NOTE: The MIT Press released a second printing of Musimathics Volume 1 midSeptember, 2008 that fixed all errata in the first printing of Volume 1 as of 7/15/2008. I have started a new table for errata that have appeared since the Second Printing. of V1, below. Though there have been additional printings since that time, there have been no further corrections to the printed text. So if you have a more current printing, you should still consult the Second Printing errata.
Errata for Musimathics Volume 1 are different depending on which printing you have. Look at the bottom of the Copyright page. If the bottom of the Copyright page reads:
“10 9 8 7 6 5 4 3 2 1”, then you have the First Printing,
“10 9 8 7 6 5 4 3 2”, then you have the Second Printing.
The third, forth, etc. printing simply lop off the final digit. For any higher printing you should still consult errata for the Second Printing.
Volume 1 – Musical Elements 
Volume 2 – Musical Signals 

Dr. Gareth Loy Rewards for Reporting Errata! 
Musimathics Volume 1 Errata – First Printing 

Note: All of these errata were corrected in the second printing, and so this table is no longer updated. Refer to the table above for errata discovered after the second printing. 

Location 
Problem 
Correction 
p. 5 
Incorrect statement 
The sentence leading from the bottom of page 4 to page 5 is incorrect. It reads, “The spring begins to go slack as the mass rises, and when the mass reaches the equilibrium point, the spring no longer lifts the mass upward.” It should read instead, “The spring begins to go slack as the mass rises, and when the mass reaches the equilibrium point, the force due to gravity is greater than that of the spring.” 
p.
14 
Sentence should read as corrected. 
“Real numbers include all integers and all possible fractional and irrational values.” 
p. 20 
Correction 
Text reads, “The Lochrian mode is purely a theoretical mode, considered unusable by conventional music theory because of the tritone that exists between its final (7) and its fourth degree.” Text should read, “The Lochrian mode is purely a theoretical mode, considered unusable by conventional music theory because of the tritone that exists between its final (I) and its fifth degree (V).” 
p.22 
Penultimate note in F major scale is E not E#. Remove “#” after E. 

p.
26 
Whole note rest is incorrect 
The way I learned it in school, the wholenote rest “hat” is “off”, and the halfnote rest “hat” is “on”. Evidently, I need to go back to school on this one. The wholenote rest should look like this:

p.29 
Plus signs in figure are incorrect 
The plus signs in the list “f + 2f + 3f + ...” are mathematically meaningless. Fig. 2.27 better represents partials. A simple fix is to replace “+” with “,”, as shown below, representing the overtone frequencies as a series of integer multiples of a fundamental frequency. Alternately, to express the waveform that results from the combination of these sine components, let p(x) = sin(2 pi x t) where t is time, and then write p(f) + p(2f) + p(3f) + ....

p.
29 
Change heading 
(Not really an erratum, but it needs a better title.) Heading reads: “Partials, Fundamentals, and Overtones” Change to read “Fundamental, Partials, Harmonics, and Overtones” 
p. 42 just before Sec. 3.2.2 
Arguments to function f are reversed 
The function is defined on page 41 bottom as f(k,v), but the examples of use on page 42 reverse the arguments: f(v,k). For example, the text reads, “A0 = f(0,9)” but should read “A0 = f(9,0)”. Here are the function arguments as shown in the text:
Here are the corrected function arguments:

p. 43 
Incorrect statement 
Text reads: “We can use equation (3.1) to add and subtract intervals. If x = 2 in that equation, then frequency f_{x} will be two octaves above frequency f_{R}.” Text should read: “We can use equation (3.3) to add and subtract intervals. If v = 2 in that equation, then frequency f_{k,v} will be two octaves above frequency f_{R}.” 
p.43 
Plus signs are mathematically wrong 
This is similar to the error on p. 29, Fig 2.19. The simplest fix is to remove the “+” signs. Alternately, the waveform can be expressed as a sum of sine components using the function p(x) as described above.

p. 45 top Figures 3.2, 3.3, also p. 49 at the bottom figure 3.5 at the bottom 
The ratio for C is given as 1/2 
The ratio for lowest pitch C should be 1/1, so that the octave above (2/1) would be correct. 
p. 46 first full paragraph 
Correction 
Text reads, “So the tempered fifth is almost 2 cents flat of a perfect fifth.” Text should read, “So the tempered fifth is almost 2 cents flat of a just fifth.” 
p. 48 bottom, 7th sentence up from the bottom starting with "2. The fifth is found by taking....." 
Ratio is shown as 12:9 
Ratio should be 9:6. 
p. 104 Equation (4.12) and (4.13) Average Acceleration 
Formulas (4.12) and (4.13) and their discussion are incorrect and misleading 
Please replace the entire section “4.10 Acceleration” with this PDF file. 
p. 111 
Section 4.14.1 presents a wrong and a misleading derivation of kinetic energy. Equation (4.28) is missing a factor of 1/2 
Please replace the entire section “4.14.1 Kinetic Energy” with this PDF file. 
p. 111 last line 
Misspelling 
“posses” should be spelled “possess”. 
p. 119 
Improvement 
Text reads: “Let P be the power of a wave at distance r_{1} propagating along direction V. This means that an amount of energy P is flowing through surface a_{1} each second. If no energy is lost, then the same power will flow through a_{2} each second as well, and P/a_{1} = P/a_{2}. Since the areas of surfaces a_{1} and a_{2} are proportional to the squares of their distances from the source S, the intensity I varies inversely as the square of the distance to the source...” Substitute: “Let P be the power of a wave at distance r_{1} propagating along direction V. This means that an amount of energy P is flowing through surface a_{1} each second. If no energy is lost, then the same power will flow through a_{2} each second as well, so total power P at a_{1} would be the same as at a_{2}. However, since the areas of surfaces a_{1} and a_{2} are proportional to the squares of their distances from the source S, the intensity I varies inversely as the square of the distance to the source...” 
p. 119, Equation (4.36) 
Incorrect equation 

p. 132 Equation (5.8) 
Angular Acceleration Equation (5.8) is incorrect and misleading, similar to problem on p. 104. Equation (5.8) should read as corrected. 

p.134, bottom, in bolditalic text: 
Improvement in wording to clarify concept. 
Should read, “Circular motion is the result of a centripetal force applied at right angles to the instantaneous velocity.” 
p. 139 Figure 5.10 
Extraneous character 
An extraneous character “1” appears just to the left of the axis origin. It should be removed. 
p. 143, sec. 5.5 
2^{nd} para. 1^{st} sentence: remove ref. to E_k. Also, E_k=mv^2 is missing a factor of 1/2 Sentence and equation should read as corrected. 

p.144 
Equation (5.27) and the sentence enclosing it are missing a factor of 1/2. Sentence and equation should read as corrected. 

p. 152 
Typo (terms are switched) 
First sentence on page reads in part: “... known as the hammer (incus), the anvil (malleus) ...” but should read, “ … known as the hammer (malleus), the anvil (incus) ...” 
p. 174 Figure 6.14 caption text 
Citation for Roederer is incorrect. 
Text should read, “Adapted from Roederer 1973.” 
p. 181 Table 6.1 
Incorrect values for Bark #0 Lower Band Edge and Critical Bandwidth 
Per [Zwicker 1961]: Bark 0 should be 0Hz for Lower Band Edge, 100Hz for Critical Bandwidth. 
p. 181 Table 6.1 
Incorrect bandwidth for Bark #13 
The Critical Bandwidth for Bark #13 should be ~ 320 Hz, not 100 Hz. 
p. 181 Fig. 6.19 
Incorrect Q for band 15 
Critical Band #15 sticks out like a long pole in a tent. It should be ~ 6.44 (because 2900/450 = ~6.44). 
p.193
– 194 
Incorrect text 
Text incorrectly implies that if a speaker has intensity I, then at twice the distance the intensity is sqrt(I), whereas in fact it will be I/4. This mistake affects each equation on this page: sqrt should be replaced by a factor 1/4, and squaring by a factor of 4. Please replace the entire section “6.13.5 Distance Cues” with this PDF file. 
p. 203 
Incorrect word 
The penultimate paragraph it reads “whether we hold pressure or velocity constant”. It should read “whether we hold pressure or volume constant”. 
p. 204 paragraph immediately below Eq. (7.5) 
Incorrect text 
Text states “To determine the speed of sound we must determine the mass density of air...” However, I just gave the mass density of air in the previous sentence! What we really need is the average molecular mass of air. “To determine the speed of sound, we must determine the average molecular mass of air...” 
p. 210 sec. 7.8 2^{nd} para. 4^{th} sentence 
Incorrect spelling 
“lightening” should be “lightning” 
p. 226 Figure 7.24 
The arrow labeled “Wavelength increases” should point up, not down. 

p. 232 Equation 7.26 
Incorrect formula 
Formula for Doppler Shift in Two Dimensions is incorrect. The term u appearing inside the parentheses in the denominator should be outside the parentheses, as shown here.
I have created a Mathematica notebook that explains this equation better which is available here. A free player of Mathematica notebooks can be downloaded here. For a vector formulation of Doppler shift, see http://ccrma.stanford.edu/~jos/pasp/Doppler_Effect.html 
p. 246 midpage (5th paragraph) 
Incorrect formula V = 7.5^ –4 m^3, should be V = 7.5 x 10^ –4 m^3. 

p.
257 
Figure subheadings are incorrect. Should read as corrected. 
a) Fundamental (first harmonic) b) Second harmonic c) Third harmonic 
p. 260 sentence before runin heading “Bar with Free Ends” 
Sentence reads, “Plugging these values into (8.18) for n = 1,2,3,4,5 yields a fundamental and partials shown in the last column of table 8.3.” 
Sentence should read, “Plugging these values into (8.18) yields a fundamental and partials shown in the last column of table 8.3.” Delete “for n = 1,2,3,4,5”, there is no n in eq. 8.18. 
p. 346, second line 
Typo 
"corresponding surpisal" should be "corresponding surprisal" 
p. 373, first sentence of the second bulleted paragraph 
Typo 
"themeslves" should be "themselves" 
p. 401, last line 
Typo 
"truely" should be "truly" 
p. 359 Third sentence into section 9.17.5 
It would be better to say that Voss & Clark used a Gaussian noise generator than a uniform noise generator. 
Text reads: “To test this hypothesis, they synthesized melodies of three types using a computer: the first type made tone and rhythmic selections with a uniform...” But should read: “To test this hypothesis, they synthesized melodies of three types using a computer: the first type made tone and rhythmic selections with a Gaussian...” 
p. 396 
Description of the Fusion Petri net structure is misleading 
The text reads, “Only one of the two input places can trigger M1,” however this does not agree with the structure of the diagram to the right of that paragraph, nor with the definition of the Petri net firing rule given on page 391. The text should read: “Both of the two input places are needed to trigger M1.” 
p. 403 
Misspelled word: “answsers”. Text should read: 
EMI’s analysis database is essentially a compendium of answers to the question, What would Mozart have done in this situation? 
p. 460 
End Note 4 in “Chapter 5” last sentence states incorrectly, “Instead, circular motion is the vector sum of centripetal force and linear velocity.” 
Corrected text should read: “Circular motion is the result of a centripetal force applied at right angles to the instantaneous velocity.” 
Musimathics Volume 2 Errata(Volume 2 errata cover the hardbound and softbound editions. Some errata appear only in the softbound version.) 

Location 
Problem 
Correction 

p.
23 
Clarification 
The text names “the ones place”, “the tens place”, etc., without defining them. When referring to the representation of decimal numbers using place notation, the convention is to say “the ones place” for the rightmost digit, “the tens place” for the digit to its left, “the hundreds place”, etc., for subsequent digits to the left. 

p.
30 
Incorrect units 
Text reads: “In this example, values of x in the range –1/4 to nearly 1/4 map to the reproduction level value of 0.” Text should read: “In this example, values of x in the range –1/8 to nearly 1/8 map to the reproduction level value of 0.” 

p.
39 
Incorrect calculation 
In the section titled “Storage Requirements of Digital Audio” I discuss how much space is needed to store a certain amount of audio. Part of the calculation is correct, up to the point where I say that we require about 0.1764 megabytes/s. However, then I claim that this amounts to about 340 megabytes for one hour. This is incorrect. At 0.1764 mb/s, this means about 10.584 mb/minute (multiplying by 60). Multiplying this by 60 again, we end up with about 635.04 mb/hour, the correct figure. (It seems that what I did to arrive at 340MB/hour is to multiply the 5.67 seconds per byte by 60!) Doh! 

p.
65 
Incorrect term 
The last term in the expression reads: \frac{7^{7}}{7!} but should read: \frac{z^{7}}{7!} Please substitute the following.


p.
77 
Textual improvement 
Text reads: “We can see by inspection for sine and cosine that adding π to any angle is the same as negating it.” Text should read: “We can see by inspection for sine and cosine that adding π to any angle is the same as negating the amplitude of the waveform at that angle.” 

p. 79 
Bold italic remarks are incorrect 
First ¶ text reads: A real cosine consists of the vector sum of two halfamplitude phasors of opposite frequency. Text should read: A real cosine consists of the vector sum of two halfamplitude phasors in conjugate symmetry. Last ¶ text reads: A real sine consists of the vector difference of two halfamplitude phasors of opposite frequency. Text should read: A real sine consists of the vector difference of two halfamplitude phasors in conjugate symmetry. 

p.
79 
Incorrect equation 
There are two problems: 1) The equation shows the sum of two phasors on the left side, but should show the difference. 2) There is misuse of parenthesis on the right side. Corrected equation 2.53:


p.
83 
Incorrect definition of odd function 
Second sentence of second full ¶ in §2.6.12, the text reads: “For any x, the positive and negative functions are equal, and sin x = –sin x. Because of this, the sine function is an odd function. In general, a function f is odd if f(x) = –f(x).” The text should read: “For any x, the positive and negative functions are equal, and sin –x = –sin x. Because of this, the sine function is an odd function. In general, a function f is odd if f(–x) = –f(x)”. 

p.
84 
Incorrect definition of odd function 
The text, figure, and table on this page assert that odd functions are defined as f(x) = –f(x), but again, this is incorrect. The correct definition of an odd function is one in which f(–x) = –f(x). Here are corrected Figure 2.25 and Table 2.3:


p. 89 
Incorrect unit of measure 
In the final sentence of the page, f_{2} is given in millihertz (mHz); it should be in megahertz (MHz). 

p. 96 
Bullet points near bottom of page are incorrect 
Bulleted text reads, “Rotate the positivefrequency components of x(t) counterclockwise 90° by multiplying them by i. Rotate the negativefrequency components clockwise 90° by multiplying them by –i.” Text should read: “Rotate the positivefrequency components of x(t) clockwise 90° by multiplying them by –i. Rotate the negativefrequency components counterclockwise 90° by multiplying them by i.” 

p. 101 
Incorrect text in ^{2nd} full paragraph 
Text reads: “By investigating conjugate symmetrical phasors, we found that a real cosine is made up of the vector sum of two halfamplitude phasors of opposite frequency. Similarly, a real sine is made up of the vector difference of two imaginary halfamplitude phasors of opposite frequency.” Change to: “By investigating conjugate symmetrical phasors, we found that a real cosine is made up of the vector sum of two halfamplitude phasors of conjugate symmetry. Similarly, a real sine is made up of the vector difference of two imaginary halfamplitude phasors of conjugate symmetry.” 

p.
105 
Typo 
For “clarinetlike”, please substitute “clarinetlike”. 

p.
106 
Wording improvement 
For “spectrogram”, please substitute “spectrum”. 

p.
109 
Wording improvement 
Text reads, “If the product signal is all positive, then the signals being multiplied must be identical. If the product signal is mixed positive and negative, then the signals being multiplied are not identical.” Replace with: “The more positive the product signal is, the closer to identical are the source signals. The more mixed positive and negative the product signal is, the less identical are the source signals.” By “identical”, I mean that the source signals are exactly the same at every point, and do not differ in any property. This means, for example, there is no phase difference between the signals, no difference in DC offset, no amplitude difference, no timeshift difference, no difference in shape. For example, if some x(t) and y(t) signals are not identical, their product will be mixed positive and negative. But if x(t) and y(t) are the same at every point, their product will be all positive (or all 0 if either or both are all zero) at every point. 

p.
133 
Clarification 
Text reads: “Spectra of negative frequencies and positive frequencies of real signals are mirror images.” Please substitute: “Magnitudes of the spectra of negative frequencies and positive frequencies of real signals are mirror images.” 

p. 134 
Incorrect acronym 
The inverse discrete Fourier transform is at one point incorrectly given as IFDT; it should be IDFT. 

p.
135 

Top and bottom rows are correct. Middle rows show the superscript k+1 in the second column instead of k. Please substitute the corrected matrix figure below.


p.
136 
Incorrect assertions 
Text reads: “The imaginary part of x(n) will be zero if the imaginary part of X(k) was zero.” This is incorrect; x(n) will be real if the real part of X(k) is even and the imaginary part is odd relative to frequency 0. Please substitute the following text: “The imaginary part of x(n) will be zero if the real part of X(k) was even and the imaginary part was odd relative to frequency 0.” 

p.
136 
Unclear statement 
Text reads: “If the spectrum being processed by the IDFT came from a complex signal, the output of the IDFT may have a significant nonzero imaginary part. But we can still separate the real and imaginary output data in a meaningful way.” While not really an error, this is unclear, and not really what I wanted to say. What I meant to say is: “Even if the output of the IDFT has a significant nonzero imaginary part, we can still meaningfully separate the real and imaginary output data.” 

p.
137 
Incorrect assertion 
Text reads: “(If the imaginary part of the input spectrum X(k) is zero, computing the imaginary part can be skipped.)” This is incorrect. The imaginary part being zero is not sufficient. The real part must also be even. Please substitute the following text: “(If the imaginary part of the input spectrum X(k) is zero and the real part is even, then computing the imaginary part can be skipped.)” 

p.
140 
Typo 
Text reads: “... increasing the sampling rate R and/or the fundamental analysis frequency N until ...” Please substitute: “... increasing the sampling rate R and/or the fundamental analysis frequency f_{N} until ...” 

p.146 
Incorrect index for odd function of x 
The index for function x in the second summation term should be 2n+1, not 2n–1. Please substitute the following formula for Eq. 3.36.


p. 147 unnumbered equation at top of page 
Incorrect index for odd function of x 
As with the error on page 146, the index for function x in the odd part should be 2n+1, not 2n–1. Probably a copy/paste error. Please substitute the following formula.


p.
155 
Incomplete thought 
Text reads: “The Hilbert transform of a signal is another signal whose frequency components are all phase shifted by 90º (–π/2 radians).” Please substitute: “The Hilbert transform of a signal is another signal whose frequency components are all phase shifted by 90º (–π/2 radians for positive frequencies, π/2 for negative frequencies).” 

p.
156 
Typo 
Text reads: “Form the elementwise product of X(k) and h(t):” Should be: “Form the elementwise product of X(k) and h(k):” 

p.
163 
Improvement 
This is not technically an erratum, but some readers have tripped over this point. Please append the following additional explanation to the end of the second paragraph that now ends with the sentence, “This rule even works even when N_{j} ≠ N_{g}”: “Recalling that we’ve defined as zero any values that lie outside the range of functions f and g, note that we will get the same result if we set the limit of summation in equation (4.1) to N_{f} + N_{g} – 1 or to any larger value, even to infinity.” 

p.
164 
Convolution sequences are incorrect 
The convolution sequences shown in eq. 4.8 and eq. 4.9 are incorrect. Row 1 is OK but rows 2 through 5 should add a leading zero and delete the trailing zero. The summation line has 5+5 elements but should only have 5+5–1 elements. Delete the trailing zero.


p.
168 
Incorrect font face in equation 
First equation in §4.4.2, the font face for F(k) on right side of the equation is incorrect; it should be standard italic font face, as shown corrected here:


p.
168 
Incorrect term in text 
End of first full ¶ in § 4.1.1, text reads: “... indicates that the inverse Fourier transform of spectrum S ( k ) is signal S ( t ).” Should read: “... indicates that the inverse Fourier transform of spectrum S ( k ) is signal s ( t ).” 

p.
171 
Some values of sampled functions f(n) and g(n) and their product function are incorrect. The graphical functions in this figure are correct, including the spectral plot. 
There are really two problems here. First, there are transcription errors in the sampled functions f(n) and g(n). More fundamentally, the figure implies that the graphical functions can be reproduced from the sampled functions, which is not true, given how highly truncated the sample values are. Basically, this figure and accompanying text do not provide enough information to adequately demonstrate what is being discussed. The reader interested in a fuller understanding of the steps shown in this figure and corrected sampled functions are referred to this PDF file. If you prefer, here is the Mathematica notebook. A free Mathematica player for this notebook is available for download here. 

p.
172 
Text is in the wrong section 
Text reads, “Multiplying in the frequency domain convolves in the time domain.” True, but this ended up in the wrong section. (A following section talks about multiplying in the frequency domain). Substitute “Convolving in the frequency domain multiplies signals in the time domain.” 

p.
173 
Text and equation should both refer to the inverse Fourier transform 
Text immediately above Eq. 4.15 and in that equation should both refer to the inverse Fourier transform. Text should read: “Using the notation F ^{–}^{1}{ } for the inverse Fourier transform, we can express this operation as:


p. 175 
Incorrect function call 
In the For loop just under the comment “// Force the spectrum to be conjugatesymmetrical around 0 Hz” the text reads: RealSet(X[N  k], Im(X[k])); but should read ImagSet(X[N  k], Im(X[k])); 

p.
177 
Incorrect word in title 
Title reads: “Product of lowpass spectrum and the signal under test.” Should read: “Product of bandpass spectrum and the spectrum of the signal under test.” 

p.
181 
Caveat 
Please add this text to end of first paragraph after Eq. 4.19: “(For simplicity, this analysis ignores possible scaling asymmetries that may arise, depending precisely upon the particular implementation of the Fourier transform utilized in practice.)” 

p.
183 
Improvement 
Text reads: “It is a basic premise of physics that periodicity and frequency are ...” Substitute: “It is a basic premise of physics that period and frequency are ...” 

p.
183 
Typo 
Equation begins: “ f ( t ) ”, but it should just be: “ f ”. 

p.
191 
Typo 
Figure 4.36 title reads: “Sinc squared function and its triangular spectrum.” Please substitute: “Sinc squared function and its triangular waveform.” 

p.
203 
Incorrect description 
Text reads: “The components between the test signals are the result of phase delays introduced by the filter.” Please substitute: “The components between the test frequencies are the result of artifacts of the analysis method (windowing of the transform with a nonharmonic test signal).” 

p. 215 unnumbered display equation just after eq 5.31 
Incorrect term in equation 
In the unnumbered display equation just after eq. 5.31, three equalities are shown separated by commas. The rightmost term of the middle equality, the term –πT, should be – π / 4T, as shown corrected below.


p.
215 
Incorrectly labeled vertical axis in figure 
The tick marks on the vertical axis labeled “Phase Shift” should read from top to bottom 0, –π / 4, –π / 2, as shown corrected below.


p.
216 
Incorrect term in equation 
The coefficient t in eq. 5.32 should be capitalized, T.


p.
217 
Missing parentheses 
Equation 5.37 consists of three lines providing alternate definitions of y(n). The second term of the second equation needs parentheses around the imaginary exponent of e, as shown below. (The third equation line shows parentheses correctly.) As written: Ae^{iωnT}^{ }^{+ }^{ϕ} Should be: Ae^{i}^{(}^{ωnT}^{ }^{+ }^{ϕ}^{)} 

p. 217 – 248 in some editions 
Duplicate pages 
It appears that pages 217 to 248 are duplicated in some editions. That's OK, we won't charge you for the extra pages! 

p.
218 
Improper range of index 
In the ¶ immediately following equation 5.39 the text reads: “... parameter a is a set of M coefficients ...”, however, because the index r ranges from 0 to M in the equation, the text should read “... parameter a is a set of M + 1 coefficients … ”. 

p.
227 
Incorrect number in inline equation 
In the first ¶, the text says “If we set … ω = 2π / 96 …”, but should say “ ω = 2π / 16”. 

p.
227 
Incorrect figure 
Figure 5.18b should show a spiral shrinking counterclockwise.


p.
227 
Incorrect figure 
Figure 5.18c should show a spiral expanding counterclockwise.


p.
227 
Incorrect
descriptions for z^n < 1 case 
In the twocolumn inline table just above §5.11.2, the descriptions for the top and bottom relations are incorrect. The top righthand entry incorrectly says: “z^{n} < 1 Contracting counterclockwise spiral”, but it should say “clockwise”: z^n < 1 Contracting clockwise spiral The bottom righthand entry incorrectly says: “z^{n} > 1 Expanding counterclockwise spiral”, but it should say “clockwise”: z^n > 1 Expanding clockwise spiral 

p.
228 
Reference to wrong figures 
Mid first paragraph, references to figures 5.18a and 5.18b should instead reference 5.18b and 5.18c, respectively. Text should read: “If we sum the sequence shown in figure 5.18b, ...” “On the other hand, the sum of the sequence in figure 5.18c diverges, ...” 

p.
229 
The range of n is misstated 
The range of n is misstated in the text following eq. 5.53. The offending sentence reads in part: “For n = 0, 1, 2, …, the sum of all previous terms ...” but it should read “For n = 1, 2, 3, …, the sum of all previous terms ...”. 

p.
229 
Textual improvement 
Second sentence of the ¶ reads: “It follows that if r < 1, then a/(1–r) is closely approximated by S_{n} ...” But given that S_{n} is what we're approximating with a/(1–r), it would be a little clearer if the text said, “It follows that if r < 1, then S_{n} is closely approximated by a/(1–r) ...” 

p.
231 
Terms incorrectly identified in text 
The text incorrectly states, “Y(z) is the spectrum corresponding to the impulse response y(n).” Because the impulse response is defined as h(n), therefore its corresponding spectrum is H(z), and Y(z) is actually the spectrum corresponding to the filter output y(n) (that is, the spectrum of the input signal convolved with the impulse response). 

p.
234 
Extra term in inline expression. 
Text reads, “By the shift theorem of the Z transform, we can rewrite this term as Y(z) = z^{–}^{m}Y(z).” Text should read, “By the shift theorem of the Z transform, we can rewrite the shaded term as z^{–}^{m}Y(z).” 

p.
238 
Incorrect definition of z 
The exponent of e should be positive. The sentence reads: “Now consider what happens when we take equation (5.70), which represents the factored form of H, and set e^{–}^{iωT}...” Please remove the minus sign from the exponent. Text should read: “Now consider what happens when we take equation (5.70), which represents the factored form of H, and set e^{iωT}...” 

p.
239 
Incorrect assertion about e to imaginary powers 
Text reads: “and e^{anything} = 1”. It should add i to the exponent. Text should read: “and e^{i }^{anything} = 1” 

p.
240 
Incorrect subscript in formula 
The offending formula reads: “e^{iωT} – Q_{1} ” It should read: “e^{iωT} – Q_{n} ” 

p.
241 
Definition of Θ(ω) is incorrect, misplaced parenthesis 
Second line of text reads: “... we can define Θ(ω) = ∠H(e) ^{–}^{i}^{ }^{ωT}.” Text should read: “... we can define Θ(ω) = ∠H(e ^{i}^{ }^{ωT}).” Note the changed location of parentheses and the removed minus sign. 

p.
241 
Missing a phrase 
Text reads: “... minus the sum of the angles of the vectors to the point e ^{i}^{ }^{ωT}.” Should be: “... minus the sum of the angles of the vectors from the poles to the point e ^{i}^{ }^{ωT}.” 

p.
242 
Incorrect equation setup 
Equation 5.78 is correct, but its setup is not. The last ¶ reads, “Setting a = 1 and substituting r = gz, we get” ... But defining r = gz throws off the derivations shown in eq. 5.78, because this would imply that we'd need parentheses around the term “gz”, which would then change how the signs of the exponents distribute, leading to perdition. Instead, what I should have said is: “Setting a = 1 and substituting r = z, and carrying along our new variable g to scale the position of the pole, we get” … This allows the derivation shown in eq. 5.78 to work as written. 

p.
246 
Incorrect definition 
Text reads: “... we know we can say H(e^{–}^{iω}) = a_{0} + a_{1} e^{–}^{iω}.” (note the minus sign in the exponent) Should be: “... we know we can say H(e^{iω}) = a_{0} + a_{1} e^{–}^{iω}.” (note removed minus sign in the exponent) 

p. 246 paperback edition only 
Typesetting error in inline equation 
Only in the paperback version, there is a typesetting error in the text 6 lines from the bottom. Text reads in part: “Plotting this, we see that …”, then there is a malformed expression. The entire sentence with corrected relation should read: “Plotting this, we see that for 0 ≤ ω ≤ π we have the cosine curve shown in figure 5.25.” 

p.
248 
Incorrect z axis values. 
The z axis ranges of Figs. 5.27 and 5.28 should be 0 – 2, not 0 – 1, in order to agree with Eq. 5.83 and Figs. 5.25 and 5.26. 

p. 248 paperback edition only 
Typesetting error in inline equation 
In the paperback edition only, below Figure 5.28, some terms of the inline equation for the phase response function Θ(ω) are typeset incorrectly: the sin and cos terms are given in Greek. To wit, “σιν” should be “sin” and “χοσ” should be “cos”. 

p.
258 
Improvement 
The last sentence of the first full ¶ reads: “So we see that the behavior of the filter when the poles are on the real axis is identical to the onepole filter.” Substitute: “So we see that when the poles are on the real axis, this filter behaves similarly to the onepole filter.” 

p.
258 
Simplify, remove extraneous concepts 
Third full ¶ from bottom, the text reads: “Scaling power by ½ is the same as attenuating power by –3 dB SIL or by –6 dB SPL.” Please substitute: “Scaling power by ½ is the same as attenuating power by approximately –3 dB SIL.” 

p.
262 
Missing a phrase 
In the last sentence of the nexttolast ¶, the text reads: “... minus the sum of the angles of the vectors to the same point.” Should be: “... minus the sum of the angles of the vectors from the poles to the same point.” (This is another manifestation of the erratum on page 241.) 

p.
271 
Incorrect term in inline equation 
In the bottom inline equation on that page, the incorrect term “e^{x} – 1” is given in the rightmost numerator. The term should be “e^{∆}^{x} – 1”. 

p.
272 
Incorrect title text 
Text reads: “Plot of e.” Substitute: “Plot of e^{x}.” 

p.
274 
Typo 
Just before eq. 6.15 the text reads: “... then the second derivative of the sine is the negated cosine.” Should read: “... then the second derivative of the cosine is the negated cosine.” 

p.
275 
Incomplete formula 
In the sentence after eq. 6.20 the text reads: “Thus the acceleration of the air packet is –a^{2}.” Substitute: “Thus the acceleration of the air packet is –a^{2} sin aθ.” 

p. 278 – 279 
Improper value in equations 
Setting k = –a in eq. 6.22 was not a good idea. It led to problems in other equations on p. 278 – 279. As the text points out (p. 277), the value of k must be negative “because the slope of the energy curve is negative at all times” for the plucked string example under discussion. But negating k in eq. 6.22 was not the right way to handle it didactically, and doing so also introduced errors. The simplest fix is to change “–k” back to “k”. Thus, eq. 6.22 should be y(t) = ce^{kt}. Then the term “–k” in eq. 6.23 becomes just “k”, which corrects the error in applying the power rule in that equation. Also please change “–k” to “k” in eq. 6.24, and also in the first (unlabeled) equation at the top of page 279. Finally, in the first ¶ of §6.2.2, change “y(x) = ce^{–}^{kt}” to “y(x) = ce^{kt}”. Then I think everything works out fine. 

p.
281 
Alternative explanation 
The text states in part, “Realistically, an educated guess is our only option.” But we can repurpose the text in the immediately preceding section to find the general solution here by way of some complex arithmetic, which (as usual) is more direct than its “simpler” brethren. Start with the Harmonic Equation 6.31: mx''(t) + kx(t) = 0. By 6.29 we can write the solutions for κ_{1} and κ_{2} immediately as κ_{1} = sqrt ( –4mk) / 2m = i sqrt (k/m) and κ_{2} = –κ_{1}. Then eq. 6.32 the Trial Solution is x(t) = e ^ κ_{1}t and x(t) = e ^ –κ_{1}t. We know that e^it = cos t + i sin t, so x(t) = cos sqrt (k/m) t + i sin sqrt (k/m) t. This shows more directly where the resonant frequency definition comes from, i.e., sqrt (k/m), and leads directly into the discussion surrounding damped harmonic motion. But if this isn't your cup of tea, the existing text is just fine. 

p. 290 
Incorrect term in equations 
In Section 6.3.3, Driven Harmonic Oscillator, equations 6.53, 6.54, and 6.56 show term cy''( t ), but should show cy'( t ). Correct as follows:


p.
295 
Add sentence for clarity 
Add this (parenthetical) sentence for clarity immediately after Eq. 6.63: “(Though I've dropped the square root from the definition of δ( ω ), the derivative of the positive root is still valid.)” 

p.
305 
Incorrect terms in equations 7.16 and following 
In eq. 7.16, the 2^{nd} function call in numerator: “f (x_{1} + t_{1})” should be “f (x_{1}, t_{1})”, and “lim x → ∞” (limit as x approaches infinity) should be “lim Δx → 0” (limit as Δx approaches zero). The unnumbered equation directly under eq. 7.16 has a similar problem: “lim t → ∞” should read: “lim Δt → 0”. 

p. 311 
Incorrect count for matrix 
Text reads: “... we end up with an N × M matrix ...” Substitute: “... we end up with an N + 1 × M + 1 matrix ...” 

p.
312 
Incorrect subscript 
The subscript of the rightmost term in the numerator of eq. 7.28 and eq. 7.30 is incorrect. Eq. 7.28 should be:


p.
314 
Incorrect term in equations 
From the ¶ beginning “To evaluate the impact ...” over to and including the first equation on the next page, the ordinary derivative equations should all be of y with respect to t instead of u with respect to t because we are differentiating the function y = A sin ωt. For example, the first ordinary differential equation (shown midpage) would be:


p.
348 
Incorrect sentence 
Just after eq. 8.34, the text says, “This is the distance a wave travels over a unit length D.” Just delete that sentence. The sentence before eq. 8.34 says it better: “The velocity of a wave over some distance D may be found with ...”. 

p.
353 
Incorrect word 
Text reads: “Since the amount of power available to the loudspeaker from the battery...” Substitute: ““Since the amount of power available to the loudspeaker from the oscillator...” 

p.
356 
Fontosis in figure 
The music fonts in the two staves at the top right of the figure are incorrect. The correct figure is shown here.


p.
368 
Improvement 
The range for t on the lower line of stacked eqations 9.2 is given as t ≥ a, but it should be given as a ≤ t ≤ a+d. 

p.
369 
The x axis maximum is incorrect 
In Figure 9.7, the point at which the waveform goes to zero should be 0.7, not 1. Here is just the corrected righthand part of the figure.


p.
374 
Inline table midpage incorrectly gives frequency as DC 
Midpage, for k = 0 the inline table incorrectly gives the frequency as DC. The correct frequency for k = 0 is 1. The text should read (only first 4 columns shown here):


p.
375 
Inline table at bottom of page incorrectly gives frequency as DC 
Bottom of page, for k = 0 the inline table incorrectly gives the frequency as DC. The correct frequency for k = 0 is 1. Text should read (only first 4 columns shown here):


p.
376 
Improvement 
The equation shown for the nonbandlimited triangular wave, while correct, does not have the same phase as Eq. 9.11, the bandlimited triangular wave. The following formula provides the same phase as Eq. 9.11: f ( t ) = 2  t – π  / π – 1, but it is limited in range to 0 ≤ t ≤ 2π. To remove these limitations, if desired, we can replace Eq. 9.12 with the following: f ( t ) = 2  2 ⎣ t / ( 2 π ) ⎦ – 1  – 1 where ⎣...⎦ is the floor function. 

p.
376 
Typo 
In eq. 9.15, t(t) should instead be f(t). 

p. 380 
Incorrect formula for linear interpolation 
There are two errors on this page. First, near the end of the last full ¶ the text reads: “In this example, the interpolated result would be y = 0.7 y_{14} + (1 – 0.7) y_{15 }= 4.11”, The text should read, “In this example, the interpolated result would be y = 0.7 y_{15} + (1 – 0.7) y_{14 }= 4.11”. So the subscripts 14 and 15 should be swapped. Second, just below this the text reads: “Step 2: y = σy_{⎣}_{s⎦ }+ 1 – σy_{⎣}_{s+1⎦}”, but text should read: “Step 2: y = σy_{⎣}_{s+1⎦} + (1 – σ)y_{⎣}_{s⎦}”. 

p.
381 
Questionable assertion 
Text says, “Spectrally, linear interpolation is equivalent to a secondorder lowpass filter with a triangular impulse response.” Substitute: “Spectrally, linear interpolation is similar to a secondorder lowpass filter with a triangular impulse response.” 

p.
383 
Missing subscript on equation term 
In Step 1 of eq. 9.24, term A should be A_{n}. 

p.
385 
Printing bug 
The first cycle of the function in fig. 9.20 has an unexpected straight segment followed by a corner. This looks to have been an error in the software I used to create the figure. 

p.
390 
Bounding arrows are too small to see 
The negativefrequency spectrum of Fig. 9.29a shows two tiny bumps to the left and right of the line for –1000 Hz. These bumps are actually the heads of a doubleheaded arrow that is showing the extent of the modulating frequency f_{m} , but they came out too small to see on the printed page. The figure fragment shown here corrects this appearance by replacing the doubleheaded arrows with two singleheaded arrows bracketing the modulating frequency f_{m}.


p.
416 
Extraneous term in matrix 
Remove the term b_{0} from the middle bracketed term. Equation 9.47 should look like this:


p.
421 
Incorrect term in equation 
In the middle stacked equation, the term “1 – cos ( β t )”, should be “1 – cos ( β n )”. 

p.
422 
Incorrect subscripts in figure 
Subscripts on the parameters for the 2^{nd} patch bank should be subscripted “2” in order to show a progression to N in the bottom row of the patch. So β_{1} α_{1} ω_{1} and ϕ_{1} should be β_{2} α_{2} ω_{2} and ϕ_{2}. 

p. 423 
Printer's error in paperback edition 
Only in the paperback edition of V2: the last two lines of p. 422 are repeated at the top of p. 423. This printer’s error was introduced by the MIT Press when they put out the paperback edition; it does not appear in the hardbound edition. 

p.
441 
Incorrect subscript in equation in figure 
The equation for R0,1 is missing term Z_{0.} The equation should be written as (Z_{1 }– Z_{0}) / (Z_{1 }+ Z_{0}), as shown here.


p.
441 
Offbyone error 
The 2^{nd} paragraph of that section, 3^{rd} sentence reads: “If we sample these traveling wave components at N equidistant points ...” This should read: “If we sample these traveling wave components at N+1 equidistant points ...” The 3^{rd} paragraph of that section: “Using the terminology of stringed instruments … and the other end the nut termination at position x = N – 1” This should read: “Using the terminology of stringed instruments … and the other end the nut termination at position x = N since it's length is N.” 

p.
462 
The upper “board” is timereversed 
The figure doesn't show x(–1)h(1) as described. Flipping the top row about h(0) (timereversing) fixes it, as shown here:


p. 463 
Misspellings 
In the 4^{th} line below eq. 10.7, “analized” should be “analyzed”. In that same line, “monotically” should be “monotonically”.


p.
465 
Incorrect spectrum in figure 
Spectrum should be shifted by k/N so it's modulated to 0 Hz. Here's the corrected figure.


p.
465 
Incorrect derivations 
The
last two equations on page 465 and first equation on 466 should
contain h(r), not h(–r) This makes things a little easier on page 466. We can delete the 2^{nd} paragraph, beginning “The negative index for h(–r)...” through and including the following equation that ends “if and only if h(r) = h(–r).” Also, in equation 10.11, we can delete the text “if and only if h(r) = h(–r).” 

p.
479 
Improper substitution 
Equation 10.18 is supposed to be the same as equation 10.4 with the substitution n = sR. However, I got carried away and also substituted r = sR in the superscript of e. The superscript shown, – I 2 π k ( s R ) / N, is incorrect, and should be – I 2 π k r / N. The corrected formula is shown here:


p. 521 unlabeled display equation above eq. A.20 
Missing parentheses 
Parentheses are missing after the coefficient i. The equation should be:

Errata Rewards for Musimathics 
I have adopted Donald Knuth's reward system to encourage
readers to report errors in Musimathics. The first finder
of any error receives $2.56 (one hexadollar). Significant
suggestions are also worth $0.32 each. 
You can also communicate with me by snail mail: 
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language besides English to discuss anything more complicated than
travel directions. Though I can usually make out French and
Spanish if I work at it, it will probably be best for all
concerned if you communicate with me in English.
Sorry. 