You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
211 lines
6.6 KiB
211 lines
6.6 KiB
<!DOCTYPE html>
|
|
|
|
<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
|
|
original version by: Nikos Drakos, CBLU, University of Leeds
|
|
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
|
|
* with significant contributions from:
|
|
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
|
|
<HTML>
|
|
<HEAD>
|
|
|
|
<meta name="viewport" content="width=device-width, initial-scale=1.0">
|
|
|
|
|
|
<TITLE>Frequency</TITLE>
|
|
<META NAME="description" CONTENT="Frequency">
|
|
<META NAME="keywords" CONTENT="book">
|
|
<META NAME="resource-type" CONTENT="document">
|
|
<META NAME="distribution" CONTENT="global">
|
|
|
|
<META NAME="Generator" CONTENT="LaTeX2HTML v2002-2-1">
|
|
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
|
|
|
|
<LINK REL="STYLESHEET" HREF="book.css">
|
|
|
|
<LINK REL="next" HREF="node12.html">
|
|
<LINK REL="previous" HREF="node10.html">
|
|
<LINK REL="up" HREF="node7.html">
|
|
<LINK REL="next" HREF="node12.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
<!--Navigation Panel-->
|
|
<A ID="tex2html707"
|
|
HREF="node12.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A ID="tex2html701"
|
|
HREF="node7.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A ID="tex2html695"
|
|
HREF="node10.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A ID="tex2html703"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A ID="tex2html705"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A ID="tex2html708"
|
|
HREF="node12.html">Synthesizing a sinusoid</A>
|
|
<B> Up:</B> <A ID="tex2html702"
|
|
HREF="node7.html">Sinusoids, amplitude and frequency</A>
|
|
<B> Previous:</B> <A ID="tex2html696"
|
|
HREF="node10.html">Controlling Amplitude</A>
|
|
<B> <A ID="tex2html704"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A ID="tex2html706"
|
|
HREF="node201.html">Index</A></B>
|
|
<BR>
|
|
<BR>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H1><A ID="SECTION00540000000000000000">
|
|
Frequency</A>
|
|
</H1>
|
|
|
|
<P>
|
|
Frequencies, like amplitudes, are often measured on a logarithmic scale, in
|
|
order to emphasize proportions between them, which usually provide a better
|
|
description of the relationship between frequencies than do differences
|
|
between them. The frequency ratio between two musical tones determines
|
|
the musical interval between them.
|
|
|
|
<P>
|
|
The Western musical scale divides the
|
|
<A ID="1095"></A><I>octave</I> (the musical interval associated with a ratio of 2:1) into
|
|
twelve equal sub-intervals, each of which therefore corresponds to a ratio
|
|
of <!-- MATH
|
|
${2 ^ {1/{12}}}$
|
|
-->
|
|
<IMG
|
|
WIDTH="37" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img110.png"
|
|
ALT="${2 ^ {1/{12}}}$">. For historical reasons this sub-interval is called a
|
|
<A ID="1098"></A><I>half-step</I>.
|
|
A convenient logarithmic scale for pitch is simply to
|
|
count the number of half-steps from a reference pitch--allowing fractions to
|
|
permit us to specify pitches which don't fall on a note of the Western scale.
|
|
The most commonly used logarithmic pitch scale is
|
|
<A ID="1100"></A>"MIDI pitch", in which the pitch 69 is assigned to a frequency of 440 cycles
|
|
per second--the A above middle C. To convert between a MIDI pitch <IMG
|
|
WIDTH="17" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img111.png"
|
|
ALT="$m$">
|
|
and a frequency
|
|
in cycles per second <IMG
|
|
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img112.png"
|
|
ALT="$f$">, apply the
|
|
<A ID="1101"></A>Pitch/Frequency Conversion formulas:
|
|
|
|
<P>
|
|
<A ID="eq-pitchmidi"></A>
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
m = 69 + 12 \cdot {\log _ 2} (f/440)
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="178" HEIGHT="28" BORDER="0"
|
|
SRC="img113.png"
|
|
ALT="\begin{displaymath}
|
|
m = 69 + 12 \cdot {\log _ 2} (f/440)
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
f = 440 \cdot {{2} ^ {(m - 69) / 12}}
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="136" HEIGHT="27" BORDER="0"
|
|
SRC="img114.png"
|
|
ALT="\begin{displaymath}
|
|
f = 440 \cdot {{2} ^ {(m - 69) / 12}}
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
Middle C, corresponding to MIDI pitch <IMG
|
|
WIDTH="54" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img115.png"
|
|
ALT="$m=60$">, comes to <IMG
|
|
WIDTH="86" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img116.png"
|
|
ALT="$f=261.626$"> cycles per
|
|
second.
|
|
|
|
<P>
|
|
MIDI itself is an old hardware protocol which has unfortunately insinuated
|
|
itself into a great deal of software design. In hardware, MIDI allows only
|
|
integer pitches between 0 and 127. However, the underlying scale is well
|
|
defined for any "MIDI" number, even negative ones; for example a "MIDI pitch"
|
|
of -4 is a decent rate of vibrato. The pitch scale cannot, however, describe
|
|
frequencies less than or equal to zero cycles per second. (For a clear
|
|
description of MIDI, its capabilities and limitations, see
|
|
[<A
|
|
HREF="node202.html#r-ballora03">Bal03</A>, ch.6-8]).
|
|
|
|
<P>
|
|
A half-step comes to a ratio of about 1.059 to 1, or about a six percent
|
|
increase in frequency. Half-steps are further divided into <A ID="1107"></A><I>cents</I>, each cent being one hundredth of a half-step. As a rule of
|
|
thumb, it might take about three cents to make a discernible change in the
|
|
pitch of a musical tone. At middle C this comes to a difference of about 1/2
|
|
cycle per second. A graph of frequency as a function of MIDI pitch, over a
|
|
two-octave range, is shown in Figure <A HREF="node10.html#fig01.04">1.4</A>.
|
|
|
|
<P>
|
|
<HR>
|
|
<!--Navigation Panel-->
|
|
<A ID="tex2html707"
|
|
HREF="node12.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A ID="tex2html701"
|
|
HREF="node7.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A ID="tex2html695"
|
|
HREF="node10.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A ID="tex2html703"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A ID="tex2html705"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A ID="tex2html708"
|
|
HREF="node12.html">Synthesizing a sinusoid</A>
|
|
<B> Up:</B> <A ID="tex2html702"
|
|
HREF="node7.html">Sinusoids, amplitude and frequency</A>
|
|
<B> Previous:</B> <A ID="tex2html696"
|
|
HREF="node10.html">Controlling Amplitude</A>
|
|
<B> <A ID="tex2html704"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A ID="tex2html706"
|
|
HREF="node201.html">Index</A></B>
|
|
<!--End of Navigation Panel-->
|
|
<ADDRESS>
|
|
Miller Puckette
|
|
2006-12-30
|
|
</ADDRESS>
|
|
</BODY>
|
|
</HTML>
|
|
|