second position in which the amplitude of the at least one tone the instrument produces
in response to the commands to produce is not adjusted.
[0048] Further illustratively according to this aspect of the invention, the processor is
a processor for adjusting the amplitudes of more than one of the tones the instrument
produces in response to the commands to produce in response to the identified note
when the first switch is in the first position.
[0049] According to another aspect of the invention, a method of operating a musical instrument
capable of producing tones, the intervals between which are equal tempered intervals
of a twelve note octave, and tones, the intervals between at least some of which are
determined by identifying at least selected ones of the notes the instrument is being
commanded to produce, includes identifying the at least selected ones of the notes
the instrument is being commanded to produce, providing a map for mapping the identified
notes to a chord type, identifying a note in that chord type, and substituting a frequency
closer to a harmonic of the identified note for the frequency of at least one harmonic
of at least one other note the instrument is being commanded to produce.
[0050] Illustratively according to this aspect of the invention, the method further includes
providing at least two different maps, and selecting one of the at least two different
maps by which the identified intervals are mapped to a chord type.
[0051] Further illustratively according to this aspect of the invention, the method includes
providing at least two different chord type decision engines, and selecting one of
the at least two different decision engines by which the instrument identifies a note
of the chord type.
[0052] Illustratively according to this aspect of the invention, substituting a frequency
closer to a harmonic of the identified note for the frequency of at least one harmonic
of at least one other note the instrument is being commanded to produce includes substituting
a frequency within a predetermined range of a harmonic of the identified note for
the frequency of at least one harmonic of at least one other note the instrument is
being commanded to produce.
[0053] Further illustratively according to this aspect of the invention, the method includes
substituting frequencies closer to at least two harmonics of the identified note for
the frequencies of harmonics of at least two other notes the instrument is being commanded
to produce.
[0054] Additionally illustratively according to this aspect of the invention, the method
includes substituting frequencies closer to at least two harmonics of the identified
note for the frequencies of at least two harmonics of at least one other note the
instrument is being commanded to produce.
[0055] Illustratively according to this aspect of the invention, providing a map for mapping
the identified notes to a chord type includes providing a map for mapping the identified
notes to at least one of a major triad, a minor triad, a triad suspended by a second,
a triad suspended by a fourth, a major sixth, a minor sixth, a major seventh, a minor
major seventh, a dominant seventh, a minor dominant seventh, a half diminished chord,
a full diminished chord, and an augmented chord.
[0056] Further illustratively according to this aspect of the invention, the method includes
resolving contention among competing ones of a major triad, a minor triad, a triad
suspended by a second, a triad suspended by a fourth, a major sixth, a minor sixth,
a major seventh, a minor major seventh, a dominant seventh, a minor dominant seventh,
a half diminished chord, a full diminished chord, and an augmented chord, and mapping
according to the contention resolution.
[0057] Additionally illustratively according to this aspect of the invention, the method
includes providing at least two different chord type contention resolutions, and permitting
selection of one of the at least two different chord type contention resolutions by
which the instrument identifies the chord type.
[0058] Illustratively according to this aspect of the invention, providing a map for mapping
the identified notes to a chord type includes providing a map for mapping the identified
notes to an inversion of the chord.
[0059] Further illustratively according to this aspect of the invention, the method includes
providing a substitution decision engine, and selectively enabling the substitution
decision engine.
[0060] Additionally illustratively according to this aspect of the invention, the method
includes providing as an input at least one of: how long the instrument is commanded
to sustain one of the twelve notes; the history of accumulated time of uninterrupted
sustainment of a sustained note; the position a sustained note occupies in a chord;
the position a sustained note occupied in a chord on at least one prior occasion;
and how much the note's current assigned frequency varies from equal-tempered tuning.
[0061] Illustratively according to this aspect of the invention, the method includes providing
a lookup table by which the identified notes are mapped to a chord type, by which
a note of the chord type is identified, and/or by which a frequency closer to a harmonic
of the identified note is substituted for the frequency of at least one harmonic of
at least one other note the instrument is being commanded to produce.
[0062] Illustratively according to this aspect of the invention, the instrument includes
a keyboard having multiple keys for producing tones which are octaves of the at least
one harmonic of the at least one other note the instrument is being commanded to produce.
The method includes substituting octaves of the frequency closer to a harmonic of
the identified note for the octaves of the frequency of at least one harmonic of the
at least one other note the instrument is being commanded to produce.
[0063] Further illustratively according to this aspect of the invention, the method includes
providing a substitution decision engine having as an input how long the instrument
is commanded to sustain one of the twelve notes, and reassigning the keys to producing
tones which are octaves of the at least one harmonic of the at least one other note
the instrument is being commanded to produce when the instrument is no longer commanded
to sustain one of the twelve notes.
[0064] Additionally illustratively according to this aspect of the invention, the method
includes adjusting the amplitude of the frequency closer to a harmonic of the identified
note which is substituted for the frequency of at least one harmonic of at least one
other note the instrument is being commanded to produce.
[0065] Illustratively according to this aspect of the invention, the method includes providing
at least two different amplitude decision engines, and selecting one of the at least
two different amplitude engines by which the instrument adjusts the amplitude of the
frequency.
[0066] Further illustratively according to this aspect of the invention, the method includes
adjusting the amplitudes of more than one of the tones the instrument produces in
response to the commands to produce.
[0067] According to another aspect of the invention, a method of operating a musical instrument
capable of producing tones, the amplitudes of which are determined by identifying
at least selected ones of the notes the instrument is being commanded to produce,
includes providing a map by which the identified notes are mapped to a chord type,
identifying a note in that chord type, and adjusting the amplitude of at least one
of the tones the instrument produces in response to the commands to produce in response
to the identified note.
[0068] Illustratively according to this aspect of the invention, the method includes selectively
maintaining unadjusted the amplitude of the at least one tone the instrument produces
in response to the commands to produce.
[0069] Further illustratively according to this aspect of the invention, the method includes
adjusting the amplitudes of more than one of the tones the instrument produces in
response to the commands to produce in response to the identified note when the first
switch is in the first position.
[0070] According to another aspect of the invention, notes being played on a keyboard are
classified into one of two categories: members of a sustained chord; or, passing notes.
[0071] A keyboard which incorporates the methods of this invention when used to accompany,
or be a member of, an ensemble of tunable instruments, for example, bowed instruments
such as violins and cellos, brass instruments, reed instruments, and human voices,
will reduce clashes/inconsistencies between the harmonies the keyboard produces and
those produced by the musicians who naturally tune to each other to reduce some of
the most undesirable dissonances, generate brilliant overtones, and produce harmonies
consistent with those produced by ensembles. When such an instrument is used to perform
solos, it will produce music which is more pleasing because certain undesirable beat
notes will be eliminated and the harmonies produced will be like those typically found
by discriminating musicians to be more pleasing. Such an instrument uses an equal
tempered scale as an underlying basis, as a point of departure and as a point of return.
Brief Description of the Drawings
[0072] The invention may best be understood by referring to the following detailed description
and accompanying drawings which illustrate various aspects of the invention. In the
drawings:
Fig. 1 illustrates a flowchart of an algorithm to identify, tune and blend sustained
chords;
Fig. 2 is a chord spiral illustrating a method and algorithm by which the type of
chord being produced, the positions occupied by the notes of the chord, and the way
the chord is voiced can be determined; and
Fig. 3 illustrates a set of loudness contours useful in understanding an aspect of
the invention.
Detailed Descriptions of Illustrative Embodiments
[0073] "Voicing" is the term sometimes used in this description to indicate the order, lowest
to highest, of the interval positions in a chord, and their spread, for example, their
separation by skipping octaves. An asterisk (*) is generally used to indicate a skipped
octave. A "cent" is generally used to describe 1/1200 of an octave or 1/100 of a semitone
or (2 × S)
1/1200 . The symbol "¢" is often used as an abbreviation for this. "Maj" is the term sometimes
used in this description to indicate a major triad. "Mi" is the term sometimes used
in this description to indicate a minor triad. "Dim" is the term sometimes used in
this description to indicate a diminished triad. "Dim 7" is the term sometimes used
in this description to indicate a full diminished 7
th. "½ Dim" is the term sometimes used in this description to indicate a half diminished
7
th. "Dom 7" is the term sometimes used in this description to indicate a dominant 7
th. "Ma 6" is the term sometimes used in this description to indicate a major 6
th. "Mi 6" is the term sometimes used in this description to indicate a minor 6
th. "Aug" is the term sometimes used in this description to indicate an augmented chord.
"dom 7 + 9" is the term sometimes used in this description to indicate a dominant
7th with added 9
th. "9" is the term sometimes used in this description to indicate a 9
th chord. The person of ordinary skill in the art will immediately appreciate that other
chords are possible, and that there are common, immediately recognizable symbols which
designate many of those. Wherever any such chord is mentioned herein, I have endeavored
to use a common description of it.
[0074] Where the role played by a note in a chord is that of the root or its octaves, the
note is sometimes designated in this description with a Roman numeral "I." Where the
role played by a note in a chord is that of second or its octaves (including the ninth),
the note is sometimes designated in this description with a Roman numeral "II." Where
the role played by a note in a chord is that of minor third or its octaves, the note
is sometimes designated in this description with a Roman numeral "IIIb." Where the
role played by a note in a chord is that of major third or its octaves, the note is
sometimes designated in this description with a Roman numeral "III." Where the role
played by a note in a chord is that of fourth or its octaves (including the eleventh),
the note is sometimes designated in this description with a Roman numeral "IV." Where
the role played by a note in a chord is that of the fifth or its octaves, the note
is sometimes designated in this description with a Roman numeral "V." Where the role
played by a note in a chord is that of augmented fifth or its octaves, the note is
sometimes designated in this description with a Roman numeral "V +." Where the role
played by a note in a chord is that of sixth or its octaves, including the thirteenth,
the note is sometimes designated in this description with a Roman numeral "VI." Where
the role played by a note in a chord is that of flatted, or dominant, seventh, or
its octaves, the note is sometimes designated in this description with a Roman numeral
"VIIb." Where the role played by a note in a chord is that of major seventh, or its
octaves, the note is sometimes designated in this description with a Roman numeral
"VII."
[0075] An instrument constructed and operated according to the invention starts from an
equal tempered scale and retunes the whole keyboard virtually in real time based on
the type of chord which is being played and the way the chord is voiced. It returns
to equal tempered tuning when the particular chord to which it has tuned itself is
no longer being sustained.
[0076] The keyboard is initially tuned to equal tempered stretch tuning. Starting from a
base frequency such as A
4 = 440 Hz, every semitone in the scale is set equal to its predecessor multiplied
by (2 x S)
1/12, where S is the stretch constant, typically set so that 1 ≤ S ≤ 1.003. Whenever a
chord is sustained for a threshold amount of time, then notes in the sustained chord
are retuned together with all like notes in the entire keyboard. The threshold value
depends on the history of sustained notes. The longer a note or chord has been sustained,
then the longer a new note added to the chord must be sustained before it is considered
to be more than a passing note. Passing notes do not affect the retuning of the keyboard.
Sustained two-note, 3-note, 4-note and 5-note chords are retuned. Retuned sustained
chords will always contain one note (typically the root) which is in equal tempered
tuning.
[0077] The user can choose from among a number of optional tuning strategies, each developed
to closely match tunings actually created by different kinds of ensembles for different
kinds of music.
[0078] A number of systems/methods have been devised for retuning an equal tempered scale
during a performance. But these systems have produced harmonics based on structured
systems such as just tuning. Instruments according to the present invention retune
to closely approximate the way musicians and ensembles tend to tune to each other
to eliminate undesirable dissonances and create brilliant overtones, while keeping
harmonic relationships consistent with their interpretation of the music and the consistency
of the tuning with the type of music being played.
[0079] The keyboard is retuned, that is, the whole scale is reconstituted, almost instantaneously,
whenever two or more notes are sounded together for an amount of time, for example,
1/5 of a second. For example, if middle G and the D above it are sounded together
and sustained for the specified amount of time, then in order to eliminate the dissonances
that exist in the equal tempered scale between G and D, either all Gs in the keyboard
will be flatted or all Ds will be sharped, and the whole spectrum of harmonics associated
with those tones will also be sharped or flatted proportionally.
[0080] The 3
rd harmonic of G
3 is 588.00 cycles per second. The 2
nd harmonic of D
4 is 587.34 cycles per second. When G and D are sounded together, these two harmonics
produce a beat note of .66 cycles per second. A slight retuning can make these two
harmonics coincide exactly, eliminating the beat note and reinforcing the harmonics.
One tuning adjustment often causes other harmonics (not simply octaves apart) to coincide
and reinforce. In the case cited above, the 9th harmonic of the G, which is an A,
and the 6th harmonic of the D coincide as the result of retuning to make the 3rd harmonic
of G, which is a D, coincide with the 2nd harmonic of D. The scale can be retuned
for this interval by sharping all Ds and all the harmonics generated by those notes
by the ratio 588 ÷ 587.34.
[0081] Table I illustrates the equal tempered frequencies of the fundamentals of the notes
in a G dom 7 chord, together with harmonics, and indicates harmonics which can be
made to coincide by retuning the other notes of the chord. The frequencies of each
note are shown for three octaves, so that combinations of different rows can represent
different voicings of the chord. Some frequencies which could be retuned to eliminate
dissonances are underlined. For example, the 11
th harmonic of the lowest octave of B and the 7
th harmonic of the middle octave of G differ by only 17¢.
[0082] Fig. 1 illustrates a flowchart of an algorithm to identify, tune and blend sustained
chords. In decision block 10, the algorithm determines which keyboard keys are being
sustained, for example, by being depressed and held, or by (a) sustaining pedal(s),
or by rapid repetition. The specific notes struck, the time they were struck, and
the time they were released, that is, no longer sustained, are computed and recorded.
This information is sent to decision blocks 12, 13 and 14.
[0083] In decision blocks 12, 13 and 14 an algorithm accumulates future pitch-holding priority
points as time of uninterrupted sustainment of a sustained note increases, and as
the percentage of that time that the sustained note was the I or V of a chord. The
priority points may be assigned, for example, as follows. Each note accumulates the
number of milliseconds since its uninterrupted current sustaining period began. Also
recorded are the milliseconds it accumulated while the I of a chord, the milliseconds
it accumulated while the V of a chord, and the position it last occupied in the chord.
Every pair of sustained notes and every triplet of sustained notes, and every quadruplet
of sustained notes constitute a sustained chord. Each sustained chord accumulates
sustained milliseconds and the milliseconds sustained when one of its members was
the I of the chord, or the V of the chord and the milliseconds when both members of
the chord were the I or the V. As two or more notes accumulate pitch-holding points,
the chords they form accumulate pitch-holding points which can build to the point
that a variety of short-duration changes can pass by or through these 2-note chords
without affecting their pitch by more than a threshold value.
[0084] When a new chord is formed and a previously sustained note is part of that chord,
its accumulated pitch holding points are a factor in determining whether its pitch
will be held, and thus whether other notes will be tuned to it. Other factors which
will influence whether its pitch will be held through a new chord of which it is a
part include the role it plays in the new chord, how much its current assigned pitch
varies from equal-tempered tuning, and its voicing position in the new chord, for
example, whether it is the lowest note, the next next lowest note, and so on to the
highest note. The overall effect will be that while a chord is being sustained, one
note in the chord, for example, the I note, will always be within a desired number,
T, of cents from equal tempered tuning, where T may be set equal to, for example,
two cents.
[0085] Tuning and blending are different functions concerned with different domains. The
tuning process involves retuning an entire keyboard to the frequencies of retuned
notes in sustained chords. The blending function is concerned with the volumes of
the individual notes sounding in a chord. The blending function typically will operate
only when activated, for example, by a pedal which returns to the "OFF" position when
it is not depressed. Given a sequence of notes, both the tuning and blending functions
require that the chord type they constitute, the voicing of the chord, and the role
each note plays in the chord all be determined. This is accomplished by using an algorithm
and modulo-12 arithmetic which are illustrated in Fig. 2, the chord spiral and the
methods disclosed herein.
[0086] As previously noted, a method according to the invention starts from, and returns
to, equal tempered tuning with natural sharping which means that the frequency of
each semitone is equal to (2S)
1/12 times its predecessor semitone, where S is a "stretch," or sharping, constant close
to unity, typically set between 1 and 1.003, for example, 1.002. Such a stretch constant
is used, for example, to progressively sharp the tones in the scale as frequency increases,
to counteract the tendency of tones to sound progressively flatter as frequency increases.
When sustained chords are encountered, the notes of that chord and all like notes
in the entire keyboard are retuned to make the shared harmonics coincide. For example,
if the chord is a 2-note open fifth, then the frequency of the I is held at its original
equal tempered tuning, while the frequency of V and all its octaves on the keyboard
are retuned so that its 2
nd harmonic coincides precisely with the 3
rd harmonic of I. When the chord is no longer sustained, the note that had been V and
all its octaves on the keyboard return to equal tempered stretch tuning.
[0087] These algorithms, software, firmware and other devices implementing them, can generate
notes where the harmonics are in the relationship f
n = (2 × S)
log2n, where n is a positive integer 1, 2, 3... T, and T is a threshold that depends on
the instruments which can be simulated by the keyboard, but is generally set ≤ 17.
This method of generating harmonics permits the user to select a value of S which
will determine to what extent higher harmonics are sharper than lower ones.
[0088] For any value of S ≥ 1, the function produces harmonics within a given note which
are consonant in the same way harmonics are consonant with the function, which is
often assumed, f
n = f
1 x n where f
n is the n
th harmonic of a given note, n is a positive integer, and f
1 is the fundamental frequency of the note. In other words, using the formula f
n = f
1 x (2 x S)
log2n the harmonics of a given note reinforce and do not produce annoying sounds because
f
n / f
m = f
2n / f
2m = f
3n / f
3m = ... f
kn / f
km where f
n and f
m are the n
th and m
th harmonic and k is a positive integer that takes on the values 1, 2, 3, 4.... Equal
tempered tuning, when S ≥ 1, is such that the frequency of every semitone is equal
to its predecessor multiplied by (2 x S)
1/12.
[0089] To tune and/or blend a sustained chord, a method and apparatus according to the invention
must identify the kind of a chord and the interval position each of the notes in the
chord occupies. The chord spiral illustrated in Fig. 2 is intended to help clarify,
simplify and illuminate an algorithm which will determine sustained chord types and
the interval position occupied by each note in the chord. The chord spiral illustrates
the relationships among notes along a scale of semitones and their relationships in
an octave. The chord type and the interval position each note occupies in the chord
are deduced from these relationships. The first position in the chord spiral, 1, represents
the lowest note in a chord. On a chord spiral, relative ascending semitone positions
are depicted on a spiral that successively passes through rays, indicated by curved
brackets, {}, each of which represents the notes which are octaves above the semitone
represented by the first intersection of the spiral with that ray. For example, ray
{4} in Fig. 2 contains intersection positions for notes which are octaves above the
note which is 3 semitones (a minor third) above the lowest note. The semitone positions
along the spiral relative to the lowest note (position 1) are tallied with the appropriate
note in the chord. Every ray that contains one or more specific note tallies is itself
tallied. In the example illustrated in Fig. 2, tallied rays are: {1}, {4}, {6}, {10}.
These rays correspond to semitone positions 1, 10, 16, 18. Semitone differences between
tallied rays are then computed going around the spiral in a clockwise direction. The
differences, or step lengths, in semitones, going around the rays clockwise starting
from ray {1} are: 3 , 2 , 4 , 3, a sequence, or signature, which indicates a particular
order of the interval positions of a dom 7 chord. The lowest note is the V, the next
higher is the III, followed by the VIIb, and finally, the I. The voicing of the chord
as indicated by the positions tallied on the chord spiral illustrated in Fig. 2 is
V, III, VIIb, I, with no skipped octaves illustrated. The absence of skipped octaves
is indicated by the positions tallied on the chord spiral itself.
[0090] The sequence of intervals and the voicing information obtained from the chord spiral
are used to determine the chord type and the interval each note occupies in the chord.
Tables II and III below indicate how the same set of notes, voiced in different ways,
can be interpreted as different chord types, and how the notes themselves can be interpreted
to occupy different positions in a chord when they are voiced in different ways. One
voicing, illustrated in Table II, implies a mi 6 chord. The other voicing, illustrated
in Table III, implies a ½ dim chord.
TABLE II
ONE VOICING OF F, Ab, C and D WITH F BEING THE LOWEST NOTE |
Notes and Voicing |
F |
* |
Ab |
C |
D |
(F) |
Spiral Numbers |
1 |
|
16 |
20 |
22 |
(25) |
Ray Numbers |
{1} |
|
{4} |
{8} |
{10} |
|
Interval Sequence |
|
3 |
4 |
|
2 |
3 |
Implied Chord Type |
|
|
Minor 6th |
Implied Interval Position of Each Note |
I |
|
IIIb |
|
V |
VI |
Voicing: I, * IIIb, V, VI
* indicates skipped octave. ( ) indicates same note as 1st column. |
[0091] The signature of a chord type is the sequence of intervals, or differences, going
around the chord spiral in a clockwise direction with the position 1 representing
the lowest note. For example, the signature of a ma 6 chord with voicing V, I, III,
VI (V being the lowest note) is 2, 3, 4, 3. The signature of a maj with voicing III,
V, I (the lowest note being III) is 3, 5, 4. A chord table which illustrates the interval
sequences, or signatures, for many types of chords is Table III.
[0092] The invention contemplates a keyboard which tunes itself the way musicians tune to
each other, yet keeps equal tempered tuning as a point of departure and return. When
musicians tune to each other, they take advantage of the tendency of harmonics which
nearly coincide to lock together in sympathetic vibration. Therefore the tuning method
herein employed searches for harmonics that contain threshold amounts of energy that
almost coincide, thus providing an option to tune the notes to make those harmonics
coincide exactly. Often there are choices. It is sometimes possible to flat a given
note to make one of its energetic harmonics coincide with an energetic harmonic of
another note in the chord, and it is also possible to sharp the given note to make
one of its other energetic harmonics coincide with a different energetic harmonic
of that other note or yet another note in the chord. Illustratively, the keyboard
deviates only a tolerable degree from the expected harmonic ratios that arise from
equal tempered, or other traditional tuning algorithms. To eliminate a beat note would
otherwise sometimes require such a great deviation from traditional harmony that the
dissonances will be preferred over the retuning that would eliminate them. Since the
energy contained in higher harmonics is generally less than the energy of lower ones,
dissonances produced when higher harmonics do not coincide, yet tuning to eliminate
dissonances caused by lower harmonics may require a greater degree of sharping or
flatting. Thus conflicting objectives must be resolved.
[0093] When a sustained chord is detected, the chord type being sustained and its voicing
are determined, for example, maj, dom 7, mi, ½ dim, and so on. An algorithm then determines
which note, for example, the I note, in the chord is to be held at equal tempered
tuning. All other notes are tuned with respect to that note. Any time any note(s)
in the keyboard is (are) sharped or flatted, all of that (those) note's(s') octaves
across the entire keyboard are sharped or flatted proportionally. The way the notes
in sustained chords are retuned, that is, to vary from equal tempered tuning, is determined
from, for example, a lookup table which classifies chords as to type and voicing.
When a chord is no longer sustained, all notes in the entire keyboard return to their
equal tempered relationships. When the type of chord being sustained changes, all
notes are returned to equal tempered tuning, and then retuned to the next identified
chord.
[0094] Different voicings of the same chord offer different tuning options and enhance those
different tuning options in different ways. The high amplitude harmonics which are
close in pitch change as the voicing of a chord changes. For example, if the I is
above the III of the chord, then there are multiple options for tuning the III. For
example, the III can be tuned 13.7¢ flat, so that its 8
th harmonic coincides with the 5
th harmonic of the I. Another alternative is to tune the III 17.5¢ sharp so that the
11
th harmonic of the III coincides with the 7
th harmonic of the I. If the I is below the III, the option to sharp the III 17.5¢ is
not as good, since the 11
th harmonic of the III would have to coincide with the 14
th harmonic of the I. The 14
th harmonic naturally is considerably lower in amplitude than the 7
th.
[0095] The I - III and the I - VIIb are both intervals which present a number of tuning
options. Voicing affects the desirability of different tuning options. For example,
a dom 7 chord voiced V, III, VIIb, I places the 7
th harmonic of I close to the 11
th harmonic of III and produces a dissonance of moderate energy. If the III is sharped
17.5 cents, then its 7
th harmonic and the 11
th harmonic of I will coincide. If the VIIb is flatted 31.16¢ at the same time that
III is sharped 17¢, then the 2
nd harmonic of VIIb, the 7
th harmonic of I, and the 11
th harmonic of III will all coincide. For some styles of music this tuning may be more
desirable than so-called "just" tuning, wherein III is flatted 13.7 ¢. In other voicings
such as I, *, III, V, VIIb where the * indicates a skipped octave, the option of flatting
III by 13.7¢ may be preferred because with this voicing the sharping option aligns
the 14
th (not the 7
th) harmonic of I with the 11
th harmonic of III, thus producing a less energetic overtone. Table VI illustrates some
options for tuning the maj III interval when it is voiced I, III, when it is voiced
III, I, and when it is voiced I * III (skipped octave). Table VII illustrates some
options for tuning the I - VIIb interval when it is voiced: I , VIIb; VIIb, I; and
I * VIIb.
Table VI.
Major III Interval Tuning-Voicing Options and Consequences |
Interval Pair and Voicing |
Tuning Option |
Harmonics Aligned |
Tendency to Lock and Prominence of Overtones |
Note Augmented |
Possible Consequences |
I, III |
Flat III by 13.7¢ |
5th of I w/4th of III |
Very High |
V |
May sound |
III, I |
" " " |
5th " " 8th " " |
High |
V |
slightly |
I * III |
" " " |
5th n " 2nd " " |
Very High |
V |
minor |
|
I, III |
Sharp III by 17.5¢ |
14th of I w/11th of III |
Very Low |
VIIb |
May sound |
III, I |
" " " |
7th " " 11th of III |
Medium |
" |
brightly |
I * III |
" " " |
28th " " 11th " |
None |
" |
major |
|
I , III |
Sharp III by 34.3 ¢ |
9th of I w/7th of III |
Low |
IX |
May sound |
m , I |
" " " |
9th " " 14th " |
Very Low |
" |
annoyingly |
I*III |
" " " |
18th " " 7th " |
Very Low |
" |
sharp |
All Voicings Leave Equal-tempered Tuning |
|
None |
-0- |
None |
Consistent with eq. temp. tuning |
Table VII.
VIIb Tuning-Voicing Options and Possible Consequences |
Interval Pair and Voicing |
Tuning Option |
Harmonics Aligned |
Tendency to Lock an Prominence of Overtones |
Note Augmented |
Possible Consequences |
I, VIIb |
Flat VIIb by 31.2¢ |
7th of I w/ 4th of VIIb |
Very High |
VIIb |
May sound |
VIIb, I |
" " " |
7th " " 8th " " |
High |
" |
flat in some |
I * VIIb |
" " " |
7th " " 2nd " " |
Very High |
" |
voicings |
I, VIIb |
Flat VIIb by 3.9¢ |
16th of I w/ 9th of VIIb |
Low |
I |
May sound |
VIIb, I |
" " " |
8th " " 9th " " |
Medium High |
" |
right on |
I *VIIb |
" " " |
32nd " " 9th " " |
- 0 - |
" |
pitch |
I, VIib |
Sharp VIIb by 17.6¢ |
9th of I w/ 5th of VIIb |
High |
IX |
May sound |
VIIb, I |
" " " |
9th " " 10th " " |
Medium |
" |
brightly major or |
I * VIIb |
" " " |
18th " " 5th " " |
Very Low |
" |
slightly sharp |
All Voicings Leave Equal-tempered Tuning |
|
None |
- 0 - |
None |
Consistent with eq. temp. tuning |
* Indicates an octave has been skipped ¢ = cents. 1¢ = (2x S)1/1200 |
[0096] When tuning a dom 7 chord, combinations of options, for example, those shown in Tables
IV and V, can be selected. The combinations selected will likely be different for
different styles of music. For blues, early jazz, gospel and other music heavily influenced
by African tuning, the options selected for most voicings and spreads may emphasize
flatting the III by 13.7 cents and flatting the VIIb by 31.2 cents. For classical
music, for most voicings and spreads, the tendency may be to sharp the III by 17.6
cents, or keeping it equal tempered, and either keep the VIIb at equal-tempered tuning
or flat it by 3.9 cents. For barbershop harmonies voiced with I below III, the choice
may be to flat III by 13.7 cents and flat VIIb by 31.2 cents or by 17.6 cents. For
barbershop harmonies voiced V, III, VIIb, I, the choice may be to sharp III by 17.5
cents and flat VIIb by 31.2 cents.
[0097] A device or devices together with an algorithm will play synthesized, naturally produced
and/or recorded music and will permit the notes of music to be sharped or flatted
by specified amounts as chord types with various voicings and spreads are sounded.
Expert musicians, music critics, music conductors and the like, listen to various
optional tuning strategies developed for various styles of music, for example, gospel,
blues, nineteenth century classical, modern jazz, and so on, and for various types
of ensembles, for example, choral groups, string quartets and so on. Strategies developed
from such critical listening are implemented in tuning/blending databases, for example,
for each of such styles of music. Such a database will contain tuning and blending
strategies for each voicing, including spread voicings, of each chord type. All of
the eleven 2-note chords, including the common voicings and spreads which might be
tuned by expert ensembles; all triads and their voicings; all 4-note chords and their
voicings; and all the more common 5-note chords and their more common voicings are
included in the tuning/blending database. The tuning options described after Tables
IV and V are some options which apply to a dom 7 chord. Hereafter a dom 7 chord will
be used to illustrate a tuning/blending database.
[0098] There are many possible voicings of a dom 7 chord. When the root (I) is the lowest
note of the chord, there are 6 compact voicings, that is, voicings in which no octaves
are skipped between notes of the chord. These compact voicings are:
I |
III |
VIIb |
V |
I |
III |
V |
VIIb |
I |
VIIb |
III |
V |
I |
VIIb |
V |
III |
I |
V |
III |
VIIb |
I |
V |
VIIb |
III. |
[0099] There are eighteen more compact voicings with HI, V and VIIb being the lowest note,
and there are quite a few spread versions of these voicings (that is, voicings in
which an octave is skipped between adjacent notes of the chord), such as
I * III V VIIb
and I * III * V VIIb
Consequently, there may be as many as 100 voicings of the dom 7 chord, and each is
a separate entry in the database. A tuning strategy is provided for each entry in
the database. That tuning strategy includes which note is to be held at equal tempered
tuning, and the ratios of all notes with respect to the note that is held at equal
tempered tuning. For example, the strategy for tuning a dom 7 voiced I * III V VIIb
, for the blues being sung by a vocal group may be to set I (the root) equal tempered,
III 13.6 cents flat with respect to its equal tempered frequency, V 2 cents sharp
with respect to its equal tempered frequency, and VIIb 31.2 cents flat with respect
to its equal tempered frequency. It should be understood that, as used here, equal
tempered tuning includes equal tempered stretch tuning as previously described.
[0100] Each tuning/blending database entry also contains a blending strategy, which again
may be arrived at, for example, by experts listening to synthesized and/or modified
recorded chords. Each blending strategy will indicate how many dB above or below some
reference level, for example, equal loudness, the amplitude of each note should be
set. There is a control, for example, a pedal, to activate and deactivate the blending
function. When the blending function is not activated, the volume of each note will
be controlled in a conventional manner, for example, by the force applied to the key,
a volume setting, or the like. When the blending function is activated, the volume
of each note in a combination of sustained notes is set by the instrument to blend
the chord, that is, to adjust the amplitudes of the various notes of the chord so
that no individual note(s) dominate(s) the sound. When the blending function is activated,
the blending device/algorithm takes into account the following parameters in adjusting
relative amplitudes of the various notes of the chord which is to be blended. Loudness
is the listener's subjective response to the energy and frequency of a note. The psychoacoustics
of perceived loudness have been the subject of considerable study, including that
leading up to the publication of the equal loudness contours, illustrated in Fig.
3 ("the Physics of Musical Instruments", p. 162, 2
nd Ed.). This phenomenon has been studied in depth and the equal loudness contours have
been developed to illustrate the relationship among perceived loudness (in phons),
sound pressure level (in dB) and frequency (in Hertz). Using these, or similar, curves,
the relative amplitudes of two notes of different frequency can be established so
that neither note dominates. The equal loudness contours, or similar curves, may be
stored in the instrument and employed in calculations by the instrument to determine
the desired amplitudes of the blended notes of a played chord when the blending function
is selected on the instrument.
[0101] The positions occupied by the various notes in a chord also affect the blending of
the notes. Certain intervals in certain chords voiced in certain ways will blend only
when their volumes are adjusted, beyond even the observations exemplified by the equal
loudness contours. In general, it is frequently desirable to reduce substantially
the volume of a minor seventh, to reduce a major third a moderate amount, and to reduce
a sixth and a minor third lesser amounts. These reductions may be mediated by the
way the chord is voiced.
[0102] Voicing of the chord also affects the blending of notes. In general, if two notes
are located less than three semitones apart, then their volumes should be substantially
equal. Thirds which are internal to a chord can be reduced in volume. Minor sevenths
which are internal to a chord and separated from other notes by at least three semitones,
and minor sevenths at the top of the chord can be substantially reduced in volume.
The volumes of major and minor thirds can be reduced even more when they are within
or at the top of a chord and widely separated from other notes.
[0103] The blending device/algorithm will utilize a table, such as Table VI, containing
deviations from, for example, the equal loudness contours, to which the instrument's
processor will refer to blend the notes of a played chord once the loudnesses, note
positions and voicing have been determined. In the context of tuning, once it has
been determined that a chord is being sustained, the notes in a newly sustained chord
are identified. The chord type is identified and the position of each note in the
chord is determined, for example, by looking it up in a lookup table. The amplitude
of the note having the lowest frequency in the sustained chord is recorded. A loudness
curve by which the amplitudes of the various notes of the chord are to be blended
is selected. Such a loudness curve may be, for example, an equal loudness contour
based upon the frequency and amplitude of the lowest frequency note in the chord and
established by interpolation between curves in Fig. 3. The amplitude of each other
note in the chord is then set relative to the amplitude for the lowest frequency note.
As another method for blending, the contents of the equal loudness contours or some
other suitable amplitude adjusting algorithm can be stored in a lookup table with
an appropriate interpolation engine, with the amplitudes of the notes of the chord
being adjusted as dictated by the contents of the table with the aid of the interpolation
engine. Table VIII illustrates one method for adjusting the amplitudes of the various
notes of several chords voiced in several different ways relative to the equal loudness
contour amplitude, v, of a reference note of the chord. Notes of the illustrated chords
whose amplitudes are adjusted downward by some number of dB relative to v are indicated,
for example, "-2.0" indicating a downward adjustment of amplitude by 2 dB relative
to v. This blending of amplitudes will be maintained as long as the chord is sustained
or until the blending pedal is released.
[0104] The entries in Table VIII are for the purpose of illustration only. Musicians who
are chord blending specialists, for example, barbershop chorus or quartet directors
and coaches, and string quartet instructors and advisors, can listen to the suggested
blendings in Table VIII and adjust values, or suggest adjustments to values, such
as those contained in Table VIII to produce chords with notes that, in their judgment,
blend well. Consensus among experts can be used to establish blending values for the
notes of various chords voiced in various ways. These consensus values can be incorporated
into blending tables, like Table VIII, which are incorporated into instruments constructed
according to this invention.