GNU Emacs @value{EMACSVER}.
@end ifnotinfo
-Copyright @copyright{} 1990, 1991, 2001, 2002, 2003, 2004,
-2005, 2006, 2007, 2008, 2009, 2010, 2011 Free Software Foundation, Inc.
+Copyright @copyright{} 1990-1991, 2001-2011 Free Software Foundation, Inc.
@quotation
Permission is granted to copy, distribute and/or modify this document
@end quotation
@end copying
-@dircategory Emacs
+@dircategory Emacs misc features
@direntry
* Calc: (calc). Advanced desk calculator and mathematical tool.
@end direntry
* The Units Table::
* Predefined Units::
* User-Defined Units::
+* Logarithmic Units::
+* Musical Notes::
@end menu
@node Basic Operations on Units, The Units Table, Units, Units
@c Describe angular units, luminosity vs. steradians problem.
-@node User-Defined Units, , Predefined Units, Units
+@node User-Defined Units, Logarithmic Units, Predefined Units, Units
@section User-Defined Units
@noindent
is replaced by the new set. (@xref{General Mode Commands}, for a way to
tell Calc to use a different file for the Calc init file.)
+@node Logarithmic Units, Musical Notes, User-Defined Units, Units
+@section Logarithmic Units
+
+The units @code{dB} (decibels) and @code{Np} (nepers) are logarithmic
+units which are manipulated differently than standard units. Calc
+provides commands to work with these logarithmic units.
+
+Decibels and nepers are used to measure power quantities as well as
+field quantities (quantities whose squares are proportional to power);
+these two types of quantities are handled slightly different from each
+other. By default the Calc commands work as if power quantities are
+being used; with the @kbd{H} prefix the Calc commands work as if field
+quantities are being used.
+
+The decibel level of a power
+@infoline @math{P1},
+@texline @math{P_1},
+relative to a reference power
+@infoline @math{P0},
+@texline @math{P_0},
+is defined to be
+@infoline @math{10 log10(P1/P0) dB}.
+@texline @math{10 \log_{10}(P_{1}/P_{0}) {\rm dB}}.
+(The factor of 10 is because a decibel, as its name implies, is
+one-tenth of a bel. The bel, named after Alexander Graham Bell, was
+considered to be too large of a unit and was effectively replaced by
+the decibel.) If @math{F} is a field quantity with power
+@math{P=k F^2}, then a reference quantity of
+@infoline @math{F0}
+@texline @math{F_0}
+would correspond to a power of
+@infoline @math{P0=k F0^2}.
+@texline @math{P_{0}=kF_{0}^2}.
+If
+@infoline @math{P1=k F1^2},
+@texline @math{P_{1}=kF_{1}^2},
+then
+
+@ifnottex
+@example
+10 log10(P1/P0) = 10 log10(F1^2/F0^2) = 20 log10(F1/F0).
+@end example
+@end ifnottex
+@tex
+$$ 10 \log_{10}(P_1/P_0) = 10 \log_{10}(F_1^2/F_0^2) = 20
+\log_{10}(F_1/F_0)$$
+@end tex
+
+@noindent
+In order to get the same decibel level regardless of whether a field
+quantity or the corresponding power quantity is used, the decibel
+level of a field quantity
+@infoline @math{F1},
+@texline @math{F_1},
+relative to a reference
+@infoline @math{F0},
+@texline @math{F_0},
+is defined as
+@infoline @math{20 log10(F1/F0) dB}.
+@texline @math{20 \log_{10}(F_{1}/F_{0}) {\rm dB}}.
+For example, the decibel value of a sound pressure level of
+@infoline @math{60 uPa}
+@texline @math{60 \mu{\rm Pa}}
+relative to
+@infoline @math{20 uPa}
+@texline @math{20 \mu{\rm Pa}}
+(the threshhold of human hearing) is
+@infoline @math{20 log10(60 uPa/ 20 uPa) dB = 20 log10(3) dB},
+@texline @math{20 \log_{10}(60 \mu{\rm Pa}/20 \mu{\rm Pa}) {\rm dB} = 20 \log_{10}(3) {\rm dB}},
+which is about
+@infoline @math{9.54 dB}.
+@texline @math{9.54 {\rm dB}}.
+Note that in taking the ratio, the original units cancel and so these
+logarithmic units are dimensionless.
+
+Nepers (named after John Napier, who is credited with inventing the
+logarithm) are similar to bels except they use natural logarithms instead
+of common logarithms. The neper level of a power
+@infoline @math{P1},
+@texline @math{P_1},
+relative to a reference power
+@infoline @math{P0},
+@texline @math{P_0},
+is
+@infoline @math{(1/2) ln(P1/P0) Np}.
+@texline @math{(1/2) \ln(P_1/P_0) {\rm Np}}.
+The neper level of a field
+@infoline @math{F1},
+@texline @math{F_1},
+relative to a reference field
+@infoline @math{F0},
+@texline @math{F_0},
+is
+@infoline @math{ln(F1/F0) Np}.
+@texline @math{\ln(F_1/F_0) {\rm Np}}.
+
+@vindex calc-lu-power-reference
+@vindex calc-lu-field-reference
+For power quantities, Calc uses
+@infoline @math{1 mW}
+@texline @math{1 {\rm mW}}
+as the default reference quantity; this default can be changed by changing
+the value of the customizable variable
+@code{calc-lu-power-reference} (@pxref{Customizing Calc}).
+For field quantities, Calc uses
+@infoline @math{20 uPa}
+@texline @math{20 \mu{\rm Pa}}
+as the default reference quantity; this is the value used in acoustics
+which is where decibels are commonly encountered. This default can be
+changed by changing the value of the customizable variable
+@code{calc-lu-field-reference} (@pxref{Customizing Calc}). A
+non-default reference quantity will be read from the stack if the
+capital @kbd{O} prefix is used.
+
+@kindex l q
+@pindex calc-lu-quant
+@tindex lupquant
+@tindex lufquant
+The @kbd{l q} (@code{calc-lu-quant}) [@code{lupquant}]
+command computes the power quantity corresponding to a given number of
+logarithmic units. With the capital @kbd{O} prefix, @kbd{O l q}, the
+reference level will be read from the top of the stack. (In an
+algebraic formula, @code{lupquant} can be given an optional second
+argument which will be used for the reference level.) For example,
+@code{20 dB @key{RET} l q} will return @code{100 mW};
+@code{20 dB @key{RET} 4 W @key{RET} O l q} will return @code{400 W}.
+The @kbd{H l q} [@code{lufquant}] command behaves like @kbd{l q} but
+computes field quantities instead of power quantities.
+
+@kindex l d
+@pindex calc-db
+@tindex dbpower
+@tindex dbfield
+@kindex l n
+@pindex calc-np
+@tindex nppower
+@tindex npfield
+The @kbd{l d} (@code{calc-db}) [@code{dbpower}] command will compute
+the decibel level of a power quantity using the default reference
+level; @kbd{H l d} [@code{dbfield}] will compute the decibel level of
+a field quantity. The commands @kbd{l n} (@code{calc-np})
+[@code{nppower}] and @kbd{H l n} [@code{npfield}] will similarly
+compute neper levels. With the capital @kbd{O} prefix these commands
+will read a reference level from the stack; in an algebraic formula
+the reference level can be given as an optional second argument.
+
+@kindex l +
+@pindex calc-lu-plus
+@tindex lupadd
+@tindex lufadd
+@kindex l -
+@pindex calc-lu-minus
+@tindex lupsub
+@tindex lufsub
+@kindex l *
+@pindex calc-lu-times
+@tindex lupmul
+@tindex lufmul
+@kindex l /
+@pindex calc-lu-divide
+@tindex lupdiv
+@tindex lufdiv
+The sum of two power or field quantities doesn't correspond to the sum
+of the corresponding decibel or neper levels. If the powers
+corresponding to decibel levels
+@infoline @math{D1}
+@texline @math{D_1}
+and
+@infoline @math{D2}
+@texline @math{D_2}
+are added, the corresponding decibel level ``sum'' will be
+
+@ifnottex
+@example
+ 10 log10(10^(D1/10) + 10^(D2/10)) dB.
+@end example
+@end ifnottex
+@tex
+$$ 10 \log_{10}(10^{D_1/10} + 10^{D_2/10}) {\rm dB}.$$
+@end tex
+
+@noindent
+When field quantities are combined, it often means the corresponding
+powers are added and so the above formula might be used. In
+acoustics, for example, the sound pressure level is a field quantity
+and so the decibels are often defined using the field formula, but the
+sound pressure levels are combined as the sound power levels, and so
+the above formula should be used. If two field quantities themselves
+are added, the new decibel level will be
+
+@ifnottex
+@example
+ 20 log10(10^(D1/20) + 10^(D2/20)) dB.
+@end example
+@end ifnottex
+@tex
+$$ 20 \log_{10}(10^{D_1/20} + 10^{D_2/20}) {\rm dB}.$$
+@end tex
+
+@noindent
+If the power corresponding to @math{D} dB is multiplied by a number @math{N},
+then the corresponding decibel level will be
+
+@ifnottex
+@example
+ D + 10 log10(N) dB,
+@end example
+@end ifnottex
+@tex
+$$ D + 10 \log_{10}(N) {\rm dB},$$
+@end tex
+
+@noindent
+if a field quantity is multiplied by @math{N} the corresponding decibel level
+will be
+
+@ifnottex
+@example
+ D + 20 log10(N) dB.
+@end example
+@end ifnottex
+@tex
+$$ D + 20 \log_{10}(N) {\rm dB}.$$
+@end tex
+
+@noindent
+There are similar formulas for combining nepers. The @kbd{l +}
+(@code{calc-lu-plus}) [@code{lupadd}] command will ``add'' two
+logarithmic unit power levels this way; with the @kbd{H} prefix,
+@kbd{H l +} [@code{lufadd}] will add logarithmic unit field levels.
+Similarly, logarithmic units can be ``subtracted'' with @kbd{l -}
+(@code{calc-lu-minus}) [@code{lupsub}] or @kbd{H l -} [@code{lufsub}].
+The @kbd{l *} (@code{calc-lu-times}) [@code{lupmul}] and @kbd{H l *}
+[@code{lufmul}] commands will ``multiply'' a logarithmic unit by a
+number; the @kbd{l /} (@code{calc-lu-divide}) [@code{lupdiv}] and
+@kbd{H l /} [@code{lufdiv}] commands will ``divide'' a logarithmic
+unit by a number. Note that the reference quantities don't play a role
+in this arithmetic.
+
+@node Musical Notes, , Logarithmic Units, Units
+@section Musical Notes
+
+Calc can convert between musical notes and their associated
+frequencies. Notes can be given using either scientific pitch
+notation or midi numbers. Since these note systems are basically
+logarithmic scales, Calc uses the @kbd{l} prefix for functions
+operating on notes.
+
+Scientific pitch notation refers to a note by giving a letter
+A through G, possibly followed by a flat or sharp) with a subscript
+indicating an octave number. Each octave starts with C and ends with
+B and
+@c increasing each note by a semitone will result
+@c in the sequence @expr{C}, @expr{C} sharp, @expr{D}, @expr{E} flat, @expr{E},
+@c @expr{F}, @expr{F} sharp, @expr{G}, @expr{A} flat, @expr{A}, @expr{B}
+@c flat and @expr{B}.
+the octave numbered 0 was chosen to correspond to the lowest
+audible frequency. Using this system, middle C (about 261.625 Hz)
+corresponds to the note @expr{C} in octave 4 and is denoted
+@expr{C_4}. Any frequency can be described by giving a note plus an
+offset in cents (where a cent is a ratio of frequencies so that a
+semitone consists of 100 cents).
+
+The midi note number system assigns numbers to notes so that
+@expr{C_(-1)} corresponds to the midi note number 0 and @expr{G_9}
+corresponds to the midi note number 127. A midi controller can have
+up to 128 keys and each midi note number from 0 to 127 corresponds to
+a possible key.
+
+@kindex l s
+@pindex calc-spn
+@tindex spn
+The @kbd{l s} (@code{calc-spn}) [@code{spn}] command converts either
+a frequency or a midi number to scientific pitch notation. For
+example, @code{500 Hz} gets converted to
+@code{B_4 + 21.3094853649 cents} and @code{84} to @code{C_6}.
+
+
+@kindex l m
+@pindex calc-midi
+@tindex midi
+The @kbd{l m} (@code{calc-midi}) [@code{midi}] command converts either
+a frequency or a note given in scientific pitch notation to the
+corresponding midi number. For example, @code{C_6} gets converted to 84
+and @code{440 Hz} to 69.
+
+@kindex l f
+@pindex calc-freq
+@tindex freq
+The @kbd{l f} (@code{calc-freq}) [@code{freq}] command converts either
+either a midi number or a note given in scientific pitch notation to
+the corresponding frequency. For example, @code{Asharp_2 + 30 cents}
+gets converted to @code{118.578040134 Hz} and @code{55} to
+@code{195.99771799 Hz}.
+
+Since the frequencies of notes are not usually given exactly (and are
+typically irrational), the customizable variable
+@code{calc-note-threshold} determines how close (in cents) a frequency
+needs to be to a note to be recognized as that note
+(@pxref{Customizing Calc}). This variable has a default value of
+@code{1}. For example, middle @var{C} is approximately
+@expr{261.625565302 Hz}; this frequency is often shortened to
+@expr{261.625 Hz}. Without @code{calc-note-threshold} (or a value of
+@expr{0}), Calc would convert @code{261.625 Hz} to scientific pitch
+notation @code{B_3 + 99.9962592773 cents}; with the default value of
+@code{1}, Calc converts @code{261.625 Hz} to @code{C_4}.
+
+
+
@node Store and Recall, Graphics, Units, Top
@chapter Storing and Recalling
and @code{calc-embedded-open-close-plain-alist}.
@end defvar
+@defvar calc-lu-power-reference
+@defvarx calc-lu-field-reference
+See @ref{Logarithmic Units}.@*
+The variables @code{calc-lu-power-reference} and
+@code{calc-lu-field-reference} are unit expressions (written as
+strings) which Calc will use as reference quantities for logarithmic
+units.
+
+The default value of @code{calc-lu-power-reference} is @code{"mW"}
+and the default value of @code{calc-lu-field-reference} is
+@code{"20 uPa"}.
+@end defvar
+
+@defvar calc-note-threshold
+See @ref{Musical Notes}.@*
+The variable @code{calc-note-threshold} is a number (written as a
+string) which determines how close (in cents) a frequency needs to be
+to a note to be recognized as that note.
+
+The default value of @code{calc-note-threshold} is 1.
+@end defvar
+
@defvar calc-highlight-selections-with-faces
@defvarx calc-selected-face
@defvarx calc-nonselected-face
@r{ v x@: k T @: @: @:utpt@:(x,v)}
@r{ v x@: I k T @: @: @:ltpt@:(x,v)}
+@c
+@r{ a b@: l + @: @: @:lupadd@:(a,b)}
+@r{ a b@: H l + @: @: @:lufadd@:(a,b)}
+@r{ a b@: l - @: @: @:lupsub@:(a,b)}
+@r{ a b@: H l - @: @: @:lufsub@:(a,b)}
+@r{ a b@: l * @: @: @:lupmul@:(a,b)}
+@r{ a b@: H l * @: @: @:lufmul@:(a,b)}
+@r{ a b@: l / @: @: @:lupdiv@:(a,b)}
+@r{ a b@: H l / @: @: @:lufdiv@:(a,b)}
+@r{ a@: l d @: @: @:dbpower@:(a)}
+@r{ a b@: O l d @: @: @:dbpower@:(a,b)}
+@r{ a@: H l d @: @: @:dbfield@:(a)}
+@r{ a b@: O H l d @: @: @:dbfield@:(a,b)}
+@r{ a@: l n @: @: @:nppower@:(a)}
+@r{ a b@: O l n @: @: @:nppower@:(a,b)}
+@r{ a@: H l n @: @: @:npfield@:(a)}
+@r{ a b@: O H l n @: @: @:npfield@:(a,b)}
+@r{ a@: l q @: @: @:lupquant@:(a)}
+@r{ a b@: O l q @: @: @:lupquant@:(a,b)}
+@r{ a@: H l q @: @: @:lufquant@:(a)}
+@r{ a b@: O H l q @: @: @:lufquant@:(a,b)}
+@r{ a@: l s @: @: @:spn@:(a)}
+@r{ a@: l m @: @: @:midi@:(a)}
+@r{ a@: l f @: @: @:freq@:(a)}
+
@c
@r{ @: m a @: @: 12,13 @:calc-algebraic-mode@:}
@r{ @: m d @: @: @:calc-degrees-mode@:}
@printindex fn
@bye
-
HCoop / bpt / emacs.git / blobdiff