# Musimathics Errata

### Last Updated: 04/17/18

NOTE: The MIT Press released a second printing of Musimathics Volume 1 mid-September, 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.

#### Determining the print version you own

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

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

# Musimathics Volume 1 Errata for Second and Subsequent Printings

Note: The second printing of Musimathics V1 corrected all known errata from the first printing. This table shows only the errata found in the second and subsequent printings.

#### Correction

p. 5

Correction

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. 17, 2.4.2 1st para. & Fig. 2.3

Solmisation syllable “ti” vs. “si”.

Solmization syllables in English countries use “ti” for the 7th degree while many other countries use “si”. The text and figure should have a footnote explaining this regional difference. (Strictly speaking, not an erratum.)

p. 19 Table 2.1

The number of semitones should not be shown for Augmented & Diminished intervals.

Yes, the augmented fourth and diminished fifth each have 6 semitones, but because these terms can be applied to other intervals as well, this is confusing. It would be better either not to indicate the number of semitones for Augmented and Diminished, or perhaps to give “+1” and “-1” for them to indicate that they add or subtract a semitone to the named interval, respectively.

p. 20

Correction

“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.”

“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 Fig 2.10.b

Typo in figure

Pitch labeled “B” should be “Bb” (“B-flat”).

p. 23 Fig. 2.12 (b)

Alignment in figure.

The bar on F Major has the flat in the position of A instead of B. The flat symbol should be nudged up a bit.

p. 24 Fig. 2.14

The printed spelling of the pentatonic scale shown in Fig. 2.14 is just one of several modes (circular permutations) of the scale's intervallic sequence, {2, 3, 2, 2, 3} discussed on page 23; there is no “official” pentatonic mode, and one is as good as another so long as the interval order is preserved. If I had it to do over again, I might have chosen the mode {2, 2, 3, 2, 3}, as it outlines a major triad in root position.

p. 24 Fig. 2.15

Typo

Pitch labeled “A” should be labeled “Ab” (“A-flat”).

(Note, this erratum incorrectly used to refer to Fig. 2.14. Heavens, an erratum in my errata!)

p. 26 Table 2.3

Whole note rest is incorrect

The way I learned it in school, the whole-note rest “hat” is “off”, and the half-note rest “hat” is “on”. Evidently, I need to go back to school on this one. The whole-note rest should look like this:

p. 31 Fig. 2.21

Typo

p. 34

Fig. 2.24

Instrument plot from Grey 1975 is probably an Eb clarinet tone, not a stringed instrument tone.

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

Correction

Text reads:We can use equation (3.1) to add and subtract intervals. If k = 2 in that equation, then frequency fk will be two octaves above frequency fR.”

Text should read:We can use equation (3.3) to add and subtract intervals. If v = 2 in that equation, then frequency fk,v will be two octaves above frequency fR.”

p. 46

first full paragraph

Correction

“So the tempered fifth is almost 2 cents flat of a perfect fifth.”

“So the tempered fifth is almost 2 cents flat of a just fifth.”

p. 59

Fig. 3.11

Correction

Last interval (bottom left of figure) is labelled as “Maj. 6th 8/5”. Should be “Min. 6th 8/5”.

p. 78

Table 3.9

Correction and clarification

Text on p. 78 describing Table 3.9, “Partch's 43-Tone Scale” incorrectly describes the highlighting and outlining shown in the table. The interval groups are correct, but the description of their highlighting and outlining is not.

For example, “Intervals of approach” (#2 - #10 and #35 - #43) are said to be “shown with a light outline”, but are not outlined in the table. Likewise, the “Intervals of suspense” (#19 - #26) are incorrectly described as having light shading, however the first and last row in the region is white, while the rows in between are shaded the same as all shaded rows. I suspect this is a typesetting problem.

p. 80

Fig. 3.25

Correction

The 9th degree of the 22-sruti scale is mis-labeled as an 81/84 ratio, when this should actually read 81/64. The ratio is given correctly in Table 3.10 on the previous page.

p. 80

Correction

End of first full paragraph under Figure 3.26, the text says “the size of the murchana interval corresponds to the Pythagorean comma”, however the text should read, “the size of the murchana interval corresponds to the Syntonic comma”.

p. 118 first ¶ after sec. 4.22, 2nd sentence

Incorrect word

Text reads: “Intensity I is the energy E per unit of time t that is flowing across a surface of unit area a

Delete “unit”, since intensity is the power in the area a, whatever its size.

Text should read: “Intensity I is the energy E per unit of time t that is flowing across a surface of area a

p. 118 Eq. 4.34

Incorrect term in equation

The denominator should be a, not a2. We divide power by the area, not the square of the area. The equation could then be more simply expressed:

I = E/t/a.

p. 118 Text line just below eq. 4.34

Incorrect term in equation

Inline equation reads: “I=P/a^2”, but should read “I=P/a” to correct the derivation from the revised Eq. 4.34.

p. 119 first ¶

Improvement

Text reads: “Let P be the power of a wave at distance r1 propagating along direction V. This means that an amount of energy P is flowing through surface a1 each second. If no energy is lost, then the same power will flow through a2 each second as well, and P/a1 = P/a2. Since the areas of surfaces a1 and a2 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 r1 propagating along direction V. This means that an amount of energy P is flowing through surface a1 each second. If no energy is lost, then the same power will flow through a2 each second as well, so total power P at a1 would be the same as at a2. However, since the areas of surfaces a1 and a2 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 Eq. 4.37

Incorrectly identifies unit for unitless ratio

Eq. 4.37 expresses the ratio of (limit of hearing)/(threshold of hearing) in units of W/m2, but this ratio is dimensionless and so has no units. Think of it this way: since the limit of hearing and threshold of hearing are both expressed as W/m2, the ratio of these units cancel, resulting in a dimensionless ratio.

p. 123

Units for average pressure variation incorrect.

Mid-page, in the paragraph beginning “Using standard values...”, the expression at the end of that paragraph should be

p. 123

Values for threshold and limit of hearing are incorrect.

Second to last sentence gives incorrect values for the threshold of hearing and limit of hearing. The value for the threshold of hearing should be 20 micropascals, not “0.1 Pa”, and the value for the threshold of hearing should be 20 pascals, not “1 Pa”.

p. 134 “rocket ship” example

Correction and clarification

In the first full paragraph, change the second sentence (“To get it to turn...”) to say, “If it is moving with linear velocity v, we would need to fire an engine on its side, pointing toward the circle's center, having force proportional to v2/r.”

p. 139 Figure 5.10

Extraneous “1” near origin

An extraneous character “1” appears just to the left of the axis origin.

p. 139

Incorrect formula

In the formula at the very bottom of the page, the first three “sin” terms should be removed. (The 4th is to be preserved.) The formula should read:

p. 140

Deprecated passage

Section 5.4.1 introduces definitions of time t (5.20) and frequency f (5.21) that are confusing, unnecessary, and unused in subsequent text. Please disregard ¶2 through ¶4 of that section, from the ¶ beginning, “It will often be more convenient...” to the ¶ ending, “we are measuring periodicity” (found just above the last ¶ of the page).

p. 143

Incorrect terms in equations

In the last full sentence of the page, v = d/t should be v = d/T and v = 2πA/t should be v = 2πA/T.

p. 144

Incorrect terms in equations

In the first sentence of the page, v = 2πA · (1/t) should be v = 2πA · (1/T), and f = 1/t should be f = 1/T.

p. 145

Incorrect terms in equation

In the first full paragraph, the text reads, “so in general l +l ≅ 0”.

The text should read: “so in general | l + | – | l | ≅ 0”.

In English the meaning is as follows: because sound waveforms are usually evenly balanced above and below ambient pressure, so in general, the magnitude of the highest peak level minus the magnitude of the lowest trough level are approximately equal to zero.

p. 147

Text is incorrect and insufficient.

First sentence reads: “Decibels of sound pressure level (SPL) correspond to twice the equivalent decibels of sound intensity level (SIL).”

Please replace it with: “Because they expresses ratios of quantities having the same unit, the dB and bel scales are dimensionless measures. Thus, to have meaning, the thing being measured (for example, SPL or SIL) must be specified explicitly. Saying 3 dB by itself is meaningless; however saying 3 dB SIL indicates a doubling of intensity.”

Second sentence begins, “Where ...”; please replace with, “Note that...”.

p. 152

Typo (terms are switched)

First sentence on page reads in part:

“... known as the hammer (incus), the anvil (malleus) ...”

“ … known as the hammer (malleus), the anvil (incus) ...”

p. 161
Figure 6.3a

Inverted axis.

The labels on the positive y axis of Figure 6.3a are inverted: “2” above, “4” below; but should be “4” above, “2” below.

p. 174 Figure 6.14 caption text

Citation for Roederer is incorrect.

p. 177
Fig. 6.16

Typo

Vertical axis says “phons”, but should say “phon”.

p. 181
Table 6.1

Incorrect bandwidth for Bark #13.

The Critical Bandwidth for Bark #13 should be ~ 320 Hz, not 100 Hz.

p. 182
Eq. 6.9

Unclear equation.

Equation 6.9 “Bark Number” should have another set of brackets for definiteness:

z ( f ) = 13 atan (0.00076 f ) + 3.5 atan ( ( f / 7500 )2 )

p. 203

Incorrect word.

The penultimate paragraph reads “whether we hold pressure or velocity constant”.

It should read “whether we hold pressure or volume constant”.

p. 204

Disambiguation.

Variable “m” is given two definitions on this page. To clarify, substitute M for m in Eq. 7.5 and surrounding text. Specifically, in the second full ¶, change the text to read “... gas is its mass M per unit of volume V”. Then in Eq. 7.5 “Mass Density” also change m to M:

ρ = M / V

Then the definition of m in Eq. 7.6 will not conflated with the “m” in Eq. 7.5. Note, the definition of m in 7.6 is used again in section 7.4.5, page 206.

p. 205

Correction to text.

At the end of ¶2, the text reads, “... between 10–7 N/atm and 10–5 N/atm.”

Replace with: “... between 10–6 N/atm and 10–5 N/atm.”

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. 249

Sec. 8.4.2

Missing preposition

Fourth sentence from bottom of that section reads in part “... creating an infinite variety sinusoidal motions ...” but should read “... creating an infinite variety of sinusoidal motions ...”.

p. 259

Incorrect number in text.

End of the penultimate paragraph, the text reads, “... to an increase in k by a factor of about 186.”

Please substitute, “... to an increase in k by a factor of about 108.”

p. 287 Fig. 9.3

Fig. 9.3 does not show the original “Ut queant laxis...” chant melody. The original chant melody is as shown.*

*This assumes the output of Guido's Method given in the figure would reproduce the original melody exactly. Strictly speaking therefore, it is not an erratum not to show the original, but because the original is historically important, I make an exception.

p. 301

Typo

In the inline table above §9.7.3 showing shifting f = 0.2615 by b = 10, there is a typo in the second-to-last line.

The text reads “26.15 10 = 26 1.15” but should read “26.15 10 = 26 1.5”.

p. 310

Typo

Last sentence at bottom of page should refer to formula 9.4 not 9.5.

Sentence should read: “So we can use the partitioning formula (9.4), as follows:”

p. 315

Typo

Missing curly-brace.

Text reads: “So we can write I ({4,6,7,10}) = (8,6,5,2}

Should be: “So we can write I ({4,6,7,10}) = {8,6,5,2}

p. 335

Textual inaccuracy

Second full ¶, the text states “… the distribution of the sum … is the convolution ….”, however this is true only if the random variables being summed are independent; it is not generally true. Please make the following substitution.

A fundamental insight of probability theory is that if a random variable x has distribution f ( x ) and an independent random variable y has distribution f ( y ), then the distribution of the sum of the two random variables f ( x+y ) is the convolution of f ( x ) and f ( y ). (Intuitively, two random variables x and y are independent if knowing the value of x does not change the probabilities of y.)”

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. 377 first ¶ after §9.22.2

Incorrect citation

In the network shown in figure 43b”,

In the network shown in figure 9.42b”.

p. 398 fifth ¶ after §9.23.2

Incorrect formula

M0(PianoImprov) = (2 · A4) + (3 · B4) + (2 · G4)”

M0(PianoImprov) = (2 · C4) + (3 · E4) + (2 · G4)”

p. 401, last line

Typo

"truely" should be "truly"

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?

# 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.

#### 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, last line

“Real numbers include all integers and all possible fractional and irrational values.”

p. 20

Correction

“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.”

“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 Fig 2.10.b

Penultimate note in F major scale is E not E#.

Remove “#” after E.

p. 26 Table 2.3

Whole note rest is incorrect

The way I learned it in school, the whole-note rest “hat” is “off”, and the half-note rest “hat” is “on”. Evidently, I need to go back to school on this one. The whole-note rest should look like this:

p.29, Fig 2.19

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 to the right, 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 sec. 2.8.1

Change heading. (Not really an erratum, but it needs a better title.)

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 fx will be two octaves above frequency fR.”

Text should read:We can use equation (3.3) to add and subtract intervals. If v = 2 in that equation, then frequency fk,v will be two octaves above frequency fR.”

p.43, Fig. 3.1

Plus signs are mathematically wrong, 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

“So the tempered fifth is almost 2 cents flat of a perfect fifth.”

“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 r1 propagating along direction V. This means that an amount of energy P is flowing through surface a1 each second. If no energy is lost, then the same power will flow through a2 each second as well, and P/a1 = P/a2. Since the areas of surfaces a1 and a2 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 r1 propagating along direction V. This means that an amount of energy P is flowing through surface a1 each second. If no energy is lost, then the same power will flow through a2 each second as well, so total power P at a1 would be the same as at a2. However, since the areas of surfaces a1 and a2 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

Intensity of a sound does not increase with the square of the distance from a sound source but with its inverse. Corrected equation is shown here. Note terms r1 and r2 have been switched.

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 bold-italic 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

2nd para. 1st 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) ...”

“ … known as the hammer (malleus), the anvil (incus) ...”

p. 174 Figure 6.14 caption text

Citation for Roederer is incorrect.

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, Section 6.13.5

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)

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 2nd para. 4th 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

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 mid-page (5th paragraph)

Incorrect formula V = 7.5^ –4 m^3,

should be V = 7.5 x 10^ –4 m^3.

p. 257 Fig. 8.13

a) Fundamental (first harmonic)

b) Second harmonic

c) Third harmonic

p. 260 sentence before run-in 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.

“To test this hypothesis, they synthesized melodies of three types using a computer: the first type made tone and rhythmic selections with a uniform...”

“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 as corrected.

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.”

“Circular motion is the result of a centripetal force applied at right angles to the instantaneous velocity.”

# Musimathics Volume 2 Errata First Printing

(Volume 2 has not had a second printing yet.)

#### Correction

p. 30 sentence before Fig. 1.24

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. 65 Eq. (2.31)

The last term in the expression reads: \frac{7^{7}}{7!}

Please substitute the equation to the right.

p. 77 the sentence just before §2.6.8

Textual improvement.

Text reads: "We can see by inspection for sine and cosine that adding Pi to any angle is the same as negating it."

Text should read: "We can see by inspection for sine and cosine that adding Pi to any angle is the same as negating the amplitude of the waveform at that angle."

p. 79

Bold printed remarks are incorrect.

A real cosine consists of the vector sum of two half-amplitude phasors of opposite frequency.

A real cosine consists of the vector sum of two half-amplitude phasors in conjugate symmetry.

A real sine consists of the vector difference of two half-amplitude phasors of opposite frequency.

A real sine consists of the vector difference of two half-amplitude phasors in conjugate symmetry.

p. 79, Eq. (2.53)

Two problems:

1) the equation shows the sum of two phasors on the left side, but should show the difference.

2) Misuse of parenthesis on the right side.

Corrected equation 2.53:

p. 83, Section 2.6.12

Incorrect example and incorrect definition of odd function in next-to-last paragraph.

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).

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, Figure 2.25 and Table 2.3

Text continues to assert that for odd functions, f(x) = –f(x), but again, this is incorrect.

The correct definition of an odd function is one in which f(–x) = –f(x).

p. 89

Incorrect unit of measure.

In the final sentence of the page, f2 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 positive-frequency components of x(t) counterclockwise 90° by multiplying them by i. Rotate the negative-frequency components clockwise 90° by multiplying them by i.”

Text should read (for clarity, I've underlined the text that must change): “Rotate the positive-frequency components of x(t) clockwise 90° by multiplying them by i. Rotate the negative-frequency components counterclockwise 90° by multiplying them by i.”

p. 101

Incorrect text in 3rd paragraph.

By investigating conjugate symmetrical phasors, we found that a real cosine is made up of the vector sum of two half-amplitude phasors of conjugate symmetry. Similarly, a real sine is made up of the vector difference of two imaginary half-amplitude phasors of conjugate symmetry.

p. 105 first

Typo

p. 106 Last line of section 3.1.6

Wording improvement

p. 109 2nd ¶, bold-italic text

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 time-shift difference, no difference in shape. For example, if some x(t) and y(t) signals are uncorrelated, their product will be mixed positive and negative. But if x(t) and y(t) are the same point for point, their product will be all positive (or 0) at every point.

p. 133 first bullet point in 3.3.8

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 Matrix figure in Section 3.4.1

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 to the right.

p. 136, Section 3.4.3, second paragraph

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.

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 Section 3.4.4 first paragraph

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 follows below.

Substitute the following text, “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, last parenthetical sentence on page

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.

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 § 3.5.1, 2nd

Typo

Text reads: “... increasing the sampling rate R and/or the fundamental analysis frequency N until ...”

Substitute the following text: “... increasing the sampling rate R and/or the fundamental analysis frequency fN until ...”

p.146

Eq. 3.36

Incorrect index for odd function of x.

The index for the odd part 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 the odd part should be 2n+1, not 2n–1. Probably a copy/paste error. Please substitute the following formula.

p. 155 1st

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).”

Substitute the following text: “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 Item 3

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 add sentences to end of 2nd

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 NjNg”:

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 Nf + Ng1 or to any larger value, even to infinity.”

p. 164 (§4.8) and p 165 (§4.9)

The convolution sequence is incorrect. Row 1 is OK but rows 2 through 5 should have a leading zero and delete the trailing zero.

The summation line has 5+5 elements and should only have 5+5-1 elements. Delete the trailing zero.

p. 168, first equation after section heading 4.4.2.

Font for F(k) on right side of equation is incorrect, should be standard italic font face.

p. 171 Figure 4.13

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, however, 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 4th line

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 immediately above Eq. 4.15 and that equation, should both refer to the inverse Fourier transform.

Using the notation F 1{ } for the inverse Fourier transform, we can express this operation as

p. 181

Add text to end of first paragraph after Eq. 4.19

(For simplicity, this analysis ignores possible scaling asymmetries that may arise, depending precisely upon which Fourier transform is utilized.)”

p. 183 first sentence

Improvement

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 Equation 4.21

Typo

Substitute:f

p. 191 Figure 4.36 title

Typo

Figure 4.36 title reads: “Sinc squared function and its triangular spectrum.”

Substitute: “Sinc squared function and its triangular waveform.”

p. 203 last sentence

Incorrect description

Text reads: “The components between the test signals are the result of phase delays introduced by the filter.”

Substitute: “The components between the test frequencies are the result of artifacts of the analysis method (windowing of the transform with a non-harmonic test signal).”

p. 215 unnumbered display equation just after Equation (5.31)

The term - pi T, should be -pi over 4 T.

p. 215 Figure 5.14

The Radians vertical axis should read from top to bottom 0, -pi over 4, -pi over 2.

P 216 Equation (5.32) Real Frequency Response

The coefficient t should be T.

p. 227 first line

Typo

Text says “ω=2π/96”, but should say “w=2π/16”.

p. 227 Figure 5.18 b.

Figure 5.18 b should show a spiral shrinking counterclockwise.

p. 227 Figure 5.18 c.

Figure 5.18 c should show a spiral expanding counterclockwise.

p. 227 table just before section 5.11.2.

|z^n| < 1 Description is incorrect, should be clockwise.

|z^n| > 1 Description is incorrect, should be clockwise.

|z^n| < 1 Contracting clockwise circle

|z^n| > 1 Expanding clockwise circle

p. 228 1st paragraph, section 5.11.3

Reference 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 text following Equation 5.53

The range of n is misstated.

The range of n is misstated in the text following equation (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. 234, about 2/3rd of the way down the page

Extra term in inline expression.

Text reads, “By the shift theorem of the Z transform, we can rewrite this term as Y(z) = z–mY(z).”

Text should read, “By the shift theorem of the Z transform, we can rewrite the shaded term as z–mY(z).”

p. 238 paragraph before Eq. 5.71.

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 eiwT...” (note the minus sign in the exponent)

Text should read: “Now consider what happens when we take equation (5.70), which represents the factored form of H, and set eiwT...” (note the minus sign no longer in the exponent)

p. 239 paragraph above equation (5.72)

Incorrect assertion about e to imaginary powers.

Text reads:and |eanything| = 1”. It should add i to the exponent.

Text should read: and |ei anything| = 1”

p. 240 3rd line

Incorrect formula

p. 241, 2nd line.

Definition of Θ(ω) function is incorrect, misplaced parenthesis.

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 text in bold-italic.

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. 246 text above equation (5.82)

Incorrect definition.

Text reads: “... we know we can say H(eiw) = a0 + a1 eiw.” (note the minus sign in the exponent)

Text should read: “... we know we can say H(eiw) = a0 + a1 eiw.” (note removed minus sign in the exponent)

p. 248 Fig. 5.27 and Fig. 4.28

Incorrect z axis values.

The z axis range of Figs. 5.27 and 5.28 should be 0 – 2, not 0 – 1, as shown, to agree with Eq. 5.83 and Figs. 5.25 and 5.26.

p. 258, first paragraph following the equations, last sentence

Improvement

Text reads: “So we see that the behavior of the filter when the poles are on the real axis is identical to the one-pole filter.”

Substitute: “So we see that when the poles are on the real axis, this filter behaves similarly to the one-pole filter.”

p. 258 3rd from last ¶

Simplify, remove extraneous concepts

Text reads: “Scaling power by ½ is the same as attenuating power by –3 dB SIL or by –6 dB SPL.”

Substitute: “Scaling power by ½ is the same as attenuating amplitude by approximately –3 dB.”

p. 262 next to last paragraph, last sentence

Missing a phrase (like on page 241)

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.”

p. 272 Fig. 6.8

Incorrect title text

Substitute: “Plot of ex.”

p. 274 just before Eq. 6.15

Typo

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, sentence after equation 6.20

Incomplete formula

Text reads: “Thus the acceleration of the air packet is –a2.”

Substitute: “Thus the acceleration of the air packet is –a2 sin aθ.”

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:

 Eq. 6.53 Eq. 6.54 Eq. 6.56

p. 295 after Eq. 6.63

Add 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 § 7.1.4

Incorrect terms in equations 7.16 and following

Equation (7.16) the 2nd function call in numerator: “f(x1 + t1)” should be “f(x1,t1)”, and the limit “as x approaches infinity” (x->inf) should be “as delta x approaches zero” (delta x->0)

Unnumbered equation directly under (7.16): limit “as t approaches infinity” (t->inf) should be “as delta t approaches zero” (delta t->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. 353 second from last line

Typo

Text reads: “... from the battery...”

Substitute: “... from the oscillator...”

p. 356 Figure 8.26

Fontosis

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

p. 368 Eq. 9.2

Improvement

Qualify the valid range of t on the second line of Eq. 9.2 as a ≤ t ≤ a+d.

p. 369 Figure 9.7

Typo: 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 part.

p. 374

Mid-page, for k = 0 the table incorrectly gives frequency as DC. Correct frequency for k = 0 is 1.

Text should read (only first 4 columns shown here):

p. 375

Bottom of the page, for k = 0 the table incorrectly gives frequency as DC. Correct frequency for k = 0 is 1.

Text should read (only first 4 columns shown here):

p. 376 Eq. 9.12

Improvement

The equation shown for the non-band-limited triangular wave, while correct, does not have the same phase as Eq. 9.11, the band-limited triangular wave. This formula provides the same phase as Eq. 9.11: f(t) = 2 | tπ | / π – 1, but it is limited in range to 0 ≤ t2π. To remove these limitations, if desired, we can replace Eq. 9.12 with the following:

f(t) = 2 | 2 t /(2π) – 1 | – 1

where ...is the floor function.

p. 380

Incorrect formula for linear interpolation. Two errors.

Last full paragraph, text reads, “the interpolated result would be y = 0.7 y14 + (1 – 0.7) y15 = 4.11”,

but text should read, “the interpolated result would be y = 0.7 y15 + (1 – 0.7) y14 = 4.11”.

Just below this the text reads “Step 2: y = σys+ 1 – σys+1,

but text should read “Step 2: y = σys+1 + (1 – σ)ys.

p. 381

Questionable assertion

Text says, “Spectrally, linear interpolation is equivalent to a second-order lowpass filter with a triangular impulse response.”

Substitute: “Spectrally, linear interpolation is similar to a second-order lowpass filter with a triangular impulse response.”

p. 390 Fig. 9.29a

Bounding arrows too small to see.

The negative-frequency spectrum of Fig. 9.29a shows two tiny bumps to the left and right above –1000 Hz. These are actually the heads of a double-headed arrow that is showing the extent of the modulating frequency fm , but they are too small to distinguish on the printed page. The figure fragment shown here corrects this appearance by replacing the double-headed arrows with two single-headed arrows bracketing the modulating frequency fm.

p. 416 Eq. 9.47

Extraneous term in matrix formulation.

Remove the term b0 from the middle bracketed term. Equation 9.47 should look like this.

p. 421 Section 9.5.3

Incorrect term in equation 9.49

In the second piecewise equation, “1–cos(beta*t)”, should be “1–cos(beta*n)”.

p. 422

Incorrect subscript in figure 9.52

Subscripts on parameters for the 2nd patch bank should be subscripted “2” in order to show a progression to “N” in the bottom row of the patch.

p. 440 Fig. 9.71

Incorrect subscript in equation in figure 9.71

The equation for R0,1 is missing term Z0. The equation should be written as (Z1-Z0)/(Z1+Z0), as shown here.

p. 441 § 9.7.3 2nd

Off-by-one error

In 2nd paragraph: “If we sample these traveling wave components at N equidistant points ...”

Should read: “If we sample these traveling wave components at N+1 equidistant points ...”

In 3rd paragraph: “...other end the nut termination at position x = N – 1”

Should read: “...other end the nut termination at position x = Nsince it's length N.

p. 462 Fig. 10.10

Upper “board” is time-reversed.

The figure doesn't show x(–1)h(1) as described. Flipping the top row about h(0) (time-reversing) fixes it, as shown here.

p. 463

Typo in 4th line

analized” should be “analyzed”.

p. 465 Fig. 10.13.a

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 last 2 equations, p. 466 first equation

Incorrect derivation

The last two equations on page 465 and first equation on 466 should contain h(r), not h(–r)
based on the substitution r = nr.

This makes things a little easier on page 466. We can delete the 2nd 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 equation 10.18

Improper substution.

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, –i2πk(sR)/N, is incorrect, and should be –i2πkr/N. The corrected formula is shown here:

p. 521 unlabeled display equation immediately before Equation (A.20)

Parentheses are missing after the coefficient i.

# 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.
Knuth notes that, “If you are really a careful reader, you may be able to recoup more than the cost of the books this way.”

You can also communicate with me by snail mail:

Gareth Loy
POB 151185
San Rafael, CA 94915.

I will attempt to reply to all such messages within six weeks of receipt.

Please do not send email to the above address except to report errors in Musimathics.  If you report an error via email, please do not include attachments of any kind, and use plain ASCII  text.  For representing mathematical equations in plain text, I suggest http://mathforum.org/dr.math/faq/faq.typing.math.html or http://www.karlscalculus.org/email.html.  The character set should be Western.

I'm very sorry, but I am not sufficiently fluent in any 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.

Regards,
Gareth Loy