 |
Description  |
|
|
BACKGROUND OF THE INVENTION
The present invention relates in general to tuning musical instruments and,
more particularly, to digital aural tuning of musical instruments having a
plurality of adjustable frequency tone generators, such as strings in
pianos, for generating a like plurality of musical notes. While the
present invention is generally applicable to a variety of musical
instruments including, for example, harpsichords, organs and pianos, it
will be described herein with reference to tuning pianos for which it is
particularly applicable and initially being applied.
Aural tuning techniques have been used to tune pianos since the earliest
introduction of these instruments in the seventeen hundreds. In
conventional aural tuning, a human tuner listens to a reference note and
adjusts another note of the piano until that note sounds consonant with
the reference note. Consonance can be indicated by a specified beat rate
between the note being tuned and the reference note. Beat rate tuning is
possible because an equally tempered scale is based upon simple
mathematical relationships. In actuality, the frequencies which make up
given notes of a piano and other instruments, do not correspond exactly to
simple mathematic relationships.
For example, while "harmonics" denote integer multiples of a base frequency
of a musical note, the overtones actually produced by a piano string are
not harmonics and, to distinguish the overtones from harmonics, are called
"partials". Each note of a piano includes a plurality of partials which
are referred to as a "partial ladder" which can be used to represent all
partials of a note or at least all partials which are required to tune an
instrument. Partial ladders can be the relative pitches of the included
partials for a note; however, more commonly they are listed as the
deviation of the included partials from their corresponding harmonics and
are quantified in "cents" where one cent is the amount of pitch difference
that is equal to one per cent (0.01) of a semitone.
The difference between a given partial and its ideal harmonic is caused in
part by "inharmonicity" which causes the partials of a vibrating piano
string to be sharper or higher in frequency than would be expected from
the harmonics for the string. Inharmonicity is due to the inherent
stiffness of the metal wire which makes up the strings. While the
inharmonicity theory presumes that all partials of a vibrating piano
string are sharper than expected, in most instances, the partials may be
either sharper, i.e., higher in frequency, or flatter, i.e., lower in
frequency, than would be predicted by inharmonicity. This phenomenon,
which is not accounted for by the inharmonicity theory and is believed to
be due to the construction of the instrument, is referred to herein as
"para-harmonicity". Every string or note of a piano can have a unique
partial structure or partial ladder. To add to the complexity, each piano
is different and even two pianos which are made side-by-side will require
slightly different tuning or pitch for at least some and more often many
of the notes of the pianos.
While manual aural tuning is the standard and produces excellent results,
it is much more of an art than a science requiring substantial training of
highly skilled and experienced persons. Further, manual aural tunings can
vary from tuner to tuner and the manual aural tuning process can take a
substantial amount of time. To reduce tuning time and the level of skill
required for tuning instruments, other tuning techniques, such as tuning
calculations, have been proposed. The concept of calculating a theoretical
tuning for a piano has been known for many years, and was addressed widely
in the Piano Technician's Journal and other publications throughout the
1970's and 1980's. The tuning calculations revolved around creating a
perfect tuning using theoretical models. Unfortunately, the calculation
techniques have not proven to be satisfactory since the calculations are
very complex and the results do not match aural tuning results.
To improve upon the calculation techniques, measurement methods for
determining the pitches of partials for the notes of an instrument to be
tuned have been explored. One of the earliest attempts measured the
difference between two partials of one note in the middle of the piano to
determine the inharmonicity of the instrument. Unfortunately, the note
chosen may or may not be representative of the notes around it and the
measurements are time consuming and often inaccurate. This method is
referred to as the partial-pair measurement method.
Another technique uses a calculated "inharmonicity constant" (Ic) which is
derived from a physical measurement of the length and diameter of a
vibrating string. This technique is referred to as the scale measurement
method. Once the Ic is determined, equations including the Ic are used to
calculate the partial structure for the notes of an instrument. A series
of equations for calculating a tuning for 88 piano notes using an Ic were
published in July, 1990 and further documented in the Piano Technician's
Journal in 1991-1992. Unfortunately, this method requires scale
measurements which normally take more time than the average aural tuner
requires, around 2 hours, making it impractical.
Another scale measurement method is used in a product available from the
inventor of the present application and sold under the trademark
"Chameleon". In Chameleon, now Chameleon 1, the physical characteristics
of five strings are measured to derive an Ic and then to calculate an 88
note tuning based on the Ic and equations which are somewhat simplified
when compared to the equations found in the Piano Technician's Journal in
1991-1992.
Another technique measures the inharmonicity between two partials on each
of three notes and calculates an 88 note tuning. This technique is an
expansion of the partial-pair method mentioned above. Because the F, A and
C notes are commonly used, this method is also referred to as the "FAC"
method and is more fully described in U.S. Pat. No. 5,285,711. In this
patent, the calculation of the 88 note tuning is performed using equations
which rely on the Ic. The equations are either directly solved or utilized
to prepare look-up tables which reduce the computing power required by a
system embodying the invention. In either event, the calculations rely
upon solution of the equations disclosed in the patent.
Unfortunately, all of the above methods presume that the inharmonicity
theory is inviolate and that the inharmonicity constant (Ic) is accurately
calculated by standard formulae, neither of which is true. The scale
measurement methods use one of several standard formulae to convert wire
type, diameter, and length into an inharmonicity constant (Ic). The
partial-pair measurement methods use two measured partials of one or more
notes, such as three notes, to calculate the inharmonicity constant with
standard formulae. In either case, the inharmonicity constant determined
is either not accurate or is not accurate for the entire instrument being
tuned due, for example, to a failure to consider para-harmonicity.
Applicant's experience and research in aural, electronic measurement and
calculated tuning has shown that the prior art tuning methods, while able
to produce tunings that are acceptable to some tuners and musicians, are
inadequate to produce tunings that rival the best aural human tuners.
Expert aural tuners can detect pitch changes of as little as
one-thousandth of a semitone, i.e., 0.1 cent again where one cent is the
amount of pitch difference that is equal to one per cent (0.01) of a
semitone. Such tuning precision is not within the capabilities of prior
art techniques. Thus, if an expert aural human tuner is given enough time,
he can produce a tuning that excels even the best prior art electronic or
calculated tuning.
Accordingly, there is a need for an improved tuning method which can
produce improved tuning results when compared to prior art methods.
Preferably, the improved tuning method would not only produce improved
instrument tunings but also would permit persons of less skill and
experience than an expert aural tuner to produce improved instrument
tunings in less time than either an expert aural tuner or a tuner using
prior art tuning techniques. The tuning method would be further improved
by use of an improved graphic and dynamic display of a pitch difference of
an unknown pitch relative to a desired pitch which would provide highly
accurate macro and micro tuning information in a single display.
SUMMARY OF THE INVENTION
This need is met by the methods and apparatus of the present invention
wherein at least three musical notes of an instrument are sounded and
recorded to generate directly partial ladders representative of the
sounded notes. The partial ladders are equalized with respective to a
reference frequency or one another to determine tuning frequencies for the
sounded notes. Tuning frequencies for the remaining notes of the
instrument are then determined from the equalized partial ladders. Tone
generators, such as strings on a piano, are then adjusted to conform the
musical notes which they generate to the tuning frequencies. Preferably,
the tone generators are adjusted using a display which provides highly
accurate macro and micro tuning information in a single display by
graphically and dynamically displaying pitch differences of the musical
notes generated by the tone generators relative to pitches of the tuning
frequencies. Reference to the display facilitates adjustment of the tone
generators to make the pitch differences substantially zero.
In accordance with one aspect of the present invention, a method for tuning
a musical instrument having a plurality of adjustable frequency tone
generators for generating a like plurality of musical notes, each tone
generator producing a plurality of different order partials with the first
partial for each note corresponding to the lowest frequency of the note
comprises the steps of: digitally recording a partial ladder for at least
three musical notes produced by at least three corresponding adjustable
frequency tone generators of the musical instrument, the partial ladders
including all partials needed to tune the musical instrument; equalizing
the partial ladders to determine tuning frequencies for each of the at
least three musical notes; determining tuning frequencies for musical
notes of the musical instrument from equalized partial ladders; and,
adjusting the plurality of adjustable frequency tone generators to conform
their musical notes to the tuning frequencies.
Preferably, the step of adjusting the plurality of adjustable frequency
tone generators comprises the step of graphically and dynamically
displaying pitch differences of the musical notes of the adjustable
frequency tone generators relative to pitches of the tuning frequencies
until the pitch difference is displayed as being substantially zero.
In accordance with another aspect of the present invention, a method for
tuning a musical instrument having a plurality of adjustable frequency
tone generators for generating a like plurality of musical notes, each
tone generator producing a plurality of different order partials with the
first partial for each note corresponding to the lowest frequency of the
note comprises the steps of: digitally recording a partial ladder for at
least three musical notes produced by at least three corresponding
adjustable frequency tone generators of the musical instrument, the
partial ladders including all partials needed to tune the musical
instrument; equalizing one of the partial ladders as a starting partial
ladder; equalizing the remaining partial ladders with respect to the
starting partial ladder; calculating digital tuning frequencies for the
remaining notes of the plurality of musical notes from equalized partial
ladders of the at least three musical notes; and, adjusting the plurality
of adjustable frequency tone generators to conform their musical notes to
the tuning frequencies.
In accordance with still another aspect of the present invention, a method
for tuning a musical instrument having a plurality of adjustable frequency
tone generators for generating a like plurality of musical notes, each
tone generator producing a plurality of different order partials with the
first partial for each note corresponding to the lowest frequency of the
note comprises the steps of: digitally recording a partial ladder for at
least three musical notes produced by at least three corresponding
adjustable frequency tone generators of the musical instrument, the
partial ladders including all partials needed to tune the musical
instrument; equalizing a first partial ladder as a starting partial ladder
by setting one partial of the starting partial ladder equal to a nominal
frequency for the one partial and adjusting all other partials of the
starting partial ladder relative to the one partial; equalizing a second
partial ladder relative to the starting partial ladder by setting one
partial of the second partial ladder to a corresponding partial of the
starting partial ladder less a widening offset; equalizing a third partial
ladder relative to the starting partial ladder or the second partial
ladder by setting one partial of the third partial ladder to a
corresponding partial in the starting partial ladder or the second partial
ladder less a widening offset; calculating tuning frequencies for the
remaining notes of the plurality of musical notes from equalized partial
ladders of the at least three musical notes; and, adjusting the plurality
of adjustable frequency tone generators to conform their musical notes to
the tuning frequencies.
In accordance with yet another aspect of the present invention, apparatus
for tuning a musical instrument having a plurality of adjustable frequency
tone generators for generating a like plurality of musical notes, each
tone generator producing a plurality of different order partials with the
first partial for each note corresponding to the lowest frequency of the
note comprises recorder means for digitally recording a partial ladder for
at least three musical notes produced by at least three corresponding
adjustable frequency tone generators of the musical instrument. The
partial ladders include all partials needed to tune the musical
instrument. Equalizer means provide for equalizing the partial ladders to
determine tuning frequencies for each of the at least three musical notes.
Means are provided for determining tuning frequencies for musical notes of
the musical instrument from equalized partial ladders.
Preferably, the apparatus for tuning a musical instrument further comprises
display means for graphically and dynamically displaying pitch differences
of the musical notes of the adjustable frequency tone generators relative
to pitches of the tuning frequencies.
In accordance with an additional aspect of the present invention, a method
for graphically and dynamically displaying a pitch difference of an
unknown pitch relative to a desired pitch comprises the steps of:
determining an unknown pitch; comparing the unknown pitch to a desired
pitch to determine a pitch difference; displaying a spinner at a center of
a display if the pitch difference is within a first defined pitch window
relative to the desired pitch; maintaining the spinner stationary if the
pitch difference is equal to zero; rotating the spinner clockwise if the
pitch difference is greater than zero but less than an upper boundary of
the first defined pitch window; rotating the spinner counterclockwise if
the pitch difference is less than zero but greater that a lower boundary
of the first defined pitch window; setting the rate of rotation in
proportion to the extent the unknown pitch is different than zero; moving
the spinner in a first direction off of the center if the pitch difference
exceeds the upper boundary of the first defined pitch window; moving the
spinner in a second direction off the center if the pitch difference
exceeds the lower boundary of the first defined pitch window; and, setting
the amount of movement of the spinner proportional to the extent the
unknown pitch exceeds the upper and lower boundaries of the first defined
pitch window. The method for graphically and dynamically displaying a
pitch difference may further comprise the step of modifying the spinner
toward a solid image as the unknown pitch increasingly exceeds the upper
and lower boundaries of the first pitch window.
In accordance with yet an additional aspect of the present invention, a
method for automatically switching notes in an electronic instrument
tuning device comprises the steps of: defining a current note; sounding a
note which can be the current note or a note adjacent to the current note;
determining the pitch difference of a sounded note relative to the current
note; using the next higher note if the pitch difference is greater than a
defined first pitch difference; using the next lower note if the pitch
difference is less than a defined second pitch difference; and, using the
current note if the pitch difference is within a current pitch difference
window between the second pitch difference and the first pitch difference.
In accordance with still an additional aspect of the present invention, a
method for pitch raise tuning comprises the steps of: setting up a table
of pitch raise overpull percentages for the musical notes of an instrument
to be tuned; and, using the table of pitch raise overpull percentages to
determine pitch raise tuning frequencies for musical notes of an
instrument to be tuned.
It is, thus, an object of the present invention to provide improved methods
and apparatus for digital aural tuning of musical instruments having a
plurality of adjustable tone generators; to provide improved methods and
apparatus for digital aural tuning of musical instruments having a
plurality of adjustable tone generators by digitally recording musical
notes sounded by at least three of the generators and determining and
recording partial ladders for the notes recorded which are then used for
tuning the instruments; to provide improved methods and apparatus for
digital aural tuning of musical instruments having a plurality of
adjustable tone generators including a display which provides highly
accurate macro and micro tuning information in a single display by
graphically and dynamically displaying pitch differences of musical notes
generated by the tone generators relative to pitches of determined tuning
frequencies; to provide improved methods and apparatus for digital aural
tuning of musical instruments having a plurality of adjustable tone
generators including automatic note switching; and, to provide improved
methods and apparatus for digital aural tuning of musical instruments
having a plurality of adjustable tone generators wherein the tuning
provides for pitch raise tuning using a table of pitch raise overpull
percentages for the musical notes of an instrument to be tuned to
determine pitch raise tuning frequencies for musical notes of the
instrument to be tuned.
Other objects and advantages of the invention will be apparent from the
following description, the accompanying drawings and the appended claims.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a listing of the partial ladders for notes measured in an
illustrative embodiment of the present invention;
FIGS. 2-4 are flow charts for operation of the illustrative embodiment
represented in FIG. 1;
FIG. 5 is a flow chart illustrating sampling, recording and filtering
aspects of the present invention;
FIG. 6 is a flow chart illustrating an automatic note switching aspect of
the present invention;
FIG. 7 is a pull-down screen available in a C program which performs the
tuning operations of the present invention;
FIG. 8 is a view of the screen of an Apple Macintosh PowerBook Duo model
No. 2300C programmed to operate as a tuning system of the present
invention;
FIGS. 9-12 show a unique display aspect of the present invention for
graphically and dynamically displaying macro and micro tuning information
in a single display;
FIG. 13 illustrates a pull-down screen available in a C program which
performs the tuning operations of the present invention and permits user
setting of various aspects of the unique display shown in FIGS. 9-12;
FIG. 14 illustrates a pull-down screen available in a C program which
performs the tuning operations of the present invention and permits
customization of the tuning operations performed; and
FIG. 15 illustrates a display of the present invention wherein the target
has been moved to the right for a sharp overpull pitch raise tuning.
DETAILED DESCRIPTION OF THE INVENTION
The aural instrument tuning of the present application will now be
described with reference to the drawings wherein FIG. 1 is a listing of
the partial ladders which are recorded for one embodiment of the present
invention. A large variety of embodiments of the aural tuning method are
possible, many of which will be described herein and others will be
apparent to those skilled in the art from a review of this description.
While the present invention is generally applicable to a variety of
musical instruments including, for example, harpsichords, organs and
pianos, it will be described herein with reference to tuning pianos for
which it is particularly applicable and initially being applied.
In the description, the following conventions are followed. The following
shorthand is used to represent partials of the piano notes described:
piano note name.fwdarw.partial number, i.e., A4.fwdarw.2nd represents the
second partial of the note A4. For ease of calculation and familiarity to
piano tuners, partial ladders are converted to cents deviation from the
standard frequencies of the musical notes they represent. Some
calculations are represented in a modified form of the C programming
language. The calculations described using this "pseudo-code" will be
readily apparent to persons familiar with programming in C and also to
those who have never programmed in C. However, it is believed that this
form of description is best in enabling those skilled in the art to
practice the invention. It is noted that the terms calculate, calculation
and the like are intended to cover any form of determination whether by
calculation performed in realtime, by pre-calculation and storage in a
look-up table or by other appropriate techniques for determining the
values referred to herein.
A brief overview of the operation of the present invention will now be
provided to facilitate a better understanding of the invention from the
detailed description which follows. In the present invention, partial
ladders are recorded digitally for at least three notes of a piano which
is being tuned. The partial ladders can be complete partial ladders
including all partials of each note which is sounded. Preferably, however,
the partial ladders include less than all the partials but do include all
the significant partials which are necessary to tune the piano. In a
working embodiment of the present invention, partial ladders including
four partials each are recorded for five notes on the piano as shown in
FIG. 1; however, any number of notes can be selected from three up to all
the notes of the piano. Each partial ladder is obtained directly as one
unit using digital filtering to filter the recorded notes at the
appropriate partial frequencies.
Thus, each partial ladder is obtained directly from the piano by sounding
the notes to be recorded without ever determining an inharmonicity
constant (Ic). In this way, both the inharmonicity and para-harmonicity
are inherently included in the partial ladders in the same way that an
aural tuner includes them as the piano is manually tuned, i.e., by
listening to the notes produced by the piano. One of the partial ladders
is then standardized by setting one of its partials to a defined frequency
for that partial with the ladders then being equalized within each ladder
and relative to the other ladders. Tuning frequencies are then determined
from the equalized ladders and the tone generators or strings of the
instrument are adjusted to conform their musical notes to the tuning
frequencies. With this introduction, a detailed description of an
embodiment of the invention corresponding to FIG. 1 will now be made.
In this embodiment, five notes, A1, A2, A3, A4, and A5, are sounded on the
piano and recorded using digital sampling techniques, see FIG. 2, block
110. The digitally recorded notes are filtered using well known digital
filtering techniques to determine and the partial ladders for those notes,
see FIG. 1 and block 112. Preferably, the notes are filtered as they are
being recorded to conserve time; however, the time for filtering depends
upon the operating speed of the recording device. The recording should be
performed using an accurate electronic tuning device, preferably one that
is accurate to within 0.01 cents. In the working embodiment of the present
invention being described relative to FIGS. 1 and 2, the entire tuning
method is performed using an Apple Macintosh PowerBook Duo model No.
2300C. In this way, the exact frequency of each of the partials is
recorded as the note is recorded and accurately extracted using a digital
bandpass filter as will be described, see block 114. The partial ladders
are converted to cents deviation from the standard frequencies of the
musical notes they represent for ease of calculation.
Preferably, the recording and filtering of each note is performed at least
two times, three times for the working embodiment being described, with
the resulting partial ladders being averaged to arrive at the partial
ladders which are recorded. The averaging operation increases the accuracy
of the recorded partial ladders by reducing possible loss of resolution
due to room noise interference and averages the effects of changes in
inharmonicity and para-harmonicity due to the user playing the piano at
different volumes and sustain lengths.
Once all partial ladders for the notes to be sounded on the piano have been
determined and recorded, see block 116, the partial ladders must be
equalized. For sake of clarity, a determination of a representative tuning
for a Steinway model D 9, grand piano will be described. This illustrative
tuning begins from the partial ladder for the note A4 with a typical
partial ladder as originally recorded for the note A4 of a Steinway model
D 9, grand piano being:
original partial ladder for A4
A4.fwdarw.4th=+8.24.cent.
A4.fwdarw.3rd=+5.07.cent.
A4.fwdarw.2nd=-0.15.cent.
A4.fwdarw.1st=-1.16.cent..rarw.must be converted to zero. (A440 hertz)
The cents offset of the primary partial, the lowest partial in the case of
the recorded A4 ladder, is subtracted from each of the partials to result
in an equalized A4 partial ladder, see 118. The subtraction of the cents
offset is necessary since the piano will most likely be out of tune when
it is recorded. The primary partial may be the fundamental, as in the A4
ladder, or the lowest partial that is strong enough to be used for tuning
if a partial ladder for a note other than A4 is used as the beginning
partial ladder. The primary partial is now represented by zero, and each
of the other partials is represented by a number that is its cents
deviation from the standard frequency of the corresponding partial of the
musical note represented by the ladder.
original A440 equalized ladder
A4.fwdarw.4th=+8.24.cent.-(-1.16.cent.)=+9.40.cent.
A4.fwdarw.3rd=+5.07.cent.-(-1.16.cent.)=+6.23.cent.
A4.fwdarw.2nd=-0.15.cent.-(-1.16.cent.)=+1.01.cent.
A4.fwdarw.1st=-1.16.cent.-(-1.16.cent.)=0.00.cent..rarw.440 hertz
The equalized partial ladder for A4 thus tunes A4 and the equalized partial
ladder for A4 is then used without modification since the primary partial
is A4 itself, which will be normally tuned to 440 hertz, zero cents
deviation. The remaining partial ladders are then equalized to tune their
corresponding notes. To equalize the ladders/tune the other recorded
notes, next an octave is tuned; however, different tuners have varying
tastes as to how "wide" to tune the octaves on a piano. Therefore, before
calculations to tune or equalize the remaining partial ladders are
performed, the user decides how much to "stretch" or widen the octaves.
Three octave width variables are specified to define the stretch:
T: how much to widen the A4 to A3 octave, and the treble;
B: how much to widen the A3 to A2 octave, and the bass; and
Dmax: the maximum double octave width for A2 to A4.
For purposes of describing the present invention with respect to T and B,
the pass/treble break is between G#3 and A3, i.e., all notes below and
including G#3 are considered bass and all notes above and including A3 are
considered treble. Typical values for T and B are between 0.0 and 1.2
cents. Although values of up to 4.0 cents have been used by some tuners.
Typical values for Dmax are 1.5 to 6.0 cents, with 12.0 cents being the
maximum acceptable as shown from empirical testing.
The second note tuned is A3, see 120. Aural tuners will normally match the
fourth partial of A3 with the second partial of A4, tuning a "4:2 octave".
The present invention performs this function by calculating a cents offset
and subtracting the offset from the whole A3 partial ladder. Initially, a
new value is determined for the fourth partial of A3 by setting it equal
to the second partial of A4 less T. That is:
new .sub.-- A3.fwdarw.4th=A4.fwdarw.2nd-T
For example, selecting a value for T of 0.66 cents, a commonly used value,
the calculation for the example piano is:
new.sub.-- A3.fwdarw.4th=+1.01.cent.-(0.66.cent.)=+0.35.cent.
resulting in new.sub.-- A3.fwdarw.4th being equal to 0.35 cents.
Original partial ladder for A3:
A3.fwdarw.4th=+0.11.cent.
A3.fwdarw.3rd=+0.60.cent.
A3.fwdarw.2nd=-2.77.cent.
A3.fwdarw.1st=-3.24.cent.
All partials of the original partial ladder for A3, except for the 4th
partial, have the original A3.fwdarw.4th value subtracted and the
new.sub.-- A3.fwdarw.4th added to equalize the ladder:
A3.fwdarw.nth=A3.fwdarw.nth-A3.fwdarw.4th+new.sub.-- A3.fwdarw.4th
where n is equal to the integer values except for four. The resulting A3
equalized partial ladder is:
A3.fwdarw.4th=+1.01.cent.-(0.66.cent.)=+0.35.cent..rarw.tuned partial
A3.fwdarw.3rd=+0.60-(+0.11.cent.)+(+0.35.cent.)=+0.84.cent.
A3.fwdarw.2nd=-2.77.cent.-(+0.11.cent.)+(+0.35.cent.)=-2.53.cent.
A3.fwdarw.1st=-3.24.cent.-(+0.11.cent.)+(+0.35.cent.)=-3.00.cent.
Notice that the 4th partial of A3 is now the same as the 2nd partial of A4,
expanded, i.e., flattened, by the amount T (0.66 cents) as specified by
the user: +1.01.cent.-0.66.cent.=0.35.cent..
The third note tuned is A2, see block 122. Aural tuners will normally match
the sixth partial of A2 with the third partial of A3, tuning what is
called a "6:3 octave". The present invention performs this function by
calculating a cents offset and subtracting the offset from the whole A2
partial ladder. Initially, a new value is determined for the sixth partial
of A2 by setting it equal to the third partial of A3 less B. That is:
new.sub.-- A2.fwdarw.6th=A3.fwdarw.3rd--B
For example, selecting a value for B of 1.00 cents, a commonly used value,
the calculation for the example piano is:
new.sub.-- A2.fwdarw.6th=0.84.cent.-1.00.cent.=-0.16.cent.
resulting in new.sub.--A2.fwdarw. 6th being equal to 0.16 cents.
Original partial ladder for A2:
A2.fwdarw.6th=-1.05.cent.
A2.fwdarw.4th=-4.74.cent.
A2.fwdarw.3rd=-3.07.cent.
A2.fwdarw.2nd=-5.26.cent.
All partials of the A2 partial ladder, except for the sixth partial, will
have the original A2.fwdarw.6th value subtracted and the
new.sub.--A2.fwdarw. 6th added to equalize the ladder:
A2.fwdarw.nth=A2.fwdarw.nth-A2.fwdarw.6th+new.sub.-- A2.fwdarw.6th
where n is equal to the integer values except for six. The resulting A2
equalized partial ladder is:
A2.fwdarw.6th=+0.16.cent..rarw.tuned partial
A2.fwdarw.4th=-4.74.cent.-(-1.05.cent.)+(-0.16.cent.)=-3.85.cent.
A2.fwdarw.3rd=-3.07.cent.-(-1.05.cent.)+(-0.16.cent.)=-2.18.cent.
A2.fwdarw.2nd=-5.26.cent.-(-1.05.cent.)+(-0.16.cent.)=-4.37.cent.
Notice that the 6th partial of A2 is now the same as the 3rd partial of A3,
expanded, i.e., flattened, by the amount B (1.00 cents) as specified by
the user.
The invention of the present application next checks the double octave A2
to A4 to make sure it is not wider than the variable Dmax, the maximum
double octave width for A2 to A4, see block 124. If this double octave
width is narrower than Dmax, then tuning A4, A3 and A2 is finished. A
typical value for Dmax is 4.0 cents. The double octave
width=A2.fwdarw.4th*(-1.0). If the double octave width is wider than Dmax,
then a proportional amount of the excess stretch above Dmax is added to
both the A2 and A3 partial ladders. In this way, the two single octaves,
A2 to A3 and A3 to A4, are narrowed by an equal amount in hertz which is
just enough to bring the double octave, A2 to A4, to the maximum double
octave width for A2 to A4 which is the value selected for Dmax.
In the illustrative tuning example, the double octave width is selected as
3.85 cents, which is less than the maximum 4.00 cents. No further
calculations are needed for A2, A3 and A4. If the double octave width were
greater than Dmax, then double width compensation or narrowing is
performed by performing the following steps. First, the overstretch cents
are calculated using the equation:
Double octave overstretch=Double octave width-Dmax.
Second, the calculated double octave overstretch is added to each partial
in the A2 partial ladder. Third, 1/3 (or 4/10) of the double octave
overstretch is added to each partial in the A3 partial ladder. The
invention of the present application then calculates the actual octave
width variables, see block 126, for later use as will be described:
T.sub.-- ACTUAL=A4.fwdarw.2nd-A3.fwdarw.4th
B.sub.-- ACTUAL=A3.fwdarw.3rd-A2.fwdarw.6th
D.sub.-- ACTUAL=A2.fwdarw.4th
If all the notes between A2 and A4 had been sounded, recorded and filtered
to record partial ladders for those notes, each note could be tuned in
turn as an aural tuner would, using the virtual equivalents of aural
tuning as described above. In this case, the described illustrative
embodiment wherein only five notes are recorded, the invention fits a
curve to the three notes already tuned. The 4th partial will be the
listening partial for this part of the tuning, although the 3rd partial
would be a logical choice also.
The following calculations are listed in pseudo-code to describe the
technique of filling in the missing notes between | | |
