Manipulating Numbers A

A.1 Units our unit of length. However, if we choose to use the foot as our unit of length, most adults have a height that is on the order of magnitude of ten Physics introduces many concepts that help us feet. Yet 10-ft is approximately 3 meters. understand and analyze physical processes— An atom has a diameter on the order of magni- sound and light phenomena, in particular. tude of 1-Ångstrom ≡ 1-Å≡ 10−8-cm. The size However, ultimately we come to trust the claims of the observable universe is 1027-cm. The weight of Physics by comparing observed numerical of most adults is an order of magnitude of 100- values of important parameters—such as sound lbs. The number of hairs on an adult with a full or light intensity and sound or light frequency— head of hair can have an order of magnitude as to how they are related to each other according high as 100-thousand.1 to the laws of Physics. We must note that a numerical value depends upon the choice of units. For example, my height is about 5-ft 10- A.3 Significant Figures in, or 70-in. In the metric system, my height is 1.78-m. Another unit of length is the Ångstrom, The order of magnitude is a very crude estimate abbreviated with the letter Å. Given that − of the value. We often have to express how 1-Å=10 10-m, my height is 1.78 × 1010- precisely we know the value of a parameter. Ångstroms. The relationship between various For this purpose, let us now introduce the term units for a given physical parameter is listed in significant figures. A value of my height as 70- Appendix D on the conversion of units. in provides me with two significant figures. A height expressed as 184-cm has three significant A.2 Order of Magnitude figures. Over the course of one day, my height certainly wouldn’t vary by as much as a centimeter. Therefore, I might wake up with a height We begin by noting that while Physicists would of 181-cm and go to bed with a height of 180- like to know a value as precisely as possible, they are also extremely concerned with what 1 is referred to as the order of magnitude of a We can arrive at this number as follows: The dimen- sions of a scalp is on the order of 30-cm by 25-cm, parameter. Typically it is expressed by giving corresponding to an area of 750-cm2. For a thick head the value to the nearest power of ten. Thus, for of hair, the distance between hairs might be about 1-mm. example, any value ranging from 5 to any value Each hair takes up an area of about 1-mm2, equivalent to (10−1cm)2 = 10−2-cm2. Therefore the number of hairs is less than 50 is rounded off to ten. We say that − 750/10 2 = 75, 000. On the other hand, if the distance a value of 21 has an order of magnitude of ten. between neighboring hairs in 3-mm, the area per hair An adult has a height on the order of magnitude would be 9-mm2; in this case, the number of hairs would of one meter, if we choose to use the meter as be 75, 000/9 ∼ 8, 300 and the order of magnitude would be 10,000.

© Springer Nature Switzerland AG 2019 409 L. Gunther, The Physics of Music and Color, https://doi.org/10.1007/978-3-030-19219-8 410 A Manipulating Numbers cm due to the compression of the disks in my Problem Confirm the above result. NOTE: One spine. If I were to express my height to only kilogram is equivalent to about 2.20-lbs and one two significant figures, I wouldn’t be taking into meter equals about 39.37 inches. account this change. Here is an ambiguity that can arise. Consider The lesson we learn from these examples is the following: If I were to tell you that my height that we must be careful not to take seriously is 180-cm, you would not know whether or not the number of significant figures listed for an the “0” digit is significant. The use of scientific important parameter. notation takes care of this problem as follows: If the zero is significant, I would express my height as 1.80×102-cm; this expression provides A.4 Relative Changes my height to three significant figures. If the zero is not significant, I would express my height as The relative change of a parameter gives us a 1.8 × 102-cm and we see my height expressed sense of the magnitude of the change that is to two significant figures. One foot is about one- independent of the units. Specifically third meter, which is, to one significant figure, 0.1-m. |change in value| relative change ≡ (A.1) Rounding off to a given number of significant |original value| figures can lead to some curious numbers. Con- sider the following: Normal body temperature is Here the vertical bars surrounding a number said to be 98.60 F. However, a body temperature represent the absolute value of the number.2 that is anywhere between about 970 F and 990 F is regarded as “normal” or acceptable. Given this Sample Problem A.1 Above we discussed the fact, how could such a strange number arise? change of my height over the course of a day. My Why should the temperature be specified to three height changed from 181-cm in the morning to a significant figures!? height of 180-cm at the end for the day. What is To answer this question note that 0C=(5/9)(0F- the relative change in my height? 32). Then we find that 98.60F=370Ctothree significant figures. In fact, doctors realize that Solution Letting the heights be h1 and h2,and people’s body temperatures are not expected to h be the absolute value of the change in height, be precisely 370C. This value was chosen as a we have to one significant figure rounded off number. h 181 − 180 Here is another example of how a choice relative change = = = 0.0.006 of units introduces uncanny constants. The so- h1 180 (A.2) called body mass index (BMI) is defined in British units (feet for length, seconds for time, or about 0.6%. and pounds for weight) as Suppose that you know that the frequency of × 2 703 weight in lbs/(height in inches) . a sound has changed by 4-Hz. Is this change significant? The relative change will be extremely Here again, we have a strange number informative. “703” occurring with three significant figures. Suppose that the original frequency is f1 and It wouldn’t seem to matter had the number been the new frequency is f2.Then chosen to be 700 precisely. Using metric system, with kilograms for weight and meters for length, |f − f | relative change ≡ 2 1 (A.3) changes the coefficient to unity (one). |f1|

2Thus, |2|=|−2|=2. A.4 Relative Changes 411

If we set f = f2 − f1, we can express the Here is an example wherein we can use al- relative change of the frequency as gebra to simplify a calculation. Suppose that we know x1 and its corresponding y1,sothaty1 = |f | bx1. Next, we know another value of x, say x2, relative change ≡ (A.4) |f1| and want to know its corresponding value y2.We could determine the constant b and then calculate Thus if the original frequency is 400Hz, the y2. Alternatively, we can use the following trick: relative change is 4/400 or one part in a hundred We have (0.01). [Note that the relative change as we have y2 bx2 x2 defined it would be 0.01 whether the change is = = (A.7) y bx x ±2 Hz, that is, from 400-Hz to 396-Hz or 400- 1 1 1 Hz to 404-Hz, since our definition of the relative or, change includes the absolute value.] y2 x2 = (A.8) In the context of music, if the relative change y1 x1 is much less than unity, the relative change in and finally sound frequency is closely related to the change in pitch: As we will learn in Chap. 12, a change = x2 in frequency by about 6% or one part in 16 y2 x1 . (A.9) x1 corresponds to a change by what is referred to 3 as a half-step. For example, if x1 = 4, x2 = 12, and y1 = 2, One might ask whether a change of one part y2 = (12/4) × 2 = 6. in a hundred significant. The answer depends We might want to know the relative change in upon the ear of the listener. For a person with y in terms of the relative change in x. We will a “good ear,” the change would certainly be now prove that ifthetwoareproportionalto significant. For many people the change would each other, the relative changes are equal. not be noticed. Then

y (bx) x Problem Suppose that the change in frequency = = b . (A.10) is 4-Hz and the original frequency is that of y bx bx the lowest note on a double bass, about 41-Hz. Determine the relative change in frequency. Thus y x = . (A.11) A.4.1 Proportionalities y x

Suppose that a variable y is proportional to a Sample Problem A.2 Here is a simple common variable x.Wehave example. Suppose that two people run the same distance d at different velocities. Given that the y ∝ x. (A.5) relative change in the velocities is 2%, what is the relative change in the times of travel? Alternatively we can write Solution The time of travel is proportional to the × = y = bx, (A.6) velocity: Since distance = velocity time, or d vt,wehave where b is a constant. v t = . (A.12) d 3For example, going from note B to note C or from A to A#. 412 A Manipulating Numbers

Distance is then proportional to velocity (at much less than unity. What is the relative change ﬁxed distance). Therefore, the relative change in in the frequency? the time is also 2%. It can be shown that the result is extremely simple.4 Now consider a string on a stringed instrument T y 1 T whose tension is changed. In Chap. 2, we will = . (A.24) learn that the frequency of a wave is proportional y 2 T to the wave velocity v. For example, if the tension is reduced by 2%, f ∝ v. (A.13) the frequency will be reduced by 1%. Generally, for small changes, if y ∝ xn, In addition, for a given string under variable y x ≈|n| . tension, as with a string on a violin, the wave y x velocity is proportional to the square root of the tension T . Therefore, the frequency is proportional to the square root of the tension: Sample Problem A.4 In Chap. 4, we discuss the √ intensity I of a source of sound or of light f ∝ T. (A.14) at various distances r. We pointed out that the intensity is inversely proportional to the square In terms of exponents, we have of the distance: 1 f ∝ T1/2. (A.15) I ∝ (A.25) r2

Sample Problem A.3 Suppose that we have two What is the relation between the correspond- strings identical except that they have differ- ing relative changes? ent tensions, T1=100-N and T2=110-N. The frequency of string #1 is 300-Hz. Find the frequency Solution In this case, n =−2. Therefore, of the second string. I 1 r ≈ (A.26) Solution We can use the relation f = bT 1/2. I 2 r Then 4We have f bT 1/2 T 1/2 y = bx1/2. (A.19) 2 = 2 = 2 1/2 1/2 , (A.16) f1 bT T where b is a constant. 1 1 Then or, = 1/2 − 1/2 = 1/2 − 1/2 1/2 y bx2 bx1 b(x2 x1 ). (A.20) f2 = T2 . (A.17) Now x = x + (x − x ) ≡ x + x. And, if x << x , f1 T1 2 1 2 1 1 1 we can use the binomial theorem to show that Finally, we have 1 x x1/2 ∼ x1/2 + . (A.21) 2 1 2 1/2 1/2 1/2 x1 T2 110 f2 = f1 = 300 × T1 100 Then = 300 × 1.05 = 315-Hz. (A.18) y b(x1/2 − x1/2) (x1/2 − x1/2) = 2 1 = 2 1 . (A.22) y 1/2 1/2 bx1 x1 Suppose that the tension is changed by a rela- y 1 x 1 1 x tively small degree. That is, the relative change is ∼ × = . (A.23) y 2 x1/2 x1/2 2 x A.5 Problems for Appendix on Numbers 413

Note that in this case the relation holds only Solution We use the same approach as in for relatively small changes in distances. Eq. A.18. We have (with four place accuracy) 3/2 3/2 Kepler’s Third Law R2 227.9 P2 = P1 = 1-yr × In the late 1500s the astronomer, Tycho R1 149.6 Brahe,5 compiled extremely precise data for = 1.88-yr (A.28) the orbits of the then known six planets about the Sun—Mercury, Venus, Earth, Mars, Saturn, and Jupiter. After Brahe’s death, his data became available to his assistant, Johannes Kepler, A.5 Problems for Appendix on so that he was able to analyze the data with Numbers extreme diligence, obviously without the help of 0 a computer or a pocket calculator. He discovered 1. Normal body temperature is said to be 98.6 F. that the data can be summarized in terms of three It is well known that in fact there is a range − mathematical statements. These statements are of “normal” body temperatures a range that referred to as Kepler’s Three Laws.However,it is considered quite acceptable. Given this fact, is important to keep in mind that these statements how is it that the temperature is given to three were not derived on the basis of any then-known significant figures? theory. It was Isaac Newton’s Theory of Gravity To answer this question express this that allowed Newton to give Kepler’s Laws a temperature in degrees Celsius, given that 0 0 theoretical basis. C=(5/9)( F-32). Note how the number “98.6” We will consider only Kepler’s Third Law. gives us the incredibly incorrect sense of what It deals with the relationship of the planetary is a normal range of healthy body temperature. period P of a planet’s orbit around the Sun and 2. Here is another example of how a choice its distance R from the Sun.6 As an example, the of units introduces uncanny numbers. earth’s period is one-year, while its distance from The so-called body mass index—BMI is the Sun is 150-million=1.5 × 108-km. defined in British units (feet for length, Kepler’s Third Law can be expressed as a seconds for time, and pounds for weight) × 2 proportionality: as 703 weight in lbs/(height in inches) . Here again, we have a strange number P ∝ R3/2 (A.27) “703” occurring with three significant figures. It wouldn’t seem to matter had the number Alternatively, we can write P = bR3/2,where been chosen to be 700 precisely. However, b is a constant. using kilograms for “weight” and meters for length changes the coefficient to unity (one). Sample Problem A.5 Consider the orbit of the Confirm the above result. planet Mars. Its distance from the Sun is 2.28 × NOTE: One kilogram is equivalent to about 108-km. Find the period of its orbit. 2.20-lbs and one meter equals about 39.37 inches. 3. Consider a pen that has a reservoir of ink containing 1-cm3 of ink. Estimate how many words you can write with the pen,giventhe following information: 5The Wikipedia article on Brahe life and work is in- – The width of a line of writing is 0.7-mm. credibly fascinating. See https://en.wikipedia.org/wiki/ – The thickness of a line of writing is 2.5- Tycho_Brahe. microns. 6The orbit is an ellipse, though typically very close to – A letter has an average length of 1-cm. being circular. Here, R refers to what is called the “semi- major axis” of the ellipse. We need not be exact here. 414 A Manipulating Numbers

– The average number of letters in a word is 4. The radius of the planet Jupiter is 7.786 ×108- ﬁve. km. Find the period of its orbit. How long a line of writing, including all punctuation, can the pen produce? Symbols B

– Å Ångstrom – di image distance for a lens – α attenuation per distance in dB per km – dmin minimum diameter of an image as a – γ gamma (used to characterize ultra high result of diffraction frequency electromagnetic radiation) – dnp near point of vision – λ wavelength – do object distance for a lens – μ linear mass density – E energy or electric field – φ loudness in phons – EM electromagnetic – ρ mass density – Ef final energy of a quantum system – σ optical activity (angle of rotation of axis of – Eind induced electric field polarization per distance through medium) – Ei initial energy of a quantum system – τ time of measurement of measurement – Eph energy of a photon – θ angle – f frequency – θc critical angle for the absence of refraction – f focal length of a lens – θinc angle of incidence of a ray of light – fB beat frequency th – θrf l angle of reflection of a ray of light – fn frequency of the n mode – θrf r angle of refraction of a ray of light – F force – a acceleration – g green color coordinate – a0 Bohr radius (∼ 0.53Å) – g(λ) green color coordinate for monochro- – A area or amplitude of a wave or of oscilla- matic light of wavelength λ tion – G green tristimulus value of a spectral inten- – b blue color coordinate sity − – b(λ) blue color coordinate for monochro- – h Planck’s constant (∼ 4.15 × 10 15eV/Hz) matic light of wavelength λ – hi height of image – B bulk modulus or magnetic field or blue – ho height of object tristimulus value of a spectral intensity – electric current – Bind induced magnetic field – I intensity – c speed of light in vacuum – I0 lowest audible sound intensity – C musical interval in cents – I(f ) spectral intensity with respect to the – d distance between two sources of a wave frequency – dB decibel –I(λ) spectral intensity with respect to the – die image distance for an eye (distance from wavelength the center of the effective lens of the eye and – IB intensity of blue primary the retina) – IG intensity of green primary – dim diameter of an image as a result of – IR intensity of red primary diffraction – k spring constant

– KE kinetic energy – R reflectance or red tristimulus value of a – L horizontal distance from source(s) of a spectral intensity wave – RT reverberation time – a length variable, not specific – s loudness in sones –mmass – S sum of tristimulus values (S = R +G+B) – M magnification of an object by a lens – SL sound level in dB’s – M magnifying power of a lens – t time or time interval –maxdo maximum object distance for the eye – T period of a sine wave or sinusoidally – max f maximum focal length of the eye oscillating object – min f minimum focal length of the eye – T for tension in a string – n index of refraction –T(λ) transmittance of a color filter as a – NB signal to the brain from the blue cones function of wavelength – NG signal to the brain from the green cones – v speed or velocity – NR signal to the brain from the red cones – v wave velocity – p pressure –Vvolume – ps sound pressure – x displacement or CIE color coödinate – p0 lowest audible sound pressure – X CIE tristimulus value –Ppower – y displacement of an SHO or CIE color – PE potential energy coödinate – q electric charge – Y CIE tristimulus value – r radius or red color coordinate correspond- – z CIE color color coödinate ing to a spectral intensity or distance from a – Z impedance or CIE tristimulus value point source – r(λ) red color coordinate for monochromatic light of wavelength λ Powers of Ten—Prefixes C

–103—one-thousand kilo as in kilogram (kg) –10−12—one-trillionth pico as in picogram or kilometer (km) or kilohertz kHz) (pgm) or picosecond (psec) [a bacterium has a –106—one-million mega as in megahertz mass of about 250 pgm; there exist chemicals (MHz) [frequency for WGBH FM radio waves such that merely 1 pgm can be fatal(!); ﬁber is 89.7MHz] optics signals can be made as short as a psec –109—one billion giga as in gigahertz (GHz) in duration] [the frequency of microwaves in microwave ovens is 2.5 GHz] Note that a cube, one-cm on a side, has a volume –10−2—one-hundredth centi as in centimeter of 1 cm3=(10 mm)3=1000 mm3. (cm) –10−3—one-thousandth milli as in millimeter Problem Suppose that an elemental device stor- (mm) ing each BIT of a computer hard drive has a vol- –10−6—one-millionth micro as in micrometer ume of 1 nm3, with the bits stored in a compact (µm); 1 µm ≡ 1micron [monochromatic red way. Suppose, too, that the total volume occupied light has a wavelength of ∼ 0.7 µm] by the bits is 1 cm3. How many bits are stored by –10−9—one-billionth nano as in nanometer this hard drive? (nm) = 10 Ångstroms =10Å (the size of an atom is typically a few tenths of a nanometer)

Constants Angle π = 3.14159 ... 1 radian (rad) = 57.3 degrees (deg) e = 2.7183 ... 1 deg=60 minutes c = 2.998 × 108 m/s . . . speed of light in 1 minute (min)=60 seconds (sec) vacuum Force h = 4.14 × 10−15 eV per Hz = 6.63×10−34 JperHzPlanck’s constant 1 Newton (N) = 0.224 pound (lb) . Length Weight equivalents (symbol =) on the Earth’s surface 1 Å ngstrom (Å) = 10−8 centimeter (cm) . 1micron(μ)=10−6 meter (m)=10−4 cm 1lb=454 gram (gm) . 1 cm = 0.39370 inch (in) 1 kilogram (kg)=2.2 lb . 1 in = 2.540 cm 1 ounce (oz)=28.350 gram (gm) 1 foot (ft) =30.480 cm Pressure 1 mile (mi)=5280 ft=1.61 kilometer (km) 1 atmosphere (atm)= 1.0×105 Pascals (Pa) = Time 14.7 lb/sq-in 1 day (d)=86,400 s Energy 1 year (yr)=3.15×107 s 1 joule (J) = 10,000,000 ergs Speed 1 electron-volt (eV)=1.6×10−19 J 1 mph=0.448 m/s 1 calorie (cal)= 4.19 joule 1m/s=2.23mph 1 Calorie=1 kilocalorie (kcal)=1000 cal=1 food calorie Area 1 foot-pound ft-lb)= 1.3549 J 1 sq-in = 6.4516 sq-cm 1 British Thermal Unit (Btu) = 252.00 cal = 778 ft-lb 1 sq-ft = 929.03 sq-cm Power Volume 1 horsepower (hp) = 746 watts (W) 1 liter (lit)= 1000 cu-cm 1 gallon = 3.785 lit

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We start with the three Laws of Dynamics that ﬁrst object with a force of equal magnitude but Isaac Newton (1642–1727) proposed in order to opposite in direction. account for the observed motion of the planets about the sun, the motion of the moon about Thus, if I am pushing on a wall with a force earth, and the motion of projectiles (like bullets of 450 N (∼ 100lbs), the wall is pushing back on or baseballs) just above the earth’s surface. To me with a force of 450 N. Likewise, if the mass these three laws he had to add his law for the of an SHO is pulling on the spring with a force gravitational force. F downwards, the spring is exerting a force F on the object upwards. Newton’s First Law If an object experiences no We can now combine Hooke’s Law with New- net force, its velocity will remain constant, be it ton’s 2nd and 3rd Laws as follows: Because the zero or otherwise. force on the massive object of an SHO is opposite to the direction of the displacement, we insert a Newton’s Second Law If an object does experi- minus sign and write Hooke’s Law as ence a net force, its velocity will change, being reﬂected by a rate of change of velocity with F =−ky Hooke’s Law (F.3) respect to time—the acceleration—that is given by In this equation, F is the force on the object. It is called the restoring force of the SHO because F a = or F = ma (F.1) it tends to bring the object back towards the m equilibrium position. Since F = ma,

Note that acceleration is to velocity as velocity k ma =−ky or a =− y is to position: m (F.4) change in position velocity = time interval (F.2) This equation can be analyzed mathemati- change in velocity cally. The analysis reveals that once the position acceleration = time interval and velocity is given at some time, the initial time, the motion is determined forever after. This Newton’s Third Law When an object exerts characteristic of Newtonian dynamics is referred a force on a second object, the second object to as determinism. In particular, the equation can automatically must be exerting a force on the

© Springer Nature Switzerland AG 2019 423 L. Gunther, The Physics of Music and Color, https://doi.org/10.1007/978-3-030-19219-8 424 F Crude Derivation of the Frequency of a Simple Harmonic Oscillator be shown to lead to the sinusoidal behavior of the We ignore the minus sign since we are inter- SHO.1 ested only in magnitudes. We then obtain Now we return to our expression for the period T . During the ﬁrst quarter cycle, the total k am = A (F.9) displacement is A, while the speed changes from m zero to a maximum value v . The average speed m We estimate that a ∼ 2a. Thus we obtain over 1/4 of a cycle is given by m k v 64A displacement A 4A A = a ∼ 2a=8 m = (F.10) v= = = (F.5) m m T T 2 time interval (T /4) T We ﬁnally arrive at the approximate relation Since the maximum speed is smaller that the average speed, we will use the estimate: m T 2 ∼ 64 (F.11) k vm ∼ 2v (F.6) so that Thus, m m T 2 ∼ 64 T ∼ 8 (F.12) 8A k k v = (F.7) m T This result compares very√ favorably with Next, the average acceleration over 1/4ofa the exact relation T = 2π m/k. cycle is given by The two expressions differ only by the numerical prefactor, 2π vs. 8, respectively. Their ratio change of velocity vm 4vm is 2π/8 ∼ 0.8. Most important is the agreement a= = = time interval (T /4) T between the two expressions with respect to the (F.8) mass m and the spring constant k.

The maximum acceleration can be obtained from Eq. (F.4) by setting y = A.

1The reader might be interested in carrying out the exer- cise in the appendix to this chapter, entitled “Numerical Integration of the Equation of Motion of an SHO.” In simple terms, the initial displacement y determines the initial acceleration a through Eq. (F.4). The initial acceleration determines the change in the velocity v from its initial value and hence its value soon after. The initial velocity determines the change in the displacement and hence the displacement soon after. This cycle is repeated on and on to yield the displacement, velocity, and acceleration for all future times. Numerical Integration of Newton’s Equation for a SHO G

This mathematical procedure for analyzing Now suppose we want to know y and v after Newton’s equation of motion (the Second Law) one second. Recall that for an SHO shows us dramatically one of change of displacement the most important characteristics of classical velocity = (G.4) physics, namely, that nature is such that if we are time interval given all the information about a system at some This will be strictly so only if the velocity instant of time, the behavior of the system in is constant. Otherwise, the expression gives us the future is fully determined. In the case of the the average velocity over the one second time SHO, the initial position x and velocity v of the interval. mass of the SHO determine the future behavior Because one second is small compared to of x and of v. We call this property determinism. the period (though not very much smaller) the We seek to show how the equation velocity doesn’t change very much in one second. Then the above equation becomes a reasonable k a =− y (G.1) ﬁrst approximation, albeit a crude one. m We then have (with a subscript 1 referring to a generates a sinusoidal function of time. For time t = ls): simplicity, we’ll set k/m = 0.1/s2,sothat − ∼ y(at 1 s) y0 v0 = = y1 − y0 (G.5) a =−0.1y (G.2) 1 s or ∼ where y is in meters and a is in m/s per second. y1 = y0 + v0 = 0 + 1 = 1(G.6) This value of k/m can be shown to√ correspond to a period of oscillation equal to 2π 10 =∼ 19.9s. Next, we seek the velocity v1 after one second. We will drop the units in what follows. We Recall that = will assume that initially, when the time t 0, change in velocity acceleration = (G.7) change in time y0 = 0andv0 = 1.00 e.g., v = 1m/s (G.3) This will be strictly so if the acceleration is a constant. Otherwise, the expression gives us the We have put a subscript 0 next to the letters, average acceleration over the one second inter- = so as to refer to t 0. val. However, if the acceleration doesn’t change much during that interval, we can use Eq. (G.7) as an approximation.

The initial acceleration can be expressed ap- You see how knowledge of y and v at any time proximately as allows you to ﬁnd y and v one second later. This process is known as numerical integration. v − v a =∼ 1 0 = v − v (G.8) For homework: 0 1s 1 0 a. Make a table, listing y and v after 1 s, 2 s, so that etc., at least up to the time when you obtain 1 − 1/2 oscillations. Round off all numbers v =∼ v + a (G.9) 1 0 0 to the nearest hundredth. Prepare your table as follows: But y0 a0 =− = 0 (G.10) 10 Time (s) y(m/s) v(m/s2) so that 0 0 1.00 ∼ y2 = y1 + v1 (G.11) 1 1.00 1.00 2 2.00 0.90 Similarly, after 2-s 3 2.90 ? y 4 ? ? v = v − 0 = 1 − 0 = 1 (G.12) 1 0 10 and WARNING: Any error you make is propa- y1 gated on to the numbers which follow. So, be v2 = v1 + a1 = v1 − (G.13) 10 careful. After 3-s: b. Make a graph of your results for both y vs. ∼ y3 = y2 + v2 (G.14) t and v vs. t. Compare your results with the plots below (Figs. G.1 and G.2). and y2 Below is a plot of the displacement vs. the v =∼ v − , (G.15) 3 2 10 time n. etc. Next we plot the velocity vs. n. c. Note that because of the approximations made, the numerical integration is unstable. That is, the displacement y oscillates with

Fig. G.1 Resulting plot of the displacement: a crude numerical integration G Numerical Integration of Newton’s Equation for a SHO 427

Fig. G.2 Resulting plot of the velocity: a crude numerical integration

ever increasing amplitude. The displacement Question: What do you suppose is the basis ultimately diverges to inﬁnity. for these equations? These equations, along with an =−yn/10, may be solved for yn and vn. They lead to: An Improved Approximation ∼ yn = 0.95yn−1 + 0.98vn−1 (G.21) Let n be the time in seconds. That is, n = 1 refers to one second, n = 2 refers to two seconds, etc. and Our previous approximation can be expressed as ∼ vn = 0.95vn−1 − 0.096yn−1 (G.22) ∼ yn = yn−1 + vn−1 (G.16) Tabulate and graph these equations. Use the and initial conditions ∼ vn = vn−1 + an−1 (G.17) y = 0andv = 1.00. along with 0 0 y a =∼ − n (G.18) 10 Just to show you how the above equations This last equation is exact. work. We have: We obtain a much better approximation if we = + use the following approximation, which replaces y1 0.95y0 0.98v0 Eqs. (G.16)and(G.17): = 0 + 0.98 = 0.98 (G.23) + = − ∼ vn−1 vn v1 0.95v0 0.096y0 yn = yn−1 + (G.19) 2 = 0.95 − 0 = 0.95 (G.24) = + = and + y2 0.95y1 0.98v1 0.95(0.98) ∼ an−1 an v = v − + (G.20) n n 1 2 +0.98(0.95) = ... (G.25)

v2 = 0.95v1 − 0.096y1 = ... (G.26) 428 G Numerical Integration of Newton’s Equation for a SHO

Fig. G.3 Resulting plot of the displacement: an improved numerical integration

Fig. G.4 Resulting plot of the velocity: an improved numerical integration

Compare your results with the plots shown in Instead, let the force be proportional to the (Figs. G.3 and G.4), which were obtained using cube of the displacement. Thus, the acceler- the above equations. ationisnowgivenby Then repeat the above with the initial values: 3 y0 = 1.00 and v0 = 0.0(Figs.G.3 and G.4). a =−0.1y (G.27) Problems with Numerical Integration and Find the displacement vs time for the two Oscillators of the velocity vs. n. initial conditions used for the SHO. For both Hooke’s Law (linear restoring 1. In this problem you will check the indepen- force) and the cubic restoring force, the force dence of the frequency of an SHO with am- increases with increasing displacement. How- plitude. To do so, use the initial conditions ever, the cubic restoring force increases faster y = 0andv = 4.00. Use the reﬁned set 0 0 with increasing displacement. This leads to a of equations to produce a plot of the displace- suggestion: Before obtaining your graphs, try ment vs. time and compare. to predict whether the frequency for a “cubic 2. In this problem you will study an oscillator oscillator” should increase or decrease with whose restoring force is not Hooke’s Law. increasing amplitude. Magnifying Power of an Optical System H

In Chap. 9 we introduced the parameter magnifi- If we have a compound lens system, such as cation of a lens as the ratio of the optical image a telescope or a microscope, the magnification size of a lens to the object size. However, the would be the ratio of the final optical image size magnification depends on both the focal length of to the input object size. On the other hand, the the lens as well as the position of the object. It is ultimate purpose of an optical instrument such as not a property of the lens alone. The maximum a magnifying glass (a single converging lens), a magnification is realized with an object posi- telescope, or a microscope is to increase the size tioned at the near point of an eye. However, the of the image of the object on the retina over the near point varies from person to person. It is stan- size of the image on the retina in the absence dard practice to define the magnifying power of the instrument. The maximum possible ratio of a lens to be the maximum magnification, as for an optical instrument is referred to as the achieved with a near point of 25-cm. magnifying power, to which we will assign the symbol M:

image on the retina with the instrument M = (H.1) image on the retina in the absence of the instrument

By increasing the image size on the retina, we An instrument that provides magniﬁcation have two effects: produces an “effective object” for the eye to view. This lens object is called the ultimate eye (1) The object appears to be larger. object. The greatest magniﬁcation is achieved (2) The details of the object are clearer: there is increased resolution.

© Springer Nature Switzerland AG 2019 429 L. Gunther, The Physics of Music and Color, https://doi.org/10.1007/978-3-030-19219-8 430 H Magnifying Power of an Optical System when the ultimate eye object is at the near point as the object for the eye to view—that is the of the eye, henceforth given the label np.1 This ultimate object for the eye-lens system. The lens is the closest that an object can be to the eye and is referred to as a magnifying glass.SeeFig.H.2. still be in focus. For the so-called normal eye, How much magnification we obtain depends this distance is about 25-cm, which is the value upon the distance between the lens and the eye. that is usually used as a standard for evaluating One specific case is shown in Fig. H.3,wherein and characterizing optical instruments. we see the magnifying glass held up against the eyeball. We also see the image of the magnifying glass, which becomes the ultimate eye object (in H.1 Image with the Naked Eye black), placed at the position of the near point. and with a Magnifying Glass For comparison, we see the actual object reposi- tioned in blue at the near point, where it would In Fig. H.1 we see the image produced on the have to be placed in the absence of the magni- retina R by a specific object at O of height ho fying glass in order to be seen clearly. Thus, we when the object is at the near point dnp .Asthe can appreciate the ability of the magnifying glass object is brought closer to the eye, the height hie to “magnify” in the applied sense. Essentially, of the image on the retina increases. The angle the magnifying has enlarged the ultimate eye θ is referred to as the angle subtended by the object located at the near point from ho to the center of the lens of the object. Clearly, this image height of the magnifying lens hi .This angle is a direct measure of the height of the height becomes the new effective object height image on the retina, since the distance die is fixed. ho (shown in the figure) for the eye. We also exhibit the central ray (dashed) from the The magnifying power of the magnifying object to the retina when the object is further glassisgivenby from the eye than the near point; we see clearly h /d d that the image height hie is reduced. M = o o = np (H.4) The maximum height of the image produced ho/dnp do on the naked eye,or“unaided eye,” is obtained when the object is located at the near point.The Sample Problem H.1 Show that the magnify- expression for the maximum height is simple to ing power of a magnifying glass that is held up obtain. We have against the eye is given by

h d dnp i = ie (H.2) M = + 1(H.5) ho do f where f is the focal length of the magnifying We set d = d for the naked eye, leading to o np glass. d max h = h ie (H.3) Solution We use the thin lens equation to obtain i o d np an expression for the object distance do in terms of f and the near point distance dnp . Noting that If we try to increase the image on the retina by moving the object closer to the eye than the near di =−dnp (H.6) point, we have the problem of not being able to bring the image into focus. However, by inserting we obtain a converging lens between the object and the eye, 1 1 1 1 1 the lens can create an image at the near point = − = + (H.7) do f di f dnp with an increased height ho. This image serves

1See Chap. 13 for details. H.2 The Microscope 431

Fig. H.1 The image on the retina for various object distances with the naked eye

Fig. H.2 Magniﬁcation of the potential image on the retina by a lens die ho'

ho R

d o h dnp ie

Substituting into Eq. (H.4) we obtain d 25 M = np + 1 = + 1 = 2 (H.10) f 25 dnp 1 1 dnp M = = dnp + = + 1(H.8) do f dnp f Sample Problem H.3 Determine the focal length necessary to produce a magnifying power of 40×. Note The symbol × is used to identify magnifying power. Thus, a magnifying lens that has a Solution We have magnifying power of 40 is characterized by the symbol 40×. It is understood that the near point dnp 25 M = 40 = + 1 = + 1 (H.11) is 25-cm and that the lens is held up to the eye. f f

Then Problem Use Fig. H.3 to show that the magnify- 25 ing power is given by = 40 − 1 = 39 (H.12) f h from which we obtain M = i (H.9) ho 25 f = = 0.64-cm (H.13) 39 Here hi is the image produced by the lens.

Sample Problem H.2 Assuming that dnp =25- H.2 The Microscope cm, calculate the magnifying power of a magnifying glass that has a focal length of 25- Whereas the purpose of a telescope is to produce cm. magniﬁcation of a typically huge object that is extremely far away, the purpose of a microscope Solution From Eq. (H.8)wehave 432 H Magnifying Power of an Optical System

Fig. H.3 Magnifying glass up against the eyeball

Fig. H.4 Schematic of a microscope is to produce magnification of an extremely small mum magnification by having the ultimate object object that can be brought extremely close to our located at the near point of the eye, as shown in eyes. For both devices there are two lens systems, the figure. with the final lens serving as a magnifying glass. The overall magnification of the microscope is Figure H.4 is a schematic of a microscope. the product of the magnification of the objective The first lens of a microscope is referred to as the and the magnifying power of the eyepiece: objective. It has an extremely small focal length fob that allows us to bring the lens very close to di dnp Mmicroscope = + 1 (H.14) the microscopic object. The second lens, with a d0 fe focal length fe,istheeyepiece or ocular, situated close to our eye. Since di fob, according to the thin lens for- We see that the objective of the microscope mula, 1 1 1 1 produces a real image that is situated just within = − ≈ (H.15) the focal length of the eyepiece.Wehavemag- do fob di fob nification by the objective: This first image is Thus, d0 ≈ fob and we can rewrite the above much larger than the object. Next, the eyepiece magnification as serves as a magnifying glass of the first image, producing a virtual image that is the ultimate dnp di Mmicroscope ≈ + 1 (H.16) “object” of the eye itself. One can obtain maxi- fe fob H.3 Problems on Magnifying Power 433

Unfortunately, there is a problem that we have H.3 Problems on Magnifying to confront: People don’t all have the same near Power point! Therefore, with the above design, the location of the image of the object would have to vary 1. Assuming that dnp =25-cm and that the focal from individual to individual. How can we design length of a magnifying lens is equal to 2.5- a microscope that takes this fact into account? cm, calculate the corresponding magnifying The answer is that we can change the design power. of the microscope so that the objective of the 2. The magnifying power of a magnifying glass microscope produces an image [which serves as is typically defined in terms of a standard near the object for the eyepiece] that lies between the point of 25-cm. However, in actuality, the near viewer and the eyepiece, but very close to the point depends upon the individual. focal point of the eyepiece. In this case the final Suppose that a magnifying glass is labeled image (produced by the eyepiece) moves very as having a magnifying power of 10×.My far away and is therefore easily visible to any near point has settled down to a stable value “reasonable eye”. The object height is also huge. of 60-cm. It can be shown that the magnifying power of the Calculate the effective magnifying power eyepiece is then: of the magnifying glass corresponding to my eyes by first solving Eq. (H.5) for the focal d d M = np + 1 ⇒ np (H.17) length of the magnifying glass. fe fe 3. Consider Eq. (H.17). Is the change from one expression to the other significant if M = The overall magnification of the microscope will 10×? What about M = 40×? then be dnp di Mmicroscope ≈ (H.18) fe fob Threshold of Hearing, Threshold of Aural Pain, General Threshold I of Physical Pain

If we look carefully at the set of equal loudness representing molecules within a box. To the right curves in Chap. 11, we find two curves that is plot versus time of the pressure produced by bound the curves from above and below. The the collisions of the molecules with the surface curve at the top is referred to as the threshold of the walls of the box. of hearing; the curve at the bottom is referred to In Fig. I.3 we depict the force on a sub- as the threshold of pain. The former represents microscopic area of an ear drum due to these the minimum intensity of a pure tone that can collisions over a short interval of time. Each spike be heard as a function of frequency. The latter represents a collision of a single molecule in corresponds to intensities that produce a sense the air, lasting about one-trillionth of a second.2 of pain as opposed to sound. We will refer to In Fig. I.3a we see the force on the outside of this latter curve as the threshold of aural pain the eardrum, while in Fig. I.3bweseetheforce so as to distinguish it from a threshold of a on the inside of the eardrum. The latter force more general sort of physical pain that one might is shown downwards in a negative direction to experience in a body structure such as one’s leg represent the fact that the above two forces have or back and normally find annoying. This curve opposite directions. is therefore a boundary between two types of In Fig. I.3c we see the two forces together sensation.1 See Fig. I.1 below. on one graph. Because each collision is so mi- We will begin our discussion by character- nuscule in its effect and because this rate over izing a bit the threshold of hearing in physical the entire eardrum is so huge, we experience terms. We have pointed out in Chap. 3 that a force that is uniform over the surface of the sound corresponds to a variation of pressure in eardrum and is extremely steady over time. To a medium. In the case of air, this pressure is pro- appreciate this fact, consider a short time interval duced by a huge rate of collisions of molecules of one-hundred trillionths of a second. Imagine of air against a surface. At a pressure of one that there were one hundred collisions during this atmosphere, there are about one trillion-trillion time interval and that they were spread uniformly (1, 000, 000, 000, 000, 000, 000, 000, 000) colli- over this time interval. The spikes would then sions each second on an eardrum, which has a touch each other, as shown in Fig. I.4. [Since col- surface area of about one square centimeter. The lisions are actually random, many would actually sound that we hear reflects a difference in the overlap each other.] forces on the two sides of an eardrum. In Fig. I.2 we see the results of a computer simulation of the pressure fluctuations. To the 2In order to get a feeling as to what the number one- left we see the positions of an ensemble of dots trillionth is, suppose we were to cover an area the size of a football field with the dots over the letter “i” in this print. There would be about one-trillion such dots. Therefore, 1The closest analogous threshold for a large body part that each dot takes up one-trillionth of the area of a football I can think of is the boundary between a tickle and an ache. field.

Fig. I.1 Hearing thresholds enclosing the set of equal loudness curves

Pressure vs Time 10.00

5.00 Pressure

0.00 90.00 100.00 Time

Fig. I.2 Computer simulation of the ﬂuctuations of the pressure on a surface (source: Wolfgang Christian and Gregor Novak, http://webphysics.davidson.edu/applets/Molecular/Pressure.html)

Fig. I.3 Force due to collisions of individual molecules on an eardrum I Threshold of Hearing 437

Fig. I.4 A segment of adjacent collisions

In fact, during an interval of one-hundred tril- sounds that are at the threshold of hearing. It lionths of a second there are one hundred trillion is not too difficult to calculate the sound that is collisions, not merely one hundred. Imagine how produced as a result of the variation of the force dense the spikes actually are! It is then easy to on the eardrums as a result of the discreteness appreciate why the force is extremely close to of the collisions. There is a frequency spectrum being constant in time. to this force that ranges from zero frequency to Because of the discreteness of the collisions, a frequency corresponding to the duration of a the force produced is not exactly zero or constant single collision (∼ one-trillionth of a second)— in time. In Fig. I.3, representing the collisions therefore, to a frequency of about one-trillion Hz. over an extremely minute area of an eardrum, The sound is a uniform mixture of frequencies (as we see that in that short time interval there are in the case of white light) and is often referred to thirteen collisions on the outside along with four- as white noise. It is not straightforward, if at all teen collisions on the inside, making a difference possible to compare the resulting intensity with of one collision. The situation for the force on the intensities corresponding to the threshold of an entire eardrum is different. Over a period hearing among the equal loudness curves since of about one second, we might have a total of these curves correspond to sounds having single about one trillion–trillion collisions on each side frequencies. Ideally, one should test people for of the eardrum. The difference in the number their threshold of hearing white noise. The best of collisions will be on the order of one part I can think of doing is to calculate the total inten- in one trillion—that is the still huge number of sity of the white noise over a range of audible one trillion collisions! The resultant variation in frequencies—say in the most audible range of the overall pressure is referred to as pressure 1000Hz to 3000Hz. The result is a total intensity fluctuations.3 As a result, even when there is no of about 10−12W/m2, equal to the threshold of sound wave present, the forces on the two sides hearing at 2000Hz. Thus, it is likely that the of an eardrum do not cancel each other. How sensitivity of the ear is as small as possible, does this net force compare to the force that is being close to being able to hear the collisions necessary to produce an audible sound? of individual molecules of air on the ear drums! In principle, with sufficient aural sensitivity, The inaudible sound due to the randomness of we should well wonder whether we can hear the collisions is referred to as background noise. the individual collisions of molecules! In fact, Our entire nervous system is wired up so that the sensing apparatus for hearing is designed normally we don’t sense the background noise so that we cannot hear these collisions. Their that tends to excite our nerves.4 presence is in fact mirrored in the fluctuations of What is the pressure at the threshold of hear- nerve impulses mentioned in chapter 10. We hear ing? It is a minimum at a frequency to which sounds that produce nerve impulses that are over we are most sensitive: about one-ten-billionth of and above these fluctuations. an atmosphere at a frequency of about 3000 Hz. Still, we would like to get back to the ques- Georg von Békésy, in his study of cats’ ears, tion as to how the fluctuations compare to the found that at the threshold, the eardrum has an amplitude of about one-tenth of an Ångstrom, 3For comparison sake, imagine rain drops colliding with a window pane and pitter-patter sound they produce. Now 4The same situation holds for vision. The eye is capable imagine what would happen if the density of raindrops of detecting the incidence of only about one-hundred were to increase greatly and their rate of collision increase photons over a period of a few seconds. Any greater greatly. Ultimately, we would describe the sound produced sensitivity could lead to problems due to noise within the as a steady continuous sound. nervous system for vision. 438 I Threshold of Hearing a size that corresponds to about one tenth the deterioration might well be belated in life since diameter of a single molecule! they revealed at my then current age of fifty- What about pain in general? Recently, I visited five the back of a forty year old. What should a physiatrist to deal with back pain that had be my response? I will paraphrase his response. developed over the past few months—another “Simple,” he said. “Stop worrying about your one of my many episodes. The main issues were back. People are fortunate to have a unique organ whether the source of the pain was due to a in the brain called the amygdala. One of its relatively simple problem of sprained ligaments functions is to control the response in your brain or strained muscles or a more serious problem to nerve impulses that can generate sensations of a herniated disc of the spine, or an even more of pain. When we have a steady or continual serious disease such as cancer. The fact was that input of such impulses, the amygdala can change I had no recollection of having had any incident the brain’s response to these sources of pain by that might have done damage to my back. causing the nervous system to treat them as our My visit to his office informed me of a fas- new level of background noise! And, if we are cinating phenomenon of the human body. The fully fortunate, we will eventually not sense any brain can reset the level of background noise that pain. If you dwell upon the pain, you will weaken can produce a sense of pain. My physiatrist’s the amygdala’s ability to perform this function diagnosis to my backache was simple: I am and therefore prolong your pain. So ignore the getting old! My discs are wearing down. Old pain and move on with your life.” MRIs of my back were evidence that my level of MAPPINGS as a Basis for Arriving at a Mutually Agreed-Upon J Description of Our Observations of the World—Establishing “Truths” and “Facts”

This book addresses the subject of physics, sound provides us with a wonderful example of how and light along with their relationship to our own people learn how to share a common mapping experience of sound and light. We have intro- of human experience. The parent shows the baby duced many concepts and equations that provide an object with a uniform surface of color, points, relationships among various physical quantities. and says the word for the color of the surface— Physics is all about relationships. And so is a for example, red. The term color is technically piece of music or work of fine art: These are better referred to as the hue, the term that we the relationships we perceive about the compo- will henceforth use in this appendix. The parent nents of a given piece of music or of a given then points to another object and says green.The work of art. In addition, there are relationships baby must learn that it is the hue that the word of these components and their sum total that is distinguishing and not another aspect of the produce the full composition with our personal object such as its shape or size. How is this aspect emotional responses to these compositions. All provided?—by using a number of objects that of these relationships are examples of what are hopefully differ essentially in all ways except for more generally called mappings. the hue of their surfaces. All forms of communication involve map- Note that, as we pointed out earlier in pings. Moreover, physical laws are mappings of Chap. 15, there is no way to tell how the actual observations—that we share—onto mathematical sensations compare among people. The same equations. Satisfactory communication, as would be true for the baby and the parent. well as satisfactory laws of physics, requires There are cases when a baby will be confused: agreement among those who share them. In this The parent shows the baby two surfaces, e.g. appendix, we will explore this subject a bit and one red, the other green, and assigns these two relate mappings to the complex philosophical different hues to the surfaces. The baby, on the questions of truth and fact. According to my other hand, seems to jump around, randomly colleague George Smith of the Department assigning one or the other hues to both surfaces. of Philosophy at Tufts University, who is an The baby doesn’t seem to differentiate. As you expert on Isaac Newton, my orientation towards might guess, the baby is color blind. How does the nature of scientific investigation is within the baby handle this confusion?1 I bring up color the framework of Newton’s proposed system thereof. 1In color blindness, two words, red and green, are per- I was led to consider mappings seriously be- ceived to represent the same experience—perceived color. cause of my study of color vision, as this subject Ultimately, the child will be told that the two words represent different colors that he/she is incapable of dis- compels us to think with great clarity about the tinguishing. I had problems of confusion of two words nature of mappings. One of the first sets of words of a different sort in hearing Yiddish as a child. In a parents teach their babies is colors. This process number of cases, two different pronunciations or even words were randomly assigned to what appeared to me

© Springer Nature Switzerland AG 2019 439 L. Gunther, The Physics of Music and Color, https://doi.org/10.1007/978-3-030-19219-8 440 J Mappings blindness here just to point out that there are J.1 MAPPINGS being Central to situations wherein people are not always able Organizing Human to establish mappings that they can agree on. Experience Imagine what the situation would be like if the prevalence of various types of colorblindness Essentially all human experience is dominated by were close to 100%! mappings of one kind or another. In the context of these notes, a mapping involves an association Note Suppose that an infant is fitted with a between two aspects of human experience. device placed over its eyes that inverts all im- Examples are: ages throughout infancy. Consider how the infant would map observations onto language: 1. written letters that spell words and their verbal counterpart as expressed words; 1. Can you think of situations where there might 2. words that refer to classes of objects referred be confusion in communication having to do to as nouns; with up and down? 3. images that we perceive in our conscious- 2. How would the child draw itself as it sees ness and the scenes; that produce physical itself in a mirror. Would the child draw an responses on the retina of an eye; image that is upside down to us? 4. printed musical notes and the tones produced by a musical instrument or the human voice; The important message that we learn from our 5. frequency and intensity of a pure tone and a study of color vision is that a chromaticity dia- sense of pitch; gram tells us only how various color sensations 6. spectral intensities and the corresponding sen- are distinguishable. The diagram is a mapping of sations associated with color (hue and satura- color sensations onto a set of pairs of numbers, tion) and brightness; the color coordinates. In technical terms it is 7. a sequence of positions of an object and the more specifically referred to as a one-to-one perception by the eye and brain as “motion” mapping of color sensations onto the diagram; of the object; “one-to-one” means that for each color sensation 8. the memory a person has of various percep- there is a specific point in the diagram and for tions of past inputs that correspond to actual each point in the diagram there is a specific color physical inputs to a person’s senses; sensation. 9. words that classify many objects that produce an experience that is common in some respect to represent the same idea. I was confused and blamed or respects—such as the appearance of tigers, my difficulty on my own inability to remember or learn lions, humans, or apples or love, or anger. the correct pronunciation or to distinguish between two Sometimes there is disagreement as to how “different” words. For example, the word for the number “two” was pronounced as either “tsvay” (as in the English objects are related to the words we ascribe word “say”) or “tsvy” (as in the English word “my”). I to them. Severe arguments can arise, often heard them as two different words. Ultimately, as an adult, merely as a result of people having different I was told that the reason that my relatives were jumping back and forth between two pronunciations was that they mappings. In these cases, ultimately what is naturally spoke with a “Galitsianer” accent (close to a important is how such classifications affect German accent). However, Litvaks (from Lithuania), with the way we use them—that is, how they are their Litvak accent, were regarded as being more cultured. mapped onto other actions or attitudes. The As a result, my relatives were sometimes embarrassed about their natural Galitsianer accent. important thing is for people to clarify as best 2 Of course there is a difference between the case of they can the mappings they are using. color blindness and a confusion between two dialects: While both involve a mapping of two different words onto what is conceived as representing the same experience, 2Recently (2008) the International Astronomical Union contrary to the latter situation, the former involves an decided to demote Pluto to the status of being a “dwarf intrinsic deficiency in perception that cannot be cured by planet”. See the article in the National Geographic explanation. J.3 The Concept of TIME as a Mapping 441

J.2 NUMBERS as a Mapping J.3 The Concept of TIME as a Mapping A number of years ago, my wife, my then nine- year-old son Avi, and I were in Grenoble, France What is time? The first level of consideration for one of my sabbaticals. Avi went to l’École and observation regarding time is the existence Houille-Blanche, a public school whose student of an ordered sequence of observations. We refer body was 50% French and 50% foreigners from to this observation as time order. This order- all over the world. Avi was placed in a class with ing is preserved in the patterns that our minds foreign children ranging in age from about 6 to provide in what we call memory. Imagine what 10, none of whom knew French. Few shared any would happen if our brains destroyed the order particular language. How are such children to be or direction of this sequence!! The next level in taught and be prepared to join the rest of the establishing or characterizing our sense of the so- student body in classrooms that used essentially called “time” requires that the physicist observe only French? All I will mention here is the a system behaving in a cyclic way: A pattern is following: The very first subject that students observed to repeat itself again and again, with were taught was mathematics—numbers being negligible observable change in the pattern. A the first of this subject. Why was this so? Because sense of equality in the evolving pattern leads one it is relatively easy to teach and discuss the to associate a time interval to a single occurrence concept of numbers without using a particular of the pattern and to then assign a numerical verbal language. All one has to do is to present value to an evolution of patterns—we number a number of objects such as one’s fingers and as- and order the patterns. The patterns are observed sign a word to each finger: “One,” “two,” “three,” to be occurring simultaneously with other phys- “four,”.... Or: “un,”“deux,”“trois,”“quatre,” ical observations so that we can assign a value ....Weareobservingtheestablishmentofaone- to the time interval of a sequence of physical to-one correspondence between an ordered set of observations. This special cyclic system becomes objects (such as our fingers) and words expressed our “clock.” Any time an event takes place, such verbally or in written script. This numerical one- as hearing a pulse of a sound or noting the posi- to-one correspondence is perhaps the simplest tion of a car on the highway, we can correlate, example of a “mapping.” that is map, that event onto the numerical value Most of us would appreciate the probability of the number of cycles of a clock has made that the first elements of communication between since we assigned an initial time. We can express an earthling and an extraterrestrial would be the the time interval between events by noting how sharing of our “words” for numbers. The reason many cycles took place between the two events. is the simplicity of this mapping and the small Here we have a mapping between two events and chance that the mapping will not be correctly number of cycles of a clock. communicated. Now consider that astronomers used the rotation of the earth and its revolution about the sun in order to measure time. These processes were believed to be periodic. People could thus count the days or years by making reference to the position of the sun or moon or stars in relation to the earth. Galileo is understood to have studied News, July, 2008. http://news.nationalgeographic.com/ the motion of objects with respect to time at one news/2006/08/060824-pluto-planet.html. It seems to me point by relying on his trust of good time keeping too ludicrous to regard astronomers of the past as having by a musician acquaintance.3 Later, Galileo used been mistaken in labeling Pluto as a planet. All we can say is that this new label allows astronomers to make statements about the now regarded “true” planets that will 3See Drake, S., The Role of Music in Galileo’s Experi- not be applied to Pluto. ments, Scientific American, p. 98, June 1975. Also see 442 J Mappings clocks that were still quite crude compared to It is because of the solidity of this edifice that what we would demand today. On the basis of the physicists have such a high degree of faith in the measurements of astronomers and scientists like laws of Physics—yes faith in the laws.6 How did Galileo, Newton was led to his three laws of me- we end up with quantum theory? The answer is chanics and his Universal Theory of Gravitation. that new experiments destroyed our total trust by Christiaan Huygens was able to improve on the revealing that the edifice was flawed in a domain limitations of the pendulum clock by extending that takes into account the behavior of systems the observations of Galileo on an inclined plan the size of atoms or smaller. Classical physics through his contributions to mathematics.4 Pure misses certain fine details and therefore had to mathematics, along with Galileo’s experiments, be refined. Ultimately, Quantum Theory wove an justified the trust he had in his clock. intricate edifice that became the basis for a new Later developments in the improvement of level of trust as did the classical laws—hence clocks with respect to precision and accuracy the trust in the accepted accuracy of the Cesium depended upon the Laws of Mechanics. Their clock. Nevertheless, physicists still use Classical quality was based on theory.Wenowhavethe Laws to account for or describe most behavior in Cesium clock, which is understood to have an the large, recognizing that corrections sometimes accuracy of one nanosecond (10−9 s) per day, have to be made to take into account Quantum or about one part in 100-trillion!5 The quoted Theory.7 accuracy is based upon an application of quantum Note: Ihaveraisedtheissueoftimehere theory. because it is an example wherein the nature of The logic behind clocks is a bit confusing: Ex- a mapping can be complex and obscure. periment based upon crude clocks led to theory; theory then led to more accurate clocks. Where lies the ultimate basis of evidence? Experiment or J.4 Mappings as the Essential theory? Is the logic circular and therefore flawed? Goal of Physics It might seem as if we use theory as our ultimate judge, so that circular reasoning is not We observe the world about us. These observa- present. But that is not exactly so. The situation tions are summarized by mappings within our is more complex. brains. We communicate with others in words While Newton proposed his laws on the basis that represent these mappings, hoping to summa- of a restricted domain of observations, his laws rize these mappings in such a way that we can have been ultimately applied to a vast set of in- establish a one-to-one correspondence between terconnected physical phenomena—for example, our words and our observations that are shared all the developments in engineering and medicine among our fellow human beings. I will repeat a and in sending rockets to the moon. The Laws statement to be found in Chap. 5: of Physics weave a network, an edifice, such The essential goal of physics is to establish that if any component were to fail to fit the a theoretical framework for describing in a theory, the structure would lose its reliability. quantitative way what we decide to and are the website (2-4-2011): http://www.joakimlinde.se/java/ 6Reader beware: the faith to which I am referring is not the galileo/, which contains a beautiful applet that enables us same as the faith in religion, which has no such edifice and to appreciate Drake’s conjecture as to how Galileo might yet has its great benefits in helping some people handle the have used a musician to arrive at his law that when a ball complexity of life’s experience. rolls down an inclined plane, its speed increases linearly 7I must warn the reader that the above view of Physics with time. as being ultimately dependent upon faith is not shared by 4 A pendulum bob moves along a curved path that can be many physicists. Interestingly, this issue does not seem to analyzed in terms of an infinite sequence of infinitesimal arise among mathematicians because they recognize that inclined planes having different angles of inclination. a mathematical theory is dependent upon a set of axioms 5(2-5-2001): http://en.wikipedia.org/wiki/Atomic_clock. that is not provable. J.4 Mappings as the Essential Goal of Physics 443 able to measure. That framework makes use the photon is a conceptual tool that is represented of models, concepts, and images. However, mathematically in physical laws that are mapped its ultimate content is a set of mathematical onto observations. Interestingly: While physicists equations, which we call laws. The laws are might debate and disagree about the issues of as simple and all-encompassing as possible, truth, fact, and reality, these disagreements don’t and provide relationships among measurable seem to affect their ability to conduct the disci- quantities. pline of physics. One of the most remarkable examples of such a mapping is the following: We observe an enor- AFinalRemark mous variety of materials made out of a relatively small number of different kind of atoms (fewer Let us recall the opening chapter of this book, than 100) arranged in a multitude of ways. We wherein we exhibited a graph of a wave for a have millions of different organic compounds, piece of music. It can be a joy to contemplate metals, and alloys, complex materials like wood that this curve has all the content that maps and so on. They have a variety of physical prop- onto our incredibly rich, sensual, and emotional erties with respect to pliability, density, color, experience when listening to the sound asso- texture, and so on. And yet, it is understood by ciated with the wave pattern. The graph maps physicists that this entire variety of properties is onto a sound wave that ultimately produces nerve describable, that is, can be mapped onto a set of impulses that are processed in our programmed a small number of mathematical equations. brains so as to produce our musical experience. For comparison sake: Note that a finite set of The processing in the brain is quite complex. If coupled algebraic equations are incredibly simple content were to be measured by, say the number in content. For example, suppose that we have of megabytes stored on a computer, we would to solve the two equations, x + y = 1and find that the ultimate content of our wonderful x − y = 3 for x and y. The solution is x = 2and musical experience is vastly greater than the y =−1. Such equations cannot provide us with content of the input sound wave.9 the richness of content that is associated with the behavior of materials. Having discussed mappings, we need to answer the question of the relationship of mappings ory of the so-called photoelectric effect, which involves to questions of truth or fact. In my opinion, these electromagnetic radiation knocking electrons out of a metal. Einstein’s theory assumes that discrete units of issues are not subject to being defined by sci- radiation collide with the electrons. As a result it has been ence. They are purely philosophical. In practice, commonly understood that this experiment along with I would say that we tend to use these terms in Einstein’s theory provides proof as to the photon’s exis- science to describe mappings for which there is tence. Someone in the Harvard audience asked whether the photoelectric effect does indeed prove the photon’s essentially universal agreement under the rules existence. The vote was overwhelmingly in the negative, that are used by scientists to test for acceptability. but not unanimously. Interestingly, no one who voted in Any person has a right not to accept these rules, the negative proposed an experiment that does prove its often to their own detriment. To add to this list of existence. Another example of existence questions has to do with atoms. Planck rejected their existence until what I consider a non-scientific issue is the ques- as late as the early 1900s. Brownian motion (the motion tion of reality. “Do photons really exist?” I heard of micron sized particles in a water suspension, due to recently this question argued at a colloquium at collisions of water molecules with the particle) is given Harvard University, during which Nobel prize credit for proving their existence. 9 winners couldn’t agree!8 At best, we can say that The reader is invited to see the book by Abraham Moles entitled Information Theory and Esthetic Perception, [University of Illinois Press, Urbana, IL, 1969]. The 8I ought to be specific: While Max Planck based his author discusses the how one might assign a megabyte theory of Black Body radiation on the assumption that value to a piece of music. You can read a review electromagnetic radiation is absorbed and/or emitted by of the book on this website: https://www.academia. atoms in discrete units, he didn’t believe that the radiation edu/31546315/Information_Theory_and_Esthetic_ itself was quantized. In 1905, Einstein produced a the- Perception_Abraham_Moles. The Mysterious Behavior of the Photon K

Abstract that measure time by counting the number of The goal of this Appendix is to demonstrate repetitions of a measured parameter that are that the behavior of photons and all matter assumed to behave in a periodic way1; at the quantum level led to a decisive change 2. Mathematical Relationships in the discipline of Physics. Whereas prior Representing the measured parameters by to the quantum era, physicists were able to mathematical variables such as x for position account for observations in terms of mathe- and t for time; then using the laboratory matical models as well as images, quantum measurements to summarize relationships systems revealed that we have to give up among the physical parameters; the use of images as part of a theory. The The combination of the above two compo- photon exhibits both wave and particle like nents leads to Laws of Physics. behavior, each according to the experiment 3. Representation of Objects by Images performed. We discuss in detail an experiment Note that Kepler did not see a planet move with photons that cannot be explained in terms in an elliptical orbit about the Sun. He proof either a particle or a wave like nature. Our duced this image by examining the data he conclusion is that it is impossible to ascribe a had for the position of a planet over time. nature to the photon that mirrors anything we Astronomers and astrophysicists accumulate can observe in our macroscopic world. data that enable astrophysicists to use theory to deduce the physical attributes of objects in outer space. For example, due to the detailed data accumulated about the Sun, we can de- The goal of this Appendix is to demonstrate that duce the physical state of the Sun extending the behavior of photons and all matter at the from its surface down to its center. We can quantum level has led to a decisive change in deduce the nature of a neutron star. the discipline of Physics. To understand what this change is about we need to review a bit of the In the realm of quantum phenomena, which history. Great physicists like Galileo and Newton includes the behavior of photons, the third com- and numerous others introduced a methodology ponent, the ability to represent objects by images that involved the following components: is denied to us. In order to demonstrate how this is so I will summarize the laboratory observations 1. Measurements in the Laboratory for a number of experiments involving photons. Choosing parameters that are measured by laboratory devices: e.g., the position of planets seen in a telescope; and the position of objects in motion; clocks 1See Sect. J.3 for a discussion about the complexity of an essential device as a clock.

Fig. K.1 Interference with a low intensity beam of photons

Fig. K.2 Schematic drawing of the action of a E I /2 calcite crystal 0 V I 0

o I /2 H 0

calcite

The results are not deniable. The issue is what are calcite crystals on a beam of light. In particular, we to conclude about the nature of the behavior we described the behavior of a beam of light that of photons. passes through a single calcite crystal as shown in Fig. K.2. In this ﬁgure, we see the exit of the ordinary K.1 Experiments with Calcite ray labeled with the letter O as well as the Crystals extraordinary ray, labeled with the letter E.Ihave modiﬁed the original image in Sect. 8.6.2 by In Sect. 7.3.3 we described two-slit interference replacing the angle 00 by a letter H indicating that with such a low light intensity that one can detect the axis is HORIZONTAL. I have also replaced individual photons. With a high intensity we see theangle900 by the letter V for VERTICAL. a continuous interference pattern on a screen. Finally I have indicated the action of the calcite At very low intensity, photons strike the screen crystal on an incoming unpolarized beam. The individually but in time, produce a speckled repli- incoming intensity I0 is divided equally between cation of the interference pattern (Fig. K.1). The the two outgoing polarized beams. behavior of photons is impossible to imagine in Suppose that we place two photon detectors, terms of what we observe at our macroscopic one in each of the paths of the two outgoing level.2 beams. We can then reduce the intensity to such For this Appendix we will examine what ex- a low level that only one detector registers a periments with calcite crystals reveal to us. In photon at a time. The result will be that the Sects. 8.6.2 and 8.6.3 we described the action of two detectors will detect photons in a random way—as in the tossing of a coin. On the average, 2I strongly urge you to view the lecture by the great the detectors will register an equal number of twentieth century physicist, Richard Feynman, on this photons. In Fig. K.3 weseeabeaminaso-called website: http://www.cornell.edu/video/richard-feynman- messenger-lecture-6-probability-uncertainty-quantum- DIAGONAL polarization state, with the label mechanical-view-nature. I cannot improve upon this D. The angle of polarization is 450.Weseetwo presentation, unless I were to try to shorten it, albeit with detectors indicating that one or the other registers a great loss. the detection of a photon. K.1 Experiments with Calcite Crystals 447

Fig. K.3 A beam in the diagonal polarization incident upon a calcite crystal detectors register the incidence of photons

calcite calcite

Fig. K.4 Calcite loop with a diagonally polarized incoming beam

The Calcite Loop At low enough intensity, the detectors In Sect. 8.6.3 we described the behavior of will enable us to be certain that only a beam that passes through a calcite loop,as one photon is in the loop at a time. shown in Fig. K.4. We noted that the outgoing Let us now assume that a photon is incident beam always has the same polarization state as upon the first calcite crystal and behaves as the incoming beam. In the figure we show an a particle, and therefore moves along a well- incoming beam in the DIAGONAL polarized defined path, we would conclude that there are state. A fourth polarization state that we will two possible scenarios, each with equal probabil- refer to is the SKEW polarized state, designated ity: First, the photon follows the path of the or- by the letter S. Its angle of polarization is 1350. dinary ray, having its polarization state changed The four polarization states that we will make use from DIAGONAL to HORIZONTAL and emerg- of in our study are shown in Fig. K.5. ing in the horizontal state. Or second, the photon Suppose that we have a diagonal polarized follows the path of the extraordinary ray, having beam pass through a calcite, so that there is an its polarization state changed from DIAGONAL outgoing diagonal polarized beam, as shown in to VERTICAL and emerging in the vertical state. Fig. K.6. We then place a SKEW polarizer in There is no way to understand how the outgoing front of the outgoing beam. Since the DIAGO- ensemble of photons is in the diagonal state. It NAL and SKEW polarizations are perpendicular is the wave theory of light that explains the state to each other, there is no ultimate intensity. 448 K The Mysterious Behavior of the Photon

Fig. K.5 Four polarization states: horizontal, vertical, H V D S diagonal, and skew

0° 90° 45° 135°

Fig. K.6 Diagonal input beam into calcite loop followedbyaSKEW polarizer; no output D D S I=0

calcite calcite

Fig. K.7 Destroying the calcite loop of Fig. K.6 by blocking the output from the vertical output of the first calcite crystal leads to an output from the SKEW polarizer of the outgoing state. The ensemble of photons final result will be that one-quarter of the pho- appears to reflect a mysterious incomprehensible tons incident upon the first calcite crystal will exit combination of wave-like properties and particle- the skew polarizer. We can refer to an intensity of like properties. the beam of photons and note that the ultimate A Paradox output of photons has an intensity that is one- We will assume that there is only one photon in quarter of the input intensity. That is, I = I0/4. The a calcite loop at a time. Next, suppose that we complete scenario is shown in Fig. K.7. Thus, by block the path of any photon that exits the first reducing the number of photons that reach the calcite crystal in the extraordinary ray, as shown SKEW polarizer we will increase the outgoing in Fig. K.7. In this case, all photons that reach intensity! the second calcite crystal will be in the horizontal We might ask how a photon incident upon state and pass out of the second calcite crystal the skew polarizer can “know” that the path of in that state. The number of these photons will the extraordinary ray from the first calcite crystal be half the number of photons incident upon the was blocked between the two calcite crystals, so first calcite crystal. They will all pass through the that it ends up horizontally polarized and has horizontal polarizer. In turn, half of these pho- a chance to pass through the skew polarizer? tons, which strike the SKEW polarizer, will pass How can a photon be in two places at once? through this polarizer, since the angle between Our conclusion is that the photon is unknowable, the horizontal and skew directions is 1350.3 The or alternatively, indescribable, in terms of our own experience at the macroscopic level. This 3According to Malus’ Law, the outgoing fractional inten- is also the case for the electron. In contrast to sity is cos2(1350)=cos2(450)=1/2. K.1 Experiments with Calcite Crystals 449 the summary at the beginning of this Appendix you” and they ask me, “What is his name?” about the nature of the discipline of physics as What shall I tell them?” God said to Moses, “I developed by Newton and Galileo, we have to AM WHO I AM. This is what you are to say to give up our ability to represent objects by images the Israelites: “I AM” has sent me to you.” in the quantum world. Similarly, suppose that we could speak to There is an interesting comparison in the an electron and ask the electron to help us Torah, wherein it is written4: Moses said to God, understand its seemingly incomprehensible “Suppose that I go to the Israelites and say to behavior. The electron might reply by saying, them, “The God of your fathershas sent me to “I am what I am.”

4https://injil.org/TWOR/30.html. Fusion of Harmonics: A Marvel of Auditory Processing L

We take for granted that when a musical in- Fusion in the Taste of Food strument plays a note we will hear the sound It might be unclear to some readers what I mean of but one source of sound. We have already by fusion of a mixture of harmonics. We can get learned in Chap. 2 that the frequency spectrum some idea of fusion by considering the taste of of the periodic wave of a musical instrument a homogeneous dish of food. I recall many years is a harmonic series, with the frequency of the ago, finding Lobster Cantonese an extremely de- wave. We have also learned that the timbre of licious dish. When I first attempted to prepare the an instrument is partly determined by the relative dish myself, I was amazed to learn that the essen- amplitudes and phases of the Fourier components tial ingredients were lobster and garlic. How well associated with the instrument. On the other I knew the taste of garlic and yet how surprised I hand, more than three decades ago, I began to was that garlic was an essential ingredient in the use a device made by the PASCO corporation recipe. Somehow, the blend of garlic and lobster for producing an electronic signal consisting of produced a taste all its own—that of Lobster a periodic wave having up to nine harmonics, Cantonese—with the flavor of neither ingredi- with a fundamental of 440 Hz. Such a device is ent standing out. And so it seems to be with called a synthesizer. The electronic signal was most superb dishes—as long as they are prepared fed into an amplifier, which was connected to properly. As another example, we can consider a loudspeaker. All listeners reported that they curries. Most are such that the ingredients are could hear the individual harmonics in the sound. individually recognizable; there are fortunately Each harmonic was audibly separated, as if the some that have a wonderful homogenized flavor harmonic had come from a separate source of all their own. And so it seems to be with the sound. What is the difference between the wave fusion of a mixture of harmonics from a musical produced by the Pasco synthesizer and the sound instrument! wave produced by a musical instrument? There is It is interesting to consider and to try to per- no visible characteristic on an oscilloscope trace ceive what the world of music would be like that indicates a difference. Perhaps there is a in the absence of fusion: The sound of a musi- significant difference that is too small to see on the trace with one’s eyes. We will discuss in this section the fact that the Brian Roberts of Aston University (Birmingham, Eng- land). Professor Bregman is the author of a book entitled brain processes the sound so that we do not hear Auditory Scene Analysis [MIT Press, Cambridge, 1994], the individual harmonics. This process is referred in which he discusses how the brain processes an en- to as fusion of harmonics.1 semble of sound inputs and organizes them according to sources. In particular, he explains how the brain is able to focus on one source of sound and ignore or become almost 1I have been greatly helped in my attempt to weed oblivious to other concurrent sources. As a result we are out the known understandings of fusion by two audio- able to hear one person speak in the midst of a dense crowd psychologists: Alan Bregman of McGill University and at a party.

© Springer Nature Switzerland AG 2019 451 L. Gunther, The Physics of Music and Color, https://doi.org/10.1007/978-3-030-19219-8 452 L Fusion of Harmonics: A Marvel of Auditory Processing cal instrument would be heard as an ensemble Therefore, the string’s length and its of harmonics that would be superimposed with tension are modulated. those of other musical instruments. We would Wind instrument such as a flute lose our ability to separate out the sounds of Here too—there is a modulation of the the ensemble of instruments. Vibratos might lose frequency due to the vibration of the their sweetness of tone. And so on. mouth of the musician. 3. Are there strong variations in the auditory Fusion of Harmonics is responsible processing of people such that sounds that are for the rich beauty of musical instruments. fused for some people are not fused for others. Here are some important interesting questions It is reported that some people can sometimes to be investigated. distinctly hear the individual harmonics produced by a musical instrument. 1. What accounts for the ubiquitous fusion of the sound produced by musical instruments? 2. What are the conditions under which a sound L.1 Mathematica File wave with a superposition of harmonics will be fused, will be perceived as having a single Below we provide a Mathematica file for produc- source? Factors that seem to be important ing a variety of waves that can be heard using the include: PLAY command of Mathematica. The reader can (a) the frequency of the fundamental— use this file to test his/her auditory processing a studies indicate that the lower the superposition of proportionally frequency modu- fundamental frequency, the greater is lated harmonics. When you run the commands a the degree of fusion; window will appear that allows you to listen to (b) the number of harmonics present and thewaveform. their relative amplitude; SYMBOLS: (c) the presence of “proportional modula- Text: fn = central frequency for a given har- tion,” sometimes referred to as “parallel monic,n=1,2,3,...[f2=2f1,...,f6=6f1] modulation.” Consider frequency mod- ≡ f = maximum change in frequency ulation, which is characterized by two hn = δf n/f n = modulation index parameters. There is a rate at which An = amplitudes the frequency is modulated—label it fm. Next, there is an amplitude of variation We begin with all modulation indices hn ≡ of the frequency—label it f . Propor- f /f n the same: tional frequency modulation would in- h = fv/f1. volve each harmonic being modulated with the same frequency of modulation INPUT LINES f but with a variation f in proportion m a f1 = f to the harmonic number n. f = 440 Here are some possibly relevant f2 = 2*f sources of frequency modulation: f3 = 3*f Acoustic stringed instrument f4 = 4*f Normally we think of the vibrating f5 = 5 f string as having two fixed ends. f6 = 6 f However, the transmission of sound fv = 10 waves involves the string moving phi = f*t + (fv/(2 Pi f))*Cos[2 Pi f * t] the bridge. Therefore, the string is phi2 = f2*t + (fv/(2 Pi f2))*Cos[2 Pi* f2 * t] not absolutely fixed at the bridge. L.1 Mathematica File 453 phi3=f3*t+(fv/(2Pif3))*Cos[2Pi*f3*t] INPUT LINES phi4=f4*t+(fv/(2Pif4))*Cos[2Pi*f4*t] phi = f*t + (fv/(2 Pi f))*Cos[2 Pi f * t] phi5 = f5*t + (fv/(2 Pi f5))*Cos[2 Pi* f5* t] phi2a = f2*t + (fv/(2 Pi f))*Cos[2 Pi* f2 * t] phi6 = f6*t + (fv/(2 Pi f6))*Cos[2 Pi* f6* t] phi3a = f3*t + (fv/(2 Pi f))*Cos[2 Pi* f3 * t] A1 = 1 phi4a = f4*t + (fv/(2 Pi f))*Cos[2 Pi* f4 * t] A2 = 1 phi5a = f5*t + (fv/(2 Pi f))*Cos[2 Pi* f5* t] A3 = 1 phi6a = f6*t + (fv/(2 Pi f))*Cos[2 Pi* f6* t] A4 = 0.01 A5 = 0.01 INPUT COMMAND A6 = 0.001 Play[ A1*Sin[2 *Pi* phi] + A2*Sin[2* Pi * phi2a] + A3*Sin[2* Pi * phi3a] + A4* Sin[2* Pi INPUT COMMAND * phi4a] + A5* Sin[2* Pi * phi5a] + A6*Sin[2* Play[ A1*Sin[2 *Pi* phi] + A2*Sin[2* Pi * phi2] Pi * phi6a], t, 0, 5, SampleRate → 40000] + A3*Sin[2* Pi * phi3] + A4* Sin[2* Pi * phi4] + A5* Sin[2* Pi * phi5] + A6*Sin[2* Pi * phi6], t, 0, 5, SampleRate → 40000]

Next, the modulation indices are proportional to the harmonic: hn = n ∗ fv/f1 Transformation Between Tables of Color Matching Functions for Two M Sets of Monochromatic Primaries *

In Chap. 15 we provided a Table 15.2 of color TCMFs. We will see shortly that in fact, only matching functions—henceforth referred to 3×3 = 9 numbers are sufficient to determine the as TCMF—that was produced by Judd and relationship.2 The nine numbers are exhibited in Wyszecki by studying the color vision of a the following three equations: set of individuals with normal color vision for r (λ) = URRr(λ) + UGRg(λ) + UBRb(λ) a specific set of primaries, λB = 435.8nm, = = λG 546.1nm,andλR 700.0nm.The g (λ) = URGr(λ) + UGGg(λ) + UBGb(λ) table consists of three columns of numbers, r(λ), = + + g(λ),andb(λ)—the color matching functions. b (λ) URBr(λ) UGB g(λ) UBBb(λ) What if we have a different set of monochromatic (M.1) primaries? How should we mix these primaries The nine numbers are represented by the sym- so as to produce the same colors? In this bols URR, UGR, UBR, URG,UGG, UBG, URB, appendix, we will derive a set of nine numbers UGB ,andUBB. They are commonly exhibited as that will allow us to determine a corresponding an array of numbers, together forming what is TCMF—starting with the original TCMF—that referred to as a matrix. A common symbol for should allow the same set of individuals to a matrix is U, here shown for the letter U. The match a spectral intensity using any other set matrix U is exhibited below.3 of monochromatic primaries. We will label their ⎛ ⎞ URR UGR UBR wavelengths as {λR, λG,andλB }. ⎝ ⎠ In the TCMF, there are sixteen different wave- URG UGG UBG lengths for each of the three primaries.1 One URB UGB UBB might assume that each of the set of 3 × 16 = 48 numbers of the new TCMF depends upon the 2In fact, if we don’t care about maintaining a specific entire set of 48 numbers of the original TCMF. requirement on the unit intensities, we need only eight numbers. We would then need 48 × 48 = 2304 num- 3For those who are familiar with matrices, we will rewrite bers to specify the relationship between the two Eq. (M.1) in a simpler form. We introduce two vector functions, (λ) and (λ). They represent the color matching 1We realize that there are an infinite number of possible functions for the respective two sets of primaries which wavelengths spanning the range 400nm to 700nm. Our have the following components: table essentially samples a given spectral intensity at a dis- cR(λ) = r(λ), cG(λ) = g(λ), cB (λ) = b(λ) crete value simply for convenience. In fact, our table was taken from another that had doubled the number of wave- c R(λ) = r (λ), c G(λ) = g (λ), c B (λ) = b (λ) lengths, with values halfway between the values in our (M.2) table. For ideal sampling, we would need color matching We will use subscript notation, with subscripts α, β, ... = functions for the continuum of visible wavelengths. The R,G,orB. Then Eq. (M.1) can be written as tristimulus values would be integrals: R = r(λ)I (λ)dλ, = G = g(λ)I (λ)dλ,andB = b(λ)I (λ)dλ. c α(λ) Uαβ cβ (λ). (M.3) β=R,G,B

The essential reason for this extraordinary simpliﬁcation is physiological: color vision is based upon a single set of three receptors with three corresponding independent nerve impulse rates, as opposed to such a set for each possible set of primaries. This fact will be demonstrated in the last subsection of the appendix.

Explicit Expression for the Transformation Matrix U

We will later show that the transformation matrix U can be expressed in terms of a sub-matrix obtained from the TCMF. Fig. M.1 The horseshoe perimeter for the Judd and Suppose that the wavelengths of the second set of Wyszecki primaries 436nm, 546nm, and 700nm primaries are given by λR , λG ,andλB . Then this sub-matrix, here symbolized by V, is given by: area that encompasses negative primaries. The ⎛ ⎞ ⎛ ⎞ process we will use involves a bit of trial and VRR VRG VRB r(λR ) g(λR ) b(λR ) error. We ﬁrst note that a change of primaries ⎝ ⎠ ⎝ ⎠ VGR VGG VGB ≡ r(λG ) g(λG ) b(λG ) can be looked at as taking a piece of rubber on VBR VBG VBB r(λB ) g(λB ) b(λB ) which we draw the TCMF and stretching it in various directions so that the points correspond- = Note, for example, that VRG g(λR ) ing to the desired primaries lie at the corners of We will show that the triangle entirely encompassing positive color coordinates. The TCMF is best seen as a whole U = V −1/u (M.4) αβ αβ β through the corresponding chromaticity diagram where the unit intensities are given by encompassed by the horseshoe perimeter.See Fig. M.1. = −1 We notice that the bulge to the upper left rep- uβ Vαβ (M.5) α=R,G,B resents a large region having negative red coordinates. Consequently, this choice of monochro- so that matic primaries is far from ideal in enabling −1 one to match colors entirely with positive color Vαβ Uαβ = (M.6) −1 coordinates. On the other hand, the perimeter Vσβ from the green primary to the red primary is = σ R,G,B extremely close to being straight, so that there is an extremely small region having a negative blue primary coordinate. Finally, the red primary is at M.1 Application of the the end of the perimeter while the extreme blue Transformation: end of the perimeter—at 400nm—is extremely Determining an Ideal Set of close to the blue primary, so that we have an Primaries extremely small region having a negative green primary coordinate. We have tried to determine an ideal set of pri- Since the extreme left end, with the greatest maries starting with the TCMF produced by Judd negative red coordinate, corresponds to a wave- and Wyszecki for the primaries 436nm, 546nm, length of about 510nm, it is reasonable to study a and 700nm. Our goal is to minimize the total new set of primaries, with the green primary at a M.2 Proof of Eqs. (M.1) and (M.6) 457

Procedure

We now summarize the procedure for obtaining the new TCMF, specifically for the set 436nm, 510nm, and 700nm. We first extract out of the original TCMF of Judd and Wyszecki those rows having to do with the new primaries.4 We arrive at the 3 × 3 sub-matrix V that is shown in light blue in Fig. M.4. The wavelengths 700nm and 510nm are present in the TCMF. However, the wavelength 436nm of the blue primary, which is the same for both the original and the new Fig. M.2 The horseshoe perimeter for the primaries 436nm, 510nm, and 700nm primaries, is absent. That is not a problem for us since the red and the green color coordinates of the blue primary must be identically zero. The blue color coordinate can be obtained by interpolation between the wavelengths 420nm and 440nm. Note that while the original TCMF was orga- nized with the wavelengths running from 400nm at the top to the highest at 700nm, we have rearranged the three wavelengths in reverse order so that we have R,G, and B running from to bottom. We next produce the inverse of the matrix V, Fig. M.3 The horseshoe perimeter for the primaries shown in yellow using the Excel.5 The unit inten- 436nm, 515nm, and 700nm sities of the new primaries are shown in magenta and are the respective sums of the columns above. U wavelength of 510nm and with the same red and Finally, we see the transform matrix, ,inlight V blue primaries. The resulting horseshoe is shown green. It is the ratio, cell by cell, of the matrix in Fig. M.2. divided by the unit intensity corresponding to the The improvement is dramatic. We have sig- column. Note that the sum of each of the three U nificantly reduced the area with a negative red columns of is unity. coordinate. However, we now have a significant region with a negative blue coordinate. We therefore experiment with a green primary a bit closer M.2 Proof of Eqs. (M.1)and(M.6) to the original 546nm, hoping that the reduction in the region with a negative blue coordinate will Let us begin by understanding better the content not lead to a significant region with a negative of the tables. The color of a spectral intensity is green coordinate. We next switch to a green produced by mixing sources of the set of given primary of 515nm. The resulting horseshoe is primaries. Physiologically, the three tristimulus showninFig.M.3. The area of negative blue primary is now about the same as the area of 4This text has only wavelengths that are multiples of negative red primary. With mathematical opti- twenty, while the original TCMF found in Williamson and Cummins has all multiples of ten. mization techniques we could, perhaps, make 5We highlight a 9 × 9 block of cells. We then type in the further improvements; nevertheless, we will stop command line: =MINVERSE(C7:E9), where (C7:E9) here and accept what we have now. identifies the matrix V to be inverted - here C7 is the cell ID of VRR and E9 is the cell ID of VBB. Of course, your cell IDs might be different. 458 M Transformation of Primaries

λ(nm) RRRGGGBBB 700 0.0041 0 0 243.9 0 4E-17 0.591 0 0 510 -0.089 0.1286 0.027 168.82 7.776 -0.21 0.409 1 -0.2655 436 0 0 1100 0 0 1.2655 u’= 412.72 7.776 0.7902

Fig. M.4 Matrices for transforming from the Judd-Wyszecki primaries 436nm, 546nm, and 700nm to the primaries 436nm, 510nm, and 700nm calculated for that spectral intensity are NOT N = S (λ)I (λ) simply proportional to the corresponding rates at R R λ which nerve impulses are sent to the brain by the = cones. In Chap. 15 we introduced the following NG SG(λ)I (λ) (M.8) functions: λ

The response functions S (λ), S (λ),and R G N = S (λ)I (λ) S (λ) are the respective rates at which the cones, B B B λ R, G,andB emit nerve impulses per unit inten- where the right-hand sides are sums over all the sity of wavelength λ. wavelengths. The nerve impulse rate for a given spectral in- To produce a match by mixing the primary = = tensity I(λ) are: NR(λ) SR(λ)I (λ), NG(λ) sources, the sources have to produce the same = SG(λ)I (λ),andNB (λ) SB (λ)I (λ). set of nerve impulse rates. Therefore we next For a given spectral intensity, the total nerve need to obtain expressions for how these rates impulse rates from the respective cones are given depend upon the primary sources. We note that 6 by generally, each of the three primaries produces nerve impulse rates from all three cones. We therefore introduce the following nine quantities:

SRR = Nerve impulse rate of the R-cones per unit intensity of R-primary

SRG = Nerve impulse rate of the R-cones per unit intensity of G-primary

SRB = Nerve impulse rate of the R-cones per unit intensity of B-primary

SGR = Nerve impulse rate of the G-cones per unit intensity of R-primary

SGG = Nerve impulse rate of the G-cones per unit intensity of G-primary (M.9)

SGB = Nerve impulse rate of the G-cones per unit intensity of B-primary

SBR = Nerve impulse rate of the B-cones per unit intensity of R-primary

SBG = Nerve impulse rate of the B-cones per unit intensity of G-primary

SBB = Nerve impulse rate of the B-cones per unit intensity of B-primary

6See Chap. 15. With matrix and vector notation, we have

Nα = Sα(λ)I (λ). (M.7) λ M.2 Proof of Eqs. (M.1) and (M.6) 459

The nine quantities can be treated as a 9 × 9 To clarify the above, let us join the ﬁrst matrix S, exhibited below: Eqs. (M.9)and(M.10) and all three equations ⎛ ⎞ of Eq. (M.12). We obtain SRR SGR SBR ⎝S S S ⎠ RG GG BG NR = SR(λ)I (λ) = R SRR + G SRG + B SRB SRB SGB SBB λ = + The nerve impulse rates depend upon the tristim- r(λ)I (λ) SRR g(λ)I (λ) SRG ulus values, R, G,andB and the matrix S: λ λ + b(λ)I (λ) SRB = + + λ NR R SRR G SRG B SRB (M.13) N = R S + G S + B S (M.10) G GR GG GB We obtain a similar equation for NG and NB . Ultimately, we have for each wavelength8 NB = R SBR + G SBG + B SBB = + + We can now use the equations from Chap. 15 for SR(λ) r(λ)SRR g(λ)SRG b(λ)SRB the dependence of the tristimulus values on the SG(λ) = r(λ)SGR + g(λ)SGG + b(λ)SBG color matching functions, namely,7 SB (λ) = r(λ)SBR + g(λ)SBG + b(λ)SBB R = r(λ)I (λ) (M.15) λ There are, correspondingly, nine quantities for = S G g(λ)I (λ) (M.12) the second set of primaries, the matrix as well

λ as the second set of color matching functions, r ,

g ,andb . We also have a set of equations parallel B = b(λ)I (λ) to Eq. (M.15): λ

These expressions, substituted into Eq. (M.10), SR(λ) = r (λ)S + g (λ)S + b (λ)S provide us with an expression for the nerve RR RG RB S = + + impulse rates in terms of the matrix . Both this SG(λ) r (λ)SGR g (λ)SGG b (λ)SGB set of equations and the set of Eq. (M.9)interms S (λ) = r (λ)S + g (λ)S + b (λ)S of the functions S (λ), S (λ),andS (λ) must B BR BG BB R G B (M.16) hold for any spectral intensity and therefore must hold for any speciﬁc wavelength. This fact will allow us to show how visual physiology reduces the number of independent variables necessary to relate matching with one set of primaries with matching by another set of primaries, as expressed by Eq. (M.1).

7In matrix notation, we can deﬁne the vector representing 8 the three tristimulus values as Cα = (R,G,B).Then With matrix notation we have

= Cα = cα(λ)I (λ) (M.11) Sα(λ) Sαβ cβ (λ). (M.14) λ β=R,G,B . 460 M Transformation of Primaries

Therefore,9 + + = + + r (λ)SRR g (λ)SGR b (λ)SBR r(λ)SRR g(λ)SGR b(λ)SBR + + = + + r (λ)SRG g (λ)SGG b (λ)SBG r(λ)SRG g(λ)SGG b(λ)SBG (M.18) + + = + + r (λ)SRB g (λ)SGB b (λ)SBB r(λ)SRB g(λ)SGB b(λ)SBB

We see above that the two matrices, S and where the parameters uβ are the unit intensities S , determine the relationship between the two for the second set of primaries, given by sets of color matching functions and that the relationship is identical for each wavelength. The = −1 + −1 + −1 uR VRR VGR VBR (M.22) algebra of matrices leads to an expression for U the transformation matrix of Eq. (M.1) that in- with corresponding expressions for the other two volves the so-called inverse matrix of the matrix unit intensities. S : Thus, the second TCMF is determined − U = SS 1 (M.19) by the 9 × 9 sub-matrix V of the original TCMF. For those who aren’t familiar with these symbols, we will exhibit one of the matrix elements of U: Proof = −1 + −1 + −1 URG SRRSRG SRGSGG SRBSBG (M.20)

−1 First let us recall (M.1), which we rewrite here: Here, for example, SRG is the RG element for the S −1 matrix . r (λ) = URRr(λ) + UGRg(λ) + UBRb(λ) Note When the two sets of primaries are identi- g (λ) = URGr(λ) + UGGg(λ) + UBGb(λ) cal, we expect the transformation matrix to yield = + + the same TCMF as the original TCMF so that U b (λ) URBr(λ) UGB g(λ) UBBb(λ) should be the so-called identity matrix.Thenall (M.23) the diagonal elements (URR,UGG,andUBB )are In Chap. 15 we pointed out that if we sum any unity while the remaining six elements vanish. color matching function, r(λ), g(λ) or b(λ) over Equation (M.19) conﬁrms this result since in this all of the wavelengths, we must obtain the same case, S = S number so that a constant spectral intensity will produce equal energy white. This fact obviously We will next prove Eq. (M.4) holds true for the second set of primaries too, except that the constant common to the three U = V−1 αβ αβ /uβ (M.21) sums can be different. We now recall that if we were to multiply every color matching function in a TCMF by the same number, all tristimulus values are multiplied by that number but the color coordinates are unchanged. We therefore are free to choose the sums to be equal for the two different sets of primaries. 9 If we carry out this sum in each of the above three equations, we will obtain the = cβ (λ)Sαβ cβ (λ)Sαβ . (M.17) β=R,G,B β=R,G,B M.2 Proof of Eqs. (M.1) and (M.6) 461 following three equations for the transformation Similarly, we have a corresponding matrix for 10 matrix : the second set of primaries, λR, λG,andλB ,we have URR + UGR + UBR = 1 + + = 1 1 1 URG UGG UBG 1 (M.25) r (λ ) = , g (λ ) = , b (λ ) = R u G u B u + + = R G B URB UGB UBB 1 (M.27)

Next, from Eq. (15.15) in Chap. 15 we obtain with the remaining functions, e.g. r(λG), vanish- ing. We also deﬁne the matrix 1 1 1 r(λ ) = , g(λ ) = , b(λ ) = ⎛ ⎞ R G B = uR uG uB r (λR) r (λG) 00 (M.26) C = ⎝ ⎠ 0 g (λG) 0 with the remaining functions, e.g. r(λG), vanish- 00b (λB ) ing. ⎛ ⎞

Let us introduce the matrix C deﬁned by 1/uR 00 = ⎝ ⎠ 01/uG 0 ⎛ ⎞ r(λR) r(λG) = 00 001/uB ⎝ ⎠ C = 0 g(λG) 0 00b(λB ) AccordingtoEq.(M.1)wehave

= = 1 = + + CRR r (λR) URRr(λR) UGRg(λR) UBRb(λR) uR = = 1 = + + CGG g (λG) URGr(λG) UGGg(λG) UBGb(λG) (M.28) uG = = 1 = + + CBB b (λB ) URBr(λB ) UGB g(λB ) UBBb(λB ) uB

These equations can be rewritten so as to or exhibit the matrix V: C = V U (M.31) = + + αβ ασ σβ CRR VRRURR VRGUGR VRBUBR σ=R,G,B C = V U + V U + V U GG GR RG GG GG GB BG We know so far only the matrix V. We can obtain = + + CBB VBRURB VBGUGB VBBUBB the unit intensities uα) as follows. First, we solve (M.29) Eq. (M.31) for the matrix U: These equations can be expressed as a multipli- cation of matrices: U = V−1C (M.32)

C = VU (M.30) In matrix element form we have

= −1 = −1 Uαβ Vασ Cσβ Vαβ /u β (M.33) σ=R,G,B

10 The three equations can be expressed in matrix notation Next we recall Eq. (M.25). It can be written as as Uαβ = 1 (M.24) α Uαβ = 1 (M.34) for all β. α=R,G,B 462 M Transformation of Primaries for all β. Therefore, Table M.1 Table of color matching functions based upon the spectral primaries 444.44 nm, 526.32 nm, and = −1 645.16 nm uβ Vαβ (M.35) α=R,G,B λ (nm) r(λ) g(λ) b(λ) 400 0.0089 −0.0025 0.04 Finally we have our expression for the transfor- 410 0.035 −0.0119 0.1802 mation matrix U in terms of the sub-matrix of the 420 0.0702 −0.0289 0.467 original TCMF, Eq. (M.6): 430 0.0763 −0.0338 0.6152 − −1 440 0.0561 0.0276 0.8778 Vαβ 450 −0.0044 0.0024 1.0019 Uαβ = (M.36) −1 460 −0.097 0.0636 0.9139 Vσβ σ=R,G,B 470 −0.2235 0.1617 0.7417 480 −0.3346 0.2796 0.472 485 −0.3776 0.3428 0.3495 M.3 Problems on the 490 −0.4136 0.4086 0.2564 Transformation of TCMFs 500 −0.4452 0.5491 0.1307 510 −0.414 0.7097 0.058 1. Suppose that the matrix V has only diagonal 520 −0.2845 0.8715 0.02 matrix elements. We say that the matrix is a 530 −0.0435 0.9945 0.0007 diagonal matrix. 540 0.3129 1.0375 −0.0064 − Explain why the matrix U is diagonal, with all 550 0.7722 1.039 0.0094 − its matrix elements equal to unity. What does 560 1.271 0.9698 0.0097 − this imply about how the new set of primaries 570 1.8465 0.8571 0.0087 − is related to the old set? 580 2.425 0.6953 0.0073 − 2. Below is the TCMF produced by Stiles and 590 2.9151 0.5063 0.00537 600 3.1613 0.336 −0.00357 Burch [See Chap. 15] for the primaries λB = 610 3.1048 0.1917 −0.00208 444.44 nm, λG = 526.32 nm, and λR = 620 2.7194 0.0938 −0.00103 645.16 nm (Table M.1). 630 2.17 0.0371 −0.00044 a. According to the theory of color vi- 640 1.5179 0.0112 −0.00014 sion that we have presented, the data 650 1.007 0.000078 0 collected by the two groups, Stiles and 660 0.5934 −0.001988 0 Burch vs. Judd and Wyszecki, should be 670 0.3283 −0.002006 0 consistent. For example, if we were to 680 0.1722 −0.001272 0 apply our transformation matrices to the 690 0.0853 −0.000683 0 TCMF of Stiles and Burch, we should 700 0.0408 −0.000337 0 obtain the TCMF of Judd and Wyszecki. Carry out this process and compare the TCMFs. Beware that the two might dif- Find the TCMF for this second set fer by a constant factor that multiplies of primaries and the corresponding all the coordinates. Why is this so? On horseshoe perimeter. the other hand, the color coordinates do Compare this perimeter with the one we not depend upon such a constant factor, derived by a transformation from the so compare the horseshoe perimeters. Judd and Wyszecki TCMF of the set of b. Derive the transformation matrix U for primaries 436nm, 546nm, and 700nm.

the set of primaries 436nm, 520nm, and 3. Prove that the tristimulus values Cα for a sec- 700nm that is based upon the Stiles and ond set of monochromatic primaries is related Burch TCMF. M.3 Problems on the Transformation of TCMFs 463

to the tristimulus values Cα of the ﬁrst set by b. Prove that the color coordinates satisfy the equation the equation

= c c Cα Uαβ Cβ (M.37) α = α (M.38) β Cα Cα

4. a. Prove that the sum S = Cα is in- α dependent of the set of monochromatic primaries. Hommage to Pierre-Gilles de Gennes—Art and Science N

In July of 2007 I learned that the physi- Our meeting was very brief and yet extremely cist Pierre-Gilles de Gennes (Fig. N.1)had productive. passed away on May 18, 2007. I received the Of what relevance is this encounter to this news with deep sadness since de Gennes was book, in which our goal is to appreciate the my supervisor when I was a young Post-Doc connection between physics and the arts? It at the Laboratory of the Physics of Solids in is that some people have an extraordinary gift Orsay, France in the academic year 1966-67 to see order and simplicity in the complex. and I had crossed paths with him many times de Gennes was one of those who applied since then. While only seven years older than this ability to many fields, whose numerous me, he had already achieved fame as leader familiar applications depend upon the results of a group of physicists studying supercon- of his research. This ability to create order ductivity. De Gennes had a great influence on from a suspended blank state of our normal all those working in the field of Condensed awareness is manifest in artists as well. De Matter. Whatever subject he touched, he trans- Gennes is quoted as describing the deep formed. Frequently he attacked problem areas influence that the painter Pablo Picasso had for which physicists and chemists felt that on his own scientific studies. To appreciate they had reached the limits thought possible this point, I am including an excerpt of one conceptually. De Gennes discovered sets of of a number of obituaries written after his concepts that produced a new exciting level of death. For details, see: http://www.nature. activity of research in the fields. com/nature/journal/v448/n7150/pdf/448149a. One meeting I had with him stands out pdf. in my memory: I was working on a problem The obituary makes reference to a film on and was stuck because of not knowing one Picasso that left a great mark on de Gennes’ of the essential parameters of the problem. research as a physicist. You can view an de Gennes listened to me attentively with all excerpt of this film on the following YouTube due patience. He then proceeded to ask me site: http://www.youtube.com/watch?v= one simple question, which I will recall with 5tn5uTTCRCg. clarity: “What is the approximate wavelength Excerpt of an Obituary of de Gennes by of the electrons in bismuth in comparison to Françoise Brochard-Wyart the roughness of the surface of the sample.” “Chacun en nous a son trésor d’images entrevues dans The reader need not understand the meaning un instant mais jamais oubliées. Un exemple pour moi: Picasso peignant grands traits blancs sur une vitre et of the question. It was the utter simplicity filmé par Clouzot. Tout ce que j’ai essayé de dessiner and directness of his question that mattered. laborieusement plus tard est né de ces moments.” For that question was all I needed to hear in “Every one of us has a treasure of images caught in order for me to proceed to complete my study. glimpses but never forgotten. A personal example: Pi- casso painting white lines on glass using large strokes,

breaking direction. Calling on his vast knowledge of physics, de Gennes drew analogies between different fields. For example, he realized that laws developed to describe superconductivity phenomena could be used to understand phase transitions in liquid crystals. ... De Gennes pursued his research with extraordinary imagination, insatiable curiosity and an ability to grasp facts rapidly. But he also gave his time to others and helped them develop their ideas. A keen ambassador of science to the public, he generated passionate debates on subjects as diverse as “Physics and Medicine,” “Inventors” and “Primo Levi.” He inspired generations of students to pursue careers in physics and played an active role in establishing the L’Oréal-UNESCO Awards for Women in Science. ... Fig. N.1 Pierre-Gilles de Gennes (source: http://authors. End of excerpt of the Obituary library.caltech.edu/5456/1/hrst.mit.edu/hrs/materials/ public/DeGennes/DeGennesintro_fr1252.htm) Here is another expression of a likening of de Gennes revelations to the product of an artist in an excerpt of a description of de filmed behind the glass by Clouzot. Everything that Gennes’ book on Soft Interfaces: “[The book I tried painting laboriously later was born from such provides us with] an impressionistic tour of glimpses.” (Pierre-Gilles de Gennes, from L’émerveillement by the physics of soft interfaces by Nobel Lau- Thibaut de Wurstemberger, Saint-Augustin, 1998.) reate Pierre-Gilles de Gennes. Full of insight With his strikingly simple yet pioneering ideas, Pierre- and interesting asides, it not only provides an Gilles de Gennes drew “white lines in large strokes” accessible introduction to this topic, but also that defined the physics of soft matter—liquid crystals, polymers, colloids and surfactants. He died on 18 lays down many markers and signposts for May. interesting new research possibilities.” Educated at the École Normale Supérieure in Paris I will end this essay with a personal re- during 1951-55, de Gennes learned theoretical physics mark: from the greatest masters of his time. He obtained his PhD in 1957 while at the French Atomic Energy Over the years, de Gennes and I corre- Commission, specializing in magnetism and neutron sponded and saw each other only once every scattering. During a stay at the University of Cali- few years. Yet, in spite of his fame and his fornia, Berkeley, in 1959 he studied with the solid- multitude of activities and acquaintances, he state physicist Charles Kittel, who taught him how to communicate ideas in physics using plain language, so remembered even the most trivial of our en- avoidingtheuseofdauntingequations.... counters. Most vivid in his mind when I last De Gennes fostered a collective research effort that saw him at a talk he gave at Harvard was about is scarcely imaginable today. Papers were signed not a dinner that his wife and he had prepared with the names of individuals, but with the name of the group. Theoreticians would spend half their time con- for my family in 1967. They had forgotten to templating liquid crystals under the microscope and buy fresh cheese for our dessert; this mistake discussing practical experiments. Researchers would required deep apologies and remained in his often arrive in the morning to find a note from de mind for 40 years. Gennes that would launch them in yet another ground- Index

Symbols Analysis, 11 1-cm, 241 Angle of incidence, 224 24-bit color, 383 Angle of reflection, 224 3-D plot, 185 Angle of refraction, 229 ℵ0, the order of infinity of integers, 354 Ångstroms, 53 ℵC 0 , the infinite number of distinct spectral intensities, Anisotropic, 104 354 Anti-diagonal matrix, 367 Antinode, 23, 60, 62 Aperture, 338 A Aqueous humour, 328 Absolute pitch, 296 Archimedes, 271 Absolute value, 410 Archimedes’ principle of lever action, 271 Absorbed, 226 Atmosphere (atm), 55 Absorption, 12, 95, 143 Atoms, 12, 101, 103 Absorption coefficient, 95 Attenuate, 79 Absorption spectra, 387 Attenuation, 12, 71, 88 Accelerating charge, 132 Attenuation coefficient, 92 Accelerating mass, 197 Attenuation constant, 92 Acceleration, 132, 423 Attenuation length, 92 Accommodate, 246 Attenuation time, 89, 148 Accommodation, 326, 334 Auditory canal, 269 Action-at-a-distance, 102, 103 Auditory nerve, 275, 279 Additive mixing, 13, 348 Australopithecus bosei, 2 Additive primaries, 356 Axis of polarization, 202 Admixture of overtones, 185 Axis of the lens, 237 After images, 331, 393 .aiff audio file, 188 Air pressure, 54 B Air resonance, 65 Background noise, 437 Alychne, 378 Band pass filter, 182 AM, 179 Bandwidth, 339 Amadeus Pro, 84 Bar magnet, 104 A major triad, 295 Base, 86 AM—amplitude modulation, 42 Basilar membrane, 274 AM-demodulation, 179 Bat, 257 A-minor triad, 295 Battery, 112 Ambient pressure, 270 Beat frequency, 173, 174, 292, 320 Ampère, André Marie, 108 Beating, 173, 292 Amplifier, 58 Beats, 173, 176, 212 Amplitude, 25, 27, 30, 275 Bell, Alexander Graham, 86 Amplitude-modulated, 179, 184 Bending force, 37 Amplitude modulation frequency, 42 Bernoulli’s principle, 66 Amygdala, 438 Biconcave lens, 237

Biconvex lens, 237 Color, 12, 353, 359 Big Bang, 153, 206 Color coördinates, 13, 354, 359–361, 371, 372 Biosphere, 209 ColorCorrection SystemTM, 391 Birefringent, 201 Color filters, 13 Black body radiation, 151 Color match, 356 Bleaching of rods, 329 Color matching functions, 355, 362 Blind spot, 333 Color matching functions for Wright-Guild primaries, Blue color of sea, 294 362 Blue cone, 330, 355 Color monitors, 379, 382 Body mass index (BMI), 413 Color patch, 347, 353 Boethius, 10 Color vision, 13, 439 Bohr, Neils, 139, 140 Combination modes, 294 Bohr radius, 141 Combination tones, 12, 267, 289, 301 Brahe, Tycho, 413 Compass, 101 Brain, 267 Complement, 393 Brightness, 81, 340, 353, 361 Complementary color pairs, 357 British thermal unit (BTU), 72 Complementary colors, 331, 370 Bulk modulus, 59 Complementary hue, 370 Burning, 76 Complements, 370 Buzzer, 111 Compound lens, 242, 325 Concave lens, 237 Condensation, 56 C Conduction electrons, 103 Calcite, 200 Conductor, 103 Calcite loop, 447 Cone(s), 328 Carrier frequency, 43 response, 389 Carrier modulation, 178 Consciousness, 14 Carrier wave, 42, 179 Consonance, 298, 308 Carson’s rule, 182 Constructive interference, 160, 171 Centi, 417 Contrast, 5 Central ray, 240 Convergence, 331 Cents, 315, 320 Converging lens, 236 Cerenkovˇ radiation, 253 Convex-concave lens, 237 Cesium clock, 442 Convex lens, 237 Charged harmonic oscillator, 147 Cornea, 325 Chemical energy, 75, 79 Cornsweet, Tom N., 404 Chiral, 207 Corrective lens, 335 Chiral biosphere, 209 Correlate, 441 Chirality, 207 Coulomb’s Law, 102 Chirp, 187 Counter-clockwise rotation, 207 Chladni, Ernst, 49 Critical angle, 230, 261 Chladni plates, 49 Critical bandwidth, 289 Chromatic aberration, 237 Crystal, 168 Chromaticities, 13 C, the order of infinity of the set of real numbers, 354 Chromaticity, 353, 355, 359, 361 Cutoff frequency, 342 Chromaticity coördinates, 361 Cycle, 18 Chromaticity diagram, 355, 359 Chromatic scale, 10, 304, 311 CIE Chromaticity Space, 366 D C.I.E. table, 377 Damping, 88 CIE tristimulus values, 377 Dark adaptation, 331 Ciliary muscles, 325 Day vision, 330 Circular reasoning, 442 DB, 86 Classical physics, 139, 442 de Rossi, Salomon, 11 Clockwise rotation, 207 Debussy, Claude, 317 Closed pipe, 62 Decay, 45 Cochlea, 268, 274 Decibel, 12, 86 Cochlear fluid, 268 Degree of saturation, 370 Cochlear implant, 276, 281 Demodulation, 181 Coherent, 88, 150, 162, 165 Dennett, Daniel, 283 Index 469

Density, 60 Electron, 103, 443 Depth perception, 325, 331 Electron-volt (eV ), 72 Desaturated, 369 Electrostatic potential energy, 75 Destructive interference, 160, 171 Elliptical orbit, 199 Determinism, 423, 425 Empirical, 139 Deuteranope, 391, 395 Enantiomorphs, 207 Deuteranopia, 391 EnChroma, 391 Deutsch, Diana, 297 End correction, 64 Diagonal, 367 Energy, 12, 71 Diagonal matrix, 462 Energy level, 146 DIAGONAL polarized state, 446, 447 Energy level diagram, 146 Diatonic scale, 304, 307 Energy spectrum, 146 Dichroism, 202 Enharmonic equivalents, 312 Dichromats, 391, 396 Enso, 176 Differential and integral calculus, 277 Envelope, 45, 89, 92, 275 Diffraction, 12, 215, 216 Envelope of the waves, 277 angle, 216 Epicenter of an earthquake, 164 broadening, 218 Equal energy spectrum, 340 grating, 137, 166 Equal energy white, 362, 364, 369 Diffusely, 258 Equal loudness curves, 284, 375 Diffuse reﬂection, 224, 225 Equal temperament, 21, 296, 305, 307, 314 Diopter, 246, 334 Equilibrium, 78 Disparity of eye position, 332 state, 27 Dispersion, 138, 235, 237 erg (erg), 72 Dispersive, 235 Escher, Mauritz C, 297 Displacement, 5, 57 Estill method, 286 Displacement current, 127, 128 Eustachian tube, 270 Dissipation, 80, 89 Exchange frequency, 39 Dissonance, 308 Excited state, 146 Diverging lens, 236 Extended sources, 166 Dominant wavelength, 370 Extraordinary ray, 201 Doppler effect, 12, 149, 215, 250, 322 Eye-brain system, 225 Dull surface, 223 Eyeglasses, 246 Dwarf planet, 440 Eyepiece, 432

E F Eardrum, 268 Fact, 443 Echo-location, 257 Faith, 442 Effective focal length, 335 Faraday, Michael, 112 Efficiency, 80 Faraday’s Law, 122 Einstein, Albert, 6, 151, 443 Far-sighted, 327 Einstein’s Theory of Special Relativity, 253 Feeling, 13 Electrical energy, 75 Feynman, Richard, 3 Electrical resistance, 112 F-hole, 65, 69 Electric battery, 104 Field, 117 Electric charge, 101, 102 Fifth, 255 Electric current, 103 First harmonic, 23 Electric dipole, 118 Fix tuned, 304 Electric field, 12, 117 Flats, 311 Electric field lines, 117 Flute, 40 Electric force, 101 FM radio wave, 43 Electric generator, 112, 115 Focal length, 237, 325 Electricity, 101 Focal point, 237 Electric motor, 111 Food calorie, 72 Electromagnet, 109 Footplate, 268 Electromagnetic force, 101 Force, 33 Electromagnetic radiation, 75, 128, 146 Fourier, Jean Baptiste Joseph, 6, 40 Electromagnetic wave, 12, 101, 128 Fourier amplitude, 40 Electromotive force, 112 Fourier analysis, 40, 83, 189 470 Index

Fourier spectrum, 40 Heat, 75 Fourier’s Theorem, 40 transfer, 77 Fourier synthesis, 40 Heisenberg Uncertainty Principle, 196 Fourier Transform, 83 Helicotrema, 274 Four primary colors, xi Helmholtz resonator, 65 Fourth, 306, 309 Hertz, H., 19, 130 Fovea, 328, 330 High pass ﬁlter, 182 Free electrons, 103 Histogram approximation, 362 Frequency, 1, 18, 184 Homochiral, 210 Frequency meter, 196 Homochirality, 210 Frequency modulation (FM), 42, 43, 181 Hooke, R., 27, 59 Frequency resolution, 196 Hooke’s Law, 59 Frequency spectrum, 40 Horsepower, 80 Frequency-time plane, 185 Horseshoe magnet, 104 Fringes, 162 Horseshoe perimeter, 364, 456 Fructose, 207 Horseshoe perimeter of chromaticity diagram, 368 Fulcrum, 271 Hsb-color, 384 Function, 86 Hs-color, 384 Fundamental, 45 Huang Chung, 9 Fundamental forces, 101 Hue, 12, 340, 353, 357, 360, 370, 439 Fundamental frequency, 193, 353 Human ear, 267 Fundamental mode, 23 Huygens, Christiaan, 151, 230, 442 Fusion of harmonics, 297, 451 Hyperopic, 327

G I Galilei, Galileo, 441 Ideal gas, 59 Gamma correction, 381 Ideal polarizer, 200, 204 Gamma value, 381 Ideal spring, 290 Gamut, 366, 373, 374 Identity matrix, 460 Gene, 330 Illuminance, 377 General ﬂuorescence, 150 Image, 240 Generalized force, 291 Image point, 240 General Relativity, 154 Impedance, 228, 260, 269 Genes, 355 Incoherence, 371 Giga, 417 Incoherent, 88, 165 Glaucoma, 328 sources, 371 Grain, 5 Index of refraction, 201, 227, 260 Gram, 101 Induced electromotive force, 112 Graph, 5 Induced EMF, 189 Gravitational force, 101 Inertia, 101 Gravitational potential energy, 73 Inner ear, 274 Gravitational redshift, 254 In phase, 158 Gravitational waves, 197 Inspiraling, 197 Green cone, 330, 355 Insulator, 103 Gregory, R. L., 393 Intensity, 12, 71, 81, 83 Ground state, 143, 146 density, 338 Guitar pickup, 126 Interface, 228 Intonation, 1, 304 Intra-beam viewing of a laser beam, 259 H Inverse matrix, 460 Haiku, 267 Inverted and real image, 241 Hair cells, 274, 275 Inverted retina, 328 Half-silvered mirror, 175 Ion, 103 Half-step, 189, 411 Ionized, 103 Hann window, 191 Iris, 325 Harmonic distortion, 290 Irreversible, 246 Harmonics, 6 Ishihara test, 392 Harmonic series, 193, 353 Isotropic, 104 Index 471

J Lumen, 376 Joule, James Prescott, 72 Luminance, 353, 361 Just noticeable difference in frequency, 298 Luminosity, 82 Just noticeable difference of loudness, 298 Luminous efficacy, 376 Just noticeable difference of pitch, 324 Luminous efficiency, 375 Just tuning, 307–309 Luminous flux, 376 Lux, 377

K Kepler, Johannes, 413 M Kepler’s Three Laws, 413 MAC, 357 Key note, 305 Mach, Ernst, 278 Kilo, 417 Mach bands, 279 Kilogram, 101 Mach number, 278 KiloWatt-hour (kWh), 72 Mach one, 252 Kinetic energy, 72, 78 Mach’s Law of Simultaneous Contrast in Vision, 278, 279 Macroflow, 103 L Macroscopic bodies, 75 Lapis lazuli, 348 Macula, 328 Largest common denominator (LCD), 44 Magenta, 360, 366 Lark sparrow, 184 Magnet, 104 Laser, 151 Magnetic field, 12, 117 Laser ranging, 258 Magnetic force, 101, 116 Law of Action and Reaction, 108 Magnetic particles, 189 Laws of Physics, 445 Magnetic polarization, 126 Leakey, Louis, 2 Magnetism, 101 Left-handed, 207 Magnetized, 101, 105 Le Grand, Yves, 353 Magnification, 242, 429 Leibniz, Gottfried W., 277 Magnifying glass, 241, 430 Length scale of roughness, 223 Magnifying power, 242, 429 Length scales, 225 Main air resonance, 65 Lens(es), 12, 236, 325 Major axis, 176 Lens-humour interfaces, 328 Major second, 306 Lenz’s Law, 126 Major sixth, 307, 309 Leu Buhwei, 9 Major third, 306, 309 Lever action, 271 Malus’ Law, 200 Lifetime of the excited state, 148 MAPLE, 189 Light, 101 Mappings, 6, 337, 355, 439, 440 Light intensity, 325 Masaoka Shiki, 267 Lightness, 354 Masking, 298 Lightness L∗, 377 Mass, 15, 27, 73, 101 Light receptors, 328 density, 53 Light-touch, 318 Match of colors, 368 Linear mass density, 32, 34, 84 Mathematics, 442 Linear response, 290 Matrix, 367, 455 Line of purples, 366, 368 Maxwell, James Clark, 127 Line spectrum, 138 Mayer, Julius Robert, 277 Localized photon, 142 Mean free path, 54 Lodestone, 101, 104 Mechanical advantage, 271, 272 Logarithm, 86 Mechanical energy, 75 Logarithmic, 286 Mega, 417 Logarithmic scale, 286 Megapixels, 400 Longitudinal, 16 Mel scale, 315 Lord Rayleigh (John W Strutt), 150 Memory, 441 Loudness, 19, 71, 81, 375 Metal, 103 Loudspeaker, 91, 110 Metamers, 13, 355, 362, 395 Loudspeaker cone, 91 Micro, 417 Low-pass filter, 181, 182 Micron, 417 lRGB gamut, 386 Microphone, 115 472 Index

Microscope, 242, 431, 432 Numbers, 9 MIDI, 315 Numerology in tuning, 304 Milli, 417 Minimum image diameter, 219 Minor axis, 176 O Minor second, 306 Object distance, 240 Minor sixth, 306 Objective, 353, 361, 432 Minor third, 306, 309 Objective input, 298 Mirage, 233 Objective parameters, 353 Mirror image, 224 Observation time, 196 Mirror reflection, 224 Octave, 305 Mismatch of impedances, 270 Ocular, 432 Mistuned consonances, 298 Oersted, H., 105, 111 Modes of vibration, 6, 194 Olduvai Gorge, 2 Modulation index, 181, 452 One atmosphere pressure, 54 Modulator wave, 178 One-over-f noise, 85, 343 Molecules, 12 One-to-one correspondence, 337 Monochromatic, 365 Open pipe, 62 Monochromatic light, 339, 340 Open strings, 1, 318 Monochrometers, 344 Ophthalmoscope, 65, 328 Monotonically increasing, 390 Optically active, 207 .mp3 audio file, 188 Optic nerve, 325 “Musical instrument digital interface”, 315 Order of interference, 162 Musical interval, 254, 306 Order of magnitude, 15, 409 Musical scales, 304 Orders of infinity, 354 Musical staff, 306 Ordinary ray, 201 Myopic, 326 Oscilloscope, 58 Ossicles, 268 “Outliers”, 8 N Out of phase, 158 Naked eye, 430 Oval window, 268 Nano, 217, 417 Overtone, 24 Natural harmonics, 318 Oxidation, 76 Near point, 327, 333, 429, 430 Near point of the eye, 430 Negative charge, 102 P Negative feedback, 126 Paintbrush, 357 Negative terminal, 104 Palette of colors, 357 Neon laser light, 339 Parallel ray, 240 Nerve fibers, 275 Partial, 24, 43 Nerve signals, 267 Partial frequency analysis, 274 Neutron, 103 Partially polarized, 203 Newton, Isaac, 33, 74, 101, 102, 277, 439 Pascal unit of pressure, 55 Newton’s Third Law, 108 PC, 357 Nexium, 209 Peak response, 91 Night vision, 329 Pendulum clock, 442 Nodal lines, 49 Pentatonic scale, 10, 304, 309 Node, 23, 60, 62 Period, 18, 25, 27, 29, 30, 413 Non-dispersive, 235 Periodic, 26 Non-ideal polarizer, 202 Periodic wave, 304 Non-linear response, 290 Permanent magnet, 109 Non-scientific issue, 443 Permeability of free space, 130 Normal to a surface, 224 Permittivity of free space, 130 Normal vision, 354 Phase difference, 158 North pole, 104 Philosophy, 14 Nuclear energy, 75 Phon level, 299 Nuclear force, 101 Phon, 285, 286, 375, 377 Nuclei, 103 Phosphorescence, 150 Number density, 53 Photoelectric effect, 443 Number of distinguishable colors, 385 Photon, 141, 443 Index 473

Photon absorption condition, 143 Q Photon detectors, 165, 446 Quadratic non-linear response, 291 Photon emission condition, 142 Quantum corral, 145 Photons, 12 Quantum energy level, 146 Photopic, 331 Quantum state, 144 Photopic vision, 330 Quantum theory, 144, 442 Physical laws, 199 Quarter comma, 324 Physics, 14 QuickTimePlayer 7, 188 Piano keyboard, 46, 305 Piano keys, 46 Picasso, Pablo, 465 R Pigments, 13, 347, 387 Radiation, 128 Pink, 13 Radiation decoupling after the Big Bang, 206 Pink noise, 85, 342 Rarefaction, 56 Pinna, 268 Ravel, Maurice, 7, 317 Pitch, 1, 5, 19, 67, 184, 353, 354, 411 Raven Lite, 184 Pitch discrimination, 12, 277 Rayleigh scattering, 149 Pixel, 13 Real image, 241, 247 Place Theory of Pitch Perception, 277 Reality, 443 Planck, Max, 142, 151, 443 Recovery time, 330, 393 Planck’s constant, 142, 419 Rectangular wave, 58 Plane wave, 216 Red cone, 330, 355 Pluck, 19 Redshift, 254 Pointilism, 400 Reducing to the octave, 309 Pointillist, 355 Reel-to-reel tape recorder, 189 Point mass, 102 Reference level, 87 Point source(s), 82, 160, 166 Reference level of the sound level, 87 Polarization axis, 200 Reflectance, 226 Polarized, 12 Reflectance of light, 348, 364 Polarized light, 12, 199, 200 Reflected, 226 Polarizers, 12 Reflection, 12, 215 Polaroid, 202 Refraction, 12, 138, 201, 215, 229 Positive charge, 102 Reissner’s membrane, 274 Positive feedback, 126 Relationships, 9 Positive terminal, 104 Relative change of a parameter, 250 Potential energy, 73, 78 Relative humidity, 92 Power, 12, 71, 80, 273, 376 Relative illuminance, 376, 377 Pressure fluctuations, 437 Relative phase, 40, 158 Preucil circle, 358 Relative pitch, 296 Primaries, 355, 356 Relative probability, 144 Primary illuminants, 379 Relative value, 15 Primary units, 367 Relative velocity, 253 Principle of conservation of energy, 65, 72 Resolution, 429 Principle of Relativity, 114 Resolution in time, 196 Principle of superposition, 157 Resonance, 1, 6, 38, 91, 147 Prism, 137, 233 Resonance peak, 91 Probability density, 144 Resonant, 1 Probability mode, 145, 146 Resonant response, 91 Protanope, 391 Response, 91 Protanopia, 391 Response function, 148 Proton, 103 Restoring force, 34, 37, 55, 423 Psychoacoustics, 283 Retina, 325 Pulse, 16, 22 Reverberation time, 12, 94 Pure tone, 299, 304 Reversibility, 229, 246 Purity, 370 Rhodopsin, 329 Purple colors, 395 Rhythm Theory of Pitch Perception, 277, 278, 280 Pythagorean comma, 313 Right-handed, 207 Pythagorean intervals, 309 Rods, 328 Pythagorean tuning, 307, 309 Rods of the retina, 353 474 Index

Roederer, Juan, 284 SPD, 367 Rotatory power, 208 Specific rotatory power, 208 Rutherford, Ernest, 139 Spectral analysis, 137 Spectral colors, 340, 355 Spectral intensities with respect to wavelength, 338 S Spectral intensity, 12, 189, 371 Sabine’s Law, 95 Spectral intensity with respect to the frequency, 83, 184, Saccadic, 393 338, 353, 363 Saturated, 340, 370 Spectral intensity with respect to the wavelength, 152, Saturation, 12, 353, 357, 360, 370 338, 353 Sawtooth wave, 58 Spectral photometer, 338 Scala media, 274 Spectral power distribution, 367 Scala tympani, 274 Spectral primary, 355, 361 Scala vestibuli, 274 Spectrogram, 187 Scattered wave, 222 Specularly, 258 Scattering, 215 Specular reflection, 224, 225 Schenker, Heinrich, 8 Spherical aberration, 237 Scientific notation, 410 Spherical wave, 239 Scotopic, 331 Spontaneous emission, 141, 143, 150 Scotopic vision, 329 Spontaneous transition, 143 Seashore, Carl E., 306 Spring constant, 27, 51, 290 Second harmonic, 23 Squillo, 286 Second Law of Thermodynamics, 76 SRBG primaries, 405 Second order beats, 298 SRGB, 386 Selective absorption, 202, 344 Stacked, 357 Semi-logarithmic graph, 286 Standing wave, 24 Semi-major axis, 199 Stapes, 268 Semitone, 306 Static electricity, 101 Sense of color, 325 Steps, 305 Series, 357 Stereoscope, 332 Seventh, 307 Stiffness, 37, 235 Sharps, 311 Stiffness of basilar membrane, 278 Shepard, R.N., 297 Stimulated emission, 150 Shepard’s Staircase, 297 Stirrup, 268 Shiny surface, 223 Stretch tuning, 46 Shock wave, 252 Stretch tuning of pianos, 322, 323 Short-time Fourier spectrum, 188 String-harmonics, 317 Short-time frequency analysis , 197 Stroboscope, 97 Side-band frequencies, 179, 182 Subjective, 353, 361 Sidebands, 179 Subjective perception, 298 Significant figures, 15, 409 Sub-matrix, 456 Silk screen, 167 Sub-pixels, 400 Simple harmonic oscillator , 339 Subtractive mixing, 13, 348, 395 Sine function, 24, 25, 30 Subtractive primaries, 357 Sine wave, 25 Surface, 228 Sinusoidal, 25, 28 Symmetry, 245 SKEW polarized state, 447 Synthesizer, 296, 451 Snell’s law, 229 Syntonic comma, 313, 321, 323 Solar constant, 81 Solenoid, 109 Sone, 286, 354, 375, 377 T Sound density, 57 Table of color matching functions, 362, 455 Sound level, 86, 87, 285, 300 TCMF, 362 Sound Level in dB, 284 Tectorial membrane, 278 Sound pressure, 57, 60, 270 Telescope, 242, 243 Sound pressure level, 86 Temperament, 304 Sound pulse, 16 Tension, 5, 15, 16, 32, 36, 412 Source of gunshots, 164 Terhardt, Ernst, 297 South pole, 104 Tetrachromacy, 395 Sparkling surface, 223 Tetrachromats, 395, 396 Index 475

Tetranope, 391 Unpolarized, 200 Thales of Miletus, 101 Upright and virtual image, 241 Thalidomide, 209 Theory, 6 Thermal energy, 71, 75, 79 V Thin lens approximation, 239 Vector field, 117 Thin lens equation, 241 Velocity, 132 Three primary colors, 354 Velocity amplitude, 78 Three-primary representation, 12 Vibrato, 40, 42, 181, 298, 313 Three-Primary Theory of Color Perception, 355 Violin, 40 Three-source interference, 164 Virtual image, 225, 241, 335 Threshold of aural pain, 435 Virtual pitch, 297 Threshold of hearing, 286, 435 Viscosity, 274 Threshold of pain, 286, 435 Visual purple, 329 Timbre, 19 Visual representation of a bird song, 187 Time, 441 Vitreous humour, 328 Time order, 441 Vocal cords, 194 Time-varying spectral intensity, 185 Vocal folds, 194 Total internal reflection, 230, 233 Volt, 104 Tower of Babel, 338 Voltage, 104 Transformation between sets of primaries, 372 von Békésy Georg, 274, 437 Transmission, 12 von Helmholtz, Hermann, 277, 308 Transmission coefficient, 327 Voxel, 13 Transmittance, 202, 226, 327, 344 Transmitted, 226 W Transmitted wave, 228 Wald, George, 325, 326, 329 Transverse, 16, 22 Watt, James, 80 Transverse waves, 164 Wave(s), 5, 15 Traveling wave, 22, 24 equation, 6 Triad, 13 generator, 58 Trichromats, 396 length, 27, 30 Tristimulus values, 362, 371 packet, 142, 371 Tritanopia, 391 phenomena, 15 Tritone, 320 propagation, 16 Truecolor, 357 velocity, 16, 30 Truth, 443 Weak force, 101 Tuning, 304, 307, 317 Weight, 73 pin, 5 Well-Tempered Clavier, 314 Turntable, 212 Well tempered tunings, 314 Twang, 286 Werkmeister, A., 315, 323 Two-source interference, 164 Werkmeister I(III) Temperament, 323 Tympanic membrane, 268 Western Music, 307 Whip, 252 White, 356, 360, 369 White noise, 85, 342, 363, 437 U Whitman, Walt, 2 Ultimate eye object, 429, 430 Whole tone, 306, 309 Unaided eye, 430 Widths of the peaks, 196 Uncertainty of the frequency, 196 Work, 77, 79 Unilateral dichromats, 391 Units, 409 Universal standard table of color matching Y functions, 377 Young’s modulus, 37